Lecture Notes in Physics Edited by J. Ehlers, M~nchen, K. Hepp, Zerich R. Kippenhahn, MLinchen, H. A. Weidenmeller, Heidelberg and J. Zittartz, K61n Managing Editor: W. Beiglb6cK Heidelberg
116 Mathematical Problems in Theoretical Physics Proceedings of the International Conference on Mathematical Physics Held in Lausanne, Switzerland August 20 - 25, 1979
Edited by K. Osterwalder
Springer-Verlag Berlin Heidelberg New York 1980
Editor Konrad Osterwalder Mathematik ETH-Zentrum CH-8092 ZQrich
ISBN 3-540-09964-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09964-6 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in PublicationData.InternationalConferenceon MathematicalPhysics, Lausanne,1979. Mathematicalproblems in theoreticalphysics. (Lecture notes in physics; 116)Bibliography:p. Includes index.1. Mathematicalphysics-Congresses. I. Osterwa~der,K., 1942- It. Title. Ill. Series.QC19.2.15381979 530.1'5 80-12140 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 215313140-543210
Foreword This v o l u m e c o n t a i n s the p r o c e e d i n g s of the I n t e r n a t i o n a l C o n f e r e n c e on M a t h e m a t i c a l Physics held at the Swiss Federal ogy in L a u s a n n e
(EPFL), A u g u s t 20 - 26, 1979. This c o n f e r e n c e c o n t i n u e d
the t r a d i t i o n of the m e e t i n g s held in M o s c o w in K y o t o
I n s t i t u t e of T e c h n o l -
(1975), and in Rome
(1972),
in W a r s a w
(1977). The next c o n f e r e n c e
(1974),
is s c h e d u l e d
for the summer of 1981 in Berlin. The symbol
Mr #
w h i c h has been chosen as a trade m a r k for all these
c o n f e r e n c e s denotes the i n t e r s e c t i o n of M a t h e m a t i c s add Physics. indicates the hope and the i n t e n t i o n that these c o n f e r e n c e s only be an o c c a s i o n to c o m m u n i c a t e results, i n t e n s i f y and to d e m o n s t r a t e
It
should not
but also an o p p o r t u n i t y to
the close contact b e t w e e n v a r i o u s fields
of m a t h e m a t i c s and of physics.
To this purpose,
the m a i n lectures of
this c o n f e r e n c e were all o r g a n i z e d as r e v i e w talks, c o v e r i n g a b r o a d area of research,
and, w i t h as few e x c e p t i o n s as possible,
parallel
sessions were avoided. The m a j o r topics of this c o n f e r e n c e were S c h r 6 d i n g e r Operators, tical Mechanics,
Q u a n t u m F i e l d Theory,
S u p e r s y m m e t r y and C*-Algebras. is in M a t h e m a t i c a l Physics,
Gauge Theory,
Statis-
D y n a m i c a l Systems,
It is clear that this is not all there
and there are c e r t a i n l y other areas of
p h y s i c s w h i c h have lively contact w i t h m a t h e m a t i c s w h i c h could not be i n c l u d e d here b e c a u s e of time limitations. There w e r e two major r e v i e w talks on each of the topics S c h r 6 d i n g e r Operators,
S t a t i s t i c a l Mechanics,
Q u a n t u m Field T h e o r y and Gauge Theory.
F u r t h e r m o r e there was a long session
(half a day or more)
c o m m u n i c a t i o n s on each one of the seven topics.
of shorter
These sessions Were
p l a n n e d and set up by session o r g a n i z e r s who were c o m p l e t e l y free to invite their speakers,
to accept or turn down c o n t r i b u t e d papers,
and
to o r g a n i z e the part of these p r o c e e d i n g s w h i c h belongs to their reSpective
session.
This v o l u m e c o n t a i n s w i t h one e x c e p t i o n all the m a i n lectures and m o s t of the smaller contributions. I.M.
Unfortunately,
Singer's talk was available.
no m a n u s c r i p t of Prof.
The i n t e r e s t e d reader is advised to
look instead at S i n g e r ' s c o n t r i b u t i o n to the P r o c e e d i n g s of the 1979 Carg~se
summer school on Recent D e v e l o p m e n t s
in Gauge Theories.
The c o n f e r e n c e was s p o n s o r e d by the I n t e r n a t i o n a l M a t h e m a t i c a l Union,
IV
the U.S. N a t i o n a l
Science F o u n d a t i o n
(travel support for A m e r i c a n par-
t i c i p a n t s through a g r a n t to Rutgers University), Suisse de la R e c h e r c h e Scientifique, en Suisse Romande,
Le T r o i s i ~ m e C y c l e de la P h y s i q u e
La F o n d a t i o n H e r b e t t e de la facult~ des sciences de
l ' U n i v e r s i t ~ de Lausanne, Hochschule
Le Fonds N a t i o n a l
ZUrich,
l ' U n i v e r s i t ~ de Gen~ve,
Eidg. T e c h n i s c h e
l'Ecole P o l y t e c h n i q u e F ~ d ~ r a l e de Lausanne.
It is a p l e a s u r e to thank these o r g a n i z a t i o n s for their g e n e r o u s cial and m o r a l
finan-
support of the conference.
The m e m b e r s of the I n t e r n a t i o n a l A d v i s o r y B o a r d w e r e as follows: H. Araki, V.I. Arnold, M.F. Atiyah, R. Haag, A. Jaffe, A. Janner, Y. Neeman,
H.M. N u s s e n z v e i g ,
E. Lieb,
J. Lebowitz, A. Martin,
L. O ' R a i f e a r t a i g h ,
C. Piron, R. Raczka, D. Ruelle, W. Thirring,
F.J. Dyson, G. Gallavotti,
K. Osterwalder,
B. Simon, Y. Sinai,
G. Velo, A. Verbeure, W.
R. Stora,
Zimmermann.
The m e m b e r s of the C o n f e r e n c e C o m m i t t e e were: W. T h i r r i n g , A. Martin,
K. Osterwalder,
In the name of all the p a r t i c i p a n t s
Ph. Choquard.
I w o u l d like to thank e v e r y b o d y
who s u p p o r t e d the efforts to m a k e this c o n f e r e n c e a success: speakers and authors of the c o n t r i b u t i o n s session organizers, ference committee,
the
the m e m b e r s of the a d v i s o r y board and of the conthe o r g a n i z e r s
and all the s e c r e t a r i a l and a d m i n i s -
trative staff m e m b e r s of the host institution, man w i t h o u t w h o s e enthusiasm,
and in p a r t i c u l a r
the
tireless d e v o t i o n and great o r g a n i z a -
tional skill this c o n f e r e n c e never w o u l d have taken place: Choquard. D e c e m b e r 1979
the
to these proceedings,
Konrad Osterwalder Editor
Prof.
Ph.
T a b l e of C o n t e n t s
SCHRODINGER
OPERATORS
Main Lectures J.-M.
COMBES:
Recent
Developments
in Q u a n t u m
Scattering
Theory ................................................. W.
HUNZIKER:
Schr~dinger
Magnetic Session V.
Operators
with
Electric
or
Fields ......................................
A New M e t h o d
D.B.
PEARSON:
R.N.
HILL:
for A s y m p t o t i c
Pathological
Proof
A Review,
Spectral
that the H
C o m p l e t e n e s s ........... 45 P r o p e r t i e s .............. 49
Ion Has Only One B o u n d
a N e w Result,
and
Some R e l a t e d
State:
Unsolved
Problems .............................................. SIGAL:
Number
of E i g e n v a l u e s
and E f i m o v ' s
J. MORGAN:
of M a n y - B o d y
T. H O F F M A N N - O S T E N H O F ,
On the E x p o n e n t i a l
and E l e c t r o n
Fall
Exact
Results
K. YAJIMA:
The Q u a s i - C l a s s i c a l
Off of W a v e f u n c t i o n s
for C o n f i n i n g
Semiclassical
L i m i t of Q u a n t u m
Scattering
Quantum Mechanics
B. SOUILLARD:
Some R i g o r o u S XIA DAOXING:
Random
Schr6dinger
Problem with
83
Indefinite
Metric ................................................
STATISTICAL
78
Operators
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
On the S c a t t e r i n g
73
for C o h e r e n t
States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. KUNZ,
62
P o t e n t i a l s ........... 68
Theory ................................................ HAGEDORN:
57
R. A H L R I C H S ,
Densities ................................
H. GROSSE:
52
Hamiltonians
Effect ...................................
M. H O F F M A N N - O S T E N H O F ,
G.A.
,25
O r g a n i z e d by B. S I M O N
ENSS:
I.M.
I
88
MECHANICS
Main Lectures E. LIEB: O.E.
Some Open
LANFORD
III:
Problems
about Coulomb
Time D e p e n d e n t
Phenomena
Systems ........... 91 in S t a t i s t i c a l
Mechanics ............................................
103
VI
Session M.
Organized
JIMBO,
by E. B R E Z I N
T. MIWA,
M.
SATO:
The U n a n t i c i p a t e d of D i f f e r e n t i a l M. AIZENMAN: lation G. TOULOUSE: D. BRYDGES,
Instability
Topology
PH.A.
cal Systems DANIELS,
Deformation
-
Theory
and Q u a n t u m F i e l d s .......... 119
of P h a s e C o e x i s t e n c e
and Defects
and Trans-
in O r d e r e d
Debye
Screening
M e d i a .......... 148
in C l a s s i c a l
Mechanics ................................. MARTIN:
Equilibrium
Properties
of C l a s s i -
A.C.D.
V A N ENTER: in M o r e
Long Time Tail
On the L a c k of F r ~ c h e t
Phase
for S p a c i a l l y
Regions ............... 160 Inhomogeneous
R a n d o m Walks CH.
RADIN:
Why
J. BRICMONT,
QUANTUM FIELD
162
Is there a Solid
J.L.
the Surface
151
w i t h Long R a n g e Forces . . . . . . . . . . . . . . . . . . . . 156
Differentiability H. SPOHN:
Between
Quantum Fields
in Two D i m e n s i o n s . . . . . . . . . . . . . . . . . . . 143
P. F E D E R B U S C H :
CH. GRUBER,
H.A.M.
Link
Equations
Invariance
Statistical
Holonomic
LEBOWITZ,
State? . . . . . . . . . . . . . . . . . . . . . . 163
C.E.
PFISTER:
Some R e m a r k s
on
T e n s i o n ....................................
164
THEORY
Main Lectures J. BROS:
Analytic
Structure
Quantum Field A. JAFFE: SesSion
Constructive
Organized
A. KUPIAINEN:
of Green's
Functions
in
Theory
.166
Field
T h e o r y .......................... 200
by J. F R O H L I C H I/N E x p a n s i o n
- Some R i g o r o u s
Results .......... 208
G. BENFATTO:
The R e n o r m a l i z a t i o n
Group
in the E u c l i d e a n
Scalar
Superrenormalizable
Field
T h e o r i e s .............. 211
J. MAGNEN,
R. SENEOR:
in C o n s t r u c t i v e D.
IAGOLNITZER:
Expansion Field
and S u m m a b i l i t y
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 217
The P h y s i c a l - R e g i o n
Multiparticle
S-Matrix ........................... E. SEILER:
Quantized
Methods
Gauge Fields:
~ . . . . . . . . . . . . . . . . . 224
Results
and P r o b l e m s ..... 228
VII
D. B U C H H O L Z ,
and
T. BALABAN:
The U l t r a v i o l e t
Lattice B.S.
KAY:
G-Models
GERDJIKOV,
Stability
Aspects
Charge-Screening
Quantum Field Bounds
Theory..233
for Some
Higgs-Kibble
M o d e l s ..... 237
of Q u a n t u m F i e l d
Theory
Space-Time ................................. P.P.
KULISH,
Quantum Aspects
GAUGE
in Local
and L a t t i c e
Mathematical
in C u r v e d V.S.
Clustering,
K. F R E D E N H A G E N :
the M a s s - S p e c t r u m
M.I.
IVANOV:
of the Inverse
Classical
Scattering
241
and
M e t h o d ..... 244
THEORY
Main Lectures D.I.
OLIVE:
Magnetic
Monopoles
and N o n - A b e l i a n
Gauge
Theories ............................................. I.M.
SINGER:
Yang-Mills
Fields
(Paper not available; the
1979 C a r g ~ s e
opments Session0rganized A.
TRAUTMAN: Gauge
Introduction
to Papers
W. THIRRING:
Gauge
Theories
P. V A N N I E U W E N H U I Z E N :
Derived
Proceedings
of
Recent Devel-
Presented
at the
M.E.MAYER:
Geometric Theories
of G r a v i t a t i o n ................. 272 as a Gauge
Theory
C o u p l i n g . . . . . . . . . . . . . . . . . . . . . . . . . 276
Gravitational
Instantons
..263
of P h y s i c s . . . . . . . . . . . . . . . . . . . . 265
Supergravity
from M a t t e r
M. MULASE:
J. HARNAD,
on:
Session ...............................
The U n i f i c a t i o n
Gauge
however,
School
by A. T R A U T M A N
Theory
GIBBONS:
and P r o b l e m s
Theories)
H. FRITZSCH:
G.W.
see,
Summer
in Gauge
- a Review
Instantons:
A Survey ........... 282
and Line G e o m e t r y . . . . . . . . . . . . . . . . . . . . 288 Aspects
of S y m m e t r y
(Higgs M o d e l s
S. S H N I D E R , L. VINET:
Breaking
Without
Higgs
Invariance
in Bosons)...291
Conditions
for G a u g e F i e l d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. STERNHEIMER:
Nonlinear
Linearizability B.SCHROER:
249
Group Representations
of N o n l i n e a r
Determinants,
296
and the
E q u a t i o n ................ 300
Green Functions
and
Induced
Action ...............................................
303
Vlll
J.S.R. CHISHOLM, Electric P.G.O.
DYNAMICAL
R.S. FARWELL:
and M a g n e t i c
FREUND:
Spin G a u g e
T h e o r y of
Spinors . . . . . . . . . . . . . . . . . . . . . . . . 305
Spin S t r u c t u r e s
and Gauge
T h e o r y ............ 308
SYSTEMS
Session
Organized
D. RUELLE:
by D. R U E L L E
Recent
Results
on D i f f e r e n t i a b l e
Dynamical
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.H.
CURRY:
On Some Systems
Equations: P. COLLET,
J.P.
Numerical ECKMANN:
the Interval O.E.
LANFORD
III:
Doubling
SUPERSYMMETRY
J. SCHERK:
of C o n t i n u o u s
o n the A c c u m u l a t i o n
331
of P e r i o d -
Bifurcations ................................
340
THEORY
by J
SCHERK
Superspace
of S u p e r s y m m e t r y Aspects
and
Supergravity...343
of S u p e r s y m m e t r y
and
Supergravity ......................................... B. JULIA:
316
Maps of
to Itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Overview
S. FERRARA:
by the L o r e n t z
Results . . . . . . . . . . . . . . . . . . . . . . . . .
Properties
Remarks
AND G R O U P
Session Organized
Motivated
311
Symmetries
of S u p e r g r a v i t y
358
M o d e l s ................ 368
C~-ALGEBRAS Session
Organized
by H. A R A K I
D.W.
ROBINSON:
A R e v i e w of D e r i v a t i o n s ..................... 377
M. FANNES:
Thermal
Equilibrium
and C o r r e l a t i o n
Inequal-
ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. BRATTELI:
On the P o s s i b l e
C*-Dynamical D.E.
EVANS:
Temperatures
of a
System ..................................
A Review on Semigroups
of C o m p l e t e l y
WORONOWICZ: Pontriagin
Pseudospace,
Pseudogroups
397
Positive
Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.L.
392
400
and
Duality ...................................
407
RECENT DEVELOPMENTS IN QUANTUM SCATTERING THEORY (x)
J.M. COMBES (xx)
INTRODUCTION Let me f i r s t recall what is considered in elementary text books as the basic ansatz of quantum scattering theory. For a particle in an external field vanishing at infinity the ingoing and outgoing solutions of the Schr~dinger equation satisfy a"radiation" asymptotic condition :
IXl-~ m where
~
is the dimension of configuration space and , . ~
the angular
scattering parameter. I t was already known in 1928 b ] that for Coulomb forces
the phase of the scattered wave had to be modified as
~'~.~|
|k~ |~I ~ - l ~ "~ ~ . (A~ ~ ~X~ ) but that nevertheless i t was s t i l l legitimate to interpret # ( - ( ~ . ~ as the scattering amplitude. By stationary phase arguments the time evolution of wave-packets should then also obeys an asymptotic condition
(2)
i~JD~partement de Math~matiques, Universit6 de Toulon and Centre de Physique Th~orique I I , CNRS, Marseille
Postal Address :
Centre de Physique Th~orique CNRS Luminy - Case 907 F-13288 MARSEILLE CEDEX 2 (FRANCE) -
Although these statements are deliberately vague from a mathematical point of view, they are essential for the physical interpretation of scattering theory. I t may seem surprising that the mathematical literature is so poor concerning the analysis of (I) and (2) apart from some notable exceptions like the papers of Green and Lanford ~42~ and Ikebe ~ 3 3 . During the last decade very much more attention has been paid to the "operatorial" formulation of scattering theory in terms of wave-operators, the connection with the original scattering wave approach being more and more hidden by the abstract formalism. This divorce of the theory with physical intuition is certainly responsible for the lack of definite progress in some important domains like multichannel systems or analytic scattering theory with long-range forces. So I will not hesitate to describe as main results of the last two years the analysis by Agmon 12-~ of by Enss"--~3of
(1)
in Besov spaces and the geometrical phase space analysis
(2). The techniques developped in these works seem to provide
new powerful tools of investigation ; they reconcile a strong mathematical technology, which is more or less implicitely pseudo-differential operator theory, with our classical intuition of scattering phenomena. I t is not so surprising that their basic strategy which is essentially simultaneous localization in X and k space can be so fruitful since one expects that for large times scattering wave-packets should propagate along classical trajectories. Some abstract preliminary formulations of related properties have been known for a long time like e.g. decay properties and Ruelle's characterizations of continuum states L 4 J , or the so-called limiting absorption principle. However i t is only rather recently that powerful enough methods have been
L5316]
{9],
developped and used for more systematic investigations culminating with the works of Agmon and Enss. We will come back to these works and to some reasonablespeculations about the new possibilities they open after a more general review of the present status of scattering theory and the evolution of ideas in the last years.
II.
TIME-DEPENDENT SCATTERING THEORY
Modern formulation of the abstract scattering theory attempts to cope with larger applications than just ordinary quantum potential scattering. For this reason the usual M~ller definition of wave-operators needs to be slightly extended. We denote by ~ , H (resp.~o ' ~ 0 ) t h e Hilbert space and Hamiltonian describing states and dynamics of the perturbed (respo free) system. Let ~ " bee bounded linear mapping from ~ o to ~ playing the role of an identification operator :~mastate (:~C- --~'0 of the free system is identified with the state ~(~ ~ of the perturbed system. The choice of such an identification operator is a matter of physical consideration and is not necessarily unique ; _ ~ need not be bijective as we will see in some applications. The wave-operators ~ C ~ . H ~ * ~ - ~ are then defined by the following strong limits when they exist LIO~, L l l ] : J
In most applications the domains
of
~ ( I ~ , H . ' ~ T ' ) are contained in ~ a e . ~ '
the subspaceof absolute continuity of such that ~ , ( ~ ) ( ~ ~ denotes the spectral family of if
cH H. exist or in other words i f
consisting of those
~'~
P0,O.¢. ~ o
W~
~-
~'w~C~
C ~,Ho~
~'~
W,~ fill H 0 ~ a.~ ~') are obtained by to a. spectral subspace P¢I(~'~ ~ ,
for H 0
corresponding to a closed set ~ C. ~ The properties of generalized wave-operators (intertwining, chain rule, etc...) are similar to those of the usual ones ; however isometry only holds under the additional requirement that the identification operator is a%anptotically isometric in the following
sense :
go
=
Local wave-operators
restricting the domainof
HO
is an absolutely continuous function (here ~0~,' ~'~0 ). One says that the wave-operators exist
Onesays that W ~ ( ~ } ~'~0~") are -_. ~ ¢ , , ~ with obvious notations
complete i f
W~
C~'~)~o~T)~ ~ J
. This is knownto imply unitarity
of the S-matrix operator
SCH,Ho] 7) - WI HHo
W.(.H,
provided wave-operators are isometric. The following criteria relating existence and completeness is due to T. Kato Proposition 1 Assume ~ " is surjective. I f both W~(~O)~ ~ ' ~ exist then
V~t(H) Ho'])and_.
~IVr: ( ~ ) ~ o ~ ' ~ is complete. If fu"rthermore ~r (resp.~'~) is asymptotically J H0 (resp. H )
isometric then
fH H,
is unitary.
The f i r s t example of a scattering system in the above sense is of course potential scattering for a single-particle. Here ~ . ~o = L~C~) and H,:-~ , ~ : - ~ • If V is ,,.. ~ relatively bounded" in the sense of operators or forms then ~ can be defined as a self-adjoint operator (provided V is also real) in an almost unique way from a physical point of view. Then i t is natural to take ~ ' : ~ . However for potentials having nasty singularities in a bounded domain some other choices can be more convenient for technical purposes (see Example 2.4 below). Let us mention some other relevant
examples : ExamPle 2.1 : Obstacle Scattering L e t _ ~ . be a bounded regular obstacle in ~ ~r~: ~= L?" ~ ) a n d ~ - the orthogonal projection operator on I f one takes Ho-_-aand ~ = . - - ~ . + boundary conditions on ~ . . ~ C " ) ~ ' ~ ~'~ exist and are isometric b 2 ] .
then
Example 2.2 : Scattering by stationary external metrics and Yang-Mills potentials
(L13 b4]) Consider an external static metric on ~Y~ . ~C~1,~. ~ [ l c . ) ~ an external Yang-Mills potential t ' ~ ( ~ ) =. | ~-~(I~) ~a ~ ' ~ _ ~ , wnere of a each ~{~ is an endomorphism of an interna~ symmetry space quantum spinorial particle and a potential V . The Hamiltonian of the particle in these external fields of forces is given by :
.,. "
wi th I f one assumes that the metric is asymptotically f l a t and locally regular, e.g. :
and that the external potentials vanish at i n f i n i t y , e.g.
(5ii) (5iii)
V
e 4 cm') e g~a V-
then i t is natural to compare the large time behaviour of this system with the free one in ~mO= L I ~ I ~ ) under the time evolution with generator ~ O : - - a The fo]lowing identification operator can then be used :
•
Under assumption (5i) ~ - is asymptotically isometric. Example 2.3 : Multichannel Scattering Consider for simplicity a 3 particle non r e l a t i v i s t i c system interacting through two-body potentials suchl:...that pairs (I~) can form only ~/~ - /--Z, • The Hamiltonian acting on one bound-state with binding energy
-" ~ - -~" L t ' ( ~~l') is, denoting by - J " l the center of mass frame :
H- 1-.V Let and
the kinetic energy operator in
HI}
describe the "free" evolution in the direct sum __elJal°f channel Hilbert spaces. Here ~T-I~ denotes the reduced mass of particle II. and center of mass of L and ~ , (~;ZI'~, ~ = [ 4 I t ) ~ ~ and /'~gl the Laplaclan wlth respect to the relativecoordinates of I~ and the center of mass of L and [ The following identification operator is frequently used ( I0 ) ( 15 ) ~. Q
where ~ ; is the normalized bound-state wave function of the p a i r ( l ~ l ~ ) . This identi~cation operator is asymptotically isometric ; this follows immediately from the well-known asymptotic orthogonality of channels. The "channel" waveoperators of Ekstein and Jauch are obtained from W ~ (~'~) ~ll O ~ ' ) by
W~ where ~ , I~:'~,~I~IILI channel subsPace I~. of
** is eas
: vq~
(H,H,;T)
i~, T*
, is the orthogonal projection operator on the ~ l l and the adjoint ~'~ of T is given by
,o s e e * h a * u n i * a r i * y o f * h e rout* channel S ma*rix
is equivalent to unitarity of ~ f H ) H e ; ~ ' ) . However th "t " g" above for completeness does not apply to this situation. For i f we consider for example a state which is asymptotically in a two-particle channel, e.g. :
P-
[email protected]~ :-
:P,L
then ~ ~-" ~ ~ID has obviously no strong limit ; in other words ~VYWf~( ~ j H o ~-T~) d°es not exist for multichannel systems. So new criteria and identification operators are needed ; P. Deift and B, Simon show in [ ~ l ] t h e convenience of a "geometrical" choice. We now come to criteria for existence of wave-operators generalizing a well-known result stated originally by J.M. Cook and S.T. Kuroda. The form given below originates from works of M. Schecter b 6 ] , B. Simon Ll7)and T. Kato LI8] . The original result of Cook called "Proof of a Lemma due to Eckstein" applied to one-body scattering with a potential V ( = L L ( ~ I ) . Theorem 1
Were(H)
~ssume
~ is a linear operator from ~ 0
i i ) ~ is a linear operator from ~io If
¢
then
e ~(H,)
tO a Banach space ~ " , ~bC~f~ ) ~ ( H o ) to
~
and ~
is
~-~-bounded.
satisfies
T
=F e~CW, (H,H, ~))
ExamPle 2.4 ( 19 ) Let us consider scattering of a quantum particle by a potential V which is short range in the sense
IV(x)l
,
,
but can be nasty 1ocally in the sense that one only requires existence of a selfadjoint operator ~ satisfying
H¢-.-~4~ ÷v¢
',~)
By the above theorem W ~ C~'{, H = exists where ~ " with {~,_ a smooth characteristic functlon of the sphere of radius Now since I~ II ~(.~ ~ l " ~ , ~= ¢~I ~ @ ~ i t . follows that
Example 2.5
is ~ - ~ . i ~
: The Hack-Kuroda Theorem
Let
H be an N - particle Hamiltonian with two-particle inter¢X~ : 0 ( I ~ I ) , I~I -~ m , and " Then channel wave-operators exist. Many other applications with proofs can be found in Volume I I I of the Reed-Simon series t 2 0 ] . For potential scattering they allow to deal with more or less important local singularities and short range forces. For long-range potentials i t is well-known that the definition of wave-operators has to be modified in order to adapt divergent phases [ I ] However Cook-Kuroda method can be adapted as was shown by many authors and more recently in the culminating work of L. H~rmander
L21]whose result is
described below. H~rmander defines admissible perturbations of the Laplacian ~
as
follows
(6)
l,n.< ,i
where (denoting by operator) : I)
oL an n-index and
(4÷I'/,I~
~"
~
the corresponding differential
)
for some ~
and
2) where ~t~(~);
6,
. _ ~t~%(}j are positive and
~
L ~ •
~"
Under these conditions H~rmander shows Theorem 2 Let ~W# be an admissible perturbation of of the Hamilton-Jacobi equation ;
and ~
be a solution
Then the modified wave-operators
(-a÷w exist and are isometric operators intertwining
~
/~
and ... ~ - ~ .
H~rmander's proof uses stationary phase arguments and applies to a more general class of systems whose generator of evolution is given by a pseudo-differential operator. Despite the wide applicability of Cook-Kuroda method for existence proofs i t is not very convenient for a time-dependent proof of asymptotic completeness via the Proposition 1. The main reasoQis that i t is usually not possible to estimate decay properties of C:~ f o r a dense set of 's in a sufficiently explicit way. Enss's method remedies to this using simply the fact that according to Ruelle L4]the particle ultimately leaves the region where the interaction is important. Before Enss's discovery the only systematic investigation of completeness by time-dependent methods was the so-called Kato-Birman Theory. I t originated in 1957 with a beautiful paper of Kato L22] which can be considered as the f i r s t building stone of abstract scattering theory ; a more detailed history can be found in ~01 . We only mention here its more important statements and recent applications. The following proposition is the optimal version given by D.B. Pearson L231of the main completeness result : Proposition 2 Assume H ~ ' J - ~ o +v where V e ~ ( ~ j ~ ) (trace class operators from ~ o to ~ ). Then ~ I ~ C ~ ) , , ~ ' ~ and-~CHo exist and are adjoint to each other. In particular i f ~' is onto then " ~ ~ j ~ O ~ J ' ) i s complete.
H~'~~
In this form the result is not very useful since in most applications (e.g. potential scattering) the perturbation ~ does not even have point spectrum. This statement however is of wide applicability once i t is supplemented by the "Invariance principle" for wave-operators. The form given below is sufficient for most applications of interest : Proposition 3 t ~ be a real piecewise monotone function such that ~ I is locally posltive and of bounded variation. Then under Pearson's conditions
10 Although the mathematical meaning of the Invariance principle is quite obvious from the spectral theorem since a self-adjoint operator and its functions have the same generalized eigenvectors, its physical interpretation only rests ~ atom on a paper of J.D. Dollard L24]reiating i t to invariance of the S-matrix under change of origin and scale on time and energy. Let me recall one of its deeper consequence, namely the equality of the S-matrices for quantum and acoustical scattering ~ 5 ] by the same medium. Among applications, let us mention : Example 2.0 : Potential Scattering, S.T. Kuroda ~ 9 ~ ~f . . , _ V C ~ ) ~ L~ C ~ R ~ then for (~ large enough ,.--(~.~÷~t - - ~ . + C ) - ! i s trace-class. This result is rather weak since i t requires a stronger decay of V than necessary. Example 2.1 :
Completeness of wave-operators for obstacle Scattering
P. Deift-B. Simon 26
, Jensen-Kato 27#
~.~
and
that
~I~ -~H
For this problem one takes . ~ ( ~ ) : show -- e~ ~ H I ~" is trace-class.'A related problem concerns decoupling of f i n i t e singularities for potential scattering ~6~ L28] ; these singularities can be decoupled by an a r t i f i c i a l Dirichlet boundary condition on a bounded surface which encloses them. Such a perturbation is equivalent to an obstacle and a trace condition is satisfied for H : - ~ . ~ N ~ and ~ : - w ~ ÷ ~ plus Dirichlet boundary condition provided the singularities of V are not too bad in the sense that --~, ~ ~ 1 . ~ - a ~ f ~ C ~ . ) for some constants C~, and CL . Then only the behaviour of ~N~ outside the surface is relevant for the question. Example 2.2 : -Scattering by external static metric and Yang-Hills potentials. R. Schr~der 14 I f ~.. is the dimension of space R. Schr~der. shows that under conditions (5i) ( 5 i i ) ( 5 i i i ) , one has L ~ ) - - ~ - - ~'CHo+~)'~-e(~C~oj~) for ~ n . S~ectral shift function, Krein's Theorem One important virtue of the Kato-Birman theory is that i t leads to very useful properties of the total phase shift as expressed by a theorem of Krein. The total phase-shift T(~ is defined as follows : since the S-operator commuteswith H 0 one has a direct sum decomposition
S~j~o ~'~
11 Then formally :
e~~' ~'(~)
(7)
~b
Sc~)
Existence of ~ ( ' , ~ can be shown under rather general conditionsL12]. Although (7) defines modulo ~ i t is possible to show that under these existence conditions ~'C, ~ can be chosen as a continuous function. The following theorem of Krein relates ~ °~ to the so-called "spectral shift function" and leads to many interesting properties : Theorem (M.G. Krein 30 ) Let ~ 0 and ~ be self-adjoint operators on a Hilbert space such that ~ = ~o~-~ with q~ ~(~). Then there exists a function
~(: Lic_~÷m)such V~C~) that
2)
for all ~ ' C ~
3) -0 o 4)
~C~ ~ : T/"I t
c
5) Here again the full power of the theorem rests on an invariance principle stating that i f ~ is a twice differentiable function on the spectrum of ~A{ and ~0 such| 'that ~'~0 and ~C~'OG) = ~ - ~ then (with obvious notations) :
This implies existence of thespectral shift for pairs of operators to which Kato-Birman theory can be applied. For such pairs i t is usually possible using
)
12
property 4) to show continuity of ~ , ~ and then according to 5) the spectral shift is equal to - l l "~ [ [ , ~ plus a constant multiple of l l . Then using I) 2) 3) one can obtain very useful estimates on the total phase s h i f t and its variations and on the corresponding physical quantities, e.g. cross sections. Let us mention some applications : 1) Asymptotic expansions For the choice Krein's theorem :
, ~A)
!
:
~m~
one obtains from formula 2 ) i n
.
kcc
By a theorem of Laplace transforms of asymptotic expansions (see e.g. 131] ) asymptotic expansions of ~ ( ~ ) a t large "energies" ~ are related to asymptotic expansions of ~[~ C~,, " ~ M _ e, ' ~ M = 3 at small "temperature" t . For Laplace-Beltrami operators such low temperature expansions can be derived in particular form the works of H. McKean and I,M, singer L32] or P.B. Gilkey [33] . The following asymptotic expansions have been obtained Example 2.1
Obstacle Scattering (V.S. Buslaev
For ~
dimensions :
34
, A. []ajda-J. Ralston
(~)
35
)
m-I
"-
Here
V
is the volume of the o b s t a c ] e ~ , S the area of the boundary and H the mean curvature function o f % ~ . For~: Z one has
Example 2.2 : Scattering by external metrics and Yang-Mills potentials (R. SchrZder
14
) For ~
~
dimensions :
13
where
I ~= ~o ~- ~ II-~~i'~ ~L
0 standard notation for covariant derivatives has been used. Notice that for potential scattering ( l ~ : ~ the expansion starts at ~ " . ~. with the usual Born term. Notice also that the form of ~ given here generalizes the i
obstacle scattering f i r s t term. Finally i t is interesting to remark (as proved in L I 3 ] ) that the expansion is only in terms of gauge invariant quantities since the S-matrix is invariant i t s e l f ; for example the term j ~ i n v o l v e s the total curvature and the potential. It would be possible to push the expansion farther using Gilkey's result ~ 3 ] ; the next term ~ would contain for example 43 structurally different terms ! These asymptotic expansions are of interest in many respects. First of all from the point of view of inverse scattering theory they show how one can
14 recover some invariants of the "scatterer" (obstacle, metric or external f i e l d s ) from high-energy experiments. Second as notices by A, Majda and J. Ralston L35] there is a close analogy between the asymptotic expansion for obstacle scattering and those for the number of eigenvalues of the corresponding internal problem ; these two authors conjecture existence of a Levinson theorem for decaying modes (or resonances) and raise the academic but fascinating problem of a one to one correspondence between these decaying modes and internal eigenvalues. 2) Monotonicity and sign of the phase s h i f t s From Krein's theorem and
i t follows that the total phase s h i f t has the sign of the l e f t hand side trace. In p a r t i c u l a r the t o t a l phase s h i f t is negative for positive potentials and obstacle scattering. I refer to T. Kato L36J for more refined investigations on the sign and v a r i a t i o n of the phase s h i f t as a function of the perturbation. Let me simply mention that the determination of the sign of the above trace is a subject of great mathematical and physical i n t e r e s t which has deserved a l o t of attention these l a s t years a f t e r the appearance of Kato's d i s t r i b u t i o n a l i n e q u a l i t y , the concept of semi-group domination and i t s r e l a t i o n to Feynman path integrals. A beautiful application of these ideas has been made by R. Schr~der in the case of scattering by Yang-Mills potentials. From Kato's i n e q u a l i t y for magnetic f i e l d s and Yang-Mills potentials (B. Simon [37] , H. Hess, R. Schr~der and D. Uhlenbock L38J) ~e argues that Yang-Mills potentials act r e p u l s i v e l y on scalar p a r t i c l e s , whereas from the paramagnetic conjecture of H. Hogreve, R. Schr~der, R. Seiler
L931 ;
i t should f o l l o ~ that they act a t t r a c t i v e l y on
spin 1/2 p a r t i c l e s . This conjecture
however has been proved only up to now at very low and very high energies Enss' method
( r3~ , ~ I ]
~9]
I
Consider potential scattering with short range forces in the following sense
.
15
As (2) suggests one expects that for large positive times, ~ only outgoing waves so that
~O~.~:~ contains
Then asymptotic completeness holds by the followin~argument : assume that ~e But then :
~'CW--~/"~&C
~
. Then
~T also does
-~b.~
II W.CH, ,'5 tends to zero se that ~ ,
~,~
liW. I,,,,,)e.,
~.,~
~ .
The original Enss' proof L31 of (11) uses a decomposition by hand of phase space leading to (2) ; essential in his proof is Ruelle's characterization of continuum states ~ ] . I present here a more algebraic proof also due to Enss, using an idea of E. Mourre [ 9 ] according to which the set of outgoing states is roughly the subspace ~ C ~ ~ where ~ ) ~ . ~ P 4 " P ~ is the generator of the dilation group. The basic ingredients are i)
~ , ~ ~~
~,
,t
.--~
H
(converges in strong resolvent sense'
ii) ~W_(H)~'~O~--~,~C"~)~O) is compact LgJ when ~'~ is a positive function with compact support on an interval [ ~ i g ] J 0,~)0 So assume ~C~g,¢,~ and ~ C " 3 : H for some ~ of the above type ; this is true for a dense set in ~ . Then -
By the compactness stated in i i ) and since P C ~ o . ~ ) tends strongly to zero the f i r s t term has a zero limit. The second one tends to CV~_(~j~O) .-T" 3 o.
which is zero ; this completes the proof of (11).
Various applications of Enss' method to more general systems can be found in B. Simon's paper ~ i ] .
16
III.
STATIONARY SCATTERING THEORY The basic problem of the stationary approach is the so-called " l i m i t i n g
absorption p r i n c i p l e " that is for a suitable Banach space ~
s a t i s f y i n g the
continuous dense inclusions
the existence
in some operator topology of the l i m i t s in
~~~I.~)~I
:
-
If
~
belongs to the continuous spectrum of
right-hand side of (12) has no l i m i t in ~ ( ~ I
H
i t is well-known that the
although i t can be defined as
an unbounded operator. The relevance of this problem for Scattering Theory comes in p a r t i c u l a r from the fact that scattering cross-sections involve such boundary values. For example for one p a r t i c l e potential scattering one has at real energies :
if
then
belongs to a Banach space
such t h a t
(12) holds
the cross-section is f i n i t e . More precise estimates on the way the l i m i t
is reached in (12) can provide bounds, asymptotic behaviour and dispersion relations for amplitudes. The l i m i t i n g absorption principle also plays an essential role in the analysis of r e g u l a r i t y and asymptotic properties for solutions of the (two-Hilbert space) Lippman-Schwinger equation :
where ~ ( ~ )
is a given solution of C H Q ' ~ ) ~ { ~ )
right-ha~d side of ( J ~ ) of
~"
~-C~ . The second term on the
plays the role of the scattered wave ; i t is an element
s a t i s f y i n g in addition ingoing or outgoing
boundary conditions, which
in an abstract way correspond to belonging to the ranges of ~ { ~ O ) .
In the
special case of potential scattering such boundary conditions shoulo be equivalent to Sommerfeld's radiation condition and then lead to (1). This connection is e x p l i cited in a beautiful work of S. Agmon and L. HUrmander [4~ Among e a r l i e r analysis l e t me mention the pioneering works of J. Green and O. Lanford [4~ and T. Ikebe E43] •
17 Finally the limiting absorption principle enters the mathematical theory of "weak" wave-operators obtained by an Abel-limit procedure
Wt (H,Ho)Z)
@.eo
]
0
"
Notice that in general one simply has
(14)
~+_(H,H,]J) C Wt~H,Ho]~)
Weak local wave-operatorsVq%(HjHo)a~3)canalso be defined by restricting the ~ -integral in (14) to the interval ~ One has the following abstract existence and completeness theorem for ~ ( H } H o , ~ ) ~ ' ) which originates in" more or less different forms from works of T. Kato [10] , M. Schecter [44] and I. Sigal [45] . Its proof depends on the general theory developed by T. Kato and S.T. Kuroda [46] . Theorem Let X (rasp. X o ) ~ . ~ ( r e s p . ~ ) be an ~'~ (rasp. Ho) dense Banach space in the sense that there exists no proper closed subspace o f ~ (rasp. ) invariant under H (rasp. H~) and containing ~ (resp. ~ ). Assume that i) The limiting absorption principle holds for ~
on some ~ ' b ~
~
contain-
ing ~0 and for all ~C-a 2) Let
~
• ~:A÷,J'a. , 0~-((~)A~'~ . Then there exists a function , continuous in the strong operator topology of 6 C X j X o ~ •
~-(~)ji~ea+
Then there exists ~ " l - ( a ) strong Abel. lim.
~= B [ ~ o , ~
intertwining
C~EHT~Ho
~ + (A~ =
H
and
Ho such that
~(:~
This theory has been applied [46] to completeness of wave-operators for the one-body problem with potentials V()~):.O(l~.l "I'L) as I Y . | ~ Here i t is essential that the decay of the potential is isotropic. In a recent paper i t has been shown by D.R. Yafaev [47] that break down of completeness is possible in 2 dimensions for potentials satisfying •
18
I with -I - ~ . Q1~ & ~
1I) J'' (.
C(-~X
. D.R. Yafaev argument is based on a nice Born-Oppenheimer l i k e
analysis and is i n t u i t i v e l y
rather convincing of the generality of this phenomenon.
Among other applications of abstract Stationary Scattering Theory
l e t me
mention the attempts to t r e a t the N-body problem in this framework by K. Yajima ~ for N = ~ and by I. Sigal ~ 5 ] . However for the general method to work i t is necessary that in addition to some reasonable assumptions on bound-states. and quasi-bound-states the potentials
V~ (~)
decay l i k e
I~I'~'~
~>0
in
I
order to avoid threshold problems ; such a r e s t r i c t i o n seems to be inherent to these approaches and to indicate t h e i r limitations. I w i l l not review here the general theory of smooth perturbations which furnishes a perturbative way to verify the hypothesis of abstract stationary theory ; I refer to [2~ -- , [4~ for its review and applications. Rather I would like to i n s i s t on the constructive approach of S. Agmon ~ I ~51 and S. Agmon and L. H~rmander F41] in Sobolev and Besov spaces. Let me f i r s t recall the e a r l i e r result [5] that the l i m i t i n g absorption principle holds for one-body short-range potential scattering in the a u x i l i a r y spaces
~..:
The r e s t r i c t i o n ]
L~4
V ~
with
4%~
" 7. is strongly related to trace kkpro pert]es ' on (n-l)
s'ona] momentum hypersurface~ (e.g. the enerqy shell
dimen-
- ~ ~pO ) of Fourier
II
transforBs of elements in L 4 . For this reason one cannot reach the l i m i t / :~ So i t is natural to ask whether there is an optimal a u x i l i a r y space such that
L for which the l i m i t i n g absorption principle would hold, Another question is how to adapt long range forces to the analysis in [5]
The f i r s t
question is answered
in [41] when y - ¢) but can be immediately extended to short range forces by the method of [5] , The results are as follows : let
the dual of the Besov space
. Then
I) The l i m i t i n g absorption principle holds in ~ weak l i m i t .
; the l i m i t in
(12)
is a
19
2) The outgoing (resp. ingoing) solutions of the Schr~dinger equation
~5~
(- ~+v-~
that is elements in
~t
the following sense : Let geneous of degree zero in once in ~ , satisfying
~:
e~
~e~
of the form
) satisfy generalized radiation conditions in be a symbol i n ( ~ 6 ~ ) ~ ~" homo¢~(~ j ~ and ~ , continuously differentiable n times in ~
,
Consider the associated pseudo-differential operator
Then i f a solution only i f
l ~ g ~" of ( - - I ~ - ~ - V - ~ )
~
:
~t
~
is outgoing i f and
The Sommerfeld radiation condition corresponds to the particular choice
associated to the symbol
tXl 3) Outgoing and ingoing elements of orthogonal in the sense
-~
are (for a fixed energy) asymptotically
4) Finally outgoing solutions have a radial asymptotic expansion at i n f i n i t y in the sense that i f ~ satisfies (15) and (16), then there exists a square integrable
20
function
n ~
%
on the energy sphere such that
Notice that for potentials ~
(which require decay faster than
l~I " L
)
this last result implies the v a l i d i t y of (1) in a precise sense. For potentials decaying faster than Coulomb solutions of the Lippman-Schwinger equation (13) s t i l l obey a decomposition like (1) i f instead of pure plane waves one uses some suitable superpositions belonging to
~
The main content of the recent (unpublished) work of S. Agmon [2] is to extend these properties to long-range forces. The basic idea is that since a priori_~ L,~ estimates for ( - ~ V ' ~ ) for short range forces as derived in L5~are uniform in ~, in any compact set outside the zero energy one should be able to adapt a s u f f i c i e n t l y small long-range perturbation. This program however requires for obvious reasons pseudo-differential
operator calculus, whereas e l l i p t i c
methods and Fourier analysis were enough for the short range case. My conviction that these progress open a new era in Scattering Theory does not come only from the fact that they provide with the help of some classical and powerful P.D.E. techniques some optimal results for one-body systems ; as an example the l i m i t i n g absorption principle ii Besov space gives almost immediately the last result of K. Chadan and A. Martin [5~ on finiteness of the cross-section for potentials such that I ~ % V ( ~ ) only have logarithmic decay. The main virtue of this approach certainly lies in the fact that in contrast to the t r a d i t i o n a l integral equation approach to scattering, these new techniques are linear in both m
the unperturbed ~ 0 and the potential ~ in a way which is perfectly well adapted to an i n t u i t i v e analysis of the phase space properties of scattering systems. This is certainly a great virtue with respect to more complicated situations where the'complexity of integral equation methods has considerably slowed down any progress. This is the case of course for the N-body problem and i t is very fortunate that new geometric ideas as those which have been advocated recently by P. Deift and B. Simon [51] seem to f i t admirably with this type of analysis. To conclude this section l e t me mention another application of the theory, namely low energy expansions of the S-matrix and time decay properties. The results of Ao Jensen and T. Kato [6] (see also J. Rauch --[7]..~f°r related results) state that as mappings between the weighted Sobolev spaces ~ ' - ' . ( ~ - - " and "~~' -i '~( ~ -- ~ ) for suitable ~>~-6, order of the expansion, one has
depending on the decay of the'"~potentialand the
21
=
In these formulas ~ _ ~ ~ ~0 the projection operator on the zero eigenspace of ~ , P~' is the projection operator-~f ~'~ associated to the eigenvalue ~ and ~)_~ dependson ~ and the zero energy resonance wave-function . that is a solution of H • = 0 belonging to ~ 4 . ( ~ $ ) butnot to L ~ ) . So ~ - 2 . and ~_~ do not appear for generic couplings. As a consequence the S-matrix ~ ( ~ ) has a low-energy expansion in ~ { L ~ C ~ ) ) , where S~ is the unit sphere in ~ :
Here C..O is equal to one when there is no zero energy resonance and related as usual to the scattering length.
C..4_"
is
This concludes this review of the present status of Scattering Theory. I t contains however important omissions like the beautiful work of K. Yajima on the classical l i m i t of the S-matrix or H. Kitada and Y. Saito works on long-range potentials ; fortunatley some of these gaps will be f i l l e d in the afternoon session.
22
REFERENCES 1] [2] [~ [4] [~ [6] [I [8] [9] DO] [17 [12] [I~ [14] [i~ [16] [17] .[18] [I~ [20] [2~ [22] [2~
W. GORDON,Uber den stoss zweier punktladungen nach der Wellenmechanik Z. Phys. 48, 180 (1928). S. AGMON, Lecture Notes, Salt Lake City (1978).. V. ENSS, Asymptotic completeness for quantum mechanical potential scattering, Commun.Math.Phys. 61, 285(1979). D. RUELLE, A remark on bound states in potential scattering theory, Nuovo Cimento A 61, 655 (1969). S. AGMON, Spectral properties of Schr~dinger operators and Scattering Theory, Ann. Scuola Norm. Sup. Pisa, I I , 2, 151 (1975). A. JENSEN and T. Y~ATO,Spectral properties of Schr~dinger operators and time decay, Duke Hath. J. 46 (1979). J. RAUCH, Local decay of scattering solutions for Schr~dinger equation, Commun.~lath.Phys. 6_~1,149 (1978). L, HORMANDER,The existence of wave-operators in scattering theory~ Math. Z. 146, 169 (1976). E. MOURRE, Link between the geometrical and the spectral transformation approaches in scattering theory, Commun.Math.Phys. 688, 91 (1979). T. KATO, Two-space scattering theory with applications to many-body problems, J. Fac. Sci. Univ. Tokyo, 24, 503 (1977). M. SCHECTER, Completeness of wave-operators in two Hilbert spaces, Ann.lnst. H. Poincar#, XXX, 109 (1979). T. KATO, Some recent results in scattering theory, Univ. of California Preprint, Berkeley (1979), P. COTTA-RAMUSINO,W. KRUGERand R. SCHRADER, Quantum scattering by external metrics and Yang-Mills potentials, to appear in Ann. I n s t . H. Poincar#. R. SCHRADER, High Energy behaviour for n o n - r e l a t i v i s t i c scattering by stationary external metrics and Yang-Mills potentials, CERN preprint TH 2698 (!979). C. CHANDLER,A,G. GIBSON, N-body quantum scattering theory in two Hilbert spaces, Journ.{~ath.Phys. 19, 1610 (1978). M. SCHECTER, A new criterion for scattering theory, Duke Math. Q., 444, 863 (1977). B. SIMON, Scattering theory and quadratic forms : on a theorem of Schecter, Commun.Math.Phys. 5__33,151 (1977). T. KATO, On the Cook-Kuroda criterion in scattering theory, Commun.~ath.Phys. 67, 85 (1979). J. KUPSH,W. SANDHAS, M~ller operators for scattering on singular potentials, Commun.Math.Phys. 2, 147 (1966). M. REED, B. SIMON, Scattering Theory (Methods of Modern ~4athematical Physics, vol. I I I ) , Academic Press (1979). L. HORMANDER,The existence of wave-operators in scattering theory, Math. Z. 146, 69 (1976). T. KATO, On f i n i t e dimensional perturbations of self-adjoint operators, J.Math. Soc. Japan 9, 239 (1957). D.B. PEARSON, A generalisation of Birman's trace theorem, J. Funct.Anal. 28, 182 (1978).
23
[24]
J.D. DOLLARD, Interpretation of Kato's invariance principle in scattering theory, J. Math. Phys. 17, 46 (1976). P. LAX and R. PHILLIPS, Scattering Theory , Academic Press, New York, 1967.
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P. DEIFT and B. SIMON, On the decoupling of f i n i t e singularities from the question of asymptotic completeness, J. Funct. Anal. Z3, 218 (i~76). A. JENSEN and T. KATO, Asymptotic behaviour of the scattering phase for exterior domains, Commun. in P.D.E 3, 12, 1105 (1978). M. COMBESCUREand J. GINIBRE, Scattering and local absorption for the Schr~dinger operator, J. Funct. Anal. 29, 54 (1978). S.T. KURODA, Perturbations of continuous spectra by unbounded operators, I,II. J. Math. Soc. Japan, 11, 247 (1959). M. G. KREIN, On the trace formula in the theory of perturbation, Mat. Sbornik 33 (75), 597 (1953). G. DETSH, Introduction ot the theory and applications of the Laplace transform, Springer, Berlin (1974). H. McKEANand I.M. SINGER, Curvature and the eigenvalues of the Laplacian, J. Diff. Geometry, 1, 43 (1967). P.B. GILKEY, Recursion relations and the asymptotic behaviour of the eigenvalues of the Laplacian, Princeton University Preprint (1977). V.S. BUSLAEV, Dokl.Akad. Nauk SSSR, 19___77,999 (1971) ; Soviet I1ath. Dokl., 12, 999 (1971). A. MAJDAand J. RALSTON, An analogue of Weyl's theorem for unbounded domains, I, II. Duke Math. Journal 45, 183 and 45, 513 (1978). An Epilogue, I I I , University of California Preprint (1979), T. KATO, Monotonicity theorems in scattering theory, Hadronic Journal i , 134 (1978). B. SIMON, Kato's inequality and the domination of semi-groups, J. Funct. Anal , 32, 97 (1979). H. HESS, R. SCHRADERand D. A. UHLENBROCK,Duke Math. Jour., 44, 893 (1977). H. HOGREVE, R. SCHRADERand R. SEILER, Nuclear Phys. B142, 525 (1978). B. SIMON, Phase space analysis of simple scattering systems. Extensions of some work of Enss. To appear in Duke Math. Jour. S. AGMONand L. HORMANDER,Asymptotic properties of solutions of d i f f e r e n t i a l equations with simple characteristics, Journal d'Analyse, 1976. T. GREENand O. LANFORDI I I , Rigorous derivation of the phase shift formula, J. Math. Phys. ! , 139 (1960). T. IKEBE, Eigenfunction expansions associated with the Schr~dinger operators and their application to scattering theory, Arch. Rat. Mech. Anal., 5, 1 (1960). M. SCHECTER,Completeness of wave-operators in two Hilbert spaces, Ann.lnst. H. Poincar~, XXX 2, 109 (1979). I. SIGAL, Scattering theory for many-particle systems I, I I , Preprint ETH ZUrich T. F-~ATOand S.T. KURODA, The abstract theory of scattering, Rocky Mtn.J.i,lath. ! , 127 (1971).
24
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D.R. YAFAEV, On the break-down of completeness in potential scattering, Commun.Math.Phys. 6_55, 167 (1979). K. YAJIMA, An abstract stationary approach to 3-body scattering, J.Fac.Sc, Univ. of Tokyo, Sec.lA, 25, 109 (1978). J.M. COMBES, Scattering theory in quantum mechanics and asymptotic completeness, Proceedings of the AMP Conference, Rome 1977, Springer Lecture Notes. K. CHADANand A. MARTIN, Boundedness of total cross-section in potential scattering, Poster session of this Conference. P. DEIFT and B. SIMON, A time-dependent approach to the completeness of multiparticle quantum systems, Com.Pure Appl.blath. 3__0_0,573 (1977).
S C H R O E D I N G E R O P E R A T O R S W I T H E L E C T R I C OR M A G N E T I C FIELDS W. H u n z i k e r Institut for T h e o r e t i s e h e Physik ETH H~nggerberg, 8093 Z[rich, Switzerland
The Zeeman- and Stark effect are first examples of q u a n t u m m e c h a n i c a l p e r t u r b a t i o n theory.
Nevertheless
velop an adequate m a t h e m a t i c a l
it has taken half a century to de-
description.
Here we summarize the re-
sults of a s y s t e m a t i c effort in recent years, and Simon
notably by Avron~ Herbst
[2,35].
I. M A G N E T I C FIELDS
I. Kato's i n e q u a l i t y A key to H a m i l t o n i a n s w i t h general external m a g n e t i c is Kato's d i s t r i b u t i o n a l A;uJ
-
which holds a(x)
inequality
6 Re (sgn u)
[13,57 If]: (iV + a(x)) u
in any d i m e n s i o n if
u s LI ~oc is a real c l - v e c t o r field and
{
u--~{u
(sgn u)(x)
=
(x){
fields B(x)=rot a(x)
-i
0
and
,
(iV + a)u c L I £o0
if u(x)
~ 0
if u(x)
= 0 .
(i) . Here
With this i n e q u a l i t y the f o l l o w i n g t h e o r e m has an easy proof [13, 57 II]:
Theorem i
Let a(x) be a real cl-veetor field on
R n and
0 ~ V I s L2~oc (Rn). Then the o p e r a t o r
(iV + a) 2 + V 1 = (p - a) 2 + V I
is e s s e n t i a l l y s e l f a d j o i n t on
C~
(2)
(Rn).
0
More recently,
Kato's i n e q u a l i t y was r e c o g n i z e d as the i n f i n i t e s i m a l
form of an i n e q u a l i t y b e t w e e n semigroups
[11,18,19]:
28
Theorem
2
bounded
f r o m below.
Let
Semi$roup le-tBu[
A
and
B
be s e l f a d j o i n t
T h e n the f o l l o w i n g
operators
on L2(M~dB)
3 statements
and
are e q u i v a l e n t :
domination
~ e-tAlul
(3)
for all u and 0 .< t < Resolvent
domination -i
I(B - z)-lu I* ~ (A - z) lUi for all
Kato's
u
and
(4)
- ~ < z < ~ (A)
inequalities
(f,Alul)
~ Re
(f,
(sgn u) Bu)
(5a)
(f,AluE)
~ Re
(f,
(sgn u) Au)
(5b)
for all 0 ~ f e Q(A) u e D(A)
. Here
= form domain
(f,Aiu~)
of A and all
is u n d e r s t o o d
u e D(B)
resp.
in form sense.
Remarks (i)
L 2 is u s e d to h a v e This
is g e n e r a l i z e d
external (ii)
(3
-tA
for This (iii)
Then
(-tA)
[11,16]
is p o s i t i v i t y
domination
Vl
of
)u I ~ e -tp2
i and 2 we note that
lul
(6)
to e s t i m a t e
of r e l a t i v e
u .
[18].
(i) and t h e o r e m s
is c o n v e n i e n t
operator
in
preserving:
for all (Sb)
lul •
to p a r t i c l e s
u ~ 0 to
u ÷
.
[e -tA ul~ e-tAlul
As an a p p l i c a t i o n
(4) implies
value map
u >~ 0
Ie - t ( ( p - a ) 2 +
Resolvent
absolute
[ii] and a p p l i e d
fields
that exp
is e q u i v a l e n t
a multiplication
in
Yang-Mills
implies
e
a natural
bound
~
perturbations. with respect
Let V to
A
be .
27
~IU ( B and letting z + - ~
z)-l[l
g ~V(A-
we conclude that
V
respect to B. Combining t h e o r e m 1 with of H a m i l t o n i a n s with m a g n e t i c fields Theorem 3 If
V2
Let
a(x)
and
Vl(X)
,
has r e l a t i v e bound
< e with
(6) we thus obtain a large class
[13, 57 II]:
satisfy the hypothesis
is a real m u l t i p l i c a t i o n
respect to
z)-l~
of theorem 1 .
o p e r a t o r of relative bound < 1 with
A , then H (a) = (p - a) 2 + V 1 + W 2 = Ho(a)
is e s s e n t i a l l y s e l f a d j o i n t on
C~(R n)
+ V
(7)
, b o u n d e d below and
O
le -tH(a)
for all u
and
uI
g e -tH(°)
lul
(8)
0 .< t <
Remarks (i)
In the following we always Other results 57 II,59], X.4 of
and r e f e r e n c e s
[57 II].
orem 3 a l l o w i n g bound (ii)
< 1
assume
on s e l f a d j o i n t n e s s
In particular,
there is a f o r m - a n a l o g u e of the-
a 6 L2zoc ' C g V1 s L 1toe
r e l a t i v e to
A
and
V2
of form-
[4,20]. (quoted in [19]) that
i n s p e c t i o n from the f u n c t i o n a l (-tH(a))
to be of this type.
to earlier work in the notes to section
It was r e m a r k e d by Nelson
exp
H(a)
are found in [4, 14, 15, 20,
[1O, 20, 61].
(8) follows by
integral r e p r e s e n t a t i o n of
Closely related is a simple proof of
(8) using the T r o t t e r product formula and the fact that in one
dimension
a(x)
can be gauged away
[20].
2. D i a m a s n e t i s m For identical particles, metric wave functions.
(8) also holds
in the subspace of totally sym-
It expresses the d i a m a g n e t i c
trary m a g n e t i c field on a system of spinless Theorem 4
For
H(a)
given by
particles
[17, 18]:
(7),
inf o(H(a) Tp-~H(a) e
effect of an arbi-
~ inf ~ (H(0)) ~ Tr e
-~H(O)
(9) (i0)
28
Proof
(9)
directly states Lemma
is e v i d e n t
by K a t o ' s
is g i v e n i
Let
in
from resolvent [17].
[61].
follows
A,B
(i0)
be b o u n d e d ,
fying
IAul ~ B lu I for all
class
and
Tr A $ Tr B
We r e m a r k natively,
that
one
By the
arbitrary
fields.
cussed
[i0].
in
3. S p e c t r a l (8)
Lemma IAu~
As
2
Other
Let
A,B
a = 0
ingredient Lemma
3
in the
the
be b o u n d e d u
this
. If
is w e l l
class
(it) then
(i)
[53] that
All
sequence
[58,
there
bounds
A
is t r a c e
suitable
the
VI
. Alter-
in a box.
(field-independent)
quantum
free
energy
and c l a s s i c a l
limits
compactness
criterion
following
operators B
known
on
L2(M,d~)
is c o m p a c t ,
then
is c o m p a c t
and easy
domination
and
points
for
to prove.
lemma
E s ~ess
u
57 Iii] exist
two
n If
E CO u
with
IIu n
are
in dis-
[52,
56]:
satisfying
A
is compact.
any m u l t i p l i e r For
2. This
(H(a))
F s CO
a ~ 0 it then
is the
essential
t
n
F e CO .
and to all
+ + =
ull
arise
from
"states
at ~"
(ll)
÷ 0
= i
is o r t h o g o n a l
sequences
-it H(a) ~IF e n
for any
then
satis-
of
sense
some
L2(M,d~)
class
for
for the
II(H (a) - E) Un~ for
on
3 for p a r t i c l e s
inequality
F (i + H(a)) -I
from resolvent
ground
Theory
using
for all
operators
is t r a c e
bound
classical
Scatterin$
a consequence,
For
B
theorem
is a l o w e r
is e x p l o i t e d
be p r o v e d
for
from
is t r a c e
formulate
energy
and
~ B lu~
follows
. If
also
argument
.
Golden-Thompson-Symanzik free
Here
u
and can
An e l e m e n t a r y
positive
exp(-BH(0))
should
classical
domination
inequality
such
+
0
u n (x)
= 0
eigenvectors
for of
[xl
< n
H(a),
that
(12)
.
29 Scatterin$ Theory
(12) is the starting point in Enss' proof of asymp-
totic completeness
[54], w h i c h is e x t e n d e d in [60] to h o m o g e n e o u s mag-
netic fields. For general
Similar results
theory to the pair case
H(a), H0(a)
under the same conditions as in the
in general there is a diamagnetic
(9) on the b o t t o m
Z(a)
effect analogous
of the e s s e n t i a l s p e c t r u m of
H(a)
Z(a) ~ ~(0) Proof:
Setting
Since
lUnl ÷ 0
weakly,
~ (Un, H(a) u n)
: (13)
E = Z(a) in (ii) and using Kato's
(lUnl, H(O)lUnl) : Z
[4].
allows to apply the K a t o - B i r m a n
a = 0 [4].
Essential spectrum to
can be obtained by Agmon's methods
a(x), r e s o l v e n t d o m i n a t i o n
inequality we obtain
.
Z(O) g lim inf (lUnl , H(O)lUnl ) ~ lim (Un,H(a)u n)
(a)
A c c o r d i n g to
(ii) only the b e h a v i o u r of V(x) and a(x) at ~ is r e l e v a n t
for the e s s e n t i a l spectrum.
Since a(x) is g a u g e - d e p e n d e n t
it is desir-
able to have some c r i t e r i a in terms of the m a g n e t i c field itself, which can be i n t r o d u c e d as the c o m m u t a t o r
Bk£ = ak, Z - a£, k
=
i [~k' N£]
'
where K = p - a . By the Schwarz i n e q u a l i t y
(14)
As an application,
assume that a single field component
as Ixj ÷ = . Then by B ~ inf ~
(ii) and
ess
(H
0
(a))
,
an i n e q u a l i t y which is saturated for h o m o g e n e o u s H0(a)
has purely discrete
sophisticated
Bk£ ÷ B > 0
(14)
s p e c t r u m if some
field.
BkZ ÷ ~ as
In p a r t i c u l a r Ixl
÷ ~ . More
"magnetic b o t t l e s " are d i s c u s s e d in [4]. It is clear from
(ii) that these results V(x) v a n i s h i n g at
~
are n o t affected by the p r e s e n c e of a potential For such a case the effect of a m a g n e t i c
on the s p e c t r u m is shown in Fig. i:
field
30
a
=
0
•
t
,
,
(a)
= B for homog.field
a ~ 0 i
new bound states ,(
binding energy
In the following we summarize negative
energy bound
and enhanced binding bound states,
Here we assume
possible effect,
effects
results
of various
generality
(number,
stability,
perturbation
in magnetic
strong binding
4. Negative Energy
multiplicity
states
For any
since some eigenvalues i). However,
of a(x)
Lieb-Rosenbljum
bound
in
< - ~
~ ~ 0 of
by resolvent
let
H(a)
may decrease
(4K)-2 fd3xd3y IV (x)l e
is independent
which holds
new
N (a) be the total
. N (a) > N (0)
is
(as in first order Zeeman
domination
and Lemma 1 the
bound
Ne(a)
N0(a)
(binding energy,
Bound States
V(x) ÷ 0 at ~
Birman-Schwinger
expansion)
in high fields).
of all eigenvalues
see Fig.
fields
concerning
[4,10].
( Cn f dnx
~V(y)~
Ix-yl
In [4] the same is proved
(15)
for the Cwiekel-
Iv(x)I n/2
n ~ 3 dimensions.
(16)
The reason
is that
(apart from the
value of c ) (16) is a classical phase volume and thus independent of n a(x). An immediate consequence of these estimates was noted in [4,10]: Theorem
5
extends
to matter
FrShlieh
The Lieb-Thirring
estimate
of the stability of matter
in arbitrary magnetic
fields,
[65]
and by a remark of
(given in [4]) also to matter coupled to a quantized radiation
field. The stability (homogeneous
of any eigenvalue
field) results
E(0)
< 0
of
from the continuity
H(0)
for
properties
a(x)
i = ~ B A x
of the re -
31
solvent as
B ÷ 0 [4]:
(z - H(a)) -I = (z - H o ( a ) ) - I + (z - H(a)) -I V(z - H0(a))-i
strongly (not norm) continuous as B + 0
for
z
around
in the r e s o l v e n t E(0)
norm continuous as B ÷ 0
set of H(0).
By i n t e g r a t i o n over a small circle
it follows that the c o r r e s p o n d i n g
is n o r m continuous
as
B ÷ 0
Therefore
of a group of eigenvalues , { } E _k (_a )
of
E(0) H(a)
spectral p r o j e c t i o n is the limit as
B + 0
with the same total mul-
tiplicity. The R a y l e i g h - S o h r ~ d i n g e r
perturbation
series has been studied in
p a r t i c u l a r for H(B)
1 1 2 = ~ (p - ~ BAx)
A proof of Borel s u m m a b i l i t y
1 r
(including atoms) has been a n n o u n c e d
For the ground state energy p e r t u r b a t i o n E0(B) ~ ~
[2,7].
series
a2n B 2n
n=0 there is a c o n v i n c i n g a2n =
(but not fully rigorous)
( i)n+i(4)5/2 ~
-2n
argument
[1] that
i , (2n + ~). (i + 0(~))
which is very well s u p p o r t e d by the n u m e r i c a l values of n = i00
a2n up to
[i].
5. E n h a n c e d Bindin$ The following i n e q u a l i t y of Lieb holds i a(x) = 7 BAx [4]: inf o (H(a)) - B
for arbitrary
V
and
(17)
~ inf ~ (H(0))
In the ease V(x) + O at ~ (Fig. i) this says that the ground state b i n d i n g energy is e n h a n c e d in h o m o g e n e o u s m a g n e t i c of (17) is the " p a r a m a g n e t i c
field.
A n o t h e r form
inequality"
inf ~ (~(a)) $ inf o (~ (0))
(18)
32
where
~(a)
extended
is the
spin
in 2 d i m e n s i o n s
1/2
a counterexample
[9] in two
hold
a's
for g e n e r a l
Following cussion
Hamiltonian
to p o l y n o m i a l
and
dimensions
V's
1-particle
H(B,I)
= -
+ ~ ~
[8].
However,
which
shows
. (18) has
been
there
is a l s o
(18)
cannot
that
[12].
[4] we c o n c l u d e
of the
H(a)
B's
our r e v i e w
Zeeman
of m a g n e t i c
fields
with
a dis-
Hamiltonian
d2
2 2
+ IV
(x 2 + y
; z)
dz Axy
-
assuming [0~
B ~ 0 . In e a c h
±i,
±2... ]
purely
discrete
spectrum
and a corresponding V(x)
sector
+ 0 L
at
= m
angular
the b o t t o m
m
if
(19) B2 + ~ (x2 + y21
line
with
'
momentum
of
(19)
a lowest
subspace
Lz : m e
(as an o p e r a t o r
oscillator
, H(B,I)
. For t r i a l
V (z) m exhibits
leads
the
same
has
eigenfunction
essential
states
Zm
=
]dxdy
the
is t r u e
I>
of the
form
embedded
many
is that
[Zm, ~)
u = f (x,y)
in the
g(z)
,
with
g
(x,y)~ 2 V ( x 2 + y 2
m
' aspect
(20)
z)
of the
if V ( 0
. Then
for any
eigenvalues
ground
happen
a homogeneous ions
(20) for
corresponding
(V s h o r t - r a n g e ,
the
cannot
of n e g a t i v e
B - E0(B)
V
Z0 = B
at all
mechanism,
From
Hence
' problem
and m
not
and
for
B i 0
identically I > 0
zero,
,
if g(z) ~ exp (- ~Izl) with ~ s u f f i c i e n t l y small. u < ~m if B > 0 and k > 0 , H ( B , I ) has i n f i n i t e l y m a n y d i s c r e t e
eigenvalues
this
~f
+ I V
effect:
for all
nitely
(191
dz 2
one-dimensional
below
of
fm(X,y).
spectrum
m
eigenvalues
That
+
=
to t h e ' f o l l o w i n g
(B,I Therefore
has
m
and
L2(R2))
eigenvalue
z
which
on
= B ([m!- m + I)
harmonic =
- BLz
state
for
with
m = 0,1,2...
m < 0 - even
I small). may h a v e
V = -i/r
magnetic
to
m
~ 0
is p r o v e d
field
if
Another
, and
H(0,1)
peculiar
and may in
can produce
infihas
no
feature
be d e g e n e r a t e .
[7].
By the
extra
bound
same states
[3,7].
it a l s o
follows
V = -i/r
that
diverges
the
ground
state
as
B ÷ ~
, since
binding the
energy
Coulomb
poten-
33
tial is c a t a s t r o p h i c
in one dimension.
÷ ÷ BI/2 x ÷ x
In this case, scaling
gives the unitary e q u i v a l e n c e H(B,I) % B H(I i.e. a w e a k c o u p l i n g p r o b l e m for asymptotic
B -I/2)
B ÷ ~ . This is used to derive the
law [2,6]:
B - E0(B)
=
[~ ~n B - £n ~n B - 0,05796 in extreme m a g n e t i c
fields
+ O
(,~n ~n
B)]2
£n B
(B > > i ~ 109Gauss)
We end with the remark that the 2-particle reduce to the 1-particle p r o b l e m geneous
magnetic
Zeeman p r o b l e m does not
(19). T r a n s l a t i o n e o v a r i a n c e
in homo-
field and the a n a l o g u e to the separation of the center
of mass are d i s c u s s e d in [5]. This analysis theory of the Zeeman effect in atoms
forms the basis for the
[7].
II. E L E C T R I C FIELDS 6. Stark H a m i l t o n i a n s Qualitatively,
the Stark effect in h y d r o g e n should be r e g a r d e d as the
"free fall" in h o m o g e n e o u s potential.
electric field, p e r t u r b e d by the Coulomb
This is the point of view in
T h e o r e m 6 [57 IV] For real H0(f)
f = - A - fx I
is e s s e n t i a l l y s e l f a d j o i n t on f I 0
C 0 (R 3)
and has s p e c t r u m
If
V ~ L2 (R 3) and V(x) + 0 for Ixl ÷ ~ " ~oc pact r e l a t i v e to H0(f) Hence, if V is also real, H(f)
If
V
is tom-
= - A - fx I + V
is e s s e n t i a l l y s e l f a d j o i n t on for
(- ~, +~) if
then
C 0 (R 3)
and has s p e c t r u m
(- ~, + ~)
f ~ 0 . f ~ 0 , s c a t t e r i n g theory for the pair
fact that the electric field a c c e l e r a t e s t ÷ ± ~
T h e r e f o r e it suffices
for
H(f), H0(f)
reflects the
the particle like
x I ~ t 2 as
V(x) to fall off faster than xl/2
34
as
x I + + ~ . In particular, the Coulomb potential has short range
in this sense.
From the many results
[21, 33, 47, 49, 60, 60] we
quote an example of [60]: Theorem 7 for some
Let
f ~ 0. If
V(I + x~) g+I/4 is compact relative to
a > 0, then the wave operators itH(f)
~± = s-lim t + ±=
-itH0(f)
e
e
exist and are complete in the strong sense: by the eigenveetors of multiplicities
H(f).
(Ran ~ ± ~
The eigenvalues
of
= subspace spanned
H(f)
for
f ~ 0
was proved for h y d r o g e n by T i t c h m a r s h
and more generally in [21]. Sufficient conditions
(i)
If
have finite
and no point of accumulation.
Absence of eigenvalues [46]
H0(f)
V(x)
is smooth
in some right half space
are, essentially: a < x I < = where
the total force satisfies ~V + f > e > 0 ~x I then any solution of (it)
If, in addition,
V(x)
continuation theorem where
S
H~ = E~
vanishes
identically in a<Xl<~.
satisfies the hypothesis
of a unique
(e.g. V(x) locally bounded on
is a closed set of measure zero such that
is connected),
then
~ (x) = 0
for all
R3~S
,
R3~S
x .
As a c o n s e q u e n c e
of this result and t h e o r e m 7, H(f)
alent to
H0(f)
for a large class of potentials
potential
[33]
is u n i t a r i l y equiv-
including the Coulomb
.
7. Stark Resonances We have just seen that all bound states of into the c o n t i n u u m when the electric pearance as concentration
f ~ 0
H(0)
g e n e r a l l y dissolve
field is turned on. Their reap-
can be described not only in terms of spectral
[25, 40, 41,
57 IV] but more c o n c i s e l y by resonances
the framework of d i l a t i o n analyticity. drogen this is d i s c u s s e d
For the separable case of hy-
in [27, 28, 30]. Here we summarize the more
general result of [34] which can be extended to m u l t i p a r t i c l e
[36]. By a real d i l a t i o n
in
x ÷ e ~ x, H(f)
is u n i t a r i l y e q u i v a l e n t to
systems
35
H(8,f)
= U(8)H(f)
= e -28 p 2 - e e f x l
+ V(eSxl )
= H0(e,f)
, where
U(e)
We a s s u m e
dilation
For some analytic The
Stark effect
H(8,f)
with
t r u m of
+ v(e)
of
n < i , V(8)(I e
V
(22)
in the f o l l o w i n g
+ p2n)-i
in a s t r i p
is c o m p a c t
sense:
and
(2s)
!Im e I < Y .
is now d e s c r i b e d and
(21)
~ e 38/2 u ( e S x )
analyticity
f e R
in terms
llm e~ < y ,
of s p e c t r a
starting
of the f a m i l y
f r o m the u n p e r t u r b e d
spec-
p2 + V (Im 8 = 0, f = 0):
For s i m p l i c i t y ne x t
: u(x)
in
-I "(8)
U
E(O)
0
•
!
E(0)
is a s s u m e d
step is to take
(Fig.2)
to be a n o n d e g e n e r a t e
Im 8 ~ 0
(we c h o o s e
eigenvalue.
The
!m e > 0), still k e e p i n g
f = 0 :
E(o)
0
(Fig.3)
This
is the r e s u l t
rest
of the s p e c t r u m
values
are a n a l y t i c
alence. New
of
continuum
8
and
and
E(0)
remains R
swings
independent
and i n d e p e n d e n t
eigenvalues
but nowhere
N o w let
The c o n t i n u u m
is d i s c r e t e in
In p a r t i c u l a r ,
("resonance")
discrete
[51]:
of
be an a n g l e of
Re 8
an e i g e n v a l u e
may appear
2 Im 8. The
e , since by u n i t a r y
the eigenequiv-
as long a s I l m el< ~/2.
in the s e c t o r
swept by the
else.
0 < Im 8 < ~/3 independent
of
and 8 :
f $ 0 . Then the s p e c t r u m b e c o m e s
purely
36 /
E(f)
i I.,¢L/L''
®
While the value of Im 8
is no longer visible in the s p e c t r u m , it still
plays an important role for stability: eigenvalues are stable as functions of "Stark resonance"
E(f) ~ E(0)
pansion to all o r d e r s . i n
as
in the shaded h a l f p l a n e
S
the
f . In p a r t i c u l a r there is a
f ÷ 0 ~ w h i c h has an a s y m p t o t i c ex-
f : co
E(f) ~
Since
E(f)
E a fn n--0 n
is independent of
(24)
e , this is p r e c i s e l y the Rayleigh-
Sehr~dinger series c a l c u l a t e d for the p e r t u r b a t i o n Finally, tinuum
if we set
(-=,+~)
of the Stark H a m i l t o n i a n
p2 + V .
H(0,f). H(8,f)
However, there is a
shown in Fig.
4
and
:
(u,(z - H(0,f))-lu) Here
of
Im 8 = 0 , the s p e c t r u m jumps back to the con-
direct r e l a t i o n b e t w e e n the s p e c t r u m of H(0,f)
-fx I
u
and
u
= (U(8)u
, (z-H(8,f)) -I U(8)o)
(25)
are from a dense set of d i l a t i o n - a n a l y t i c vectors, and
(25) is first e s t a b l i s h e d for
z
in the set
S ~{Im
z > 0]
f i c i e n t l y far from its boundary, where both resolvents lytic c o n t i n u a t i o n (i)
For
in
f a R
z and
and suf-
exist.
By ana-
it then follows that: Im 8 > 0
the s p e c t r u m of
H(e,f)
is in the
lower h a l f plane and in fact strictly below the real axis if the Stark H a m i l t o n i a n (ii)
The m a t r i x elements t i n u a t i o n from eigenvalues
has no eigenvalues
(Fig. 4).
(u,(z-H(0,f)) -lu) have a m e r o m o r p h i c con-
Im z > 0
shown in Fig.
The clue to these results as
H(0,f)
into
Im z < 0
w i t h poles at the
4.
is the explicit
"free fall-propagator" w r i t t e n
37
pt = exp
(- i t e - S H 0 ( 8 , f ) )
-it3ef2/12 =
where pt
~ = e
-3e
.
t i o n to
/)
Im
~ g 0
continuous
the g e n e r a t o r
. Via L a p l a c e
z s S (Fig.
boundary
of
(i.e.
0 $ Im 8 ~ ~/3)
and real
semigroup mapping as the c l o s u r e
f
f ,
into f
of its r e s t r i c -
we o b t a i n the b o u n d
~ d(z) -I
d(z)
f i 0
itfxl/2 e
~ H 0 (e,f)
transform
4), w h e r e
S . For
e
contraction
ll(z - Ho(e,f))-ill for
-it~p 2
e
For
is a s t r o n g l y
(This d e f i n e s
itfxl/2
e
and
is the d i s t a n c e
of
im ~ < 0
is a s t r o n g e r
there
z
f r o m the esti-
mate llPt~ which
falls
H0(8,f)
t3f21me/12
off f a s t e r
has
To s t u d y
g e
empty
t h a n any e x p o n e n t i a l
spectrum
the e f f e c t
for
f ~ 0
as
and
of the p e r t u r b a t i o n
t ÷ ~
. As a r e s u l t ,
0 < Im 8 < ~/3 V(8)
.
we c o n s i d e r
compact V(8)
pt
-it3~f2/12 = e
e
itfxl/2 ~ V(8)(I
-itap 2 (i + p2n)
4_ ~, + p2n)-i
itfxl/2
e
e I
D ~ ~ const The c o m p a c t
factor turns
the
strong
norm continuity~
so that the L a p l a c e
V(e)(z
-I
for
-
H0(8,f))
compact
(ii)
analytic
(iii)
norm convergent
to zero as
(iv)
norm continuous
as
z E S, f E R
The r e s u l t
to all of Fig.
continuity
for in
transform
Im ~ < 0
f~ 8
K(z,8,f)
and
t
into
=
is
(i)
(ii) e x t e n d
(i + t -n)
in
(z,8) d(z) ÷
f ÷ 0
and
0 < Im e < m i n
(¢, ~/3).
z
by the r e s o l v e n t
equation
4 now follows
and the r e p r e s e n t a t i o n
from analytic
We add that for
(i) and
(z - H 0 ( e , f ) ) ? l
Fredholm
theory
[57.1]
38
(z - H) -I
in w h i c h f * 0
(just as
for
Borel
let
8
V
In
in the
N
[36]
(24)
- H0)-IK(I
last
is p r o v e d
satisfy
term
- K) -I
is n o r m
for hydrogen
< 0
condition
resonance p2
of
+ V
. Then,
region
- ~ < arg estimate
N-I - E n=0
an
fn
CN
I ~
(23),
,
continuous
in
[27,
28],
0 < ~ < min
corresponding
asymptotic
as
and
for
for s o m e
f < ~ + e
N!
,
(y,~/2),
and
to a n o n d e g e n e r a t e ~ 0, E(f)
R
f < R
where
is
it s a t i s -
N
Ifl
(26)
.
this
is a c t u a l l y
degenerate
8, the
4) the
(z
[36]:
Stark
E(0)
IE(f)
tain
in
be t h e
a strong
for all
of
case
eigenvalue
fies
+
Theory
Let
E(f)
analytic
- H0 )-I
in s e c t i o n
summability
general
Theorem
(z
z ~ ~ (H(e,0)).
8. P e r t u r b a t i o n
the
=
Borel
cases
proved
for m u l t i p a r t i c l e
(including
transform
B(z)
hydrogen).
= E a zn/n]
systems
and
As a c o n s e q u e n c e
is a n a l y t i e
for cerof theorem
in the u n i o n
of
n
the
circle
~z~ < C -I
For s u f f i c i e n t l y obtained
small
by i n v e r s e E(f)
with
the
f > 0
Borel
sector the
~/2
Stark
- ~ < arg
resonance
z < ~/2
E(f)
+
can then be
transform:
= ~ f dt
B(~tf)
e -~t
,
O
where
~ ~ 0
puzzling the
has
question
complex If the
is c h o o s e n the
resonance
Stark
~/2
real
- ~ < a r g ~ < ~/2.
perturbation
series
This (24)
answers ean
the
determine
E(f).
Hamiltonian
has
no eigenvalues,
then
the
resonance
than
any p o w e r
E(f)
a width F(f)
for r e a l as
how
in
f ~ 0
~ - 2 Im E(f) , which
f ~ 0 . In this
case
by
> 0
(26)
vanishes
the p e r t u r b a t i o n
faster series
(24)
of
is d i v e r g e n t
f for
39
f ~ 0 . In f a c t
any
F(f)
determines
the
leading
term of
an
as
[23, 35, 36] :
n ÷ ~
Theorem then
9
a
If = 0
V(x) for
= V(-x)
in a d d i t i o n
to the h y p o t h e s i s
of theorem
8,
n odd and
n
R i = - ~ f F(x)
a2n
x
-(2n+l)
dx + 0(
R-2n)
(27)
O
for
some
R > 0
Proof
Since
E(-~).
Let
and
n ÷
E(f) K
is n o n d e g e n e r a t e ,
be t h e
- R $ z ( R < R
semicircle
. Then
(27)
V(x)
in
follows
= V(-x)
Im z ~ 0
implies
with
by e x p a n d i n g
E(f)
=
basis
both
sides
of the
identity i : --: f dz
E(f)
z E ( z ) ( z 2 - f 2 ) -I
K in p o w e r s
of
f .
9. H y d r o g e n The
general
tative
and
theory
for t h e
ground
lations
[22,
lifetimes analytic
of
Stark
quantitative state
24,
only sense.
resonance
26,
[22,
resonances
results
32,
39,
E0(f)
43,
39] r e f e r
in t h e
44,
by d e t a i l e d
ease
, in p a r t i c u l a r
V = -i/r
. Of the m a n y 45]
directly
Rayleigh-Schr~dinger
is i l l u s t r a t e d
of
Stark
numerical
energy
to r e s o n a n c e s
coefficients
a
calcu-
shifts
in t h e
quali-
and
dilation-
have
been
com-
n
puted
to h i g h
tative [22,
order
results
23,
26,
(n ~ i00)
concern
31,
43,
the
44,
by a l g e b r a i c
asymptotic
48]
or
schemes
behaviour
f ÷ ~
[22,
23]
of
[42,
44].
E0(f)
and of
The
as a
quali-
f ÷ 0 as
n ÷
n
[22,
23,
accuracy units,
44].
We q u o t e
in the
i.e.
range
f o r the
[22], w h e r e
10 -2
< f < 1020 . T h e
ground
state
E0(f)
resonance
is c a l c u l a t e d
results
are
given
to
high
in a t o m i c
of
i -i : - ~ A - fx I - r
H(f) f÷
from
O:
2 3f
4
r0(f~" = ¥
e
[i + A f + Bf 2 + O ( f 3 ) ] A = -8.916
, B = 25.87
(28)
40
This is accurate to i part in 104 up to is known as "Oppenheimers
formula"
f = 10 -2
after an incorrect result
and can be o b t a i n e d as a t u n n e l i n g p r o b a b i l i t y in [37,48].
This p r o c e d u r e to c a l c u l a t e
tended in [31] n + ~ :
Im E0(f)
As a c o n s e q u e n c e of (27) and
which is accurate to i part in 104 for +
oo
in [38]
WKB-approximation
is justified and ex-
. (28)
6 -3 2n A + 2B a2n = - ~ ~ ) (2n)![l+ 3-~ 9n(2n'l)
f
. The leading term
+ O(n-3)]
n > 12
.
•
arg E0(f)
= - ~~ I 1
+ ~I
+ o ~r£n F F / p£nf~ ~J] (29)
i f 3/2 8 Zn £nf IE0(f)l = ~ ( ~ £nf) [i + 3 % n ~ + 0(
a c c u r a t e to ~ 1% at
f = 1020
. The a s y m p t o t i c regimes
are joined by the n u m e r i c a l values range
10-2< f < 1020
[22].
)],
for
E0(f)
(28) and
(29)
in the large intermediate
41
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Oppenheimer J.R.: Phys. Rev. 31, 68-81 (1928)
[39]
Reinhardt W.P.: Int. J. Quant. Chem. Symp. iO, 359-367 (1976)
[40]
Rejto P.A.: Helv. Phys. Acta 4~3, 852-687 (1970)
[41]
Riddel R.C.: Pacific J. Math. 2_~3, 377-401 (1967)
[42]
Silverstone H.J.: Phys. Rev. A 18, 1853-1864
(1931)
(1978)
[43]
Silverstone H.J.: "Asymptotic relations between the energy shift and ionization rate in the Stark effect in hydrogen"; preprint 1979
[44]
Silverstone H.J.; Adams B.G.; Clzek J.; Otto P.: "Asymptotic formula for the perturbed energy coefficients and calculations of the ionization rate by means of high order perturbation theory for hydrogen in the Stark effect"; preprint 1979
[45]
Silverstone H.J.; Koch P.M.: "Calculation of Stark effect energy shifts by Pad~ approximants to Rayleigh-Schr~dinger perturbation theory"; preprint 1979
[46]
Titchmarsh E.C.: "Eigenfunction Expansions Associated with Second Order Differential Equations", Part II. Oxford University Press 1958
[47]
Veseli~ K.; Weidmann J.: Math. Zeitschrift 156, 93 (1977)
[48]
Yamabe T.; Tachibana A.; Silverstone H.J.: Phys. Rev. A 16, 877-890 (1977)
[49]
Yajima K.: "Spectral and scattering theory for Schr~dinger operators with Stark-effect"; to appear in J. Fac. Sci. Univ. Tokyo
[50]
Yajima K.: "Spectral and scattering theory for Schr~dinger operators w~th Stark-effect, II"; preprint 1978
VfV
Other References [51]
Aguilar J.; Combes J.M.: Commun.math. Phys. 22, 269-279
(1971)
44
[52] Dodds P.G., Fremlin D.H.:
[53] [54] [55] [56]
[57]
"Compact operators in Banach lattices"; submitted to J. Functional Analysis Enss V.: Commun. math. Phys. 52, 233-238 (1977) Enss V.: Cormnun. math.
Phys. 61, 285-291
Lieb E.H.: Rev. Mod. Phys. 48,
553-569
Pitt L.: "A compactness condition in function spaces"; submitted to J. Operator
(1976)
for linear operators Theory
Reed M., Simon B.: !,tMethods of Modern Mathematical Academic Press, New York
[58]
Ruelle D.: Nuovo Cimento
[59]
Schechter M.: "Spectra of Partial Differential North Holland, Amsterdam 1971
[60] [61]
[62]
Simon B.: Duke Math.
61 A, 655-662
J. 46, 119-168
(1978)
Physics"
Vol. I-IV;
(1969) Operators";
(1979)
Simon B.: "Functional Integration and Quantum Physics"; Academic Press, New York 1979 Thirring W.: "Lehrbueh der Mathematischen Physik. 3. Quantenm e c h a n i k yon Atomen und MolekNlen"; Springer Verlag, Wien, New York, 1979
A N E W M E T H O D FOR A S Y M P T O T I C C O M P L E T E N E S S
V o l k e r Enss F a k u l t i t fur Physik der U n i v e r s i t a t D-4800 B i e l e f e l d i, F.R. Germany
We p r e s e n t a g e o m e t r i c proof of a s y m p t o t i c c o m p l e t e n e s s ing systems which follows w i t h m a t h e m a t i c a l
rigor the physical
tion of the s p a c e - t i m e b e h a v i o r of a state. treat here q u a n t u m m e c h a n i c a l
The d e c o m p o s i t i o n
c o r r e s p o n d s to the p o i n t - resp., the s e l f - a d j o i n t H a m i l t o n i a n
continuous
H = H
pseudodifferential
Asymptotic completeness to the bound states state, lim T÷~
i.e.,
sup t>0
scattering H =
[3].
HP(H) •
Hc°nt(H)
spectral subspaces of
+ V, H O
plication-or
intui-
As an example we will
short-range potential
The H i l b e r t space is H = L 2 ( ~ ) .
for scatter-
= -(i/2m) A, V a multiO
(velocity d e p e n d e n t force)
operator.
(strong form) means that any state o r t h o g o n a l
(HP(H))
is an a s y m p t o t i c a l l y
free s c a t t e r i n g
for any ~ E Hc°nt(H)
II [exp(-iHt)
- exp(-iHot)]
exp(-iHT)~II = 0 ;
or e q u i v a l e n t l y , ~ Ran
or
~2out, g~out = s - l i r a e x p ( i H t )
lira [I (a°ut - ~ )
exp(-iHT)~
exp(-iHot)
II = 0
;
(i)
;
(2)
T-~Oo
and s i m i l a r l y for the past. tinuous spectrum. set of vectors times
This implies absence of a s i n g u l a r con-
A c t u a l l y it is s u f f i c i e n t to show
~ e Hc°nt(H)
(2) for a dense
and for a suitably chosen sequence of
Tn; that is w h a t we will do below.
We impose on the p o t e n t i a l a local c o n d i t i o n to avoid local absorption
[8] and decay assumptions.
Let
g,g
be functions in C
t y p i c a l l y one may think of them as s m o o t h e d c h a r a c t e r i s t i c g(H)
etc.
as s m o o t h e d projections.
By
F(I~ I ~R)
the m u l t i p l i c a t i o n w i t h the c h a r a c t e r i s t i c
(~) ,
functions,
etc. we denote
function of the i n d i c a t e d
region. We r e s t r i c t the s i n g u l a r i t i e s
of the p o t e n t i a l by assuming
local
compactness F(IX I ~ R) g(H)
is compact
V g, R <
(3)
48
Th i s
assumption
ator-sum,
is v e r y weak,
form-sum,
physical
Friedrichs
discussion,
N e x t we r e q u i r e
it h o l d s
if H ° + V is d e f i n e d
extension,
[i] for s u f f i c i e n t
that
far away
or
....
See
conditions
f r o m the s c a t t e r e r
as oper-
[6] for a
on V.
the p o t e n t i a l
becomes
W~ ak :
il[(n+i) -1 - (~o+i)-l]F(IIl : ll(H+i) "I V ( H o + i ) - I F ( I x (3) and
(4) t o g e t h e r
g(H)
r&~,
- g(H O) is c o m p a c t
e.g.,
in
I > R)I 1 ÷ 0
are e q u i v a l e n t
~ o s i n g l e out short-range
F~
>_ RIll R + ~
(4)
to:
for all g
(5)
~nteractions
we h ave
(4), b u t the f o l l o w i n g w e a k e r
to s p e c i f y
requirement
the d e c a y is s u f f i c i e n t :
all g,~ II~(H) v g(~o ) F(III >R)N=:h(R)
V
as
--
÷ )-l-s Ixl
(i +
We c o n s i d e r Support
iS a s h o r t - r a n g e
the d e n s e
not
does
~ L:([0,~)) potential
set of v e c t o r s
include
(6) in any d i m e n s i o n
~ s Hc°nt(H)
zero.
Thus,
whose
9.
compact
energy
for each ~ there is a
o0
g s C O(~\{0}) ~(H)~
-- ~
An a r g u m e n t i~
such t h a t
[i0])
(7) related
that
to the e r g o d i c
for any Y e Hc°nt(H)
theorem there
shows
( c o r o l l a r y on p . 3 4 3
is a s e q u e n c e
of times
Tn÷~
such t h at w -lim
exp(-iHTn)~
= 0
(8)
n-~0
TO p r o v e
(2) we w i l l
s h o w for any 8 > 0:
II (~out _ ][) exp(_iHTn)~ll< AS a f i r s t step we have b i g region:
IIF(Ixl < 2 R )
exp(-iHTn)~ll
IIF(I~I < 2 R )
g(H)
Ilg(Ho ) e x p { - i H T n ) ~
annihilated borhood
leave
l a t er d e p e n d i n g
an a r b i t r i l y
on e)
:
(3) and
II ÷ 0
(8)
as
[ii,i].
n ÷ ~
(i0)
Furthermore,
by
(5) and
(8)
- exp(-iHTn)~II =
exp(-iHTn)~
II ÷ 0
of the s t a t e w i t h
by g(H o) i.e.,
of zero,
(9)
the state w i l l
(to be c h o s e n
exp(-iHTn)~
by local c o m p a c t n e s s
Thus the c o m p o n e n t
as n ÷ ~
to w a i t u n t i l
for a n y R < ~
II [g(H o) - g(H)]
e
as
n ÷ ~
small kinetic
(ii) energy which
the c o m p o n e n t w i t h v e l o c i t i e s
disappears.
is
in a n e i g h -
47
Choose g(e)
E Co(R)
such that g(e)
g(~)
= g(~);
then by the intertwin-
ing p r o p e r t y (~out _ ~)
g(Ho)
exp(_iHTn) ~
= g(H) (~ °ut - ~ ) g ( H o) exp(-iHTn)~ + [g(H)
- g(Ho)]
g(H o) exp(-iHTn)~
The second term on the RHS vanishes a simple extension
(12)
as n ÷ ~ by
of Cook's method to estimate
(5) and
(8). W e
use
the first term.
For
any pair g,g: II~(H) (~out _ ~) £
f2
dt fig(H) V g(H o) exp(-iHot)#
We know already g(Ho)
g(Ho)¢l I
from
exp(-iHTn)~
integral
[I
(13)
(i0) that at time Tn, t = 0, the state
is far away where
the potential
is weak.
(13) will be small only if the state moves
t > 0 under the free time evolution.
But the
farther away for
This is true for the outgoing
part ~OUtn of the state g(H o) exp(_iHTn) ~ = ~OUtn + ~in.n A classical
free particle
if p • x > 0.
moves away from the origin in the future
Guided by the classical
4epending on p, we use a phase state.
intuition we will restrict
space d e c o m p o s i t i o n
One can split a state ~ into pieces
÷
~,c
p-space
of the q u a n t u m
~, ~ having strict ÷
localization centrated in +x-space
Alternatively
in small n e i g h b o r h o o d s of {pj} and being connear $ c =~. Then G °ut = Z ~9.~ where pj.$ ~ 0. one can use a smoothed projection onto the positive
spectral halfspace w.r.t, Typically ~out in n (~n)
the self-adjoint
in any such d e c o m p o s i t i o n
operator
the outgoing
has the following properties:
x-p + p-x.
(incoming)
part
(a) the compact kinetic energy
support is contained
in E > (m/2)v 2. > 0; (b) up to rapidly decreasmln ing tails it is localized in > 2R and p-x + x.p ~ 0 (< 0); (c) II ~out/in II < 2 II~ fin The following estimate forbidden
region"
ary phase m e t h o d Mellin
transforms
on the "propagation
into the classically
can be shown using direct calculation [10,12],
spectral
theory
[3], station-
[4], commutators
[7],
[9]:
IIF(I~ I --< R + Vmint)
exp(-iHot)~°Utn II
< AN(I + t + R) -N
V N, t > 0
The cutoff at R instead of 2R takes care of the q u a n t u m tails.
(14)
48
Now we are ready to finish the estimate
Io dt
V g(H o) e x p ( - i H o t ) ~ °nu t
1]g(H)
< 2 II~ II
(13) for ~OUtn = g(Ho)~nOUt:
II
dt llg(H) V g(H O) F(I~ I >_ R + Vmint)ll
]o
+ 11 (H)vg( o)ll r dt 11F(I L < R + Vmint) exp(-iHot) °ut LIUsing
(14) and the decay assumption
than any ~ > 0 for R big enough. future time evolution It remains analogous
(6) this expression
Thus a s y m p t o t i c a l l y
~out has a free n
and it lies in the range of ~out.
to show that the incoming argument
is smaller
component vanishes
as n ÷ ~.
shows that any incoming part would have
freely in the past
An
evolved
(relative to Tn) and would have had to come in from
even farther away: [g(H O) e x p ( - i H T n ) ~ ] i n z [g(Ho)
exp(_iHoTn)
~ [g(H O) e x p ( - i H o T n ) ~ ] i n ÷ ]in F(Ix I -> R + v m l .n T n )~ ÷ 0
because IIF(Ix I >R + Vmi n Tn)~ II + 0. proof of asymptotic
completeness.
A different version
[7] uses that
This finishes
(3) and
the outline of the
(6) implies
compactness
of
(H) V g(H O) and one shows that g(H) (~out - ]l) g(H o) is compact on the positive
spectral
subspace
The method can be extended
of p-x + x-p.
to treat long-range potentials
Dirac-equation
and very general
wave equations
[12,2], two cluster
tems as a special
free Hamiltonians scattering
case of m u l t i c h a n n e l
[4], the
[12], classical
for m u l t i p a r t i c l e
scattering
sys-
[5], and many other
applications. References 1.
2. 3.
4. 5. 6. 7. 8. 9. i0. ii. 12.
W. O. Amrein, V. Georgescu: Helv. Phys. Acta 46, 635-658 (1973). P. Cotta-Ramusino, W. KrUger, R. Schrader: Scattering by external metrics and Yang-Mills potentials, preprint F.U. Berlin, 1979. V. Enss: Commun. Math. Phys. 61, 285-291 (1978). - - : Ann. Phys. (N.Y.) 119, 117-132 (1979). - - : Commun. Math. Phys. 65, 151-165 (1979). R. Haag, J. A. Swieca: Commun. Math. Phys. i, 308-320 (1965). E. Mourre: Commun. Math. Phys. 68, 91-94 (1979). D. B. Pearson: Commun. Math. Phys. 40, 125-146 (1975). P. A. Perry: Mellin transforms and scattering theory, preprint Princeton Univ., in preparation. M. Reed, B. Simon: Methods of modern m a t h e m a t i c a l physics, Vol. III. Scattering theory, New York, London: Academic Press 1979. D. Ruelle: Nuovo Cimento 61__A, 655-662 (1969). B. Simon: Duke Math. J. 46, 119-168 (1979).
PATHOLOGICAL
SPECTRAL
PROPERTIES
D. B. Pearson
A particle obstacle
moving
of Applied of Hull,
in a straight
qi + Pi = i.
qi by
to the
(i - i)'th.
qi of reflection
labelled
say 1 and 2, the joint transmission
For a sequence
of n obstacles,
are additive.
If n = ~ there will be recurrence
the reflection
coefficient
now a quantum mechanical
(-~
into
~ t
(k2>O).
particle
made up of an infinite
(
Mll
k-lM12
kM21
M22
to the right,
The transmission
moving
is a =
one)
n ~ 1
)
transforming
probability
projected
to the
if and only if
~- ~i 1
potential
For a single bump, define
~
to the left of the bump
/
and reflection
of the Schrod±nger probabilities
equation
- ~
for a single bump,
in terms of the transfer matrix
+ V~ = as a funct-
to give a reflect-
ion coefficient = q__ 1 2 2 2 2 2). p = ~(M21 + MII + M22 + MI2 -
transfer
labelled
(i) and
(2) and separated
by distance
N, the joint
matrix becomes
ii (2)
~ k M 2 1 (2)
is
oi - the o's
in a one-dimensional
series of "bumps".
for solutions
ion of energy k 2, may be expressed
For a pair of bumps,
of
- qlq2 ) "
(i.e. a particle
right will return to its starting point with probability
the transfer matrix
back to
Define the coefficient
2 P = PlP2 + Pl(q2ql ) P2 + Pl (q2ql) P2 + .... = PlP2/(I Hence the joint coefficient of reflection is ~ =q= l-p p p = ~i + q2"
V(r)
qi of
coming f r o m the
qi = qi/Pi"
For a pair of obstacles,
Consider
At the i'th
there is probability
For example,
to the i'th there will be probability
(i + i)'th and Pi of transmission
reflection
a series of obstacles.
from left to right)
and Pi of transmission;
(i + i)'th obstacle
Mathematics,
Hull, ENGLAND.
line encounters
(i = i, 2, 3, ... labelled
reflection
the
Department University
k-iM12 (
cosNk
k-lsinNk
Mll
M22 (2)
-ksinNk
cosNk
kM21 (i)
M22 (i)
/
50
This leads to a combined reflection G(k) = a(k)
+ b(k)
For large N, a(k)
coefficient
of the form
cos2Nk + c(k) sin2Nk.
is a rapidly oscillating
function of k, with mean value
(averaging
over a range of k large compared with l/N) g i v e n by ~(k)
= a Ck) = ~i + ~2 + 2~i~2"
~ = ~1 (e-I
Writing
- i) , o i = ~1 (e ~i - i) we have ~ = 71 ~ + -12 .
Since ~ = log(l + 2~), it follows that This suggests that,
(n) , 0
n
n~H
(i + 2~) is multiplicative.
for a sequence of n bumps and in the limit as n --) ~, we look at
(n) (k) lim
I.e. ~ is additive.
being the combined reflection
coefficient
for the first n
(i + 2~. (k)) 1
i=l bumps.
This limit may be shown to exist in the sense of distributions
to a measure d ~ . L p r o v i d e d sufficiently
like
and converge
N. b e t w e e n c o n s e c u t i v e bumps goes to infinity 1 In other words, for large n and on averaging over energies,
rapidly.
~(n) (k) behaves
~ i=l
the
distance
(i + 2~. (k)). i 0m
If gi
"-9 0 as i --) ~
(gi = height of the i'th bump)
the infinite p r o d u c t i=l
(i + 2 ~ (k))converges l
if and only if
~ ~. (k) converges, i=l l
and the condition for this
2
is
Z
gi < ~
If
i=l
~ i=l
measure d ~
2 (and the N. increase sufficiently gi < ~ i ~ 2
is absolutely
continuous,
but if
~
gi
rapidly)
then the
= ~ then
i=l
(i)
d ~
is singular continuous with respect to Lebesque measure.
(2)
lim~(n) (k) = ~
for almost k
(w.r.t. Lebesque measure).
For large n, the trans-
n-~ mission probability
will be very small,
except for a small range of energies
i.e. we have an almost totally reflecting barrier, "transparent" (3)
H
=
_
dX2 +
which however
-
is almost
at the right energies.
V in L2(O, ~
has purely singular continuous
spectrum.
(The same
dr applies in L2(-~, ~) if the bumps are also made to extend in the negative r direction. HZ= - ~d 2 dr 2
Regarding V(r)
as a spherically
+ V(r) + Z(~ +2 i) r
each partial wave subspace ~.
in L2(O, ~
symmetric
3-dimensional
has singular continuous
potential,
spectrum in
51
Proofs of the above results can either be found in of [ i ]
.
[ i ] or follow the ideas and methods
A further reference on singular continuous spectrum is [ 2 ], which gives
an abstract characterisation of the singular continuous subspace in potential scattering. 1 . 2 .
Pearson, D., Comm. Math. Phys. 16, 30-36 (1978). t Sinha, K., Ann. Inst. Henri Poincare 2~, 263 (1977).
PROOF THAT THE H- ION HAS ONLY ONE BOUND STATE: A REVIEW, A NEW RESULT, AND SOME RELATED UNSOLVED PROBLEMS
Robert Nyden Hill Department of Physics, University of Delaware Newark, Delaware, 19711/USA
The H- ion, made up of a proton and two electrons, has long been known to have at
least
one
bound
state
below
calculations of sufficient accuracy discovery of hound
the Schrodinger
states do
1977.1'2
not exist
The present paper
three electron system H
the
lowest
breakup
threshold.
Variational
to demonstrate this appeared shortly
equation. below the
A
rigorous demonstration
lowest breakup
after the
that additional
threshold first
appeared in
will review that proof,
prove the new result
that the
has no bound states below
the continuum in the
spin 3/2
sector, and discuss related unsolved problems. Review The internal Hamiltonian Hin t for two electron atomic systems such as
H-, He,
or Li+ can be written in the form Hin t = H 0 - 27VI'V2 + Z-Iv
(i)
H0 = - ~ i - 2rll - ~ 2 - 2r21'
(2)
where and
÷ -i V = 21r+l - r21
Here rl,
r2 give the
(3)
electronic positions relative
to the nucleus, y = m/(m
= ~/M, m and M are respectively the electronic and nuclear masses, ~ is mass, and Z
is the nuclear
charge in units of
+ M)
the reduced
the electronic charge.
The length
unit is the reduced mass Bohr radius ~%2/(NZe2) and the energy unit is the hydrogenic binding energy Z2~e4/(~52) .
The notation agrees with
reference 2 except
that the
length and energy units differ by factors of Z and Z 2 respectively. The time-independent Schrodinger equation
can be solved exactly for
H 0 . The
coulomb repulsion V between the electrons, and the Hughes-Eckart term -2yV I. V 2 must, however,
be handled
via approximate
methods which
produce rigorous
bounds.
The
basic tool used is a well known comparison theorem: Theorem
i:
Let
H (I)
and
H (2)
be
Hamiltonians whose discrete eigenvalues can be characterized
two
essentially
self-adjoint (Hermitian)
below the bottom of the
by the faniliar variational
essential spectrum
principal E -- rain <~IHI ~>/<~I ~>,
with the minimization for excited states carried out subject to the I~> be orthogonal to preceding eigenvectors.
(i) by E 1
constraint that
Denote the ordered eigenvalues of H (i)
"~i) . E (i) %< .. %< E tij" " where E ~i~" " is %< E~ %< "" %< n " ess' ess
the energy at
which the
53
essential spectrLm (if any) begins.
Assume <~ IH (I) i~>
is defined for
all vectors
I~> for which <~IH(2) i~> is defined.
Then if <~iH(1) I~>~< <~H(2) I~> holds
for all
admissable state vectors I~>, E (I) ~< E (2) holds for all n, and E (I) ~< E (2) Upper n n ess ess" bounds on the n~nber of bound states below the continutm (or lower bounds to eigenvalues) are obtained from Theorem 1 by letting H (2) be the original Hamiltonian with H (I) a more tractable approximate Hamiltonian. The lowest plus a
threshold for breakup,
free electron at
rest at
infinity, lies at
follows from Hunziker's theorem that Hin t
and H 0.
Because H^
into a hydrogen
has an
atom in its
-i for both
ground state
Hin t and
H 0.
the continuous spectrtln begins at -i infinite number
of bound
It
for both
states below
-i, the
approximate Hamiltonian H~I) must contain enough of the original repulsive V to push all
but
one
of this
approximation to
infinity
V must
of
couple to
bound
states up
all of the
to
levels of
-I;
in
particular, the
H 0 below
the continuum.
Stated another way, the approximation to V must contain enough of the original preserve shielding: If one electron is in a hydrogenic ground state with electron far out, the far out electron
V to
the second
must see, after the replacement of V
by its
approximation, a potential which cannot support an infinite number of bound states. The approximation projection operator. 3
to V
is chosen
to have the
form VI/2pv I/2
where P
The fact that a projection operator such as P cannot increase
the length of a vector such as vl/21~> implies that <~Ivl/2pvI/21@> ~< <~IVl~>. H (I) , obtained
from
H (2)
hypothesis of Theorem i. range is the
=
Hin t by
replacing
V
P is constructed as follows.
space spanned by all
hydrogenic ground state and X
states of the form
is arbitrary.
QIV-IQI . Let Pl = V-I/2GIV-I/2"
Then
Let
P is chosen to be the
by
vl/2pv I/2,
Let Q1 be a
Thus
satisfies the
projector whose
#l(rl)x(r2) where ~I
G 1 be the generalized
Pl is a projection operator.
projection operator obtained by interchanging in P..
is a
is the
inverse of
Let P2
be the
the coordinates of particles 1
projection onto the span of the ranges of P1
and 2
and P2"
vl/2~//2 then couples symmetric states of the form ->
-~
->
->
~S(rl, r2) = f(rl)~l(r 2) + #l(rl)f(r2),
(4)
and couples antisyrmnetric states of the form ~A(rl, r2) = g(rl)#l(r 2) - ~l(rl)g(r2).
(5)
The space spanned by states of the forms ~S and ~A is denoted by *~U ; the orthogonal complement of ~I| is called ~ l .
States in J l
are not coupled by vl/2pv I/2.
Consider now the case of infinite nuclear mass, for which there is Eckart term ( y= 0).
The spaces.~ll and ~ I
H (I) = H 0 + vl/2pv I/2 The spectrum continuum for
of H (I)
on ~ l
starts
the full problem
at -1/2,
at -I.
no Hughes-
are then reducing spaces for
which lies above
The Schrodinger equation
(6) the bottom for H (I)
of the on JH
reduces to the one particle Schrodinger equations (I + I#I><~II)I/2Hs(I + l~l><~ll)i/21f> = (E + l)(I + l~l><~iI)f>
(7)
54
and HAIg> = (E + i) (I - I~i><~iI)Ig>
(8)
for if> and Ig>, where H S and H A are given by 1 H S = (I + i~i><~I~)i/2{ - V2 - 2r-I + ~1%1><~iI + Z-I[uI/2(I - M)-IuI/2 - 161~i><~II]}(I35 + I~i><~i I)I/2
(9)
and by HA=
(I - I~i><%11)[ - V2 - 2r-I + Z-IuI/2(I + M)-IuI/2](I - I%1><~iI)
(i0)
Here U is the ordinary potential + - ÷r'l~~i (r')i2-3+,--i U(r) = [II ir o r ] ,
(ii)
I is the identity, and M is a certain nonnegative Hilbert-Schmidt which is defined and analyzed in reference 2. M is O(e-r) . Thus
(7) and (8), which
integral operator
For large r, U(r) = 2r-I + O(r-3) and
can be regarded as
one-particle Schrodinger
equations for the outer electron, have incorporated shielding of the by the large
inner electron to distances which
the extent that
cannot support
the outer electron
an infinite
nt~nber of
nuclear charge
sees a bound
potential at states.
potential at large distances behaves like r-3 instead of the physically because polarization is remain
in the
enough to
not properly treated when
hydrogenic ground
count the
state.
number of bound
the inner electron is
forced to
Nevertheless,
the approximation
is good
states correctly.
Standard methods
for one
particle operators show that, for Z = i, H S has one bound state and HA has none. follows
that
the H-
interactions only, problem in
ion,
in
has at most
the singlet
This
correct r-4
infinite nuclear one bound
sector (spin 0,
mass
approximation
state below the sM~metric spatial
It
with coulomb
continuum for
the full
wave function) , and no
bound states below the continuum for the full problem in the triplet sector (spin i, antisymmetric spatial wave function). When the nuclear mass is finite,
the problem is complicated by the
the Hughes-Eckart term -2y? I. V2 couples the spaces ~II and ~ l " be overcome by replacing the part of
H 0 - 2YVl. V2 which couples ~ i
lower
particle equations
bound B;
possible.
a reduction
to one
fact that
This difficulty can
like (7)
to itself and (8)
by a
is then
In this way it can be shown that there is at most one bound state for y <
TS % 0.1736263618, which includes the physical H- ion. A New Result Now that rigorous results have been obtained for the number of bound states of the two electron atomic
system H-, it is natural
to ask about the number
states of the three electron atomic systems H-- and He-.
The
of bound
relevant Hamiltonian,
in infinite nuclear mass approximation with coulomb interactions only, is H3 -- H03 ÷ z-l(vl2 C ) I3 ÷ V23 ( 9 11 ÷ V 3 1 ( 9 I2 )
(12)
where H03 = -
2 - 2r31" V2 1 - 2r]l _ V22 - 2r21 _ v3
(13)
55
Here Vij ~
÷ -i in rj~
Ik is the tensor product of the coulomb repulsion Vij = 21~ i -
the space of electrons i and j with the identity Ik in the space of electron k. lowest threshold for breakup, into a two electron system in its lowest state free electron
at rest
electron system. H
at infinity, is
at the
bottom of the
This threshold is" known exactly only
spectrum of
for the quartet
The
plus a the two
sector of
In the quartet sector (spin 3/2, antisy~metric spatial wave function), triplet
H-, which
has no
bound states below
the bottom
behind when one electron moves off to infinity.
of its continuum
at -i,
is left
Thus the continuum starts at -i for
the quartet sector of H It will sector. A n
now be shown
that H
has
no bound states
below -I in
the quartet
approximate Hamiltonian H~ I) , which has no fewer bound states
than H3,
can be obtained by replacing Vii ~) Ik in H 3 by V'.13 = (vl/2--1/2'ij ~vij ) Q
[Ik - (l~l><~ll)k] + Vij G
(l~l><~ll)k
(14)
where (~#l><~ll)k is the projector l#l><~iI in the space of electron k.
V'. couples 13 only states which have at least one electron in the hydrogenic ground state @I" In the quartet sector, these states have the form ~A(rl, r2, r3) = h(r I, r2)~l(r 3) + h(r2, r3)#l(r I) + h(r3, rl)~l(r 2)
where h(rl, r2) = - h(r2, rl) and 6] @l(rl)h(rl, [ 3 - +r2)d 3+rl spanned by states of the form ( i ) u y " ~ H A, and let complement of ~A~ in
20A3 Denote the space be the orthogonal
and ~ 3 A are then ± H~I) + Z-I , , , = H03 (VI2 + V23 + V31).
for
the quartet sector.
(15)
~A~
reducing spaces (16)
The bottom of the spectrum of H03 on ~ 3 A starts at -3/4, which is above of the continuum in the quartet
sector.
the bottom
The Schrodinger equation for H3(1)- on J~A-
reduces to the two particle Schrodinger equation
[P, IQ
+
P,12 ÷ (P,±IQ P,n)v 2(P,,l(9 P, 2)] = (E + i)(P@IIQ
P@I2 ) [h>
(17)
where HAl is the one particle triplet sector Hamiltonian HA of equation (i0)
in the
space of
and
electron i
and P¢li_~ = Ii~ - ([ @1><%1)3"
Because VI2 is repulsive
has no bound states below 0, the Schrodinger equation (18) has no bound states below -i.
It follows
that
H
, in
infinite
nuclear mass
approximation
with coulomb
interactions only, has no bound states below the continuum in the quartet sector. Related Unsolved Problems Rigorous upper bounds
for the number
sector (spin i/2, spatial wave permutation group on three
objects) , and for the number
the doublet and quartet sectors, arises from the
of bound states
fact that the
of H-- in
the doublet
function belongs to the mixed representation have not yet been obtained.
The
He- in
major difficulty
ground state energy E (2)- and the ground
function #i are not known exactly
of the
of bound states of
for the two electron system left behind
state wave when one
56 electron moves off states
of these
to infinity. systems are
appears necessary to
If
rigorous upper bounds
to be
obtained via
introduce ¢i into the
to the number
the methods
above, it
approximate Hamiltonian H (I) in
way that the continuum for H (I) starts at E~2)" just as for the H 3.
of bound
outlined
such a
original Hamiltonian
This can in fact be done, and a one particle Schrodinger equation of the form
(18)
Hlpl~> = (E - E~ 2))I~> obtained, where spectrum at
the one particle
0, has
depends on #I"
no fewer
Hamiltonian Hlp has
bound states
than the
The analysis of Hlp is now under way.
the bottom of
its essential
original Hamiltonian
H3, and
It will be interesting to see
how accurate an approximation to ~i is needed to obtain a rigorous
determination of
the nt~ber of bound states of Hlp. A nt~ber of
unsolved problems remain for
variational calculation by
bound state, with an energy no
has only
one bound
state below
at -0.25.
-i should
system.
the H- ion has
higher than -0.25070 in our units, in
parity sector, where the continuum begins H-
the two electron atomic
G. W. F. Drake 4 has shown that
the unnatural
The methods used to
also be
A
a second
effective in
show that
settling the
question of the nt~ber of bound states in this sector. Information on the number of bound states of the two electron atomic system as a
function of
continut~
Z
and
possessed
inequality
y is by
implies
<~i ~u-ll ,l><,iul ~> =
HS
also incomplete. disappears
that
for
The
< ~ (I~i><~i~)I~>
35 ~<~IUI~>.
This
single bound
sufficiently =
plus the
small
state Z-
below the
The
Schwarz
I (<~IIU-I/2) (UI/21~>)12
fact that
all eigenvalues
~<
of the
integral operator M are nonnegative and less than 1 implies that, for Z < 32/35, the 35 Hamiltonian with the potential - 2r"I + (Iz-I + ~-~)U(r) has at least as many bound states as H S.
It is easy to
see that this potential has no bound states
below the
continuum for sufficiently small Z, which implies that H S has no bound
states below
the continuum for sufficiently small Z,
and in turn that the original
two electron
atomic system has no bound state below
-i for y = 0 and Z sufficiently
small.
The
value of Z at which the bound state disappears is not known. The values Z = i, y = 1/2 arise for two electrons plus a positron. of bound states of
this system is also not
known with certainty, but
The number the question
could be settled via the methods used for H- if a less drastic approximation is used to handle the Hughes-Eckart term. term should also permit proof 5
that this
system
An improved method of handling
the Hughes-Eckart
treating the proton-electron-positron system; has no
bound state
is
incomplete because
the existing part
of the
argument rests on a numerical calculation for which error bounds were not computed. References i. R. N. Hill, Phys. Rev. Lett. 3~, 643-646 (1977). 2. R. N. Hill, J. Math. Phys. 18, 2316 (1977). 3. The method used here generalizes a method introduced in N. W. Bazley, Proc. Natl. Acad. Sci. U. S. A. 4_~5,850-853 (1959) and Phys. Rev. 12___00,144-149 (1960). 4. G. W. F. Drake, Phys. Rev. Lett. 2_~4,126-127 (1970). 5. I. Aronson, C. J. Kleirman, and L. Spruch, Phys. Rev. A 4, 841-846 (1971).
Number of ei~envalues of many-bod~ Hamiltonians and Efimov's effect
!. M. Sigal Department of Mathematics Princeton University
i.
In this talk I present a new method of obtaining estimates of the number of
bound states of many-body systems with potentials close to critical. discussed below expose a striking phenomenon:
The estimates
existence of infinite number of bound
states in certain short-range, many-body systems.
Note that for all attractive po-
tentials and many mixed ones the coupling constants can always be adjusted in such a way that the phenomenon occurs.
The effect was experimentally observed in three
nucleon scattering and in solid states. played by impurities.
In the last case the role of particles was
In the first case the forces between particles are close to or
exactly critical, while in the second case they can easily be adjusted. All our considerations are illustrated in the case of three particles. Plank constant 2.
~
is set to
The
1 .
In order to formulate rigorously our results we need
Definition.
A short-range, pair potential
h£ = ( 2 ~ ) - l A + V~ >__ 0 , where equation
h~¢ = 0
Remark. vector of
~ = (-A)-lIv£11/2¢
~ , and the
, where
¢sL2~3).
h£ >__0 , such a solution is nondegenerate and is not an eigen-
[5], we call it a quasibound state of
Theorem 1. potentials
will be called critical iff
is the reduced mass for the pair
has a solution of the form
Since
h£
~
V~
Let
H
h~ .
be the Schr~dinger operator of a three-body system with pair
V£ ~ L p (] L q ~ 3) ~ p > 3/2 > q .
Let the masses of the particles obey
the condition [7], which for the sake of space we write here only in the case m i = mj : mi/m k > .9 •
Then if the potentials
points, the number of eigenvalues of as
Vik
and
Vjk
Theorem 2.
H
Vik
and
Vjk
approach critical
increases to infinity and becomes infinite
reach critical points. Let
~
= m.j ,
spherically symmetric and satisfy
-1 mi~m~_ > .9
and the potentials
f01V£(n)In dn < ~ .
states (isolated eigenvalues counting multiplicities) of asymptotic behavior as the potentfals
Vik
and
Vjk
Vik
and
Vjk
be
Then the number of bound H
has the following
approach critical points:
N = - 0'0107 w (mi/mk - 0,9)3/2~n P + uniformly hounded term ,
(*)
58
=maXlYsk[O) I _ i
where
~
J
.i
, ya(k) = k • c o t ( 6 ( k ) + k )
, 6(k)
is the s-wave phase shift
S = I ,J
for the pair
a .
Note that
bound state; in this case Corollary.
If all three potentials
2 are true for all masses
Va
has a shallow
is the bound state energy.
approach critical points then Theorems
1 and
on the masses holds always for some
(ijk) = (123)
An expression analogous to (*) can be obtained for a nonspherically-
symmetric case as well. 3.
and small if and only if
m I , m 2 and m 3 .
Indeed, the restriction Remark.
y~(0) < 0
7a(O) = - e/~-, where -s
Here
y~(k)
should be replaced by
detp(l+V(h0+k2)-l)
O
The physical idea behind our approach is that a system of three particles
behaves in many ways as a system of two of the particles attractive interaction,
connected by an effective,
produced by the exchange of the third particle.
We show that
if this third particle has quasibound states with each of the other two particles, then the effective interaction effective interaction
is long-range,
- IR1-2 , at infinity.
namely
The
described above is somewhat analogous to the interaction via
an exchange by virtual particles
in quantum field theory with the square root of the
energy of the third particle above
(times the reduced mass) playing a role of the
mass of the virtual particles. 4.
Below
tiplicities)
N(A,k)
denotes the number of isolated eigenvalues
of an operator
placian in the variable
A , which are less than
x .
and in the Jacobi coordinates, R = x I - x 2 , has the form -1
k
A three-body Hamiltonian say, for pair (12),
and
(counting the mul-
Ax
stands for the La-
in the center-of-mass
r = mlxl+m2x2/ml+m2
H = -(2m) -1Ar - (2~) -1AR + V , where
frame
- x3 ,
m -1 = (ml+m 2 )-l+m~l '
-1 -1 = mI + m2 5.
The proof of Theorem 1 is based on the following three propositions,
are given without proofs.
Propositions
1 and 2 do not assume the potentials
which to be
critical. Proposition i.
Let
u : ~ "3
~ L2(IR3)
make the expressions below meaningful, > N(H,k) ~ U
and
have an obvious smoothness, ..~uCH)~r : 1 V R
sB 3 •
required to
Then
N(H,k)
, where ~
=
H
(2~)-lAR + ¢(R,u)
, ¢(R,u) 5 <(Hu)(R),u(R)> r
U
=
r - (2~)'I r . Here
HB0(R ) = -(2m)-iAr + V(R)
is a Born-0penheimer
Hamiltonian
on
L2~3,dr)
•
59
The next proposition describes tail.
u
for which
¢(R,u)
Note that we have shown [6] that the ground state
has a long-range negative UB0(R)
Born-0penheimer Anzatz which comes from disregarding the term in
~(R,u) , leads to
~(R,UB0)
with a positive tail.
simplicitB particles 1 and 2 to be identical.
of
~0(R)
, the
-(2p)-l r
Henceforth, we assume, for
We set
U = V13 = V23
and
h = -(2n)-lA + U , n -1 = ml I + m31 Proposition 2.
Let
(R)
u(R) = T(I/2)Rv(R) + T_(I/2)RV
, where
is the solu-
v
tion of the equation (h+E)v
Ilu(H)llr
n o r m a l i z e d i n such a way t h a t
=
where
E(R)
+ v(R)U
= 1,
and
-E(H) + cl
= ¢(r+R).
(2)
= 0
(3)
Proposition 3.
x = e
If the potential
and
K
is
Here
8
is uni-
is critical, then (3) has the unique solu-
uniformly
H(R) = -(2n)-iA
r (2m) -1 + .5(2~) -1 , which replaces Lemma 1.
U
E(R) = ~21R1-2 +
The next lemma shows that
the Hamiltonian
.
H .
tion of the form to
Then
,
[2(l+e-~)]-2(l+(l-2e+2e2/3)e-~)m3(m3+ml)-i ,~ = ~ - - ~
ca =
6.
(TR¢)(r)
is the unique solution to the consistency equation for (1):
formly bounded in
solution
(l)
,
1-2 + elHI
l + [(h+E)-lv](R) and
= 0
Let
E0(R)
bounded
u(R)
2~n ~ = .3216... in
R
is the unique
.
is, in fact, an approximate ground state of
+ V(R) , where HB0(R)
(2n) -I
comes f r o m
in our case.
be the lowest eigenvalue of
H(R) .
Then
(4)
0 ~ E(R) - E0(R) ~ const IR1-3 . Remark.
Estimates (2) and (h) can be proved without solving (1) and (3).
can be shown that only the asymptotics of 7.
v
as
IRI ---+ ~
It
is important.
The proof of Theorem 2 is obtained from Propositions i, 2 and
Proposition 3'.
If the potential
U
is spherically-symmetric
critical) then the unique solution of (3) has the form where ~ ( s )
satisfies
phase shift for
h
(not necessarily
E(R)=m(yIRI)21RI-2+~IR1-3
x = e-x - s , T = lim k cot(~(k)+k) k÷0 and ~ is uniformly bounded in R .
, 6(k)
is the s-wave
,
60
Recall that: quasibound states;
y > 0 ~
states = solutions of Remark.
h~
y > 0 ~ h$ = 0
Properties of
0
and has no quasibound states;
h ~ 0
y = 0<
and has no quasibound states.
such that
@ ~ L 2 ~ 3)
and
>h
has
Here quasibound
IVII/2@ s L2~R B) .
(h+E) -1 , needed to prove Propositions
3 and 3' , are
described in [ill ([R,10] contain related formulae) for noncentral, critical potentials and in [12] for general spherically symmetric potentials.
8.
Derivation of (i).
(~(R)+E(R))u(R)
Write (fR
=
= 0
TRf):
We begin with the exact eigenvalue equation for (~)
u(R) =
(R) .
VCll2)R
(R)
+ S_(ll2)R
(fR : T R f )
(5) becomes:(h+E)v + URV + ( ~ + E ) s R + Us R = O , where Approximation:
URV = v(-R)U R
Approximate equation:
and
•
= _(2~)-IA
+ vR
•
Us R = s(R)U .
(h+E)v + s(R)U + (hR+E)s R + v(-R)U R = 0 .
Decouple this equation and Write the consistency condition. [(h+E)-lv](±R)
H(R):
.
the terms of the higher order in
In this approximation the decoupled equations are Literature Comments.
IR1-1
Dropping from
leads to
v(r) = v(-r)=s(r).
identical to (1).
It was discovered by V. Efimov [2] that the number of
bound states of a system of three identical particles with finite-range, attractive interactions such that there are no bound states in the two-body subsystems, grows as the potentials approach critical points.
The arguments presented by Efimov are
based on replacing the potential in a three-body problem in question by a boundary condition at the origin.
R. Amado and J. Noble [1] have given an alternative mathe-
matical proof, made rigorous by D. R. Yafaev [R] (Yafaev, however, does not treat the question of the growth of the number of bound states).
Their proof is based on
the study of the homogeneous Faddeev equation and is rather mathematically involved. In [6] we give another, variational approach to the problem in question. Theorem 2 is contained in this paper. et al [3].
Similar ideas were advanced in A. Fonseca
Different methods of estimation of the ground state energy of
H(R)
(see Proposition 3 and Lemma l) were proposed by M. Klaus and B.-Simon [4] and B. Simon [8]. The author is indebted to W. Hunziker and B. Simon for useful discussions and to K. Hepp and W. Hunziker for their hospitality at the Department of Theoretical Physics, ETH, Zurich, where the first draft of this talk was written. partly supported by
USNSF Grant MCS-78-01885.
The work was
61
Appendix: tion I and
Proof of Proposition i.
fn
be eigenvectors of
orthogonal and that
= .
N(H,I) ~ N ( H u , I ) Supplement:
Let
Hu
u
satisfy the conditions of Proposi-
Then the functions
~n = u(r'R)fn(R)
are
This via the variational principle implies
.
N-body case.
There are many cases in which we expect the existence
of infinite number of eigenvalues in short-range, N-body systems.
In the simplest
situation we have a proof of this fact: Theorem.
Consider an N-body, short-range system of which the bottom,
the continuous spectrum is defined by a break up into three clusters Assume two of the three clusters, say, Further, assume that the subsystems their two-cluster threshold
~ .
cI
and
cI U c 3
and
~ , of
Cl, c2, c 3 .
c 2 , contain each just one particle. c2 U c3
have quasibound states at
Then the whole system has infinite number of bound
states.
References i.
R. Amado and J. Noble. 2002.
Phys. Lett. 35B (1971), 25; Phys. Rev. I)5 (1972), 1992-
2.
V. Efimov.
3.
A. C. Fonseca, E. F. Redish, and P. E. Stanley. Efimov Effect in an Analytically Solvable Model. Univ. of Maryland (1979).
4.
M. Klaus and B. Simon.
Ann. Inst. H. Poincar@, to appear.
5.
M. Klaus and B. Simon.
In preparation.
6.
Yu. N. Ovchinnikov and I. M. Sigal.
7.
I. M. Sigal.
8.
B. Simon.
9.
D. R. Yafaev.
Phys. Lett. 33B (1970), 563; Soy. J. Nucl. Phys. 12 (1971), 589.
Ann. Phys., in press.
Unpublished.
J. Funet. Annal., to appear. Math. Sb. 9~ (1974), 567-593. Preprint, 1979, Berkeley.
i0.
A. Jensen and T. Kato,
ii.
M. Loss and I. M. Sigal.
12.
V. De Alfaro and T. Regge.
In preparation. Potential Scattering, North-Holland (1965).
ON THE E X P O N E N T I A L F A L L OFF OF W A V E F U N C T I O N S AND E L E C T R O N D E N S I T I E S
M. H o f f m a n n - O s t e n h o f +, I n s t i t u t fur T h e o r e t i s c h e P h y s i k der U n i v e r s i t ~ t W i e n T. H o f f m a n n - O s t e n h o f , R. Ahlrichs,
Institut f~r Theoret. Chemie und S t r a h l e n c h e m i e d. Univ. W i e n
I n s t i t u t fur P h y s i k a l i s c h e Chemie der U n i v e r s i t ~ t K a r l s r u h e
J. Morgan, D e p a r t m e n t of Physics, P r i n c e t o n University, N e w Jersey, U S A
i. I n t r o d u c t i o n "
We give a s h o r t survey on o u r results o b t a i n e d on the e x p o n e n t i a l decay of subc o n t i n u u m atomic w a v e f u n c t i o n s and e l e c t r o n densities
[i] - [4]. Let ~ denote an L 2-
solution o f the S c h r S d i n g e r e q u a t i o n o f an n - e l e c t r o n atom
(H-E) ~(Xl, .... x n) = O
,
xie
~3
(1 < i < n )
(.1)
rij :
(2)
whe re nA.
H = -
[
1 ~-
i=l
n
[ ,=
Z_Z_+ ri
i I
n
~
1
r-~-- ; i,j=l lj i<j
ri =
IXil
'
IXi-Xjl
denotes the H a m i l t o n i a n in the infinite n u c l e a r m a s s approximation,
Z is the n u c l e a r
charge and E the e i g e n v a l u e c o r r e s p o n d i n g to ~. ~ is a s s u m e d to be real and normalized to i. The k - e l e c t r o n d e n s i t y Pk (i < k < n) is d e f i n e d b y
Pk(Xl . . . . .
x k)
= f
I~(x I .....
~3(nLk )
D e c a y p r o p e r t i e s of such w a v e f u n c t i o n s
Xn) 1 2 d X k + l . . . d x
(3) n
and e l e c t r o n densities are i n t e r e s t i n g f r o m a
m a t h e m a t i c a l and p h y s i c a l p o i n t of view: T h e r e is a close r e l a t i o n s h i p b e t w e e n spectral properties oft he
c o n c e r n e d H a m i l t o n i a n and asymptotic p r o p e r t i e s o f sub-
c o n t i n u u m w a v e f u n c t i o n s resp. densities. F u r t h e r m o r e it w i l l be seen that concepts like n u c l e a r s c r e e n i n g and the o n e - e l e c t r o n p i c t u r e - on w h i c h p a r t o f our unders t a n d i n g o f atomic structure is b a s e d - e n t e r in a n a t u r a l way. Unfortunately, b u t n o t unexpected,
the s i t u a t i o n w i t h r e s p e c t to c o r r e l a t i o n effects is less transparent.
O u r m a i n tool in the derivations o f u p p e r and lower bounds to w a v e f u n c t i o n s and densities are comparison theorems for d i f f e r e n t i a l inequalities. It s h o u l d be n o t e d t h a t there is a different p o w e r f u l approach for r a t h e r general situations b y D e l f t e t al.
[5]. H o w e v e r for atoms in the infinite n u c l e a r mass
+) S u p p o r t e d b y Fonds zur F 6 r d e r u n g der w i s s e n s c h a f t l i c h e n F o r s c h u n g in 0 s t e r r e i c h , P r o j e c t No.
3655
63
a p p r o x i m a t i o n the methods o u t l i n e d here lead to s o m e w h a t more d e t a i l e d results.
2. C o m p a r i s o n T h e o r e m s
The d e r i v a t i o n o f the e x p o n e n t i a l l y d e c r e a s i n g bounds to the k - e l e c t r o n densities is b a s e d on the following two comparison theorems: L e t ~ b e an o p e n s u b s e t o f ~{n and let f, g satisfy (i)
f,g ~ C°(~);
f,g > O in ~ (and f,g
+ Ixi~
(ii)
g < f
V x £ ~.
T h e o r e m 2.1. (iii)
O for u n b o u n d e d ~),
(e.g. [5]): L e t f, g obey
(i),
(ii) and let further
(- A + W l ) g + F 1 < O i n the w e a k sense in (- A + W 2 ) f + F 2 > O
(iv)
W 1 >_ W2, F 2 < F I
and
(v) W 2 >_ 0
.
Then f >__g in all o f ~. T h e o r e m 2.2. (iii)
[6]: Suppose f > O a.e. in ~, let f, g Obey
(i),
(ii) and let further
(- A + W l ) g <__O i n the w e a k sense in (- A + W 2 ) f >__O
(iv)
W 1 > W 2 a.e.
in ~ and
(v) Af, A g e
L I(~).
T h e n f > g i n all o f ~.
3. S c h r 6 d i n ~ e r I n e q u a l i t i e s and E x p o n e n t i a l De.cay of k - e l e c t r o n D e n s i t i e s
(I ! k ~ n~
It w i l l be shown that @/~k satisfies a d i f f e r e n t i a l i n e q u a l i t y w h i c h has essentially the structure o f the S c h r S d i n g e r e q u a t i o n o f a k - e l e c t r o n system, w h e r e the absolute value o f the e n e r g y is r e p l a c e d by the sum o f the first k successive ionization p o t e n t i a l s e.
(i < i < k): L e t
1
H 'n-ij'' =
n [ j=i+l
4. (_ ~ _ 2
n ~_i_) + [ 1 r.3 j, i=i+l rj£ j
(i < i < n)
(4)
denote the H a m i l t o n i a n of the i - f o l d i o n i z e d n - p a r t i c l e s y s t e m and let E (n-i)'' denote 0 the g r o u n d state energy o f H (n-i) in the appropriate s y m m e t r y s u b s p a c e (induced b y the s y m m e t r y b e h a v i o r o f ~), then
s I
=
and h e n c e
E
(n-l) o
-
n ~ si = i=l
E
,
IEI.
¢i
=
E
(n-i) o
-
E
(n-i+1) o
(2
<
i
<
n)
(5)
64
Theorem
3.1.
(SchrSdinger
inequalities)
AI
( - Y + ~i- z__) ~/~i(~i)<0 r 1
k i=l [
--
A.
k
Z ri
{-T+~i---+
1 • ' r-~') ~PPk(Xl ..... Xk) ~ O ; 3 i i]
(2 < k < n-l)
(6)
(~-E) I~l ± 0 The inequalities Proof:
are to be understood
We sketch the proof
and particularly inequality
f ~(-
for
- I~IAI~I A1 2
for PI" For the case k ~ 2 the procedure
(6) is an immediate
principle
f ~(H (n-l)
+
and
obtain
a rigorous
Theorem
(8)
~n
electronic
we suppose
.
inequality
~9~
repulsion
(6) with the aid of
For the following fact,
(7)
(5) yield
via Cauchy-Schwarz's
inequality
distributional
(4) we have
- E)~ dx2...dx n > e I Pl(Xl)
the positive
of Kato's
is analogously
~ 1 L - E)~ dx2...dx = 0 j=2 rlj n
~ h ~-< / ~ A I ~ 2 Neglecting
consequence
~ - ~A~. Due to (I) and
Z H(n-l) rI +
The variational
Furthermore
I~I
in the weak sense.
term
(8) and
(n-l)
f - -~2 dx2...dx n in r12
(7) we
(9).Q
E 1 ~ ~2 ~ "'" ~ £n" This is just an experimental
proof is so far missing.
3.2. Z
-/2e I r /Pl(x)
! C(l+r) 2 ~ I
e
(I0)
Z
~Pk(Xl,...,Xk)
where C is a constant Proof: l~ge ~le
k < C S ~ -i=l
(l+r.)
in Theorem
procedure.
e
-2/~e.r. ii
(ii)
and S acts as a symmetrizer.
2.1. Since
p/~l obeys
f of [- A + 2(e - Z/r)]f
We indicate
1
l
For pl the proof is s t r a i g h t f o ~ a r d :
solution
2/~Tai
the main steps
Choose
(6) we obtain
~ = {x e ~ 3
r ~ c}, c sufficiently
the upper b o ~ d
> O. Inequality
(ii) is proven
(iO) by a suitvia a recursive
for p~2: Let ~ = {(xl,x2)!rl,r 2 ! c},
65
c sufficiently The suitable
large in Theorem 2.1. Due to (6)
function
f obeying this inequality
2 [ [- Ai + 2(e.1 - Z/ri) ] P/~9 < 0 " i=l in the other direction yields the
desired bound
(Ii) for P/~9 in ~. To prove that f fulfills the boundary
and to verify
(ll) in the region {(xl,x2)
is used.
condition
(ii)
r I >__ c, r 2 <_ c}
(12) follows from a so called subsolution
estimate by Trudinger
[7]. For Pl
the already obtained upper bound is used. D For l-electron systems the r.h.s, of (10) shows the right asymptotic behavior. For the n-electron exponential
case one should expect an effective nuclear charge in the pre-
factors of (10) and (11). This is in accordance with the physical picture
that k electrons
at large distance
from the nucleus
"see" a nucleus screened by the
remaining n-k electrons.
4. Screening Effect in .the Decay. of the k-elgctron Densities 4.1. UPper Bounds
To obtain results on screening effects a refinement of the Schr6dinger ties is necessary,
inequali-
where the so far neglected electronic repulsion terms become im-
portant. Theorem 4.1.1.
Z-(n-l)
/P-Pl(x) ! C (l+r) 2/-----~I
i
_~
r
(13)
e
Z- (n-k)
k ~/Pk(Xl,...,x k) < C S i=l
i (l+r.) 2/~ei 1
_2/~eiri e
Proof: The proof is similar to that of Thm 3.1. To prove
, (2 < k < n)
(13) we split the Hamiltonian
by H = - AI/2 - Z/rl + ~(n-l) (Xl). The variational principle
f ~ ~(n-l)(Xl) ~ dx 2 ... dx n > ~ 0 ( X l ) P1(Xl)
with E O(xl) denoting symmetry subspace.
E0(Xl)
the bottom of the spectrum of ~(n-l)(xl)
By a perturbation
argument
= E (n-l) + _n-i _ + O(!2 ) o rI rI
(15) and (16) lead to
(14)
leads to
(15)
in the appropriate
[8] E0(Xl) has the asymptotic expansion
(16)
66
AI Z-(n-l) (- - 7 + el rI for sufficiently
large
_ d) r~
/p (x 1) < 0 ; i --
r I. Application
of Theorem
d > O
2.1 completes
the proof of
(137.
For p/~k the proof is again recursive.O A sharper ted term
result
can be shown
for p/~2 by taking into account
the so far neglec-
i . Q/~2: r12 4.1.2.
Theorem
e/P2(Xl,X2)
r i
r2]
< C s[(l+rl)e(1+r2 )B
where Z-(n-2)+i/2
z-(n-l)
4.2. Lower Bounds In the following we consider electron
atom.
the mathematical
It is well known that ~o(Xl,X2)
ground state ~o(Xl,X2)
> O a.e.. To derive
bound to ~0' ~o(Xl'X2 ) > O for rl,r 2 < ~ will be necessary. in~nediate consequence Theorem 4.2.1.
behavior Theorem
[9] Let ~(x)
denote
(x e ~3n)
the i-electron
of Pl is described 4.2.2.
lower
this is an
of be a solution
Then ~ has both signs in any arbitrary Let Pl(X)
However
of a 2-
a positive
(See also
of equation
small neighborhood
density
corresponding
(17 and let ~(x ) = O. 0
of x . 0 to C0. The asymptotic
by
[10]) Z-1 -
i
-
I~
r
e Proof: ionized
We define system,
Schwarz's
u(x I) = f~(x2)~o(Xl,X2)dx2,
A1 (- ~ + It can be shown
O < U(Xl) ~ /Pl(X1"J.
eI
-
z) u(xl ) + f -¢~o -dx rI
r12
Using
the ground state of the
By Theorem
4.2.1
and Cauchy-
(I) we get
2 = O .
(17)
that for sufficiently large r 1 Z-I
f ~ dx 2 < i _ U(Xl ) + ~ r12 rl ~I where we used the upper bound inequality
~ denotes
Z2 (- A2/2 - Z/r2)~(x 2) = - T ~(x2).
i.e.
inequality
where
for u results.
leads to the desired
c
rl
e
(13) for P/~I" Combining
Application
lower bound
(187
(17) and (18) a differential
of T h e o r e m 2.1 in its inhomogeneous
for pl.[3
version
67
Unfortunately
the lower bound to ~0(Xl,X2)
is less satisfactory
than the result
for PI" Theorem 4.2.3.
~0(Xl,X2)
with R = Proof:
>__C~ e - ( / ~
+ r29 and 6 arbitrarily
+ 6)R ;
(6 > O)
small.
Let ~ = {(Xl,X 2) R > d > 0}, d sufficiently
large. By Theorem 4.2.1 ~i~.~ > 0
and Theorem 2.2 yields the above lower bound.
References
[i] M. Hoffmann-Ostenhof,
T. Hoffmann-Ostenhof,
Phys. Rev. AI6,
[2] T. Hoffmann-Ostenhof,
M. Hoffmann-Ostenhof,
R. Ahlrichs,
1782
(1977)
Phys. Rev. AI8, 328
(1978) [3] T. Hoffmann-Ostenhof, [4] R. Ahlrichs,
[5] P. Deift, W. Hunziker, [6] T. Hoffmann-Ostenhof, [7] N.S. Trudinger, [8] J. Morgan,
J. Phys. A12,
M. Hoffmann-Ostenhof,
1181
(1979)
T. Hoffmann-Ostenhof,
J. Morgan,
in preparation
B. Simon, E. Vock, Commun. Math. Phys. 64, i (1978) J. Phys. A, in press
Ann. Scuola Norm. Sup. Pisa
(3) 27, 265 (1973)
B. Simon, submitted to Int. J. Quantum Chem.
[9] M. Hoffmann-Ostenhof, [10] W. Thirring,
T. Hoffmann-Ostenhof,
Lehrbuch der Mathematischen
B. Simon, submitted to J. Phys. B
Physik 3, Springer
(1979)
EXACT RESULTS FOR CONFINING POTENTIALS
H. Grosse Institute for Theoretical Physics University of Vienna, Austria
Abstract
We summarize recent results obtained together with A. Martin on properties of confining potentials as they are relevant in particle physics.
i. Physical Motivation
After the discovery of two of the so-called "New Particles" it was suggested that these states represent the ground state and first excited state of a quarkantiquark pair interacting via a potential going to infinity at infinity. The usual procedure is to start with a certain potential and to adjust a few parameters; one makes predictions conderning the spectrum, and magnetic transition probabilities
then
leptonic decay widths and electric
[i]. Starting from Coulomb plus linear poten-
tial the existence of P states below the 2S state and of a D state above the 2S state was conjectured and found experimentally. We asked how stable are certain features which one observes, like the level order or relations between the decay widths, against changes of the potential.
So we
determined classes of potentials showing these features and on the other hand classes giving opposite results [2,3]. Furthermore, experimentally,
since very recently a second heavy quark system has been detected
one is interested in comparing relevant quantities of two systems
assuming that the same potential acts between the constituents.
2. Energy Levels
We consider the Schr6dinger equation for a radial symmetric potential V(r) and ask for a class of potentials which give the "canonical" level order:
A) EIS ~ Eip ~
B) E2S ~
A) EID ~
E2p
69
N o t e that A) become e q u a l i t i e s for the C o u l o m b case, w h i l e B) b e c o m e s equal for the h a r m o n i c oscillator;
furthermore,
since the angular m o m e n t u m o p e r a t o r is p o s i t i v e one
has in g e n e r a I EnS < Enp < E n D < . . . . T H E O R E M i: L e t W(r) = ~r(r2V(r))
A)
(
)2W(r) ~ 0 ,
then
- ~ < W(O) ~ O ~
E2S > EIp
, E2p ~ E I D ...
The m a i n idea of the p r o o f is to s t a r t from the degenerate case e n e r g y d i f f e r e n c e as a function of a parameter. the V i r i a l theorem,
and to follow the
U s i n g d i f f e r e n t i a l e q u a t i o n methods,
the F e y n m a n H e l l m a n n t h e o r e m and a k i n d o f m o m e n t i n e q u a i i t y
(which follows from the node structure of the wave functions involved) m o n o t o n i c i t y o f that e n e r g y difference. T o have B ) f u l f i l l e d
one derives
one needs concavity in
r2:
T H E O R E M 2:
(f:]'")2W(r)
B)
< O , --
dr 2
_ = < W(r)
~
E2S > -- E I D
r=~
"
T o i l l u s t r a t e conditions A) and B) let us give e x a m p l e s 2 V(r) =
/ -1
de r @ ~((~) O(c~) ;
0 >_o
,
=
•
We have w o r k e d out a larger class g i v i n g the canonical o r d e r t o g e t h e r w i t h counterexamples
[3]. F r o m theorems i a n d 2 we learn that the d o m i n a n t c o n t r i b u t i o n should
n o t rise s t r o n g e r than r 2 w h i l e p o t e n t i a l s like -r -3/2 s h o u l d be avoided. A s s u m i n g that the same p o t e n t i a l describes two systems d i f f e r i n g o n l y in the m a s s one m i g h t c o m p e n s a t e the decrease o f levels due to an increase in mass b y inc r e a s i n g the a n g u l a r momentum.
U s i n g the M i n - M a x p r i n c i p l e t o g e t h e r w i t h the e l e m e n -
tary i n e q u a l i t y
dr u '2 ~ f
dr u 2
o one arrives at the b o u n d s
En(M,
for
u(O) = O
4r 2 (for M > m):
£ = 21--(~7m - i)) _< En(m,
~ = O)
,
V n .
U s i n g the node structure a n d c o n t i n u i t y an i m p r o v e m e n t has b e e n obtained:
T H E O R E M 3: V £ A) ~
El(m , ~Tm-
I) ~ E l ( m , O ) .
T o relate the left h a n d side to p h y s i c a l o b s e r v a b l e s one uses c o n c a v i t y properties:
7O \ From the M i n - M a x principle i(Z+l). We have o b t a i n e d
it follows that the ground state energy is concave in
concavity in ~ themselves
for special
6V e B) in first order in ~; it holds asymptotically holds for r ~, 0 < ~ ~ 2. More generally, ence of the first two energy
THEOREM
4:
(d )2(r2~r) dr 2
for V = r2+l~V,
we established m o n o t o n i c i t y
of the differ-
levels:
< O ~
~(E2(Z)
Applying the above considerations mass difference ~
cases:
for large £, V e B); and it
- El(Z))
< O .
to the J/~ and T system gives a b o u n d on the quark
- m c ~ 3.29 GeV.
3. The Wave Functions a t t h e O.rigin ~n.(O) (Z = O)
Leptonic
decay probabilities
are proportional
to
]gn(O)12;
F~+e+e_
-- 4.8 keY > F~,~e+ e_ -- 2.1 keV. A result of A. Martin
THEOREM
5:
(
)2V(r) ~> 0
-=~ 192(O) 12 ~> 191(O)
T O get this result one uses a continuity
obtained
from the Schr6dinger
is p r e f e r r e d
The generalization the WKB approximation, potential
)2V(r)
The mass dependence
6:
and the relation
So from the data an overall concave potential
convex one.
to higher states is more difficult to prove: it holds in first order p e r t u r b a t i o n
it holds within
theory around the linear
and it is true asymptotically
d (
THEOREM
the node structure
u(r) = r ~(r)
equation.
against an overall
[4] states:
12
argument,
U' 2 (0) = / dr u 2 d V dr ' o
experimentally
(
)2V(r)
> 0 ,
of ~n(O)
dV > 0 d-~
~n (O) n ~
is of obvious
< O ==~
lim
--
r+O
2 > __ m2
Ul, 2 denote the wave functiorsto mass ml,2" charge +2/3 for the b o t t o m quark
¢ ~ ~ [
O
interest:
V £ .
Application of such a b o u n d excludes
[5]. From a generalization
only partial
answers are
known. One m i g h t ask for the mass dependence
of the p r o b a b i l i t y
for a particle
p r e s e n t within a sphere of radius R. For the ground state one finds dV
THEOREM
7:
d~r > O ~
~mm - dr o
lull 2 > O
i
[5]:
to be
71
4. Inverse P r o b l e m
for Confining Potentials
For a fixed angular m o m e n t u m T H E O R E M 8: Given a sequence E
n
[6]
(especially
Z = O) we have proved:
and u~(O) b e l o n g i n g
to a confining potential which
fulfills
V ¢LIO c ,
V(r)
i dr V--~7~ V '2 < ~ ,
is uniquely
i dr
for some R ,
determined.
We have related that p r o b l e m
to a simpler one, which is the inverse p r o b l e m for
the same equation with different b o u n d a r y
- w "n +
V'V-~2 < ~
(V(r)
-
The usual G e l f a n d - L e v i t a n
e n) w n = 0
procedure
conditions:
w'n (0)
i
is applicable
=
0
.
to that problem.
The relationship
b e t w e e n both problems has been found by studying
R(E) =
for a solution
(dr d ~ ( r ) / % (r)) r=O
uE(r)
defined by a WKB decay at infinity p r o v i n g
and admits a representation N R(E) = lim { I N +~ n=l The data determine - R-I(E)
that R(E)
is Herglotz
like [u~(O) 12 E - E n
2 / EN+EN+I~ ~ ----7-----
"
R(E) b u t then :
oo
X
lWn(O)
En
n=l
gives the information necessary We have also considered
12
E for the second problem.
the inverse p r o b l e m where
there is given El(Z)
the
ground state energy as a function of i.
5. Regge Trajectories
for C o n f i n i n g Potentials
One tries to integrate
the S c h r ~ d i n g e r
Re £ > - 1/2. The p r o b l e m is that En(£) dE ~ u2 n n/ di = f dr o r2 blows up.
(/ dr u 2) n
[3]
equation
for complex angular momenta with
might be n o n - a n a l y t i c
if the derivative
72
One knows from the relation
Im %
7 dr -lUnJ2 = o
that En(£)
Im E
r2
n
(~) f dr o
fUn 12
,
I
=
£(~+i)
cannot have p o l e s or e s s e n t i a l singularities.
We have r e s t r i c t e d o u r s e l v e s to p u r e p o w e r p o t e n t i a l s and have a d a p t e d a m e t h o d d e v e l o p e d b y L o e f f e l and M a r t i n for the a n h a r m o n i c o s c i l l a t o r to exclude b r a n c h p o i n t singularities in the complex i plane: T H E O R E M 9: L e t V = r e , then E
(£) can be c o n t i n u e d to complex £ and is a n a l y t i c in n Re i > - 1/2. F u r t h e r m o r e we have a r e p r e s e n t a t i o n like
En(1) = A + B1 +
1 --
-1/4 dl' I m E (I') n J I' (I'-i)
'
Im E n
>
O
.
References
[i]
J.D. Jackson,
[2]
A. Martin, N e w ' P a r t i c l e s o r "Why I b e l i e v e in Quarks", L e c t u r e s given at the
Tokyo,
C. Quigg, J.L. Rosner, Proc. 1978
19th Int. Conf. on H i g h E n e r g y Physics,
(Physical Soc. Japan, Tokyo,
1979) p. 391
15th Int. School o n S u b n u c l e a r Physics, Ettore M a j o r a n a Centre for S c i e n t i f i c Culture, Erice
1977 ; C E R N T H 2370
[3]
H. Grosse, A. Martin, E x a c t Results on P o t e n t i a l Models for Q u a r k o n i u m Systems,
[4]
A. Martin, Phys. L e t t e r s 67B
[5]
C. Quigg, J.L. Rosner, Q u a n t u m Mechanics w i t h A p p l i c a t i o n s to Quarkonium,
CE~N T H 2674, to appear in Phys. Rep.
Fermilab Pub-79/22-THY [6]
(1979)
(1977) 3 3 0
(1979)
H. Grosse, A. Martin, Nucl. Phys. B 1 4 8
(1979) 413
THE QUASI-CLASSICAL
LIMIT OF QUANTUM SCATTERING THEORY Kenji Yajima
Department of Mathematics University of Tokyo Hongo, Tokyo, i.
Introduction,
113 Japan
Theorem.
The aim of this note is to discuss the quasi-classical
limit of quantum scattering
operator and its relation to classical mechanical scattering theory.
Let H h =-(N2/2m)A
+ V(x) be a Hamiltonian for a quantum mechanical particle of mass m and H(x,$) = ~2/2m + V(x) be the corresponding Hamiltonian for a classical particle. the Planck's constant.
E = h/2~, h > 0 is
We assume that the potential V(x) satisfies the following
assumption.
ASSUMPTION(A). (2)
(i)
V(x) is a real valued infinitely differentiable
For any multi-index
~, there exist constants C
function on R n.
> 0 and m(~) > [~] + i such
that
(i.i)
[(~/ax) ~ V(x)l ~ ca (i + [xl) -m(~).
Under this assumption the followin~ are well-known in quantum scattering theory (A~mon~ and Kuroda[6]).
(i)
H h is a selfadjoint operator in the Hilbert space
a one parameter unitary group
(2)
Let H0h = -(~2/2m)A--
W h = s-lim + t÷+ ~
exp(-itHh/~).
be the free Hamiltonian.
exp(itHh/~)
exist and are complete;
L2(R n) and hence generates
Then the wave operators
h exp(-itH0/~)
hence the scattering operator
S h = (W~)*W h_
is a unitary operator on The corresponding (I)
L2(Rn).
result for classical mechanics is proved by Hunziker
For any (a, n) c F = R n x ( R n ~ { 0 } ) ,
(t,a,~)) of the Hamilton equation
[5] and Simon ~].
there exists a unique solution
(x+(t,a,~),p+
74
such that as (1.2)
t ÷ ± =,
[x+(t,a,q)-t~-a[
The wave operator W~ 1 w~l(a,~)
÷ 0,
[p+(t,a,~)-n I ÷ 0
in classical mechanics
= (x+(0,a,n),p±
is defined on F by the equation
(0,a,~)).
w+cl
is an infinitely differentiable
canonical mapping on
(2)
There exists a closed null set
e C
F
F.
such that
w~1(r\e) C w~I (r) The scattering S~I
operator S cl
is defined as
. cl~-i cl =cw$~ w
on the initial set F \ e.
in the classical mechanics
r \ e.
S cl
is an infinitely
differentiable
canonical mappin~ on
We write as
sCl(a,~)
= (a+(a,n), ~+(a,~)).
We study the asymptotic behavior on the coherent state
of
Sh
as h ÷ 0
fh = exp(-ia.~/~)f(~)
in momentum space representation
in terms of classical mechanics
a
Scl"
= hsh h) ~.h
is the Fourier transform:
~hu)
(~) = ( 2 ~ ) -n/2
f exp(-ix.~/~)u(x)
To state our theorem, we introduce
e(a) = { ~ e R n \ { 0 } ;
e(a)
Clearly
is a closed set of R n
if
is a closed set of
and e(a) ex
for a
E R n.
R n.
and is a null set for almost all a s R n.
det(~n) Since
(a,~) = 0}.
[~+[ = In I, e(a) = e(a) ex
n = i.
THEOREM. as above. any q ~ R n = ~
e(a)
(a,~) s e } U { 0 }
e(a) ex = e(a)U{n~e(a): e(a) ex
two set
dx.
Let Assumption
(A) be satisfied.
Suppose that f s L2(R n)
Let
a s Rn
and let
has support in the exterior of
there exist only finite number of
and the following relation holds:
njs
supp. f
e(a) ex
be defined
e(a) ex.
such that
Then for
q+(a,~j)
75
lira h+o
I[ ( ~ h f h ) ( ~ ) _
det(~q+/~q)
~. {exp i(S(a,~j)/i~-a+(a,~j)'~/15J
(a,qj)1½f(qj)II
Here the summation is taken over all
the Keller-Maslov's
~IndT(a,~j)/2)}[
= 0.
~.s
such that
3
index of the path
~ = q+(a,nj);
(x_(t,a,Dj), p_(t,a,~j)),
respect to the obvious manifold being considered;
S(a,~j)
Indy(a,qj)
is
- ~ < t < ~ ,
with
is the difference of the
action integrals:
2
S(a,~j) = lira {s~j/2m - t ~ 2 / 2 m
+
L(x,x) = mx(t)2/2 -V(x(t))
For a smooth function
is the Lagrangian of the system. h fp(~) = exp(-i~(~)/~)f(~)
~(~), we write as
type of wave packet quasi-classical wave function). and Q(x)
,
S
S +-°°
where
/t L(x_(~,a,~j),x_(o,a,nj))d~}
(we call this
Then for smooth functions P(~)
with compact supports we see h h lira (f$(~),P($)f$(~)) h+o
= (f(~),P($)f(~));
lira (f~'(~),Q(x)fh(~)) ~ = (f(~),Q(~/8~)f(~)). h+o h f~
By virtue of these relations, we may think that the quasi-classical wave function represents, manifold
in the limit
(~/~,~)
If(t) 2d~
h = 0,
an ensemble of classical particles on the Lagrangian
in the phase space
(see also D i r a c ~ 3 ~ ) .
Rn × R n
with momentum distribution density
Thus by taking fha
quasi-classical wave function such that, inthe limit particles eomeentrated at the configuration If(~)I2d~
.
x(t)~t~ + a, p(t)~n , by
x = a
Then the incoming wave function at
If(~)I2d~ .
mated (as h + 0)
where
~
represents particles like
is fixed and the distribution density of
~
is given
The theorem says that then the outgoing wave function can be approxiby an incoherent superposition
the density
(with additional factor
If(nj)12d~j
t = =
of the quasi-classical wave functions
at
h = 0
with momentum distribution density
t = -=
each of which represents
at
in the theorem, we choose a h - 0, it represents classical
the particles
this incoherent superposition
x(t)~ tn + a+(a,~j), p ( t ) ~ exp(-i~Indy
with
(a,nj)/2)). Moreover
turns to coherent one, since
76
-(~I~G) (S(a,~j)-a+(a,nj) • ~) = a+(a,~j)
and
a+(a,~j) # a+(a,n k)
if
of the classical scattering operator
2.
by the canonical property
~+(a,~j) = n+(a,~k), j # k Scl.
Outline of the proof.
Starting point of the proof of the theorem is the following
LEMMA 2.1.
Let
lim h#o
a s Rn
sup
I tl~s
and
f e L2 ( R n ) .
Ilexp(_itHh/h)
Then for any
6 > 0,
~h,~fh)(x)-(m/iltl) n/2
exp(im(x-a)212tN)f(m(x-a)l t) II
=
0.
The important point of Lemma 2.1 is' that the approximation of exp(_itHo/h)h h~.*fa _~ ) given by the second term is the uniform approximation with respect to the time ItI>~. h exp(-itH0/~)(exp(ix.$/~)
This kind of approximation is impossible for the wave function
f(x)). If we assume that the support of the function
f(~)
for large
is far outside the effective range of
Itl ,
the support of
f(m(x-a)/t)
is outside of the origin, then
scattering, where classical particles move almost freely.
There the classical WKBJ-
method does work well and we can prove the following lelmma.
LEMMA 2.2. R > 0
Let
f s L2(R n)
have support
KC
R n {0}.
Then there exists a constant
such that the following statements hold.
(i)
For any
(2)
If
t
< -R,
~ = n(-R,a,x)
lim h+o
the mapping
K~n * x (t,a,n)
is determined by the equation
is a diffeomorphism. x = x_(-r,a,n), then
I I exp(iRHh/h)w~h*f~(x) - exp(-in~/4 + is-R(x)/~)_
Tdet(~x_(-R,a,~)/~)I-½f(n)II = o, where
for
n = ~(t,a,x), t St_(x) = s-+-~lim(/s L(x_(~,a,~),x_(o,a,~))do + sn2/2m).
77 From time supp f
-R
the wave packet gets into the effective scattering region.
is taken sufficiently
small, there exists time
it gets through the region and the mapping phism. [4]
such that for t > T
supp f)q+X (t,a,q)
is again a diffeomor-
In the scattering region we use Fujiwara's quasi-classical and the stationary phase method [2] ; for
again.
T > 0
t > T
But if
fundamental solution
we use classical WKBJ-method
We get
LEMMA 2.3.
Let f(~)
there exists
T > 0
diffeomorphism.
such that for
Let
lim
sup
h+o
t>T
have sufficiently
II
~e
supp f
small support in the outside of
t > T ,
the mapping
be determined by
exp(-itHh/~)wh~h*fh(x)
supp f ~÷x (t,a,n)
x = x_(t,a,n).
- exp<-inz/4
e(a) ex.
Then
is a
Then
- iIndTt/2
½ + ist(x)/~) det(Dx_(t,a,D)/D~)I- f(~)II = 0,
where
IndYt
By definition
is Keller-Maslov index of
~h
=
lim
{(x0(o,a,q),p
(o,a,~))-~ < ~ < t}
exp(it~2/2~hexp(-itHh/~)wh~h*
.
D
Then by Lemma 2.3 and
stationary phase method, we get the statement of the theorem.
REMARK 2.4.
Similar results hold for long rang e scattering for SchrSdinger equation
and the Dirac spinor in the static electro-magnetic
field [9] , [I0]
[Reference~ i. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Agmon, S.; Ann. Scu. Nor. Pisa, Ser IV, 2.2, 151-218 (1975). Asada, K. and D. Fnjiwara; Japan. J. Math. 4, 299-361 (1978). Dirac, P. A.; Principles in Quantum Mechanics, 5 ed., Oxford. Fujiwara, D.; to appear in J.d'Analyse Math. Hunziker, W.; Commun. Math. Physics 8, 283-299 (1968). Kuroda, S. T.; J. Math. Soc. Japan 25, 75-104 (1973). Simon, BI; Commun. Math. Physics 23, 37-48 (1973). Yajima, K., to appear in Commun. Math. Physics. Yajima, K.; preprint, Univ. of Tokyo. Yajima, K.; in preparation.
SEMICLASSICAL QUANTUM MECHANICS FOR COHERENT STATES George A. Hagedorn The Rockefeller University New York, New York U.S.A.
Abstract: For certain Gaussian states we present a simple approximate evolution which is asymptotic to the Schr~dinger evolution as ~ ÷ 0 . In 3 or more dimensions our error estimates are uniform in time if the potential is suitably chosen. Consequently, our methods apply to scattering theory. The approximate evolution is obtained by using the Trotter Product Formula and the second order Taylor expansion of the potential about the center of the wave packet.
In this brief note we wish to exhibit some simple ideas which can be used to study the ~ ÷ 0 limit of quantum mechanics.
Detailed proofs of theorems which
use these ideas may be found in [i]. Our analysis deals with motion of certain normalized Gaussian wave functions ~(A,B,~,a,~,x)
Definition:
which are defined as follows:
Let A and B be complex
n x n
matrices with the following
properties: A and B are invertible; BA
-i
(i)
is symmetric ([real symmetric] + i[real symmetric]);
(2)
Re BA -I = ½[(BA -I) + (BA-I) *] is strictly positive definite;
(3)
(Re BA-I) -I = AA*
(4)
Then for all
a sIR n , N el~ n , ~ g ~ ,
and
~ > 0
we define
~ (A,B,~,a,~,x) = (2~) -n/4 ~-n~/2(det A) -½ exp {-(4~2~) -I <(x-a), BA-l(x-a) > + i < ~ , ( x - a ) >
/~}.
The choice of the square root in this definition depends on the context and must always be specified. t
Partially supported by NSF Grant PHY78-08066.
79
One should think of
~(A,B,~,a,D,x)
as having its position localized near
a and its momentum near N . The position and momentum widths are heuristically given by the matrices We let
~
(AA*) ½ and ~I-~(BB*)½ , respectively.
H(~) = H0(M) + V = -(~2/2m)A + v
gradient of
V
denoted by
at
x
V(2)(x).
is denoted by
be our quantum Hamiltonian.
V(1)(x), and the Hessian
~2V .T~x" 1 j
The
is
With this notation we can state the finite time result of
[i]: Mx2
Theorem:
Suppose
Given any
R > 0, assume there exists
! BIx-Yl and
B0
1/3 < ~ < 2/3, V g C2(l~n),
whenever satisfy
Then f o r e a c h
% < Min {3~ - i, 2-3~} , there exist
, and
V > - C2 •
B > 0, such that IIv(2)(x) - v(2)(y)i [
Ixl < H and IYl < R.
(1)-(4).
IV(x) l < Cle
C
Let
a0 E ~ n , n0 E ~ n
T>0
, and let
A0
and e a c h p o s i t i v e
and @ > 0 such that
< ~ implies
Ile-itH(~)/M ~ ( A 0 , B o , ~ , a 0 , D 0 , ")
_eiS(t)/~ @ (A(t),B(t),~,a(t),D(t),-)ll < C ~ % , whenever
Itl < T
(det A(t) is never zero, and the branch
(5)
of the square root
_½ (det A(t))
is determined by continuity in
t ).
Here
[A(t),B(t),a(t),n(t),S(t)]
is the unique solution to the system of coupled ordinary differential equations: d__~D(t) = - V(1)(a(t)) dt
(6)
d a (t) = n(t)/m dt
(7)
dA (t) = iM I-2~ B(t)/2m dt
(8)
d B (t) = 2 i M 2 ~ - i dt
(9)
V(2)(a(t))A(t)
dS (t) = (q(t))2/2m - V(a(t)) dt
with
A(O) = Ao,B(O) = Bo,a(O) = a0, n(O) = n O , and S(0) = 0 .
(i0)
80
~a(t) ~a(t) ~q(t) ~N(t) Furthermore, the d i f f e r e n t i a l s - , , , and ~--~7~- exist, ~a(O) ~(0) ~a(0) oll~u7 and ~a(t) A(t) = --~TV-A-~A(0) + ½ i ~ I-2~ ~a(t) B(O) ~n(O)
(Ii)
B(t) =
(12)
Remarks:
~N(t) B(O) - 2 i M 2 ~ - i ~n(0)
i.
~q(t) A(0) . ~a(0)
Note that there are no problems with caustics in the above theorem.
The Maslov index [4] is contained in the choice of the square root (det A(t)) -½ . The index appears explicitly in Yajima's approach [5] to the classical limit. 2.
Our results are closely related to those of Heller [2] and Hepp [3].
However, for suitable potentials in 3 or more dimensions, the above theorem can be modified to handle scattering theory (see [i]). The first ingredient in the proof of our theorem is the Trotter Product Formula.
We approximate
e -itH(~)/~ ~(A(O),B(0),~,a(O),N(0),.
Ee, itH0(~)/N ~ e_itV/N~] N ~(A(0),B(0),~,a(0),q(0)," Next, we replace
)
),
where
by
for some large
e-itV/N~ ~(A(0),B(0),~,a(O),q(0),"
e-itWa(0)/N~ ~(A(0),B(O),~,a(0),n(O),.
)
)
N .
by
Wa(X) = V(a) +
+ ~i <(x-a), V(2)(a)(x-a)> We can now exactly compute e-itH0 (~)/N~ e-itWa(0)/N~ ~ (A(O),B(0),~,a(0), N (0),.)
= eiSN (i) I~
¢ (AN(1),BN(1),~,aN(1),nN(1),")
•
Iterating the approximation procedure, we replace the next factor e-itWa(1)/N~
to obtain
eiSN(2)/~ ~(~(2),BN(2),~,aN(2),qN(2),
e
-itV/N~
" ), etc.
by
81 1 < n < N, we have
In general for
n
NN(n) = N(0) -
[ V(1)(aN(J-I))t/N j=l n
aN(n) = a(0) +
~
qN(j) t/Nm ,
n
SN(n) =
I
~nN(j))2/2m - V(a(j-l))~ t/N ,
j=l n
~(n)
= A(O) +
[
i
1-20~
B(j) t/2Nm ,
j=l n
2i~2~-i
BN(n) = B(0) +
V(2)(aN(J_I))~(j_I ) t/N .
j=l
As
N ÷ ~, ~(n),BN(n),aN(n),nN(n),
a(nt/N),n(nt/N), and S(nt/N), where (6)-(10).
and SN(n) approach
A(nt/N), B(nt/N),
A('),B('),a(-),n(-), and S(') satisfy eqns.
Eqns. (ii) and (12) follow from eqns. (6)-(10) by standard ordinary
differential equations methods. Since we have fairly explicit information on the behavior of A(t) and ~ ( n ) , we can prove that our wave packets remain sufficiently narrow in It I < T.
x-space
Thus we can estimate the errors incurred by the replacement of
its second order Taylor approximations W a
for V
by
This error estimate implies the
theorem (see [i]).
To extend the theorem to infinite times we require that derivatives fall off sufficiently rapidly. the Taylor expansion errors See [I] for the details.
V
and some of its
This fall off allows us to control
even though the x-space width diverges as It[ ÷ ~.
82
Re£erences
i.
Hagedorn, G.A.; Semlclassical Quantum Mechanica I; The ~ ÷ 0 for Coherent States:. Commun. Math. Phys.
2.
(1976).
Hepp~ K.: The Classical Limit for Quantum Mechanical Correlation Functions-~
4.
(to appear),
Heller, E. J.: Classical S-matrix Limit of Wave Packet Dynamics. J. Chem. Phys. 6 5 , 4979-4989
3,
Limit
Commun. Math. Phys. 35~ 265-277 (1974),
Maslov, V.P.: Th~orie des Perturbations
e t M~thodes Asymp_totiques.
Paris ~ Dunod 1972. 5.
Yajima, K.: The Quasi-Classical Commun. Math. Phys.
(to appear).
Limit of Quantum Scattering Theory.
RANDOM SCHRODINGER SOME RIGOROUS Herv@
OPERATORS
RESULTS Bernard
Kunz
Souillard
Centre de Physique Th~orique Ecole Polytechnique 91128 PALAISEAU, France
Laboratoire de Physique Th6orique E.P.F.L. 14,Avenue de l'Eglise Anglaise LAUSANNE, Suisse
I. Introduction We report here results operators
with
stochastic
~Cv? = which
of finite
difference
Our hamiltonian
Schr6dinger
is then
+ V
~ C ~ v) ;
on
/k
is the discretized
Laplacian
by
(z~, ~c~;= for
potentials.
- J
is an operator
defined
on a class
~re
7-----
~t(~v)
~-c~)
-
2 ~
and the potential
V
act as a multiplication
operator
The potential tial features The
V(~)
, ~
an absolutely
complete
will be random.
:~
In order
and with
our results
statements
to deal with
stick here
, will be independent
continuous
will be bounded However,
~
of the problem, we will
distribution compact adapt
variables
, whose
case. with
density
support.
to more general
and proofs
random
dV
~(V)
the essen-
to the simplest
will be found
situations
and the more
in the paper by the au-
thors. This model to discuss
systems.Simple -
harmonic
is the model
transport
used by Anderson
and localization
and other
properties
authors
of various
physical
realizations
of such a model
oscillators
with nearest
neighbour
are
coupling
since,
disordered
: and random
masses. -
electrons tential There
tems
:
scattered ~C~)
by impurities
at the point
are four main problems
~
creating
an effective
~in the tight binding
of interest
random po-
approximation.
in the study of such sys-
84 -
Determine means
the s p e c t r u m
~C~)
with probability
one.
Physically,
to find the support of the density of states,
this
in particular
to
find if there are gaps or not. -
Find the nature where
of the spectrum,
with probability
it is pure point corresponds
cal litterature where
the presence
it is continuous,
to what
of localized
corresponds
one.
is called states,
The case
in the physi-
whereas
to the presence
the case
of delocalized
states. -
Determine would
-
the decay properties
of the pseudo-eigenfunctions.
like to know for example
if we have exponential
spectrum
is pure point.
Finally,
one is interested by the transport properties
tem. This depends more precisely the model.
In the case of oscillators,
the heat conduction, which matters Up to now,
whereas
the following
authors
(Ishii,
of the absence 2. In 1977,
the quantity of interest
is
it is the electrical
conductivity
as rigorous
from
point of view, which have been obtained, are concerned situation.
Apart
from some work on the presen-
results have been obtained
i. Beginning with Borland ~lJ rious
of
in the case of electrons.
the one-dimensional
ce of gaps,
of such a sys-
interpretation
the only results which can be considered
the~mathematical with
on the physical
One
decay when the
in 1963,
a series
Lebowitz-Casher,
of absolutely
of contributions
by va-
P a s t u r ) has led to a proof
continuous
Goldstein-Molchanov-Pastur
:
spectrum.
C2~ have proven
that the spec-
trum is pure point. 3. In 1978, M o l c h a n o v
~3~ proved
that the wave
functions
decay exponen-
tially. The methods
used to prove
to one-dimensional
situations,
ding of the m u'itidimensional Our method of approach brings
the problem
statistical
continuous
are h o w e v e r
all very specific
and do not give a clue to an understan-
case.
is not one-dimensional
from the beginning
into a kind of problem of continuous
mechanics
se transition
such results
and suggest
in statistical
a precise
mechanics
spin system in
connection between
and the transition
the pha-
from point
spectrum.
Presently,
however,
it is only in one dimension
and
that we have arts-
to
85 wered all the main questions mentioned before that the spectrum is pure point, nentially and furthermore
: we get a new proof
that the wave functions,
the static conductivity
decay expo-
is proved to vanish,
which is a new result. II. General results The spectrum Z ( H )
of this class of hamiltonians
is simply the alge-
braic sum of the spectrum of the Laplacian and the spectrum of the potential Theorem 1
With probability one
A simple corollary of this theorem is that the hamiltonian has no isolated eigenvalues,with
probability one.
Our next goal is to obtain useful criteria
for localization.
They
are provided by the following results. Let
ei
be the eigenvalues
of the operator H restricted to the box
Yi with zero boundary conditions and _~C=) denote the associa' II II ted normalized eigenvector. Let J)Cr; A ) ~e4 defined by
B'9"xI{" where
A
is an interval of energy.
Then we have Theorem 2 trum of
If
~Crj~)<
~
~
It is interesting opposite
~
, the~with probability one, the spec-
is pure point in the interval A.
:
to note that when ~
~Cr~ ~) ~ o
that the presence of an absolutely
~=
- ~
Y~cZ(H).
, we have on the It seems therefore
continuous part in the spectrum of
H is associated with the existence of long range order in the correla-
~(r~ ~}
tion function A criterion the following Theorem 3
for the presence of continuous
spectrum is provided by
:
If the so-called participation
ratio
86 is such that
then with probability
one, H has continuous
The proof of these two theorems which make
precise
the intuitive
spectrum
is based on Ruelle's
Concerning
Let
theorems E4~,
idea that a wave packet will diffuse
if and only if the spectrum of the hamiltonian
criterion
in A.
the transport properties,
is continuous.
we have obtained
the following
for the static conductivity.
q~A(~)
(~)
be the frequency
by the Green-Kubo
formula,
dependent
conductivity,
then if the static conductivity
given
of the infi-
nite system is defined by t
A ~_.v
,~"-~o +
a,a
o
we have Theorem 4
If
~ re ~_.Y
I~l ~ .DCwj~)
~ oo
then
,
This result shows that exponential plies
~w o
localization,
for example,
zero conductivity.
We have seen in this way that it is possible
to say something
the nature of the spectrum or the transport properties enough information ke
about some wave
<~&{~)~¢~
~6~
a finite box A expressions rally,
7^
~11 ~
correlation
functions,
li-
of the hamiltonian
restricted
to
for such quantities.
It is possible
to expectation
spins.
<~>^
<~^
= ~ ~
to do so, quite gene-
in a form which bears some re-
values of correlation
functions
in a lattice
The result is given by the following:
S
Let
about
if we can get
. We are faced now with the problem of finding explicit
system of unbounded Theorem
function
and in fact to bring the result
semblance
Then
im-
"~(ed~ _~F~ ) ~^
is given by the expression
:
87 where
JC~)=
2 IF ~ - ~ ( ~
~2)~I ~
In this last expresssion,
/7-
~
/
the sum runs over all the spanning
trees
~A of The idea behind distribution
of
the proof of this result
V
induces
an eigenvector~ of
H^CV),
III. The one-dimensional
a probability
sion.
distribution
of corresponding
eigenvalue
on
y
to be
e
case
By means of the criteria possible
is that the probability
discussed
and the formalism
to give a rather detailed analysis
The results obtained
settled, it is
of the model
can be summarized
in one dimen-
in the following
theorem.
Theorem 6
DCr~ A) ~
a)
cCA)e~-
~CA~I~I
for any compact subset A of R, with independent
of
~cA)>o
b) for almost all
V
,
c) for almost all
V
,
d) for almost all
V
, the static conductivity
~CV)
has pure point s p e c t r u m
the eigenvectors
are exponentially
The results b), c), d) follow essentially
DCrjA)
bound on
results
< ~~ e ~
tion function
~cA), cCA)
, and
A.
transfer operator,
>^
decaying
is zero.
from a). The exponential
from a detailed analysis
~ ( ~ C~xl ,l ~ )~
familiar
~
of the correla-
by means of some kind o f
in statistical
mechanics
of one-dimensio-
nal systems. Moreover, pressions value
it is possible,
for the average
using this technique,
spectral
and the participation
function,
to give explicit
its average
ex-
absolute
ratio.
References 1
R.E. Borland,
2
I. Ya Goldstein, Prilozhen
Proc. Roy. Soc. A 274, 529 S.A. Molchanov,
(1963)
L.A. Pastur,
Functs.Anal.y.
ii, 1 (1977)
3
S.A. Molchanov,
4
D. Ruelle,
Math.
USSR Izvestija
Nuovo Cimento A 61
655
42 (1978)
(1969).
Transl.12,69
(1978)
ON THE SCATTERING
PROBLEM WITH INDEFINITE
METRIC
Xia Daoxing Fudan University,
Shanghai,
China
Several
physicists
try to get over the difficulty
the theory
of quantum
fields
theory
concerning
by using
indefinite
Let ~t~ be a Hilbert
metric
be the projection
nite metric
to the space ~
Hamiltonian
H
a bounded and
n = -i
. A vector
valued
function
every
and
operator,
x £~ x(')
there
H±:~6_+ ÷~_+
which
states.
~
= x
is a function
fa(.)EL I
H
W+=W+(H',H0):
o ~+
÷ ~'
is defined
Under certain of the scattering an expression
-itH ' W±~
conditions, operator
of scattering
W±(H+,H 0)
then
in
~+
81<0,
that for
. The wave operator
we proved
=
0
.
the existence
S = W~ I W+ ; in addition operator.
We notice
SI
And we shall discuss
and the unitarity to these,
W+(H+,H0)
W[(H',H+)-Iw+(H',H+) SI
we gave
that if the wave oper-
in the Hilbert
W+(H',H 0) = W+(H',H+) , where
H+ = H 0 .
and in
such that
-itH 0 - e ~
to the indefinite
them.
vector-
by
S = W_(H+,H0)-IsIW+(H+,H 0) to consider
is denoted
as the space of all asymptotic
W±(H+,H 0)
does not relate
operator
~-I
of the pair of Hamiltonian
exist and are unitary,
81>0
is
fa (m)d~
be the free Hamiltonian
lim Ile t÷+_~
ators
f
: ~
states
+~_
is an analytic
both in
The
to the indefiF:~+
and has the property
. We shall consider ~.+
Let
, where
is the self-adjoint
is analytic
(H-1)x(1)
an indefi-
[x,y]=(Jx,y).
with respect
, iff there
i
H'=H
(x,y), x , y e ~ , ~ £ = ~ C + ~ _ ,
The space of all physical
(x(X),a)
Let
in the following.
by giving the definition self-adjoint
in
The scattering
J = P+ - P_ . We introduce
is in~}~'
the equation
a e~,
may be formulated
H = (H_+ ~F*)P_+(F+H+)P+
in this case.
by ~ '
satisfies
and
is an operator,
linear
metric.
space with inner product
P±:~6~+~_+
nite metric[x,y],
indefinite
of divergence
metric, only.
space ~ + and . But
so we didn't
need
Thus we suppose
that
89
Let
~
be the spectrum of
H+ . For
I[~
, we construct
the oper-
ator h(k) in
~_
: k - H_ - n F(k-H+)-IF *
This operator-valued
to both
H
and the decomposition
We consider the following L2(o,~)
analytic
~}~= ~ +
case.
with values
(~_
and it is useful. is the space
and square-integrable
in a separable
: I ~¢(~)~a~
Ii~ iI z
h(.) is also related
Suppose t h a t ' S +
of all strongly measurable
valued functions
function
d~
Hilbert
(H+¢)(~)
,
vector -
space ~
= m~(~)
and
,
g
for
~¢L2(0,~)
that
~(~)
, and there is an operator-valued
is an operator
from
~
F% = ~o {(~)~(m)
Theorem
If
h(mii0)
and
to d~
~
for
, for
h(m±i0) -I
function m~
C(~)
such
and
% s~C+
exist for all real
~
and
~ k(~±iO) -I { (m) ~ 2d~ < Then the indefinite the space ~ ' ators
W±
metric
[x,y] in ~ '
endowed with
exist,
are unitary operators
W±% = f h(~±i0) -I ~ ( ~ ) ~ ( m ) d ~
And the scattering
operator
S = W-Iw+
which has the form (S~)(~) = S(~)~(~) i + 2 ~ i ~ ( ~ ) *h(~+i0) -i~ (~) For Lee and Wick's model in concrete
form of
of two variables,
S
[%-in~
onto
~'
is a unitary operator
by solving a singular
and
*f h(m±i0)-l~(t)#(t) t - (~±i0)
~ where
N 0@0 sector
in
is the unknown
[I]
, we determined
~+
the
integral equation of function
,
function
and
dt]
S(~) =
- f Y(~I'~''~(~2)~(~')+Y(~''~2)~(~')~(~I) ~,.~+~I+~2_i 0
= 6(~l-al)6(~2-a2) y(',')
from~+
definite,
space, the wave oper-
i.e.
H(~-~l-~2+i0)Y(~l,~2)
where
becomes positive
[x,y] is a Hilbert
e is given,
d~
90
H(~) = ~-m+ f ~(~)2(X-~)-idm matrix elements of
S
is 3
S(ml,m2~3;al,a2,a 3) =
~(~l+m2+~3-al-a2-a3
• In this case, the explicit form of the
-iarg (H(~j +i0 )H(aj +i0 ) ) e
~(ml~2,~3;al,a2,a3)+
i=l
)2~i ~ 3 j=l
~(~j)~(a.)j. ~3 H(~.+i0)H(a.+i0)] ] i,j=l
i L(a.)]
IH(a.+i0)i 2 J ~(~j-aj)+f(~j;aj) ~(a.) z 3 where
f
and
L
If we change
can be expressed by the given function
e .
q = - i to n = i in the theorem, then the scattering
problem concerning indefinite metric becomes the scattering problem with intermediate system similar to the case considered by Lifschitz[3]. We also proved this theorem in the case applications of decay
in the case
q = i
n = i . And then we gave some
to determine the asymptotic probability
of the intermediate system and the corresponding generaliz-
ation of Wigner-Eisenbud's
formula.
REFERENCES [i] Eee, T.D.; Wick, G.C.: Nucl. Phys. Bg, 2(1969), 209-243; BI0, 1(1969), i-i0. [2] Xia Daoxing: Scientia Sinica 18(1975), 165-183. [3] Lifshitz, M.S.: Dok. Acad. Nauk II! (1956), 67-70, 799-802
SOME OPEN PROBLEMS ABOUT COULOMB SYSTEMS E.H. Lieb * Departments of Mathematics and Physics Princeton University Princeton, N.J. 08544, USA
I.
Introduction While much progress has been made in the rigorous mathematical
theory of Coulomb systems,
it hardly needs to be said that much remains
to be done. This talk is supposed to be a review, but it is not inappropriate to survey the subject by emphasizing some open problems whose solutions
seem to be not far beyond the reach of present math-
ematical technology. Naturally everyone can produce his own list, but attention should be paid to the following points:
(i) Not all the obvious problems that
hold for general many-body systems are appropriate now. They may be presently too difficult and unusual problems to be necessary. tential.
in the Coulomb case.
(ii) There exist special
connected with Coulomb systems whose solution seems
An example is Problem 9 on the boundedness of the po-
Another is the existence of screening.
ematical physicists
(iii) While some math-
appear to believe that the sole aim of their sub-
ject is to find rigorous proofs of what is already heuristically known, that is far from the truth. mathematical physics,
There is also a computational aspect to
namely to find good approximation schemes which
yield upper and lower bounds;
in the best case these bounds should be
capable of improvement to arbitrary accuracy.
This goal is especially
important for Coulomb problems because there is already a vast literature on uncontrolled approximation schemes ecular and solid state physics. compute physical parameters, perimental verification
for problems in atomic, mol-
As these schemes are widely used to
sometimes
in regimes in which direct ex-
is difficult or impossible,
sirable to set limits on their aceuracy.
it is highly de-
Some of the problems mentioned
here are of this kind.
~) Work partially supported by U.S. National Science Foundation grants INT 78-01160 and PHY-7825390.
92
II.
Lower Bounds
to Ground
State Energies
In this and the next two sections we will be concerned with the non-relativistic
Schroedinger
equation when the nuclei are held fixed
(also called the Born-Oppenheimer k
fixed nuclei of charges
electrons
(in units
lel= I,
N
U =
V(x)
Another
clei a p p r o x i m a t i o n
+ ~ Ixi-x jl-!+ i<j
(2.1)
U
(2.2)
k E z.z. IRi-Rjl -I i<j i 3
properties
important
h2/2m = I) is e
k = E z. Ix-Rjl -I i ]
One wants to find good estimates as qualitative
(2.3)
for the ground
of the dependence
problem
state energy E0, as well
of E 0 on the {R.}. l
is to find corrections
when the nuclear kinetic
energy
to the static nuis introduced.
sults in this direction have been given by Aventini, Grossman and Seller
[i], for example.
Combes,
mass,
then obviously
Re-
Duclos,
We will not say anything more
about this here except to note that if the nuclear kinetic introduced,
for
N
H = -~A -~ V(x i ) I i i
where
The Hamiltonian
Zl.. , z k > 0 located at RI,... , R k and N
in which N
approximation).
the nuclei are not fixed.
the nuclei will be predominantly
around
energy is
For large nuclear
{Ri }, the values
of the
R. w h i c h m i n i m i z e s E . I n t h e a f o r e m e n t i o n e d p r o b l e m p o s e d by ( 2 . 1 ) , 1 o h o w e v e r , a l l {R.} c a n be c o n s i d e r e d . This is not without physical inl terest. In particular, a widely used approximation to the partition function
of matter
partition
function
E0({Ri}) T
nuc
at moderately obtained
is regarded
high temperatures
from t h e n u c l e a r
as a potential,
is the nuclear kinetic
(~,Hg)
P~(x)
is
the
in which
namely H ' = Tnu e + E0({Ri}) , where
energy.
One time honored and sensible way to approach approximate
and pressures
Hamiltonian
(2.1) is to try to
in terms of the single particle
density
= N Z f [~(x, x 2 . . . , X N ; O l , . . , ~ N ) 1 2 d x 2 . . d X N
p9 given by
(2.4)
93
Here,
{o i} are the spin coordinates,
states
in general.
~
~i s{l,...,q},
is antisymmetric
in (2.1) is easily evaluated,
(%,V~) =
and
there being q spin
(~,~) = i. The second term
namely
~ V(x)p~(x)dx.
(2.5)
Thus we have to worry only about the first and third terms. Problem i (Lower bound to the kinetic equality
energy):
We want to find an in-
of the form
T~ ~ (¢, -ZA i ~) ~ Kq -2/3 f p~(x) s/3 dx .
The choice
0
5/3
possibilities
.
ms made for dimensional
exist,
reasons.
such as [f0~a] b with
classic and also the most tractable.
3b(a-l)
(2.6)
Of course, = 2, but
The conjecture
other
(2.6) is
[2] is that
(2.6)
holds with K = K c ~ 3/5
Kc
is the classical
following reasons. essentially literature lems.
If
bounds
value.
First,
gives Thomas
as will be discussed - Fermi
is correct,
(except
Hartree-Fock
term of secondary
If the HF and TF constants
What is known
Problem
bounds
[3] is that
in (2.1) is usually in the state the indirect
0~
automatically
upper bound
Second,
(the Weizsaecker
approximation
(2.6) holds with
cor-
K c , then scheme.
K > K c [1.83/4~]
2/3
~ 3
part of the Coulomb repulsion):
approximated
lower
(of. Problem 4) using
have a common value
of a powerful
prob-
become
(2.6) with K c, together
importance
also hold in dimensions
2 (The indirect
(2.6) with K c
There is an enormous
terms which are easily bounded).
conjectured
one would have the beginning
later,
for the
is applied to physical
(HF) theory which also contains
with an additional
Analogous
(TF) theory.
these calculations
for exchange
there is a c o m p l e m e n t a r y
(2.7)
The proof of (2.7) is important
in which the TF a p p r o x i m a t i o n Kc
rection).
(672) 2/3
by the average
The third term
Coulomb repulsion,
and we want a lower bound to the remainder,
part of the Coulomb energy, which we call
D ,
namely
I~ . Thus
94
N
(~, Z Ixi-xjl-l¢) i<j
D(p)
For dimensional
= 1/2
//
+ I9
= D(p~)
p(x)p(y)lx
, where
y1-1 d x d y
-
(2.8)
.
(2.9)
reasons we can seek a bound of the form
I, ~ - C SP~(X)4/3dx
(2.10)
.
This form was introduced by Dirac [4] as an approximation, with C re-l/3 placed by (0.93) q , and is widely used (TF-Dirac theory). Note that there is no q-dependence this form exists
in (2.10).
It is known that a bound of
[5] with 0.93 j C ~ 8.52. The problem
sharp C , which is probably
is to find the
not far from i .
If we now combine the various
estimates,
TFD theory is obtained,
namely
E 0 ~ inf {EL(P)ISp EL(P)
: N ] , where
: S [q-2/3Kep5/3
Next, we consider upper bounds III.
Upper Bounds to Ground
- Vp - Cp 4/3] +D(p)
+ U .
of the same form as (2.11).
State Energies
The obvious way to obtain an upper bound is to compute for some suitable normalized fectively
in practice.
(2.11)
~ . This, however,
A favorite
choice for
~
(~
Hg)
is not easy to do efis a Slater determinant,
namely
@ = (N!) -I/2 det [@i(xj;~j)]
where the
{¢i }
are
N
orthonormal
functions.
,
(8.i)
Then
N
p~(x)
= E Pi(X) i
,
Pi(X)
: Zl~i(x;a)I 2
(3.2)
N
T, : lZ T.=
,
Ti = ~ /Ivhl 2
(s.3)
95
I~ < 0
(3.4) is e s p e c i a l l y important function.
(3.4)
and is one good reason for using a HF
It is a c o n s e q u e n c e of
Ix - yl -I
being positive definite.
(3.4) suggests the following:
P r o b l e m 3 (HF Coulomb energy): zero, to
Find a useful upper bound, other than
I V . The most useful would be
I V j - C' fpv 4/3
The most important p r o b l e m is an upper bound for the kinetic energy, T V , in terms of
p V . There is none, of course, because
a r b i t r a r y large for any given however,
it becomes
sensible.
0(x) s a t i s f y i n g
Then choose
N
p V . If we turn the q u e s t i o n around,
How small can
(i)
orthonormal
fp = N
functions
T V , given by
can be
That is
P r o b l e m 4 (Lower bound to the HF kinetic energy): function
TV
and
Let a n o n n e g a t i v e
(i$) f[vpl/2] 2 < ~
{~i (x;o)} such that
be given.
(3.2) holds.
(3.3), be? The c o n j e c t u r e is that
T V can
be made to satisfy
TV I q
-2/3
Kc f p 5 / 3
+ ~[vpl/212
(3.5)
The second term in (3.5) is the W e i z s a e c k e r correction. There are two obvious advantages
to (3.5).
First, it implies that
one can do HF theory by c h o o s i n g only the single p a r t i c l e density instead of h a v i n g to choose the N o r t h o n o r m a l functions
p
{%i } . The
result is
E o ! EU(P)
for any
Eu(P)
p ~ 0
such that
= f[q-2/3K°pS/3
Second, the m i n i m i z a t i o n
f0 = N
, where
- Vp - C'p 4/3 + (vpl/2) 2] +D(p)
of ( 3 . 6 )
and
+ U .
(3.6)
(2.11) are similar problems.
96
While
(3.5) is h e u r i s t i c a l l y reasonable,
problem,
even for
in (3.5)
. For
it seems to be a very hard
N = 2 , and even if different constants are allowed
N = i
it is trivial
(take { = pl/2)
was raised by March and Young [6] who purported, it in three dimensions.
. The p r o b l e m
incorrectly,
to prove
However, they did c o r r e c t l y prove the anal-
ogous c o n j e c t u r e in one dimension, which is
T, <_. q-21
with
KlC = 2 / 3
. The c o n s t r u c t i o n for
~j(x)
q = i
= [p(x)/N] I/2 exp{ikj
with
k. = (2~/N)[j - (N + 1)/2] , j = i, 3 This is easily g e n e r a l i z e d to q > i .
IV.
Properties
(3.7)
is
...,N .
of TF and Related Theories
Let us c o n s i d e r
(2.11)
(TFD theory)
or
(3.6)
(TFDW theory).
properties of these m i n i m a are somewhat complicated. has been made
[7] but m u c h remains to be understood.
W e i z s a e c k e r terms are omitted, however, ory)
is much b e t t e r u n d e r s t o o d
lems remain
(cf
(3.8)
f~p(y)dy}
Recent progress 4/3 If the p and
the r e s u l t i n g p r o b l e m
[3a,8,9,10].
The
(TF the-
Some interesting open prob-
[3a,8,9,18]).
The TF energy for a neutral system has the p r o p e r t y that it decreases under d i l a t i o n of all nuclear coordinates
[9]. This is surely
also true for a n o n - n e u t r a l system, but it is not yet proved. theorem
(which holds g e n e r a l l y
[8]) is implied by this.
for N (which need not be an integer)
E~F(N)
Teller's
It states that
electrons
k > min E E TF j=l atom
(zj kj) '
'
(4.1)
where
E TF (z,~) is the TF energy f o r a single nucleus and electron atom number X . The m i n i m u m in (4.1) is over all kj such that E k1 ~ j = N . The fact
(4.1) is one proof of the stability of m a t t e r [2], once one
has the kinetic energy inequality when
k
becomes
large and
N
(2.6). Now c o n s i d e r what happens
is fixed.
For
~ << z ,
97
E TF (z,~) = -z21 I/3 This is q u a l i t a t i v e l y correct when i < l<
Then for large E0(N)
k
= -z2N
if
(4.1) becomes
-z2k(N/k) !/3
zk << N ; this was in fact
[ii] using a m o d i f i c a t i o n of the D y s o n - L e n a r d proof
[12]. The trouble with
TF theory is that it cannot d i s t i n g u i s h the
situation in which the e l e c t r o n number is less than one.
P r o b l e m 5 (Improvement of TF theory for zk >> N): f i c a t i o n of the TF stability proof E2,3]
Find a simple modi-
to obtain
E 0 ~ -z2N
when
zk >> N . Despite the f o r e g o i n g problem,
TF t h e o r y (and also TFD and TFDW
theories) have an important feature.
If
k
is fixed and all
then TF theory gives the correct energy a s y m p t o t i c a l l y
lim all z.÷~ 3 Note that
E TF 0 /
z. ÷ = , 3
[8].
E0 = i
(4.2)
(4.2) is true only if K c is used in (2.11); this is another
good r e a s o n for solving P r o b l e m i. The TF energy for a neutral atom is -(2.21)
z 7/3
. The next c o r r e c t i o n
ably that suggested by Scott
is p r o b a b l y order z ~ and is prob-
[13], w h i c h we state as
P r o b l e m 6 (The Scott correction):
Prove that
k E 0 = E TF + (q/8) ~ zj 2 + o(z 2) l
where
E TF is given by
(4.3)
(2.11) with C = 0
Scott's idea is that the c o r r e c t i o n to
E TF comes from the electrons
close to the nuclei and should t h e r e f o r e be additive for each atom in the molecule.
Cf. ref.
[3a], p. 560.
It is p r o b a b l y not too difficult to get an upper bound to the form
(4.3) by using a Slater determinant.
d i f f i c u l t to obtain.
E 0 of
A lower bound is more
The following r e m a r k may be useful in this con-
nection. The m i n i m i z i n g TF
p
for a neutral system satisfies the TF equation
98
(5/3)q-2/3Kep(x)2/3
Thus
= V(x)
E TF = -(2/3) q-2/3KCf@5/3
particle H a m i l t o n i a n
(~, H~)
where
D(f,g)
-D(p)
[p(y)lx
yJ -1 d y ~ ¢ ( x )
- D(p) + U . Let
-A- %(x)
and let
: (~, H09) - 2n(p,p~)
+ 2D(p,p~)
these terms are
h
(4.4)
be the single
+ U + (~,EJxi-x j -i~
)
(4.5)
.
45, the last term in (4.5) is bounded below
- (2.21)Nz 2/3 - z-2/3[ p 5/3
0(z 5/3)
Thus, to
O(z 5/3)
. The last two of
,
(4.6)
D(p)+E 0 ~ E00 = inf spec H 0
E00
.
H 0 = ZI N h i . Then
= (1/2) fff(x)g(y)Jx-yJ -I dxdy
As shown in [3a], equ. by
-
is the sum of the lowest N eigenvalues
of
h (including spin)
Suppose one can show that
E00 > -(2/5)(3/5K0) 3/2 qS ~5/2 + (q/8)E k zj
2
The first term in (4.7) is the classical approximation; term is the Scott c o r r e c t i o n for sired lower bound is obtained.
h
. Then, r e c a l l i n g
(4.7)
the second
(4.4), the de-
In other words, b a s i c a l l y the Scott
correction is the same as the n o n - c l a s s i c a l c o r r e c t i o n to the single particle H a m i l t o n i a n
H 0 , in which the p o t e n t i a l is the n e g a t i v e of
the TF potential.
V.
The Boson P r o b l e m Consider the H a m i l t o n i a n
(2.1) for bosons
As far as the ground state energy is concerned, setting
q = N
instead of fermions. this is the same as
in the fermions problem. (4.1) is still true, but the 2/3 7/3 -(2.21)q z . Taking k = N and
energy of a neutral a t o m is z = i , we thus obtain
E 0 ~ - ( c o n s t . ) N 5/3
(5.i)
99
This result was o b t a i n e d by Dyson and Lenard (5.1) is correct in that it can be shown
[12]. [14] that E0~ - ( c o n s t . ) ' N 5/3
for a suitable choice of the R. (depending on N). However, (5.1) is mis1 leading if the nuclear kinetic energy is included~ h o w e v e r large the nuclear mass
M N may be.
ergy does not alter With
MN = i
In the fermion case the nuclear kinetic en-
E 0 very much,
if
and boson nuclei,
M N >> i , but for bosons
Dyson
[15] showed that
(5.2)
E 0 ~ - ( c o n s t . ) N 7/5
by a v a r i a t i o n a l c a l c u l a t i o n with a f u n c t i o n plicated.
However,
the i n t u i t i o n behind
it does.
~
that was quite com-
(5.2) is r e l a t i v e l y straight-
forward, as follows. At high density the Coulomb i n t e r a c t i o n is weak in some sense. this regime the B o g o l y u b o v
In
a p p r o x i m a t i o n is expected to be valid;
the
wave f u n c t i o n in this a p p r o x i m a t i o n describes particles with paired momenta
k
and
-k . The B o g o l y u b o v a p p r o x i m a t i o n yields an energy
[16] ' E 0 = -(const.)
N p
1/4
, large
p
Suppose that the total energy of the bound s y s t e m is
(5.3)
E 0' + N I / 3
p 2/3
,
the latter term being the " u n c e r t a i n t y principle" kinetic energy needed to localize the system so that its density is
p . (5.2) is then ob-
tained by m i n i m i z i n g the f o r e g o i n g e x p r e s s i o n with respect to p(p ~ N 8/5)
. The mean spacing b e t w e e n particles
P r o b l e m 7 (Boson ground state energy): (5.2) when the H a m i l t o n i a n k=N,
z.=l 3
is
is
N -I/5
Find an upper bound of the form
(2.1) + (nuclear kinetic energy)
and
.
It may be w o n d e r e d why P r o b l e m 7 is p r e s e n t e d as an important pmoblem at the f r o n t i e r of the subject. r a t h e r than its p h y s i c a l
The reason lies in its c o n c e p t u a l
importance.
Sections
Ii-IV stressed the fact
that fermion systems could be p l a u s i b l y u n d e r s t o o d in terms of a fluid of e l e c t r o n density.
Quantum mechanics
that the self energy of the fluid was on the other hand,
entered through the statement q-2/3 Kelp5/3
. Boson systems,
cannot be easily u n d e r s t o o d in such a simple way.
Quantum correlations,
of a sort r e l a t e d to s u p e r f l u i d i t y and supereon-
100
ductivity, play a central role. be u n d e r s t o o d if progress problem.
It is essential that these correlations
is to be made w i t h the quantum m a n y - b o d y
Problem 7 appears to be the most tractable p r o b l e m of this
genre.
Vl.
Infinite Systems Let us first discuss the p r o b l e m of the t h e r m o d y n a m i c
the free energy in the canonical ensemble.
For th4 H a m i l t o n i a n
s u p p l e m e n t e d by the nuclear kinetic energy operator, been shown to exist role,
[17]. Q u a n t u m mechanics
limit of (2.1)
the limit has
does not play m u c h of a
except that it provides a linear lower bound on E 0 (H-stability),
because the d i f f i c u l t y comes from the long range i/r control the Coulomb potential,
fall off. To
a p r i m i t i v e form of screening was used,
namely Newton's t h e o r e m which states that two disjoint, neutral , isotropic balls have v a n i s h i n g i n t e r a c t i o n energy. There are several reasons that the use of Newton's t h e o r e m is unsatisfactory:
(i) It requires that the system be locally isotropic;
this property fails for the solid state p r o b l e m d i s c u s s e d below;
(it)
Screening is really the property that the p o t e n t i a l of an a r b i t r a r y charge inside a surface S can be c a n c e l l e d outside surface charge on
S . Screening holds
Newton's theorem does not. the Yukawa potential,
S
by a suitable
for the Yukawa potential, but
Thus, the proof in [17] does not hold for
a l t h o u g h the t h e r m o d y n a m i c
limit does exist by
more standard arguments. Apart from these criticisms,
there is the p r o b l e m that the use of
Newton's t h e o r e m obscures the main physical fact r e q u i r e d for the thermodynamic
limit.
This is that two large systems have a weak Coulomb
interaction on the average. points of view:
One can explain this from two a l t e r n a t i v e
(a) In order for two systems to have a strong interac-
tion they must have large internal charge fluctuations.
These fluctu-
ations are rare because they raise the internal energy by a large amount; (b) If large charge fluctuations occur then one can place a suitable charge density
o
on a surface
a way that the energy of
o
S
separating the two systems in such
will be muoh more negative than the pur-
ported interaetion b e t w e e n the two systems.
A suitable
o
is, in fact,
the charge density that screens out the charge f l u c t u a t i o n in one of the two systems. While it seems to be d i f f i c u l t to c o n s t r u c t a proof of the thermo-
101
dynamic limit using
(a), the use of (b) is attractive.
One can easily
argue r i g o r o u s l y that if large charge f l ~ c t u a t i o n s occur then one can introduce
n
extra p a r t i c l e s
into the system and localize t h e m to
give a charge density a p p r o x i m a t i n g ergy by an amount much more than
~ . This will lower the free en-
n , thereby giving a c o n t r a d i c t i o n
p r o v i d e d the f o l l o w i n g ean be solved.
P r o b l e m 8 (The i n s e r t i o n enersy of a particle):
(e)
Consider the
H a m i l t o n i a n d e s c r i b e d at the b e g i n n i n g of this section for a neutral c o l l e c t i o n of
N
particles
Show that the free energy,
in a box of volume V p r o p o r t i o n a l to
N .
or ground state energy, changes by an amount
of order unity if a neutral pair of p o s i t i v e and negative particles is i n t r o d u c e d into the system. tential is of order unity.
In other words, (B)
show that the chemical po-
Next, c o n s i d e r the "solid state Hamil-
tonian" which is given by periodically.
(2.1) with all z. = z and the {R.} arranged ] ] Show that the ahemical p o t e n t i a l is order unity for this
system also. If P r o b l e m 8B can be solved then, by the p r e c e d i n g argument, will be able to prove the t h e r m o d y n a m i c
one
limit for the solid state
H a m i l t o n i a n - s o m e t h i n g w h i c h has not been done so far. It is absurd that P r o b l e m 8 does not have a trivial solution. is easy to see, using H - s t a b i l i t y , the energy by 0(N).
It
that adding O(N) particles changes
What is needed is the obvious, but u n p r o v e d fact
that adding two particles
changes the energy by 0(i).
P r o b l e m 8 is e s s e n t i a l l y e q u i v a l e n t to
P r o b l e m 9 (Boundedness of the potential): equilibrium, tential
For a neutral system in
or in the ground state, show that the average Coulomb po-
<~(x)>
is bounded,
density and t e m p e r a t u r e
u n i f o r m l y in x , by a function of the
only.
Of course P r o b l e m 9 hints at a much more difficult problem, to prove the e x i s t e n c e of the c o r r e l a t i o n functions namic limit for all densities (i.e. e x p o n e n t i a l
fall off)
and temperatures.
in the thermody-
In p a r t i c u l a r screening
is e x p e c t e d to occur in the two point func-
tion - at least at high temperature.
In a r e m a r k a b l e paper
[18] Brydges
has shown this to be true in certain Coulombic, but classical, models at high temperature.
namely
lattice
102
References ~i]
Aventini, P.; Combes,J.M.; Duclos,P.; Grossman,A.; Seiler,R.: On the method of Born and Oppenheimer (to be published), cf. J.M. Combes in The Schroedinger Equation, W. Thirring and P. Urban eds. Springer, Berlin, P. 139 (1976).
[2]
Lieb,E.H.; Thirring,W.E.: Phys.Rev. Lett. 35, 687 (1975); errata ibid 3--5,1116 (1976). See also Studies in Mathematical Physics, E.H. Lieb, B. Simon and A.S. Wightman eds., Princeton Press, p. 269 (1976).
[3]
(a) Lieb, E.H.: Rev. Mod. Phys. 48, 553 (1976). (b) Lieb, E.H. Bull. Amer. Math]--Soc. 8_~;1751 (1976). Lieb, E.H.: The number of bound states of one-body Schroedinger operators. Amer. Math. Soc. Symposium on the Geometry of the Laplace operator, (1979), to appear.
[4]
Dirac, P.A.M.: Proc.Camb. Phil. Soc. 26, 376 (1930).
[S]
Lieb, E.H.: Phys. Lett. 70A, 444 (1979).
[6]
March,N.H.; Young,W.H.: Proe. Phys. Soc. 72, 182 (1958).
[7]
Benguria, R.: The von-Weizs~cker and Exchange Corrections Theory, Ph.D. Thesis, Princeton University, Lieb, E.H.; Simon,B.: Adv. in Math. 23, 22 (1977).
[8]
in Thomas-Fermi 1979.
[9]
Benguria,R.; Lieb,E.H.: Ann. of Phys. (N.Y.) ii0, 34 (1978); Commun. Math. Phys. 63, 193 (1978).
[io]
Brezis,H.; Lieb,E.H.: Commun. Math. Phys. 65, 231 (1979).
Lii]
Lenard, A.: Springer Lecture Notes in Physics 20, 114 (1973). Dyson,F.J.; Lenard,A.: J. Math. Phys. ~, 423 (1967). Lenard,A.; Dyson,F.J.: ibid. ~, 698 (1968).
[12]
[13]
Scott, J.M.C.: Phil. Mag. 43, 859 (1952).
[14]
Lieb, E.H.: Phys. Lett. 70A, 71 (1979).
[15]
Dyson,F.J.: J. Math. Phys. ~, 1538 (1967).
[16]
Girardeau,M.; Arnowitt,R.: Phys. Rev. Ii__~3, 755 (1959); Foldy, L.L.: ibid. 124, 649 (1961). Lieb,E.H.; Lebowitz,J.L.: Adv. in Math. ~, 316 (1972).
[17] [18]
Brydges, D.C.: Commun. Math. Phys. 58, 313 (1978).
Time Dependent Phenomena in Statistical Mechanics Oscar E. Lanford II1" D e p a r t m e n t of Mathematics University of California Berkeley, California 94720
O. Introduction This lecture will unfortunately not be a systematic review of the subject of rigorous results in non-equilibrium statistical mechanics. A preliminary attempt to outline such a review led me quickly to the conclusion that the field is too diverse to be summarized in a single lecture. I have therefore decided instead to discuss a few related.works in depth. The works I have chosen are: 1. The paper of J. Fritz and R. L. Dobrushin[5] on two-dimensional dynamics. 2. The paper of W. Braun and K. Hepp[3] on classical mechanics in the Vlasov limit. 3. A recent preprint by H. van Beijeren, J. L. Lebowitz, H. Spohn, and myself[2] on autocorrelations and fluctuations in the dilute equilibrium hard-sphere gas. I. Fritz and Dobrushin on Two-Dimensional Dynamics The problem is to investigate solutions to the Newtonian equations of motion:
d2~ = £ ~ ( g , - ~ ) ; dt2
r=-v~
i#j
where: a. The force function ~(g/-ga.) is small for large Ig,-gal but b. There are infinitely many particles. In contrast to the existence problem for finite-particle dynamics, this is a singular and delicate problem, in essence because it is difficult to bound the force exerted on any particle and hence difficult to rule out the development of singularities in finite time. The approach generally taken is to restrict the sat of initial phase points. On the other hand, it is undesirable to restrict them too drastically, since one wants to keep enough to form a set of probability one for at least some interesting statistical states of the infinite system. There are two broad approaches to the problem of restricting the initial phase point; I will refer to these two approaches as mechanical and probabilistic. The mechanical approach is the one which comes to mind first; one seeks explicit verifiable bounds on the initial phase point which are propagated by the equation of motion in such a way as to guarantee that solutions do not "blow up". The work of Fritz and Dobrushin is the most powerful result available in this direction; previously an existence theorem of this sort valid for a large set of initial phase points had been proved only for one-dimensional dynamics. The probabilistic approach is based on a more subtle idea, which can be traced back to a *Preparation of this report was supported in part by the National Science Foundation under grant MCS78-06718. It is a pleasure for me to acknowledge also the hospitality of the Seminar ftir theoretische Physik of the ETH, ZQrich, where the first draft was written under ideal conditions.
104
paper of Sinai[8] on dynamics in one dimension. In this approach, it is assumed that the interparticle potential ~b is thermodynamically stable, and the idea is to exploit two facts: a. Any Gibbs state/z 0 for ~b assigns probability one to a set of initial phase points with very good regularity properties. b. If the solution flow to the equations of motion exists, it ought to leave/z 0 invariant, and hence almost all solution curves should have good regularity properties not only at time zero but at all other times as well. To see how these two ideas might be used, imagine that a local existence theorem--up to time 8t--can be proved for /z0-almost all initial phase points and that the solution mapping T 8t leaves/z 0 invariant. (This is intended only to illustrate the reasoning in a simple case; I know of no system in which the argument I a m about to give is actually useful.) The idea is as follows: We are assuming that {to : T~tto is defined} is a set of/z0-probability one. But since T 8t preserves/z 0, {to : T~tto is defined and in the domain of T ~t } is also of probability one, Le., {to : T28tto is defined} is of probability one. Iterating this argument, {to : TnStto is defined for all n} is of probability one, Le., a global existence theorem holds for /~0-almost all initial phase points. A more circumspect version of this argument avoids even the necessity of having a local existence theorem to start with, and shows that, for a wide class of potentials ~b, the set of initial phase points for which global solutions exist has probability one with respect to every Gibbs state for 4) [1],[6],[7],[9]. These results are considerably less powerful than they appear at first glance. Their defect, in comparison to the results of the mechanical approach, is that they do not permit a determination, in any practical way, of whether a given initial phase point admits a solution. This is clear in the illustrative argument given above: Even if the local existence theorem is constructive in the sense that it explicitly identifies the phase points for which T 8t is defined, it is nevertheless necessary actually to know T(n-1)stto in order to determine whether TnStto exists. Similarly, such theorems tell us nothing about the existence of solutions with initial conditions describing typical non-equilibrium situations, e.g., which have one well-defined density for large positive q l and a different density for large negative ql. With these remarks as background, we turn to the work of Fritz and Dobrushin, which establishes the existence of solutions to the equations of motion in two dimensions for a set of initial phase points which is large and which can be described explicitly. We will see, however, that these results do not extend in any obvious way to more than two dimensions. Let to = (~/,V/) be a phase point, let ~ E R 2, and let o- > 0. Denote by H(to, ~, ~r) the total energy--kinetic plus potential--of those particles in to which lie inside the disk with center and radius tr. Also, let g ( u ) denote 1 +log(1 + u), u >~ 0, and define H(to) ~ sup{ H(t°' ~ ' ~r)
}
The condition H(to) < says in a precise way that local energy fluctuations in the phase point to grow no faster than
105
logarithmically with distance from the origin. What Fritz and Dobrushin prove is that, under conditions on the potential to be specified later: 1. If H(to) < ~o, there exists a solution tot to the Newtonian equations of motion, defined for all t, with too = to. 2. For this solution, -H(tot) as a function of t is bounded on bounded intervals. There is no other solution with the specified initial value which has this property. 3. The solution tot may be obtained as follows: Delete from to all particles outside a disk, centered at the origin, of radius M. Evolve the resulting finite system to time t, and let M go to infinity. The heart of the argument, as might be expected, is an a priori estimate: 4. There is a function p(T, h) such that, if tot is a solution of the Newtonian equations with H(to0) < h and with H(tot) bounded for t fi [-T,T], then H(tot) <<-p(T,h) for t E [-T,T].
Once the a priori estimate has been proved for phase points with a finite n u m b e r of particles, it as a simple matter to construct, by a compactness argument, at least one solution to the Newtonian equations with a given initial phase point to as a limit of finite particle solutions, provided only that H(to) is finite. The constructed solution will have H(tot) bounded on bounded intervals. Uniqueness is proved by a contraction argument which, while distinctly non-trivial, contains no real surprises. Finally 3. follows from uniqueness by the general principle that a sequence in a compact space with only one cluster point must converge. The main thing, then, is to prove 4. The following conditions on the potential are sufficient for the argument to work; the minimal hypotheses are actually somewhat weaker (but more complicated). 1. $ is of finite range, even, and twice continuously differentiable away from 0". 2. For some strictly positive/3, c, @ >/ [ - ~c
for sufficiently small [~']
Either ~b is non-negative or/3 is at least 4. .
lv, 4.
l
const
for small '.
For some 3',
026
~< 1~1-v for small ~:.
(This last condition is used only in the proof of uniqueness.) Note that hard cores are excluded. We wilF give a sketch of the proof assuming ~b to be non-negative. The general case is significantly more involved, but the ideas remain the same. The main steps are as follows: 1. Replace the local energy H(to, ~, 0-) by an equivalent smoothly cut-off quantity W(to, ~, 0-). Equivalent will mean: H(to, ~, 0-) ~< W(to, ~, 0-) x< const (H(to, ~, 0-) + 1). All estimates will then be made in terms of W rather than H. Thus, we also introduce W(to) = sup{ W(to'0"~' 2 0-) • ~ E R 2, 0" > / g ( l ~ [ ) }
106
Construction of W is relatively straightforward. Choose (with some care) a cut-off function
f(u), a smooth positive function on R satisfying, among other conditions: f(u)=Oifu <-3R;f(u)=lifu >/ 0, where R denotes the range of the potential. For each i, define
Wi(oJ) = 1 + ~2 + Z ¢(~--gj), and put W(~, ~, or) = E l ( o - - I ~ - ~ ' 1 )
W,(~,).
i
Thus,
W(o~, ~, o') = 2H(co, ~, 09 + N(oJ, V; or)+boundary terms. 2. Prove a "partial differential inequality" for W(oot, -g, or), where cot denotes a solution to the Newtonian equations:
-~ W(oJt, -g, o-) <~ K g([~[ + ~ ) ~
0-~ W(°Jt' "g' o').
(1.1)
Here, K is to be independent of o~t, ~, and o-. This inequality seems to me to be the real heart of the proof. It is clever in at least two respects: a. It exploits effectively the fact that--as is well known-- the energy in a region changes only through boundary effects. b. It shows how to control those boundary effects in a very clean way; they can be dominated by the effect of changing the size of the region. Thus, if we shrink the region fast enough as cot evolves, the energy in the time-dependent region will not increase. To prove the partial differential inequality (1.1) start by differentiating W(cot, -g, ~r) with respect to t and using the equations of motion. The resulting expression contains some relatively uninteresting terms coming from differentiating the cut-off functions; these we will not discuss. The remaining terms can be organized as 1 ~- • (f, - fj) (V¢) (E - ~)" (v--}+ ~). (1.2) Here ~ denotes the value of the cut-off function f ( o - - I q ' - ~ l ) at q ' = E . Observe that a pair (i, j) contributes nothing to the sum unless IV/-~1 ~< R and unless at least one of V/,Vj is in the annulus {~': cr ~ [~'-~'1 ~< cr +3R}. Now group the non-zero terms in the sum into those for which 17,- 1 is small enough so that the inequality:
I'1 Ire ® I -< const ¢ (-~ (valid for small [~'1) can be used to estimate the gradient, and those for which IV/-~-[ is not so small. We will deal explicitly only with the former class of terms. (The latter are estimated using the inequality
Iv¢l
const-~/'¢,
valid for any smooth non-negative function of compact support, and then applying the Schwarz inequality to the sum.) One of the respects in which care was exercised in the choice of the cutoff function f was to guarantee that
If,-fjl
Ig,- l
107
We also estimate, somewhat crudely,
Iv I, I~1 ~
constg([~'l+o-)~,
(1.3)
using the fact that the velocity of a particle is bounded by the square root of the energy of any cluster which contains it and that a particle at ~'is contained in a cluster with energy no larger than g(l~'l) z ~(co,). Inserting all these estimates and reorganizing, we find that we can bound the sum of the terms in (1.2) with 17,-7jl small by
constg([~'l +o)a/W(cot) ]~f/]~4,(V,-Vj) i
~< const g([~'[
j#i
+ O')'M/~ t) '~fi' Wt'(co) i
- const g(l~'l + o . ) ~
-~-~w(cot,-g, o.)
We observe for later use that the only change in these estimates if they are carried out in three rather than two dimensions is that, in the velocity estimate and hence in the final result, g([~'l +o-) is replaced by [g(]~l +0")] 3/2. 3. We exploit the fundamental partial differential inequality (1.1) as follows: Let cot be a solution of the equations of motion. Choose ~" ~ R 2, o. >/g(l~'l), and a positive time T. We want to estimate
W (coT, -if, o . ) . To do this, we note that the partial differential inequality implies that if r(t) is any (positive) solution of
dr = - K "q/'-W~t) g(lg[ + r(t) ) dt then
d W(cot, -~, r(t)) ~ O. dt Taking r (t) to be the solution satisfying
r(T) =o., we get
W(COT,-~,O.) o.2
~<
W(coO,~, r (0)) ~ [r(O)] 2 o.2 ~< W(co0)[ J -~•
Taking the supremum over ~', o. with o-/> g (l~[):
sup{ir-,12
°
/
The problem is now to estimate r (0)/o.. We have, integrating the differential equation for
r, T
r(O) =
o. + g f g(I-~l + .(t)) ~ 0 T
<<.o.+Kg(l-~l+r(Ol) f ~ d t , 0 T
or, writing z(T) for
f a / ~ ( c o t) dt, 0
at
108
r(O) <~ o- + Kg(l-~l+r(O))z An elementary (but tricky) analysis of this inequality leads to the explicit bound
r(O___~) << ~ ( l + z g ( z 2 ) ) 0"
(with K another constant) and thus to the inequality: ~/'~(~O:r) ~< ~
~" (1 +zg(z2))
This is true for all positive T, so, writing t instead of T and using the fact that
dz = ~ ,
-5 we get the ordinary differential inequality
dz <~ g ~ / ~ ( t o 0 ) ( l + z g ( z 2 ) ) , dt
z(0)=0.
(].4)
5. We are now essentially done. Solving the differential equation obtained by replacing the inequality (1.4) by an equality gives an a priori bound for
The only thing which needs to be checked is that the solution to this differential equation £ du diverges at does not go to infinity in finite time; this follows from the fact that j 1 + U g(u 2) infinity. What goes wrong in three dimensions is that the methods no longer yield a bound for r(O)/o- uniform in cr >~g(l~l). To see this, recall that the differential equation for r(t) becomes, in three dimensions,
d r = - K 4W(oJt)[g(l~l + r(t))] 3/2 dt Dropping the r (t) inside g gives the lower bound r(O) >t o" + K g(l'~l)3/2z(T), or, for (r =g(l~'l),
r(O___~) >1 l + K g(l_ffl)V2z(T). O"
Because the right-hand side becomes arbitrarily large as goes to infinity, we don't get a uniform bound for r (0)/or, and the whole technique of proof fails. 2. Braun and Hepp on the Vlasov Dynamics We look next at a different kind of infinite particle dynamics, the Vlasov or weak-coupling or effectivefieM limit. In the previous section, we let the system become infinite by letting the volume and the particle number go to infinity at least approximately proportionally, keeping the interparticle potential function 4 , @ fixed. This means that we are looking at a large system on a length scale of the order of the typical distance between particles and on a time scale of the order of the length of time it takes a particle to travel that distance. The limit we now want to look at is one in which the force exerted by any single particle on any other is extremely gentle--no roarer how close together the particles are--but in which the forces have a range which is very long compared with the typical interparticle distance, so each particle feels the effects of very many others, i.e., moves in an effective force field which doesn't depend very much on what any individual particle is doing. We should further think of the particles as spread out over the one-particle phase space approximately according to some
109 density. The net force a particle at point ~" feelsis therefore proportional to the convolution of this distribution density with the pair force F = - V 4 ) so it changes only over a distance comparable to the range of the potential. It would therefore be quite uninteresting to look at the behavior of the particle over the length of time it requires to travel a typical interparticle distance--it would be subject to a force which is in the limit constant--so we want to look instead on a length scale of the order of the range of the potential. Finally, we should arrange that the overall strength of the interparticle force is such that the length of time it takes a particle's velocity to change significantly is of the same order as the length of time it takes the particle to travel a distance of the order Of the range of the potential. We will now proceed to formalize this limit; it will turn out--not surprisingly--that the limiting dynamics is much less singular than either the infinite-particle dynamics discussed in the previous section or the dynamics in the Boltzmann-Grad limit to be discussed in the next. We consider, then, a system of N particles which we can think of either as in a fixed ( Le., N-independent) container or in free space. The space dimension doesn't matter very much; we take it to be three. We also fix an interparticle potential ~b which for simplicity we take to be C ~ and of finite range. The methods to be described work for more general potentials, but it is essential that the gradient of the potential be bounded. We take the force exerted by a particle at ~j on a particle at ~ to be
1 V c k ( ~ - ~ j ) ==- N --F(qj-qj) N and the particle mass to be unity. (Equivalently, we could take the force to be F a n d the particle mass to be 1/N, and observe the behavior on a time scale of order 1/N. The initial velocities should then be of order one in the new time scale, i.e., of order N in the original time scale.) As long as the system is finite, there is no problem with the dynamics. We will write the solution curves as % = (~.(t), ~(t))i<~i~< N To pass to the limit N - - ' ~ , we re-express the dynamics in a form which is as nearly Nindependent as possible. To each phase point ~o we associate the discrete probability measure /x~o
1
N
on the one-particle phase space R 3 x R 3. We will write/z t instead of the more awkward/z '°'. The N-particle dynamics thus defines a flow T/v on a very sparse subspace of the set M 1 of probability measures on R3×R 3. Having imbedded all the N-particle phase spaces in the single space M+1, it is natural to ask the following: Do two initial phase points, perhaps with different numbers of particles but which look similar, Le., are near to each other in the weak-* topology for measures, have similar time evolutions? The first major result of Braun and Hepp is that they do: Theorem 2.1. There is a flow T t defined on all of Ml+ which reduces on each N-particle subspace to T[v. Ttlz is weak-* continuous in IZ. We may thus think of Tt/~ for a continuous measure tz as describing the dynamics of a phase point for a system of infinitely many particles initially distributed over the one-particle phase space ~ccording to the measure be. Remarks. 1. The fact that this theorem concerns probability measures does not mean that there is any statistical element in it; it is a theorem of classical mechanics, not classical statistical mechanics. Probability measures are simply a convenient representation of the points of an (idealized) infinite system phase space. 2. This theorem describes a situation which is radically different from-- and simpler than--the Boltzmann-Grad limiting regime to be described in the next section. In the latter context,
110
modulo technicalities, the picture is that the Boltzmann equation generates a semiflow-- Le., a flOW defined only for positive times--on the space of continuous probability measures and most discrete measures weak-* approximating a given continuous measure have evolutions which, at least for a short time, approximate the evolution given by the semi-flow. There are, however, necessarily exceptions; a simple way to construct them is to take a "typical" phase point tel)resenting approximately some non-equilibrium continuous distribution; let it evolve for a short while; and then reverse all the velocities. Theorem 2.1 says that there are no such exceptions in the Vlasov limit. Their absence means in particular that the limiting dynamics is reversible and cannot describe approach to equilibrium. To prove Theorem 2.1 we rework the equations of motion for N particles into a form in which it is apparent how to generalize them for initial measures which are not normalized sums of N unit point masses. Let/x t be a solution curve for the N-particle dynamics. From it, we construct a time- and position-dependent force field on the one-particle position space:
7(7, t) - f ~(7-gl)/-6t(d~l d~l). We then consider the motion of a single particle in this force field, and write Z t x ~ (-~tx, Vtx) for the solution curve starting at x = (~, "7) at time zero. Because the force field is time-dependent, the mappings Z t do not in general form a group; we could (but won't) consider the two-parameter family of solution mappings Z ( t 1, t 2) evolving from time t 2 to t 1 and would then get the multiplication law
Z ( t 1, t2).Z(t 2, t 3) = Z ( t l , t3). Each Z t is, nevertheless, a volume-preserving mapping of R a x R a into itself. The mappings Z t contain in principle more information than the solution curve /x t = ! ~ ( t ) , V,(t)) itself, and the solution curve is reconstructible from Z t and the initial condition by
( ~ ( t ) , ~ ( t ) ) = Zt(~(O), ~(0)). (This uses the fact that F(-~)=0".) A moment's reflection shows that this last equation can also be written tZt = Z t ~ o ,
where the image of a measure/z under a mapping Z means
f g(x)( z ~ ) ( a x )
-
f g(Zx)~(ax).
Thus, we can write the equations defining Z t directly in terms of tzo rather than in terms of tzt; they are, simply,
aOtx -dt
V'; ,dVtx -- f ~o(dy) -ff(O'x - -~ty). dt
(Reminder about what the notation means: x,y are points of R 6, the one-particle phase space. Z t is a mapping of R 6 into itself; its value at a point x is a pair of three-vectors which we write Once the equations are written in this form, it is easy to see how to generalize to initial measures tz 0 which are not discrete; simply write and solve the same equations for more general tz o. Of course, existence and uniqueness have to be proved, but this turns out not to be difficult. It is not necessary that tz o be normalized or even positive, but it should have finite total variation. We get in this way, for each tz0, a family of mappings Zt(tZo) which are weak-* continuous in tz 0 in the sense that the mapping
(x, t~0) -" zt (t~0) x is weak-* continuous into R 6. The time evolution flow acting on measures is then
111
/z t = Tt/z0 = (Note the double appearance of/z0.) multiplication law
Zt(tzo) IX0 •
This time, the group law
does hold, by virtue of the
zt2(i.l,t) Ztl(g, O) = zt2+tl(lzo), which follows from the uniqueness of the solution to the equation for Zt(~0). Although the flow T t on the space of measures is the intuitively natural form in which to view the dynamics (by analogy to the solution flows on the finite-dimensional phase space in the dynamics of finite systems), the Zt(tz) have great technical advantages as objects with which to work. For example, Braun and Hepp show that --- z ' ( ~ )
is differentiable in the sense that lim
h-o
Zt(tz + h v ) - Zt(tz) h
exists (in the weak-* topology) for all signed measures v. Heuristically, this ought to mean that /x "--~Tt/~ is differentiable, but as a technical matter mappings into spaces of measures are almost never differentiable, and /x --~ Tt/x is in fact not even continuous (in the norm topology). If/~0 is of the form f0W, V')a~a~), with f0 smooth, then, since the preserving,/~t is given by the density
Zt(tzo) are volume-
ft = fo °Zt(/zO )-l" This density is again smooth, and one can write a differential equation for its time development. The differential equation is the Vlasov equation: O f = _~, Of Ot 0~"
-~fv.f F ( g - g ' ) f ( ~ , -V') d~' d-f'. We have been concerned, so far, with the classical mechanics of "pure s t a t e s " . The measures appearing in the theory do not have a probabilistic interpretation; they simply represent phase points. We now introduce some probabilistic considerations. To minimize the confusion between measures with a probabilistic interpretation and measures as phase points, we will generally denote the former by p or P. Let p be a probability measure on the one-particle phase space R6; PN the product of N copies of p as a measure on R6N; and PN as the measure on M+~ which is the image of PN under our usual representation of N-particle phase points as normalized discrete measures. Here is an alternate description of PN: Let X denote the mapping x ~ 8 x from R 6 into M, which we view as a vector-valued random variable on the probability space (R 6, p). Then is just the joint distribution of L
N
( X 1.+- . . . .+-XN
)
PN
112
where X 1. . . . . X m are independent copies of X. This description of PN suggests that the Law of Large Numbers and the Central Limit Theorem can be applied in this situation. We are now going to argue that this is indeed the case, and that all we need are the classical versions of these theorems, not refined versions valid for Banach-space valued random variables. Let g be a function on R 6. Define a function n (g) on R 6N by 1
N
n (g) (oJ) = --~ ~., g(xi) if co = (x 1. . . . .
XN).
i=1
If g is the characteristic function of a set, then n (g) measures the fraction of particles which lie in that set. For this reason, we refer to n (g) as an occupation number observable. Technically, we will usually want to consider only n ( g ) ' s with g a smooth function of compact support. If the phase point co corresponds to the measure ~,o, then
n(g)(¢o) =
f g(xl ~°~(dx).
We use this formula to define n (g) on all of M:
n(g)(tz) =
f g d~.
It is easy to check that the distribution of n (g) with respect to PN is the same as the distribution of 1--~-( G I + . . . N
+GN)
where G 1. . . . . GN are independent copies of g regarded as a random variable on the probability space (R 6, p). Thus, by the Weak Law of Large Numbers, the random variable n (g) on the probability space (M, PN) converges in probability to the constant = n ( g ) ( p ) as N - - . ~ . Re-expressed in analytic language, this means that, given any weak-* neighborhood V of p in M, the probability, with respect of PN, of finding the phase point outside of V goes to zero as N - - . oo. Heuristically,
fg ap
PN ~ ~p. Now apply the dynamics. Since T t is weak-* continuous, we have in the same sense
Pu(t) =-- TtPN ---.Sp(t) ; p(t) =Ttp. This result can be expressed as "propagation of chaos". If we translate the measure Pu(t) back onto R 6N, the resulting measure Pu(t) will generally not be a product measure. What the above argument shows, however, is that if PN(t) is integrated over all but its first j arguments and N is allowed to go to infinity, the resulting sequence of measures converges to a product o f j copies of Ttp. Braun and Hepp also analyze the more delicate question of fluctuations in the Vlasov limit. Given a function g as above, and given N, define a fluctuation observable dPN(g) = ~
[n (g) -- n--(~],
where the overbar denotes mean value with respect to PN" F r o m what has already been said about the distribution of n (g), it follows at once from the Central Limit Theorem that dPN(g) converges in distribution (as N - - - , ~ ) to a centered Gaussian with variance
More generally, any finite number of fiuctutation observables ~u(gl) . . . . . asymptotically jointly Centered Gaussian with covariance lim ~u(gi) ON(g)) = N~cx3
(PogJlgj),
~N(gn) are
113
where ( I ) denotes the inner product in LZ(R 6, p) and P0 is the projection onto the orthogonal complement of the constant functions in that space. (The reason why P0 appears is that the total particle number does not fluctuate.) All this is elementary and concerns only time zero. Given the time evolution, however, one can ask about fluctuation observables at other times. Since even the fluctuation observables at time zero converge only in distribution, not in any pointwise sense, we ask only about convergence in distribution. Thus, choose t 1. . . . . tn, gl . . . . . gn and ask what happens to the joint distribution of the ~N(gJ)t, as N---oo. Braun and Hepp show that, in the limit, the joint distribution is centered Gaussian, and they give a formula for the covariance. We can therefore think of the limiting fluctuations as defining a Gaussian stochastic process with a large and unpleasant instantaneous state space--some space of distributions, say, large enough to contain the typical sample fields for the time-zero fluctuations. I want to make the point that this stochastic process is of a degenerate type--it is deterministic. I am using the word "deterministic" here in a very simple-minded sense; intuitively, what I mean is that knowing all fluctuations at time zero permits prediction, with certainty, of all fluctuations at other times. Actually, because of the linearity of the evolution of fluctuations, the precise statement is both simpler and more powerful than the intuitive statement: Given any time t and any test function g, there is a test function ~ such that
dPN(g) t--dPN(g) 0 converges to zero in probabiliy, i.e., a fluctuation observable at time t can be predicted with certainty from the value of one properly chosen fluctuation observable at time zero. We will see in the next section that this contrasts with what apparently happens in the BoltzmannGrad limit. The deterministic character of the fluctuation process is readily checked by examination of the formula for the covariance given by Braun and Hepp. I will not reproduce this formula but will instead describe an analogous but technically simpler situation which seems to me to show clearly what must be ~going on. Recall that Pu is the distribution of an average of N independent copies of a random variable X with values in M and that time evolution is given by a flow T t on M such that /~ -'~ Tt~ is, at least heuristically, differentiable. The model system I want to discuss is one in which PN is the distribution of the average of N independent copies of a random variable X taking values in a finite-dimensional vector space E equipped with a differentiable flow T t. The analogues of the n (g)'s are just the linear mappings ~ from E to R, i.e., the elements of the dual space E* of E. Put
By the Central Limit Theorem, the ¢ u ( $ ) are asymptotically jointly centered Gaussian with covariance
E(qsl(X) qJ2(X)) - E(~bl(X) ).E(qJ2(X) ). Now look at a time t fluctuation observable
~ N(~ ) t ~-- .q~ [tOoTt-EN(qjoTt)], (where E N denotes the expected value with respect to PN)" Let p denote the expected value of X, and write the general point of E as x = p +'0. By the differentiability of T t,
Ttx =Ttp + DTt(p)'0 + O(1"012), and so
114
qj( Ttx) = O( TtP) + DTt(p) *O(~) + O(l~l 2) dPN(l~)t(X) : DTt(p) * ~ ( - 4 ~ "0) + O(-4'-N 1'012)
=
~PN(DTt(p) *~) + 0(',/~ 1"012).
By the Central Limit Theorem, ~
['012 converges to 0 in probability, i.e.,
¢~N(t~ ) t -- dPN ( D T t ( p ) *tk ) ~ 0 in probability.
This means in particular that the aPN(~)t'S are all asymptotically jointly Gaussian, and we can thus regard them, as above, as defining a Gaussian stochastic process. This time, there are no technical difficulties with the state space; it is just E. If we denote the instantaneous state of this process by F t E E, so the F t are all vector-valued random variables on some unspecified probability space, we get F t = DTt(p) F 0 with probability one. I don't recommend the above argument as a pattern for constructing a theory of the time-dependence of fluctuations in the Vlasov limit. The technical difficulties in implementing it directly seem to be formidable. The fact is, nevertheless, that if it could be applied it would lead to exactly the formula for the covariance of the limiting fluctuation observables that Braun and Hepp obtain by a different argument. We can thus think of the fluctuations in the Vlasov limit as evolving linearly, deterministically, and (properly understood) reversibly. Their motion is given simply by linearizing the solution flow T t about the particular solution under consideration.
3. Autocorrelations and fluctuations in the equilibrium hard-sphere system In this section we consider a system of many elastic spheres of diameter E, in a limiting regime in which ~--* 0 and in which the number of particles per unit volume goes to infinity like E2. We refer to this regime as the Boltzmann-Grad limit; its essential feature is that the mean free path remains of order one as the limit is taken. The work to be described here exploits the techniques used in my study of the Boltzmann equation [6], and I will begin by sketching these techniques. Let ~t(Xl .....
XN) = tto(T~f (Xl . . . . .
XN) )
be a time-dependent probability density for a finite system of N hard spheres of diameter e. Here, T[, ) denotes the solution mapping for the hard-sphere dynamics, and each x i denotes a pair (~,, V/) giving the position and velocity of one of the particles. We construct the correlation functions for /xt and scale the j-th correlation function by e2j; the resulting functions, which we will denote by rj(x 1. . . . . xj) and refer to as the rescaled correlation functions, have at least the possibility of remaining bounded and not approaching zero as e approaches zero. The Liouville equation for the time evolution of/x t translates into the BBGKY hierarchy for the time evolution of the 1). This hierarchy can be written schematically as arj(t) : Hj(, ) rj(t) dt
-F C
(') 1r - l ( t ) j,j+ j-l,
and even more schematically as dr(t) = H(,)r(t) + C(,)r(t)" dt Here Hj (~) denotes the Liouville operator for the system of j hard spheres of diameter E. Writing S f f ) ( t ) for the j-particle streaming operator ( i.e., the group generated by Hj(~)), we can formally solve the hierarchy as
115
r(t) = ~
f
(3.1)
dtl " " . dt m
S (~) (t - tm) c (~) s (~) (t m - tin_ 1) • • . C (~) S (c) (t 1) r(0).
Having arrived at this point considering a finite-,but possibly very large--number of particles, we can now if we like take a thermodynamic limit, letting the particle number and the volume both go to infinity keeping e fixed. The main step in the argument leading to the Boltzmann equation is to show that if the rj(0) satisfy some uniform bounds as E goes to zero and converge in a sufficiently strong sense to a sequence of limit functions rj (°) (0), then the convergence of the series (3.1) is uniform in E, and the series goes over term by term to a series solving the so-called B o l t z m a n n hierarchy. Because of limitations on the range of values of t for which the convergence of the series (3.1) can be controlled, we can only show that the r j ( t ) converge to a solution of the Boltzmann hierarchy for relatively small values of t--no larger than about one-third of a heuristic estimate of the mean free time. Convergence for larger values of t would follow, however, if bounds on the rj(t) uniform in c could be established by some other means. The starting point for the work to be described here is the observation that none of these arguments depends on the fact that p~0(Xl. . . . . x N) is a probability density. By looking at /~0's which are not probability densities but which are related in various ways to the hardsphere equilibrium measure, it is possible to extract information about some time-dependent phenomena in the equilibrium hard-sphere gas. Technically, we will work in the grandcanonical ensemble, so N does not have a definite value, but this causes no problems. We look first at the velocity autocorrelation of a single particle. Since the particles are treated symmetrically in the equilibrium ensemble, we may as well look at particle 1. We want to analyze the joint distribution of V 1(0) and V l(t), for fixed t, as ~ approaches zero. We can probe this distribution by choosing two continuous functions f, g of compact support on R 3 and studying
where E(~q) ( ) denotes the mean value in equilibrium. We can obtain a formula for this quantity as follows: Let t z ~ ( x 1. . . . . x N) denote the N-particle component of the probability density for the hard-sphere grand-canonical ensemble, and put
gN(O;, X 1. . . . .
X N)
= f('V1)/.t~Cq(x 1. . . . .
XN).
Then
E~q) [f(V"l(O))g(~l(t))]
=
Zf.u(t, x, . . . . .
XN)g (V"1) dXl • " • dx N.
N
where IzN(t;''" ) is obtained by applying the streaming operator S,4')(t) to / Z N ( 0 ; ' ' ' ). This latter expression can be rewritten as
f g ( ~ l ) rl(t; ~1, ~1) dql d-tip (In this formula, we are using a non-standard normalization for tion r 1. We are also assuming that the thermodynamic limit taken, then r 1 becomes independent of q'l, and the integral over Thus, determining the joint distribution of V1(0) and-¢1 (t) To do this in the limit as E goes to zero, we argue as follows:
the rescaled correlation funchas not been taken. If it is this variable is suppressed. reduces to determining rl(t).
1.
Although IZN(t; X I . . . . . x N) is not symmetric in x 1. . . . . x N, it is symmetric in X u , and this suffices to permit the derivation of the BBGKY hierarchy for the r j ( t ) . 2. We have
x 2. . . . .
[tZN(0; X1 . . . . .
XN)[
< [Iflloo t*~(xl . . . . .
XN).
116
Since the equilibrium ensemble is invariant under the time evolution, the inequality remains true if we replace t~N(0) by IZN(t) for any t. By integrating over xj+ 1. . . . . Xy, we get
xj)l ~ IlfIl~
xj). This inequality gives us a time-independent bound on the size of the rj(t). 3. rj(0; x 1. . . . . xj) -- f('71) rfq(xl ..... xj). Irj(t; xl . . . . .
4. lim rj (0; x 1. . . . .
rfq(xl . . . . .
xj) = f(71) lJI h/3(V/), where h/~ is the Maxwellian with inverse tempera-
~0
i=l
'
ture/3. 5. The solution of the Boltzmann hierarchy with initial condition as on the right-hand side of 4. is
rj(O)(t; x I . . . . .
xj) = eAt f(V1) ~I h~(Vi), i=l where A is the linear Boltzmann operator or Rayleigh-Boltzmann operator: A f(--Y) =
favlfdeoeoC:-- l) h (Vl) +
where V',-9'1' are the outgoing velocities for a collision with incoming velocities 7, V1 and momentum transfer direction &, and where means the surface integral over the half of + the unit sphere where ~'('~--'~1) > 0. NOW apply the general analysis and the time-independent bound 2. on the rj(t) to conclude that
fdCo
lim rl(t;V 1) = cAlf(V1 ) ho(V1)
~0
for all positive t, and thus lim
E~q,)[g(-:1(t)) f(Vl(O))}= (eAtflg),
a~O
where ( I ) denotes the inner product in L2(h/3 dT). I believe that this result is the first non-trivial case where the validity of a picture derived from the Boltzmann equation can be proved for all times. It has as an immediate consequence that the power-law decay of the velocity autocorrelation function which is believed to occur for all finite a must disappear in favor of exponential decay in the Boltzmann-Grad limit. It also appears that the argument can be extended to show that the stochastic process V l(t),
-co
(with distribution given by the equilibrium ensemble) converges in distribution as approaches zero to the stationary Markov process constructed from the positivity-preserving semigroup e At as in the Euclidean formulation of quantum mechanics. The second application of these ideas is to the study of time-dependent fluctuations in the equilibrium hard-sphere gas. If f is a continuous function of compact support on R 6, put N
n (f) (x 1. . . . .
XN ) = ~2 ~,, f(xi). i=1
(In contrast to Section 2, we are normalizing by dividing by E-2 rather than by the particle number N which is not constant h e r e . ) Let
¢~.)(:)= 1In f:)-j<.)}. E
117
Dobrushin and Tirozzi[4] have shown that the fluctuation observables ¢ ( , ) ( f ) are asymptotically jointly centered Gaussian with covariance given by the inner product in L2(h# (7) d~ d-C). Our objective is to study the joint distribution of fluctuations at unequal times. Our results are limited in that all we are able to analyze is the covariance ,401imE~ql[dP(e)(g ) tl dP(el (f ) ,2] and that only for small It 1- t 2 [ . Since the equilibrium state is invariant under the time evolution, we may as well take t2--0 and t 1 = t>/0. As before, we proceed by constructing an initial t~ adapted to the problem at hand. We take
ILN(O'~X 1. . . . . XN) =
f(xi)
--
(,) /£)~q(x1 . . . . .
XN).
We let IzN(t; x 1..... X N) be the corresponding time-dependent density and rj(t; x 1..... the rescaled correlation functions. A straightforward computation shows
=f
g(Xl)rl(/;
xj)
x1)dXl,
so again all we need to do is to compute rl(t). Another straightforward computation shows that
xj) = ~f(x
rj(0; x 1. . . . .
i) ryq(x 1.....
xj) + correction,
i=1
and
the
correction
would vanish if ry~l ( x I . . . . . xj+ 1) were exactly equal to This factorization does not hold for non-zero ~, but a detailed computation using a low-density expansion for the equilibrium correlation functions shows that the correction is O (~) (in an appropriate norm) and so can be ignored in the zero-~ limit. Thus,
rj(x 1, . : . ,xj) rl(xj+a).
lim rj(O;x 1. . . . .
Xj) = ~ f(x i) iXi h~l(Vi),
•~0
i=1
i=1
and what we therefore have to do is to solve the Boltzmann hierarchy with initial condition given by the right-hand side of this equation. The solution turns out to be
i:1
-
i=1
where
ft -----eLtf, with L the linearized Boltzmann operator:
(Lf) (7, V) =
-V. o f + ~V
fay1f+ a~,~.(v-v~) ho(v~)
{f(~,, -ffl') + f(~,, -Y')- f(~, -Yl)- f(E -V)] We thus get lim rl(t; X 1) = ~0
(eLtf)(x1)
h/3(V'l).
This time, we have no way of bounding the rj(t) uniformly in t, E, so the limit is proved only for small (positive) t. For such t, we have
118
E~q)tc~(,)(g)tc~(,)(f)o] = (gleLtf). It is attractive to conjecture that the ¢(,) (f) t, all f and t, become jointly Gaussian in the zero-¢ limit. So far as I know, this has not been proved. (A few higher moments can be computed without too much difficulty, and their values are consistent with the conjecture.) I want to note, nevertheless, that /f the conjecture is correct, it implies that the equilibrium fluctuations in the Boltzmann-Grad limit do not evolve deterministically as in the Vlasov limit. To see this, let
= (e Lt) *g. Then lim
c~0
E~q)[[ap(~)(g)t-~(~)(fi,) o] qb(~)(f)o}
=0
for all f. But, for centered Gaussian random variables, orthogonality is equivalent to independence, so the conjecture implies that, asymptotically, the time t fluctuation observable ~(E)(g)t can be written as the sum of a time zero fluctuation observable ~(~)(g)o and a centered Gaussian random variable which is independent of all time zero fluctuation observables. The latter term has variance [[g []2- I[g ][2, which is generally non-zero. The presence of a non-zero independent component means that time t fluctuation observables are not predictable with certainty even with complete knowledge of the time zero fluctuation observables. References
1. 2. 3. 4. 5. 6. 7. 8.
9.
Alexander,R.K.: Time evolution for infinitely many hard spheres. Commun. math. Phys. 49, 217-232(1976) van Beijeren,H.,Lanford,O.E.,Lebowitz,J.L.,and Spohn,H.:Equilibrium time correlation functions in the low density limit. Preprint(1979) Braun,W. and Hepp,K.: The Vlasov dynamics in the 1/N lirnit of interacting classical particles. Commun. math. Phys. 56, 101-113 (1977) Dobrushin,R.L. and Tirozzi,B.: The central limit theorem and the problem of equivalence of ensembles. Commun. math. Phys. 54, 173-192(1977) Fritz,J. and Dobrushin,R.L.: Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Commun. math. Phys. 57, 67-81(1977) Lanford,O.E.: Time evolution of large classical systems, in: Dynamical Systems: Theory and Applications, ed. J.Moser. Springer Lecture Notes in Physics 38, 1-111 (1975) Marchioro,C., Pellegrinotti,A. and Presutti,E.: Existence of time evolution for vdimensional statistical mechanics. Commun. math. Phys. 40, 175-185(1975) Sinai,Ya.G.: Construction of the dynamics in one-dimensional systems of statistical mechanics. Teoret. Mat. Fiz. 11, 248-258(1972) English translation: Theoret. and Math. Phys. 12, 487-494(1973) Sinai,Ya.G.: Construction of cluster dynamics for dynamical systems of statistical mechanics. Vest. Moscow Univ.,no 1,152-158(1974)
Holonomic Quantum Fields --The
unanticipated link between deformation theory
of differential equations and quantum fields - -
By Michio Jimbo, Tetsuji Miwa and Mikio Sato RIMS, Kyoto University, Kyoto 606, JAPAN and partly, Yasuko MSri Department of Mathematics, Ry~ky~ University Naha, Okinawa 903, JAPAN
§i.
Introduction Through recent study of the problems in mathematical physics, a deep, unexpected
link has emerged:
a link between the monodromy preserving deformation theory for
linear (ordinary and partial) differential equations, and a class of quantum field operators ([i] [2] [3]).
The aim of this article is to give an overview to the
present stage of development in the theory (see also [4]). The fruit of the above link is multifold. compute exactly the
n
On the one hand it enables one to
point correlation functions of the field in question in a
closed form, using solutions to certain non-linear differential equations of specific type (such as the Painlev~ equations).
On the other hand it provides an effective
new tool of describing the deformation theory by means of quantum field operators. Thus it stands as a good example of the fact that not only the pure mathematics is applied to physical problems but also the converse is true. Thefollowing
Example i.
examples will show how the theory actually works.
Th___escaling limit of the 2 dimensional Ising model.
The first example is concerned with the calculation of
n
point spin correlation
functions
of the 2 dimensional Ising model, specifically their Jlkl Jnkn scaling limit T ± at the critical temperature T ÷ T ±0. Wu et al. [5] found that n c the 2 point function T 2 admits a closed expression
(1.1) ~(~)
= const,
tsinh~ l~e
exp(~[ dt ~((dt t d_~) 2-s"inh 2~(t))) . .
L c o s ~2 where
20 = const. IT-Tcl~(jl-J2)2+(kl-k2 )2
is the scaled distance, and
~ = ~(e)
is
a solution of the following non-linear second order differential equation (an equivalent of a Painlev~ equation of the third kind ([6]))
120
d2~ + ~1 dd~@ = 2 sinh 2~.
(1.2)
dO 2
Their result is generalized as follows ([H] III, IV, [S] II-IV).
The
n
point
+
correlation function T- as a function of 2n scaled variables a = al+ia 2 al-za"2 n v ~ a* ~ ~ (~i 2 is expressed 2 ' ~ 2 , ,a ,av~ = const.,T-Tcl.(j~,k I I ) ~~=~,...,n~, £ I ± (1.3)
as
[ i/det sinh H exp
Tn(al,a~,''',an,a ~) = const.
~0),
/det cosh H where
fm
(1.4)
~ = ~1 U~ fHvf U
is a primitive of the closed 1-form da -da p ~ + m2 au-a~
n
(6Nv-g~v)a~dav + complex conjugate, p,~=l
-2H
and the matrices
F = (fH~), G = ( g u v ) = e
are solutions of a non-linear system
of partial differential equations n
aal
(1.5)
[
f
K(#U,~) PKf<~(-
aK-a
aK-a
+ m 2 (6%H-6%~) <~la~gKugKv
~f UV = m 2 ~a~ (ap-a~)g%ugIv
aa% ~gN9 = ~a~
~ gpKf<~ K(#~) [
(GFG-I)p
subject to the symmetry
Example 2.
a<-a ~%p-~XK a*-a*
tG = G* = G-I, tF = -F, F* = -GFG -I.
The reduced density matrix for the i dimentional gas of impenetrable
bosons. The second example deals with a non-relativistic bose gas described by the body problem in a periodic box
(1.6)
0 < x < L
1 N ~ -2 (- ~z[l ~ x i(. = ) + ci<j [ 6(xi-xj))~(Xl'''''XN) = E~(xI'''''XN)'
or equivalently by the quantal non-linear SchrSdinger equation (1.7)
i ~ = _ !___22@ + c@,~ 2 ~t 2 Sx2 D(0,x),
$*(0,x')]
= 8(x-x'),
N
121
with the impenetrability condition with
c = +~o. The ground state wave function of (1.6)
e = +oo takes the simple form ([7])
(1.8)
~N,L(Xl,...,XN ) = i
~ le
2~ix./L 2~ix~/L J -e m I"
N!L N j
The subject of interest here is the
pn ( x±h
(i.9)
" ,x
n
particle reduced density matrix
;xl, 'x") n = <~N,LI~*(Xl )'" ~*(x~)~(x~)...@(x")__n I~N,L >
N! IL0" . IidYn+l. (N-n)! . . .
* dYN ~N'L(XI" "'''Xn'Yn+l" "'''YN) x SN,L(X'j, • "',Xn,Yn+ " I, • • • ,yN)
in the thermodynamic limit
N,L ÷ ~, with the particle density
N P0 = L
A complete parallelism to Example 1 is exhibited in ;this Example 2. reduced density matrix
The 1 particle
Pl(X~;X~)± ± = p(Ix~-x~I)± ± admits a closed expression ([S] XVI,
[3]; we choose
P0 = n-I)
(l.iO)
p(x) = P0 exp( 0dt °t(--t)),
where
kept fixed.
fX
o = d(x)
satisfies a second order non-linear equation d2cy,2
(l.ii)
(~) dx
do do do = -4(x ~x -i-o) (x ~x + (~x)2-O)
which is an equivalent of a Painlev~ equation of the fifth kind. n particle reduced density matrix
Pn (x'''''_ 1 ,Xn, . . .Xl'" ..
For general
,x")n with
n, the
xi<.--<X'n, x~'<''.
<x" is expressible in the form ([3]) n' (1.12)
" "'" 'x") " j ,k=l,'" • ,n x exp(j~). r 'x'" n' Xl' On (x',... ± n = const. de t (RI (xj' ,Xk))
Here, denoting by (1.13)
Xl<'''<X2n
the re-ordering of
...
1 Rl(X j,x k) = 2i(xj_Xk ) (r+jr-k-r-jr+k) 1
...
l 'X Vn' we have
(j#e) + ir+jr_j
xj,-xj
~0 = -
)2 dxj-dxj, 1 .!. (r+jr_j,-r+j 7 J j, ,r_j
r±j (j=l,...,2n)
v vv 'X n' Xl'
(r+jr_j,-r+j,r_j) 2 j,(#j)
and
v Xl,
xj-xj,
(j=k),
2n + i ~ r r dx j=l +j -j J '
satisfy the system of non-linear partial differential equations
122
(1.14)
~r±j i ~x k = ~(-r±kr_kr+j
+ r..r ,r .) i_ zm *m -] xj-x k
(k#j)
1 ~ (-r±j,r_j,r+j + 1 = ±irij - ~ J'(#J) r±j,r+j,r_j)xj_xj,
(k=j).
In both of these examples the relevant non-linear differential (1.5),
(i.ii), (1.14) have a common characteristic
feature:
equations
(1.2),
they arise as monodromy
preserving deformation equations of appropriate linear differential equations
(§2,§5).
Roughly speaking the monodromy is a notion that measures the character of ramification of a multi-valued
function.
Take as an example the following system of
linear differential equations with rational coefficients
(1.15)
dY ~x = A(x)Y,
A(x):
m×m
matrix of rational functions.
Its fundamental solution matrix
Y = Y(x)
points the poles
A(x)
a closed path
al,...,a n
of
is multi-valued,
and
~.
having as its branch
Thus by analytic continuation along
y (Figure I) it undergoes a linear transformation
(1.16)
Y(x) ~-> Y(x)My .
The constant matrix
M
monodromy matrix for
depends only on the homotopy class of y, and is called the Y Y(x) corresponding to y. The correspondence
(1.17)
~l(~-{al,'..,an}) y
constitutes a representation
--> GL(m,~) e-~ M
Y
of the fundamental group
~l(~-{al,'--,an}),
the
monodromy representation.
Example 3.
The monodromy problem of Riemann-Hilbert.
In contrast with the preceding two examples, mathematical
the third one is of a purely
character.
The problem is this: find a differential equation (1.15) n A with simple poles, A(x) = ~ x a , whose solution Y(x) has the prescribed monodromy representation.
T~e answer is given in a particularly
language of quantum field theory ([H] II, [S] VI). free fermion fields
~(i)(x), ~*(i)(x)
simple form in the
By introducing one dimensional
and a class of field operators
~(a;L), we
find such a matrix as the analytic continuation of the vacuum expectation value
(1.18)
Y(x) = 2~i(x-x0)X
with the normalization
<~*(i)(x0)~(al;Ll)''" <~(al;Ll)
Y(x0) = i.
~(an;Ln)~(J)(x)> I ~(an;Ln)> i,j=l,..',n
As we vary the branch points
a , the monodromy
123
representation Schlesinger
Y(x)
is kept invariant by construction.
The classical work of
[8] states that in this case the coefficient matrices
a system of non-linear differential equations (here we choose
(1.19) The
n
(1.20)
~Av 8a
[Av'A~] av-a ~
point function of
(~#V),
~A___~= _ 8a ~ v V'(#V)
v
should satisfy
xo=~)
[Av,A~,] av-av,
~(a;L)'s is again expressed in terms of
I <~(al;Ll)'''~(an;Ln)> = const.exp(w),
A
L.
A v' s as
de_de v ~ = ~ trace ApAv ~
It should be mentioned that these correlation functions (1.3), (1.12), (1.20) in turn play an intrinsic r$1e to the non-linear partial differential equations (1.5), (1.14),
(1.19) respectively.
Namely the latter are rewritten as non-autonomous
classical Hamiltonian systems ([9] [i0] [3]) involving several "time" variables, whose Hamiltonians are given by the closed 1-form ~ ([i0] [3]), i.e. essentially the logarithmic derivatives of the correlation functions. The mathematical theory of isomonodromic deformations goes back to the classical works [8] [Ii] [12] in the dawn of this century.
In recent years another type of
deformation theory of linear differential equations has widely attracted people's attention in the branch of non-linear waves: the isospectral deformations, or the soliton theory.
The deep interrelationship between these two theories is now becom-
ing apparent ([13] [14] [15]).
Our second example stands as an indication of this
connection at the quantal level ([3] cf. [16] [17] [18]).
We expect that such non-
linear completely integrable systems at certain critical coupling (c = +~o for the present case) should fall within the category of our theory. All these models described above are those in one space (+ one time) dimension. To generalize our scheme in higher dimensions, it seems inevitable to introduce fields that depend on extended objects (§6).
Construction of the model is based on a
peculiar commutation relation similar to the one discussed in the literature [19], although its analysis is not worked out except for some special case.
The deforma-
tion equations are now taken place by a variational formula of Hadamard's type ([20], [S] XIII, XIV). This paper is organized as follows. §2 deals with the first of the above examples, the scaling limit of the 2 dimensional Ising model.
The mathematical theory of monodromy preserving deformation is
formulated in §3, where a system admitting an irregular singularity of rank 1 is discussed.
In §4 field theoretical description is given to the Riemann-Hilbert
problem (Example 3 above). in the next §5.
The second example of impenetrable bose gas is discussed
Lastly in 56 we discuss analogous constructions in space-time
dimensions more than 2.
124 An algebraic building block of our whole story is the theory of Clifford group, which we have no room to explain in the text.
See [HI I, [3] Appendix.
The authors would like to express their sincere gratitude who gave them an opportunity
to Professor E. Br~zin
to attend and to give a talk at the Lausanne conference.
+ + x -a
-x +a
"a
n base point Figure i
Figure 2
§2.
Scaling limit of the two dimensional
2.1
History.
Ever since the monumental work of Onsager
Ising model, calculation problem.
Ising model
of the spin correlation
functions has been a longstanding
After the exact computation of free energy
tization
[21] on the two dimensional
([21]) and thespontaneous
magne-
([22]) there appeared an enormous literature dealing with this model from
various points of view (see [23]), but it was only quite recently that exact infinite series expressions
for two- and multi- spin correlation
on the lattice as well as in the continuum limit([5], [27]).
Among them, however,
functions have been derived
[24],
[25],
[26],
[S]I,
T. T. Wu, B. M. McCoy, C. A. Tracy and E. Barouch
have found the following most remarkable
result:
[H]V, [5]
the scaling limit of the 2 point
correlation function
(2.1)
+ T~(0)= lim g-1/4 , T+T ±0
20=~M2~++N2" • '
E=IT-T I C
÷ 0, M ~ N + N 2 ÷
C
admits a closed expression ÷ logT~(@)
(2.2)
= (- ~
1
1 dq $ l~)d@
1 4
@ dq 2. 2 ((i~q2)2-(~) ) q
in terms of a classical
transcendental
scendent of the third.
It is defined through the following non-linear
ordinary differential
equation
(2.3)
_
d2q de2
-
-
=
i (dq~2 q "~"
--
i dq e d0
(cf. q3
The function
4(0)
was to r e l a t e
the c a l c u l a t i o n of
function
q = q(O), known as a Painlev~
tran-
second order
(3.4))
i + q --
.
in (1.2) is related to
q(0)
by
q(O) = e -@(@)
+
T2(@)
to a linear integral equation
Their approach
125
i
(24
i,x l]dx fcosh0x
f l0(°Ix x I)g(x
where
K0(x)
= Isinhex]
denotes a modified Bessel function.
This type of integral equation has
previously been studied by G. Latta [28], who converted
(2.4) into a system of linear
differential equations _ 2.df 1 (i-x)dxx = (7-o)xf + (q-l(~ +p) + 0(l_x2))g
(2.5)
(l_x2)~x = (N(I-P)
with constants
q,P
1 +p)xg, + 0(l-x2))f + (~
left to be determined.
quently determined by J. Myers
The B-dependence of
~,P
was subse-
[29], who found that
P(@) = ~7_(l_q2_A d~
(2.6)
7~ )
and that
N = N(0)
The fact is:
should satisfy the Painlev~ equation
(2.3).
all the Painlev~ equations I-VI are known to be monodromy preserv-
ing deformation equations differential equations
(in a due sense) of associated second order linear ordinary
(see §3).
The above work of Wu et al. strongly suggests that
some kind of monodromy problem should be comprised in the structure of the model itself. 2.2
We shall see below that this is indeed the case.
Field theory of the Ising model in the scaling limit ([26],
[S] I, [H] VI,V).
A key step of Onsager's original calculation of the free energy ([21]) is to introduce free fermion operators s
mn
(2.7)
Pmn' qmn
satisfies the characteristic
SmnPm,n = ±Pm,nSmn
(m,nEZ),
with respect to which the spin operator
commutation relations
(m' ~ m),
(cf. [30] [31])
Smnqm,n = ±qm,nSmn
(m' ~ m).
The theory of Clifford group ([H] I, [S] V) tells that a commutation relation of the type (2.7) with free fields uniquely determines
Smn
up to a constant factor.
In
the continuum limit (2.1) these free fermion fields are scaled to give 2 dimensional free relativistic Majorana fields (2 •8)
lim - 1 / 2 Pmn = ~ + ( x ) ~
~(x) = t(~+(x), ~_(x))
(x),
lim g-1/2 iqm n = ~+(x)$~
--
whereas the spin operator const. ~
(x=(x°,xl) E ~2 ) (x),
(T + r ±0)
--
-1/8
Smn
C
(with a slight modification
for
T < Tc)
gives ([HI V, [26]) PF(X) / 2 (2.9) with
~F(x)
= :~0(x)e
PF(X)/2 :
(T ÷ Tc+O) ,
~F(X) = :e
:
(T ÷ Tc-0)
126
(2.10)
oo du _2~--~
-im(x-u+x+u -I)
i+
~0 (x) =
e
[+°°r+°° du
Here
~(u)
du'
x@ = (x0±xl)/2, and
~(u)
-i(u-u') e-im(x-(u+u')+x+(u-lugl))~(u)~(u, )
denotes the creation
(u < 0)
annihilation
operator of the free Majorana field carrying t~e energy-momentum ~ ) .
~F(x), ~F(X)
By the Kramers-Wannier duality these fields
as the continuum version of the and Ceva [32].
(u > 0) u+u -I tP ,P ) = tm 2 ' , 0
i,
.
are also regarded
order and disorder variables introduced by Kadanoff
The peculiar commutation relation (2.7) is inherited to the continuum
so that
(2.11)
~F(a)~±(x) = s(xl-al)~_+(X)~F(a),
holds for
x
and
a
~(a)~_+(x) = s(al-xl)~_+(x)~F(a)
spacelike (Figure 2).
The normal product (2.9) of exp (quadratic form in the fields in the Clifford group ([H] I). generally their products variables.
(2.12)
~...~
~) is a characteristic of
The point is that these fields, or more
are characterized by a kernel function of two
In other words the expectatio n value
w_+~,(x,x') = <~,(x)~,,(x')~F(al)''-~F(an)>
/ <~F(al)...~F(an)>
carries all the informations concerning the product function of
x
or
x'
~(al)"'~(an).
larity of a free propagator
~ <~(x)~±,(x')>
(2.11) now has the following consequence:
at
x=x'.
The commutation relation
the Euclidean continuation
(2.12) is double-valued, changing its sign each time when around some
a
Viewed as a
(2.12) satisfies the free Dirac equation, and has a singu-
(p=l,'--,n).
x
or
x'
Euc (x,x') wi_~,
of
makes a circuit
In this sense the monodromy problem is inherent in the
P theory, and the commutation relation (2.11) determines the monodromy structure of Euc
w~_± T
•
2.3
Monodromy problem ([H] III, [S] II).
coordinate of the Euclidean space-time
Let
IR2.
z = (xl+ix2)/2
Given
n
denote the complex
points
al,''',a n arbitrarily, t Characterize the functions w = (w+,w_) with
we consider the following problem. the properties
(2.13)
(i)
Euclidean Dirac equation:
(ii) w a,
~-w_=mw+, ~ . ~ + = m w _
and
Iw[ (iii)
except at
z=al,''" ,an,
is double valued; it changes sign when prolonged once around each
= O(- I
[w I = O(e -2mlzl)
)
(z + a ), as
]z] + co.
127
Call
W
the linear space spanned by these functions. Then it turns out al,''',a n dim Wal,...,an = n. Moreover a basis Wl,''.,w n of Wal,...,an is shown to
that
satisfy, along with (i), a system of linear differential equations of the form
•
=
(B~z
~z*
+
F)
=
[MFWnj where
B,B'
and
F
are
nXn Wc~
B = A, B' = G-IA*G
t G = G* = G- 1
2.4
_
z,_{~
1
(1
_l )
n
may choose such a basis
(2.15)
~
tF
constant matrices independent of
with
= -F,
A =
, A* =
For instance one
"'a
lash] I nl
F* = -GFG -1
Deformation theory ([H] III, [S] II).
of the above normalized basis
z.
that
WCD
Now we consider the
and the matrices
G, F.
av, a~-dependence
For notational simplicity
. . . . . . a~ " a I, ,an, a ~ , u , n (In general for a function u of Xl,''',Xn, du = ~ f.dx. stands for Sx. = fj' j=l j J J j=l,''',n.) We find the following linear system
we employ the exterior differentiation
dWcl .
(2.16)
=
(_dA.~z
-
d
G-IdA*-G~
+
[dWcn j where
@ = (@By)
-f vd leg (aN-a v)
with respect to
@)
.
[WcnJ is a matrix of 1-forms related to (N#v), =0
(D=v).
F = (f v)
@~v =
Equations (2.16) express the necessary and
sufficient condition that the system (2.13)-(i)+(2.14) av, a~
through
is deformed with respect to
while preserving its monodromy structure (in our case (2.13)-(ii)).
integrability condition for (2.13)-(i),
The
(2.14) and (2.16) then gives rise to non-
linear total differential equations, the "deformation equations":
(2.17)
dF = [G,F] + m2[dA, G-IA*G] + m2[A,G-IdA*.G],
In the case of x0 ~-->x with 2.5
n=2
n=2, this reduces to the Painlev~ equation (2.3).
By rescaling
the linear system (2.5) coincides with a restriction of (2.13)-(i)+(2.14) on the branch cut between
a I = -@
and
Correlation functions ([H] III, IV, [S] IV).
(2.18)
dG = -GO-tQ*G.
WFD(X ) = <~(X)~F(al).--~(a
a 2 = 8. Set
)...~F(an)> / <~F(al)...~F(an)>
These functions arise as coefficients of the local expansion of
w(x,x')
at
x'=a ,
128
so that
their
in fact
they
Euclidean are
Euc WFv
continuations
related
to
Wc's
share the monodromy property
(2.13)-(ii);
through
Euc] (2.19)
i
.
= G(I+G) -I
,
lwEUCl ( Fn J
LWcnJ
and hence they also constitute a basis of
W
The significance of (2.18) is al,''',a n
that the logarithmic derivatives
appear as the second $-~a~ log< ~F(al)'..~F(an)> WFv(X ) at x=a . This is seen from the following
coefficients of the expansion of short distance expansion
(2.20)
i = ~(a)
@,(x)~F(a)
(m(-x +a ))$1/2 + -i (+i/2)! m -
_ ½~F(a)(m(x+_a +)).1<2
(,1/2)!
-
-
~ F ! m ( - x +a ))i$i/2 ~a- (I~i/2)!
-
+
~ F (m(x+-a+)) I*I/2 _ _i __ m 3a-~ (1,1/2)! .... "
By (2.19) it is only a little bit of algebra to relate these coefficients coefficients
F,G
of differential
the Euclidean continuation
(2.21) where
equations.
TFn (=T~
to the
Thus we obtain a closed expression of
in (1.3))
of the
n point function
d log TFn = d log /det cosh H + ~ G = e
(2.22)
-2H
and
w = _ ~1 trace(FO + Q*GFG -I) + m 2 trace(d(AA*)-G-IA*GdA-GAG-IdA*).
• Mixed correlation functions
~l'''~m rFn
= <~F(al). ..~(a
I )-.. ~ F ( a
given by (2.23)
-..
m Vl...~ m
TFn
2.6 Comments.
= TFn × Pfaffian
(i(tanh H)v~,)~,~,=~l,...,~m
The above construction admits of extensions to several directions.
It is straightforward to generalize the monodromy -i 2~i~ 2~iL e ([H] IV, IS] VII), or even a matrix e
factor
the free Dirac (instead of Majorana)
fields.
to an arbitrary phase ([S] XIV], starting from
The deformation theory in 2.3, 2.4 holds
without change, the only difference being the alzebraic conditions on
G
The relevant fields are related to the solution of the Federbush model
([H] IV
supplement,
(2.24)
are
)'-- ~F(an)>
and
[33]) whose interaction Lagrangian is
•int
(current-pseudocurrent
=
~ -gg~Jl(X)Jll(X)
coupling between two species of fermions).
constant coincides with the exponent of the monodromy:
g=2~.
The coupling
F.
129
In the massless case, our construction is directly related to the original problem of Riemann (§4).
The corresponding Lagrangian field theory is the massless
Thirring model ([H] IV supplement). Another possibility is to start from free bose fields (neutral or complex) instead of fermiens ([H] IV, [S] IX). acting fermi field ~F(a)
In the neutral case there results an inter-
~B(a ) = t(~-i(a), ~B(a)),_ to be compared with the bose field
obtained on the basis of free fermions.
share the same S-matrix
(_)N(N-I)/2
These two fields
@B(a),
~F(a)
in the N-particle sector ([H] IV, [S] I, IX,
[26]), and their n point correlation functions are inverse to each other apart from a simple factor ([H] IV, [S] IX).
§3.
Monodromy preserving deformation of linear ordinary differential equations
3.1
History.
The most typical example of a linear ordinary differential equation is
the Gauss' hypergeometric differential equation
(3.1)
(
,
__d)
= dx
x(l-x)y" + (y-(~+$+l)x)y' - ~By = O.
The coefficients of (3.1) are completely determined by the characteristic indices at the regular singularities general equation.
0, 1
and
~.
However this is not the case for a more
Even if all the singularities are regular ones, not only the local
indices but also the structure of global monodromy is needed to specify the equation. Here arises naturally the following question, first posed by Riemann [34]:
Find a
differential equation with regular singularities whose monodromy representation is the prescribed one. Schlesinger
[8] considered the first order system (1.15) with
(3.2)
A(x) =
n A [ ~=i x - a
and initiated the following approach to the Riemann's problem: AI,'..,A n
so that the monodromy representation is preserved under the variation of
the position of the branch points solution
Y(x)
equations for
(3.3)
Find conditions on
~x
at
al,''',a n.
With a suitable normalization of the
x =~, he obtained the following system of linear differential
Y(x), n = ( ~
A "~'=~ )Y'
~Y ~a
A x-a~
Y
(v=l, "'',n),
and then, as the integrability conditions for (3.3), the system of non-linear differential equations (1.19) for
A
(D=l,''',n).
The simplest nontrivial case of (i.19),m=2
and
n=4, is equivalent to a second
order non-linear ordinary differential equation, known as the Painleve equation of the sixth kind ([ii]).
130
Originally the Painlev~ equations emerged in a different context.
Painlev~
obtained the following six canonical types of equations through the classification of second order non-linear algebraic differential equations whose general solutions have no movable branch points. (3.4)
(I)
y" = 6y2+x
(II)
y" = 2y3+xy+~
(III)
Y" = y,2 _ ~ y x
(IV)
y" = ~ 2y
(V)
yd = (~y + ~
(VI)
1 + ~1 y,, = (~
+ ~y2+B + yy3 + x y
+ 3~ 2
+ 4xy2 + 2(x2_~)y +
y
1 ) ,2 y' (Y-l)2(~y+~ 8) + 26X + ~y(y+l) y - x + x y-i Y x2 i +
+
i
y-
+
1
~
+ y(y-l)(y-x) (a + ~ x2 (x-l) 2 y
,
-
+ y(x-1) + ~x(x-l) ). (y-l) 2 (y-x) 2
Subsequently Garnier [12] showed that the Painlev~ equations I through V are also obtained as monodromy preserving deformation equations for certain second order linear ordinary differential equations with regular and irregular singularities. In this section we pursue this approach by Schlesinger and Garnier (see [14] [36] for recent progress), laying stress on the Hamiltonian structure of deformation equations ([9][10][3]).
This point of view reveals the mathematical r$1e of the
"physically" introduced correlation functions appearing in our theory of Holonomic Quantum Fields. 3.2
(3.5)
Monodromy data ([14], [36]).
A(x) =
Let us consider (1.15) with
n A ~ --~-- + Am, ~=i x - a
[tl Aoo =
] ".
• .tm
It has a normalized solution which admits the following asymptotic expansion in a certain sector in the neighborhood of
x=~.
AooX
(3.6)
Y(x) % ^ Yoo(x)xDe
, D:
diagonal,
We can choose successive sectors the normalized solutions continuation of
Y%(x)
~%+i
Y%+l(X) = Y%(x)C~
=~lY ~x -~
~2' "'" (see Figure 3.1) and correspondingly
Yl(X), Y2(x), ... into
but differs by a constant matrix (3.7)
~i'
Y (x) = 1 +
in each sector.
However the analytic
does not necessarily coincide with
C%, called the Stokes multiplier: (%=l,''',k).
Y~+l(X),
131
These matrices D, CI,''',C k the irregular singularity the path encircling
(3.8)
constitute a refined notion of the monodromy matrix at x --~.
x =~
In fact the monodromy matrix
Moo corresponding to
clockwise is given by
Moo = e2~iDCk I • .-CI -I.
We choose a path joining
along this
~
to
a
and consider the analytic continuation
path of the flormalized solution
that the eigenvalues of local expression at
A
Yl(X)
of the first
are distinct, we conclude that
sector.
Y(x)
Assuming
has the following
x=a . L
(3.9)
Y(x) = Y (x)(x-a)
(3.10)
A
~,
Y^ (x) = ~ 0 Y
~(x-a )~,
det Y~0 # 0,
= Y 0L Y ~ .
The monodromy matrix
M
at
x=a
is given by
2~iL (3.11)
M
= e
Conversely, let x = al,''',an ,=. x = a
Y(x)
be an
m×m
Assume further that
(v=l,''',n)
and
matrix holomorphic and invertible except for Y(x)
x =~, respectively.
system of the type (1.15)+(3.5). Linear equation
with monodromy data al,
..-,an, LI,''',Ln, D, CI,''',C k
Deformation equation ([36]).
• ..,tm, keeping the monodromy data
• -,tm, Y
satisfy a
Y(x)
al,...,an, Al,''',An,Aoo
d
Y(x)
Multi-valued function
with singularity data
denote by
Then it is shown that
In this sense the following equivalence:takes place.
dY(x) = A(x)Y(x) dx
3.3
has local expressions (3.9) and (3.6) at
Now let us deform parameters LI,''',L n
and
al,''',a n
D, CI,''',C k
the exterior differentiation with respect to
fixed.
and
tl,
If we
x, al,''',an, and
tl,''"
satisfies the following system of linear differential equations. n
(3.12)
dY = ~Y,
(3.13)
Q.. = 0,
33
=
X A d l o g (x-a v) + d(xAoo) + @, v=l n Ojk = ~ ( A ) j k d l O g (tj-tk). ",)=1
The integrability condition
dg~ = S2A~
for
(3.13)
gives rise
to the following
132
non linear system of total differential equations.
(3.14)
dA
= -
~
[A ,A ,]d log (a -a~,) - [A ,d(a A ) + 0].
v'(#v) 3.4
Hamilton structure.
The system (3.14) can be written as a Hamiltonian system.
Let us define a Hamiltonian 1-form
(3.15)
co =
1
I
~
with time variables
al, • -., an, tl,..-,t m.
1 trace AvA ,dlog (a -a ,)+~traceA d(a A )+ ~traee ~ ~ A ~ ~ D ~ "
Then (3.14) is rewritten as
(3.16)
dA
= {A ,m}.
Here the Poisson bracket is defined by
(3.17)
3.5
{(A )jk,(A ,)j,k,} = ~
,(~jk,(A )j,k-6j,k(A )jk,).
Painlev~ equation of the fifth kind.
In the special case when
n=m=2, the
Hamiltonian system (3.16) reduces to the following system.
(3.18)
H(t,y,z) = z+(-yz(~l~z)+y-l(v2-z)(V3-z)-2z2+(~l+~2+v3)z)~
(3,19)
{z,y} = y,
(3.20)
~
,
= {y,H(t,y,z)} = -ty-2z(y-l)2-(y-l)(-~lY+~2+~3 ),
dtdZ = {z,H(t,y,z)} = -yz(~l-z)-y-l(~2-z)(~3-z).
Eliminating
z
from (3.20), we obtain the fifth Painlev~ equation (3.4)-(V) with
parameters (3.21)
1 2 ~ = 2~i'
If we set
1 2 B = - ~(v2-~ 3) , ~ = -l+Vl-~2-~ 3
o(t) = tH(t,y(t),z(t)),
second order differential
(3.22)
3.6
O(t)
and
itself satisfies the following
equation.
d2o. 2 . td_~+2.do. 2 . do -2+ do. (t ---~) = to- dt (~) -(~l+V2+~3)(d-t )) 4 ~ k V l dt
Ising model.
1 ~ = - ~.
do) ~)(~3~ (v2-
do ~)"
We show that the deformation theory of Ising model (§2) can be
incorporated into the framework of this section.
By introducing the generalized
133
Laplace transformation
(3.23)
w (z,z*) =
~
/~-i
e
w tu)
the system (2.14), (2.15) is converted into equations ^ (3.24)
d~u
^ = (-mA -
+
G-ImA*G)
u
U
n having two irregular singularities of rank
~n i
at
u=0
and
u=~.
The monodromy pre-
serving deformation equation for (3.24) is just the same as that for (2.14) ([36]). This suggests the stability of isomonodromic property under the generalized Laplace transformation ([37]). The algebraic expression for
~
(2.22) in terms of
F
and
G
gives a
Hamiltonian i form for (2.17), where the Poisson bracket is defined to be
(3.25)
{Gjk' Gj,k,} = 0,
{(FG-I)jk,
(FG-I)j,k,} = 0,
{Gjk,(FG-I)j,k,} = 6jk,6j, k.
~k ~
Figure 3
~
~
Sectors at
Riemann's monodromy problem
4.1
Problem.
Let
Here
'small'
(4.1) at
Y(x) = Yoo(x)x D,
D
(= c l o s e t o O) and imposing t h e f o l l o w i n g
x =oo and
Yoo(X) =
(4.3)x 0
oo ~ Yoo~X-~, ,%=0
(or (4.3) a ).
det Y~0 # 0 .
is the 'small' matrix uniquely determined by the following condition. 2~iL 1
(4.2)
be real numbers. We construct m×m matrix n (al,L1),-..,(an,Ln) i n t h e s e n s e of ( 3 . 9 ) , assuming t h a t
are sufficiently
additional conditions
(4.1)
Figure 4
a I < a2<'''< a
w i t h monodromy d a t a
L1,''',L n
<
x=~.
§4.
Y(x)
C
e-2~i)
e
2~iL "'" e
n
= e
" 2~iD
The meaning of (4.2) is clear:
A path encircling
homotopic to a path encircling
~.
an,-..,a I
successively is
134
The
second additional condition is the normalization;
(4.3)x0.
Y(x 0) = i
for
x0 # al'''''na , ~,
or
(4"3)a V
Y~0 = i.
By a suitable linear transformation we obtain the solution subject to the Schlesinger's normalization: (4.3)oo
4.2
Yoo0 = i.
Operator solution ([H] II).
fermion operators on ~. (4.4)
Let
~(J)(x), ~*(J)(x)
(j=l,---, m)
denote free
They satisfy the following anti-commutation relations.
[~(J)(x), @(J')(x')]+=0, [~*(J)(x),~*(J')(x')]~0, [~(J)(x),~*(J')(x')]+=~(x-x').
The vacuum
Ivac> is defined by
(4.5)
~(J)(p) Ivac> = @* (J )(-p)Ivac> = 0
for p > 0
where q~o (4.6)
+oo
~(J)(P) =I edx -ixp~(j)(x)'_
~*(J)(P)= I_codx e ixp~*(J)(x)"
Then we have the following vacuum expectation values. <~(J)(x)~(J')(x')> = <~*(J)(x~*(J')(x')>. • = 0,
(4.7)
<~(j)(x)~,(j')(x, ~ = <~,(j)(x)~(j')(x,)> = 1 i 27 x-x'+i0 For an (4.8)
mxm
matrix
L
we define a field operator +oo +co
~(a;L) =: exp p(a;L):, p(a;L) = j ,k=l
(4.9)
(T)
by
j~ I dxdx'~(J) (x)R(x-a'x'-a;L)jk~*(k)(x') ~-oo --0o
R(x,x';L) = -2i sin~L x -L x'L( i i -~iL+ i -i ~iL) (~) - - ~ x-x'--+i0 e 27 x-x'-i0 e °
~(a;L) (4.10)
~(a;L)
satisfies the following commutation relation with free fields.
~(a)~(k)(x) = ~ ~(J)(x)~(a) Mo(x-a)jk, ~(a)~*(k) (x)= ~ ~*(J)(x)~(a)(tM0(x-a)-l)jk j=l j=m
xL = 0 (x >0),
= [xlL (X <0).
135
1 where
~.x) = {e_2~i L
if
x > 0
if
x < 0
Let us consider the following matrix of vacuum expectation vaiues. I al...an] Y+ x0, x ; jk LI'''L n
(4.11)
<~*(J)(x0)~(k)(x)~(al;Ll)---
~(a ;L )> n n , <~(al;Ll) --. ~(an;Ln)>
= 2~i(x_x0 ) [
Y- Xo'X ;
al...an]
jk
LI'''L n
<~*(J)(xo)~(al;gl)
... ~(an;L)~(k)(x)>
= 2~i(x-x0) <~(al;Ll) --- ~(an;Ln )> (4.7) implies that, as a function of
x, Y+(x)
(resp. Y_(x))
is the boundary
value of a holomorphic function in the upper (resp. lower) half plane. that
Y±(x)
(4.10) implies
satisfy the following relation on the real axis.
(4.12)
Y_(x) = Y+(x) M(x), M(x) = M~'--M n
(a~_ I < x < a~), M
= e-2~iL~
From (4.12) and (4.7) we see easily that the analytic continuations of
Y±(x)
the required monodromy property with the normalization condition (4.3)x0. check the stronger requirement (3.9) about the exponent
L
satisfy
In order to
we should look into the
local behavior more closely.
4.3
Infinite series expressions.
An application of the Wick's theorem to the explic-
it formula (4.8), (4.9) and (4.11) leads us to the following infinite series expression for
Y(x0, x) n
(4.13) where
Y(x01 x) = 1 ZU~ (x0, x)
2~i(x0-x)
~ Z v(x0, x) ,~=i
is a holomorphic function defined for
(xO, x) E (~-FH) x (~-F)
by the following (see Figure 4 as forF~): 0
(4.14)
0
z~v(Xo'X)=~v~ ~ dXldX2 2~ x0-xl-a~ ~(Xl'X2;~12~ x2+a~-x -co
-0o
0 +
=
Vl"'''~%
0
i=I -oo'" " -o~dXl" " "dx2~+2 2~ x0-xl-a U
(x2'x3)
136
... R(x2%_l,X2%;L %_l)A~%_~(x2%,x2~+l)R(x2%+l,X2%+2;L
Here we set
Ap~(x,x')-~(l-@)
Assuming that
i x+a -x'-a -is
0 ' s
)~
i x2z+2+a~-x
= sgn(~-~).
Imal> ...>Iman , we can rewrite (4.13) in a form suitable to
check the local behavior. (4.15)
Y(x 0,x) = (x0-a)-LP Y~v(x0'x)(x-a )L~
(4.16)
^ Y~(x0'x)
= @p~+2~i(x0-x)
I
r 1 i L dXlJ dx2 2~ x0--xI (xl-a~) C C -L
2~r
(l_~ v) + ~
p(#D) O(¢~)
Here the contour of integration posed to be inside of
Cp
~
and
C~
Zpo.(Xl,X2)) (x2_a)) " v12~ ~--i
is given in Figure 4, and
o
-2 x
x0
and x
are sup-
C~, respectively.
We note that (3.9) follows from (4.15).
Moreover, the normalization (4.3)a~ is
achieved if we take (4.17)
Yv(x) =
L lim (x0-a~) ~)Y(x0,x).
x0+% ' Y~ (x)
(4.18) Then
is also expressible in terms of field operators. %D*(k)(a;L) =:
Y~(x)
a
~ dXl~*(J)(xl) j =i -oo
We set
I ~sin~tL x-a tL-i ~jjk exp p(a~L):.
is written as
2~i(x-a~)<~(al;Ll).-.~*(J)(a ;L ).--9)(an;Ln)~ (k)(x)> (4.19)
Y~ (X)jk = <~(al;Ll) -.-
~(an;Ln )>
We have already mentioned about <~(al;Ll)-'" ~(an;Ln)> in §i (1.20). We note that dlog<~(al;Ll)...~an;Ln)> can be expressed as an infinite series similar to (4.13)
([HI n). §5.
Density matrix for impenetrable bose gas.
5.1
Fredholm theory
([38], [39]).
cretized version of (1.7), with dimensional (5.1)
XY
c=+ °°
As was pointed out by Schultz [38], the disand finite particle density
model defined by the Hamiltonian
i x x }~Xy(y,h) = - ~ ~((l+Y)~mOm+ 1 + (l-Y)omY~Y+l + 2h~ z) m
00' is the 1
137
with
Y=0, h=cos~P0E
(~=lattice spacing).
This is one of typical models solvable
by the technique of the Clifford group ([H] V). similar to the Ising model (§2, 2.2).
The structure of (5.1) is quite
This time, going back to the continuum
p(x)
=lim~-i/2pm, q(x) = lim~-i/2qm, we obtain the one particle reduced density matrix PFF(X) for free fermion (we choose
(5.2)
as
po=~ -I)
2 sin(x-x') ~ x-x' - -2PFF(X-X')'
- ~(x-x')
while the scaling limits corresponding to
and
• -- ~F(a2n)> in (2.12) give the Fredholm determinant
<~(x)~±,(x')~F(a I)
and the first Fredholm minor
belonging to the kernel (5.2), considered on the union of finite intervals with endpoints
al,---,a n.
The
n
particle reduced density matrix (1.9) itself is the n-th
Fredholm minor determinant ([39]).
We are to characterize these quantities by defor-
mation theory. Another approach due to Vaidya-Tracy [40] is to study the double scaling limit of the
XY
model (5.1). For this we refer to [40] and [S] XV, XVI, [3].
5.2 Deformation theory.
Let
a I
be real numbers and set
I = [al,a2] U
xl'''x
) r ;% and Ri(x,x';l) Xi-.-Xt r the Fredholm determinant, the r-th Fredholm minor and the resolvent kernel for the [a3,a4]U.." U[a2n_l,a2n].
We denote by
integral kernel sin(x-x')/(x-x')
(5.3)
AI
xl'''x r Xi---X I r
(5.4)
~iog£1 (i)/~a j
(5.5)
on
gi(1),
AI
I, respectively.
. i ) = (-l)r£1(%)det(Rl(Xj'Xk;~))j,k=l, •
,r
(_)j+1% RI (aj ,aj ;%),
$21og~(%)/SajSak = (_)j+k+l 2 Ri(a j,ak;l)2
the characterization of
Since we have
d log~(%)
(j#k),
is sufficient.
We set
(5.6)
Yloo(x)
= Rti(x;%)
(5.7)
RZI(X;I)
+' iJl I +'e±'i(Xo-Xl) sin(xl-x2) sin(xz-l-X%) +ix% 2i Xl_X 2 " "" X~-l-X£ e R+~i(x;%)= ' I • • • Idx I" • "dxz(x0=x) •
YI(X) behavior of
is holomorphic and invertible except for Yioo(x) at
x=°° and
x=a. J
(j=l,''',2n)
x =c°, al,''- , a2n. are as follows.
The local
138
(5.8)
Yi~(x) = (1 + 0(i)) e(~-i)x
Yloo(x)
YI (x) ~(x_--~2 )
'
(x-a2n)
i~
J
A
Here
Yi(x)
is holomorphic and invertible except for
x--~. We note that the Stokes
multipliers are trivial in this case. These monodromy properties for
Yioo(X) leads us to the following linear system:
(5.10)
2n dYl°° = ~IYI °°' ~I = j[--IAjdlog(x-aj) +
(5.11)
Aj =
Ir+j r .] • (-r_j r+j), -j
li _i] dx,
rij = ~ ( R + + I
_ (aj ;%)+R±l(aj;%)).
As the integrability condition for (5.10) we obtain the Hamilton±an equation, an equivalent of (i. 14) : (5.12)
dr±j
with the Hamilton±an (5.13)
= m
{rij, ~} , of (1.13) and the Poisson bracket
{r+j, r+j,}= 0,
{r_j, r_j,} = 0,
{r+j, r_j,} = ~jj,.
5.3
1 particle reduced density matriE.
p(x)
for impenetrable bose gas and the level spacing probability distribution funci^L O J (0x ; ~)2 and A[0,~t](~).~ 1 Eu(t) in [41] coincide with - Z~mr~,X~ If we set
tion (5.14)
o~et = ~ t l°gAl(t)' ~ i n
=
The One particle reduced density matrix
d
0
t~
lOgAl( t ; X ),
I = [O,t],
they satisfy the non-linear equation (3.22) with (~i' ~2' ~3 ) = (0, 0, 0) and (0, i, i) (i.e. (i.ii)), respectively. We note that the one particle reduced density -i matrix P0 PFF(X) = sinx/x for free fermion also satisfies (i,i0), (I.ii). Finally, we give the small and large x behaviors of p(x) (C=I/2~ in (5.15)):
(5.15)
poiO(x)=l_~ix 2 + ~2C 1 4 - IIC x 3+--x
(5.16)
polp(x)= ~P~
•
32
5!
(i + l2( cxo s 2 x
5
1 6
61C
3-~.52x --~.tx +22.33.52.72
C2 )x8 +.-. x7 + ( ~ + ~. 3~.5 ~
i + _ _ 3 sin2x+ 3 (ii_ 31cos2x) + -'') -~) 24x3 28x4 23
p~ was determined by [40] to be ~e I/2 2-1/3 A -6 (A=Glaisher's constant). XVI, [3] for further expansions up to x I0 and to x-8
See [S]
139
5.4
Low temperature
and larse couplin$ expansions.
the case of zero temperature for finite
~
and
Each coefficient $=c=~
c
kT=B-I=0
The above analysis
concerns with
and infinite coupling
is written down in the form of
c=~. Density matrix -i and c expansions.
8 -1
contains a finite number of integrals
over the density matrix with
and simple elementary functions.
§6.
Higher dimensions
6.1
General setting.
construction objects.
dimensions
~[C'],
dependent
r (resp r').
r+r' = n-2.
When
is greater than 2, analogous
the main field
~[C]
on a closed spacelike submanifold
Here C
n
requires to introduce fields that depend on extended
Let us start with two kinds of fields,
iliar~ field
tion
When the space-time dimension
as in §2 inevitably
r, r'
and
C'
are non-negative
and the aux-
C (resp. C') of
integers subject to the condi-
are in a mutually spacelike position so that
they lie on a common spacelike hypersurface, one has the notion of the linking number ~(C, C').
We set the following
fields for mutually spacelike
(6.1)
~[C],
Up to this point
commutation C
and
C'
relation between the main and auxiliary (Figure 5):
[C'] = (_)~(C,C') ~[C'] ~[C].
(6.1) looks quite similar to the 'tHooft algebra
in order to apply our machinary we require one more point should be free in a sense or anether, main field 6.2
~[C]
(theory of Clifford group,
Construction
from local free fields
ing on extended objects" to the case ~(x).
so that
([19]).
: the auxiliary
(6.1) will completely
However, field
characterize
~[C'] the
[H] I).
([S] XII).
Unfortunately
is not a tractable notion at present.
"free fields depend-
We restrict
ourselves
r'=0, where the auxiliary field is local, e.g. the usual free Dirac field
In this case the main field
~[C]
is given by : exp (quadratic form in
~):,
having (6.2)
w(x, x') = < ~(x)~(x')
as the kernel of the quadratic the Euclidean al submanifold
continuation C
form.
wEUC(x,x ')
forward to incorporate phase factors
Deformation
The commutation
theory
([S] XIII).
closed submanifold
real analytic matrix on
(6.1) now tells that
(or matrices) e Euc w
-I. 2~i%
n-2
dimension-
More generally it is straightin place of
-i.
The problem
theory for
To this end we consider a Riemann-Hilbert
dromy problem for Euclidean Dirac equations dimensional
relation
of (6.2) is ramified around an
with the corresponding monodromy
is now to construct a deformation 6.3
~[C] > / < ~ [ C ] >
(- ~ + m)w = 0.
of the Euclidean space
]Rn.
Let Let
F, assumed to be close to the unity.
F M(~)
be an be an
mono-
n-I NxN
We pose the following
140 problem : find
(6.3)
Here
w(x,x')
such that
(i)
(-$x+m)w(x,x ') = ~n(x-x')
(x,x'~]Rn-F)
(ii)
lw(x,x')l = O(e -mlx])
(Ix I ÷ ~, x'
(m)
w(~+,x ') = M(Ow(~-,x')
w(~±,x ')
means the boundary value from inside or outside F (Figure
Conditions (6.3) uniquely determine and
M.
field
fixed)
w(x,x') = w(x,x';F,M)
Consider a small variation
F' = {~+~P(~)I~EF}
~p(~) = (6pl(~),...,~pn(~))
to be preserved in the sense that
on
F.
of
F
(~• F).
F
by specifying a vector
We assume the "monodromy"
M'(~+~p(~)) = M(~)
6 ).
as a functional of
M'(~')
on
F'
Then the following
variational formula is valid:
r
n
6w(x,x')
5w(x,x') =Ida(~)j ~=i~ 6P~(~) 6pU(t) '
(6.4)
6w(x,x')
w(x,~+).(nu(~)$_~(~)~u)M(~).w(~-,x,)
6pV(g) where
d~(~)
is the surface element and
n(~) = (nl(~),'",nn(~))
is the unit outer
normal. Our original situation is realized dimensional submanifold
C
and
as a limiting case where
F
contains an n-2
M(~)
is a step function on F (Figure 7 ). Since 6w (6.4) contains a tangential derivative of M(~), - = 0 wherever M(~) = constant.
6p~(~)
This guarantees that the variational equation does not depend on the choice of
F.
The deformation equations in 2 dimensions (§2, §4) are recovered from (6.4) along with (6.3)-(i) and the Euclidean covariance ([S]XIV).
Figure 5 n=3, r=l, r'=O. H is a spacelike hypersurface that contains C and C'.
F
~+6p(~)
Figure 6
Figure 7
141
REFERENCES i.
M. Sato, T. Miwa and M. Jimbo, a series of papers entitled "Holonomic quantum fields." I: Publ. RIMS, Kyoto Univ. 14 (1978) 223, II: ibid. 15 (1979) 201, ]]I: ibid., to appear in 15 (1979), IV: RIMS preprint 263 (1978), V: ibid. 267 (1978), supplement: ibid. 304 (1979). These papers are referred to in the text as [H] I, etc.
2.
, a series of short notes entitled "Studies on holonomic quantum fields." Proc. Japan Acad. 53A (1977) 6(I), 1 4 7 ( Z ) , 153(]]I), IS3(IV), 219(V), 54A (1978) I(VI), 3~(VII) , 221(V]II) , 263(IX), 309(X), 55A (1979) 6(XI) , 73(_XII) , ll5(Xm), 157(X~), 267(XV), 317(XVI) . Some of them are with different authors (IX: M. Jimbo, ~I~ ~ V : M. Jimbo and T. Miwa, XV: M. Jimbo, T. Miwa and M. Sato, XVI: M. Jimbo, T. Miwa, Y. MOri and M. Sato). These papers are referred to in the text as [S] I, etc.
3.
M. Jimbo, T. Miwa, Y. MSri and M. Sato, Density matrix of impenetrable bose gas and the fifth Painlev6 transcendent, RIMS preprint, Kyoto Univ. 303 (1979).
4.
M. Jimbo, T. Miwa and M° Sato, Kakuy~g@ Kenkyu, Inst. Plasma Phys. Nagoya Univ., 40 Suppl. (1978) 45. , Proc. of the "International Colloquium on Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory" Les Houches, France (1979), to appear.
5.
E. Barouch, B. M. McCoy and T. T. Wu, Phys. Rev. Lett. 31 (1973) 1409. T. T. Wu, B. M. McCoy, C. A. Tracy and E. Barouch, phys. Rev. BI3 (1976) 316.
6.
B. M. McCoy, C.A . Tracy and T. T. Wu, J. Math. Phys. 18 (1977) 1058.
7.
M. Girardeau, J. Math. Phys. 1 (1960) 516.
8.
L. Schlesinger, J. Reine u. Angew. Math. 141 (1912) 96.
9.
J. Malmquist, Arkiv Math. Astro. Phys. 17, No.18 (1922) I.
i0.
K. Okamoto, Polynomial Hamiltonians associated to the Painlev~ equations, preprint Tokyo Univ. (1979).
ii.
R. Euchs, Math. Ann. 63 (1907) 301.
12.
R. Garnier, Ann. Ecol. Norm. Sup. 29 (1912) i.
13.
M. J. Ablowitz, A. Ramani and H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type I, II, preprints (1979).
14.
H. Flaschka and A.C. Newell, Monodromy and spectrum preserving deformations I, preprint (1979).
15.
K. Ueno, Monodromy preserving deformations and its application to soliton theory, RIMS preprint 302, Kyoto Univ. (1979).
16.
L. D. Faddeev, ModUles compl~tement int~grables de la th~orie quantique des champs, preprint (French translation by Cambefort in Dijion) (1979).
17.
J. Honerkamp, P. Weber and A. Wiesler, Nucl. Phys. B152 (1979) 266.
18.
H. B. Thacker and D. Wilkinson, Phys. Rev. DIq (1979) 3660.
19.
G.'t Hooft, Nucl. Phys. B138 (1978) i.
142
20.
J. Hadamard, Oeuvre t . ~ . P. LEvy, Probl~mes concrets d'analyse fonctionelle, Gauthier-Villars, Paris (1951).
21.
L. Onsager, Phys. Rev. 65 (1944) i17.
22.
C. N. yang, Phys. Rev. 8 5 (1952) 808.
23.
B. Mo McCoy and T. T. Wu, The two dimensional Ising model, Harvard University Press, Cambridge, Mass. (1973).
24.
B. M. McCoy, C. A. Tracy and T. T. Wu, Phys. Rev. Lett. 38 (1977) 793.
25.
R. Z. Bariev, Phys. Lett. 55A (1976) 456. Phys. Lett. 64A (1977) 169.
26.
M. Sato, T. Miwa and M. Jimbo, Field theory of the two dimensional Ising model in the scaling limit, RIMS preprint, Kyoto Univ. 207 (1976).
27.
D. B. Abraham, Comm. Math. Phys. 59 (1978) 17.
28.
G. E. Latta, J. Rational Mech. Anal. 5 (1956) 821.
29.
J. M. Myers, J. Math. Phys. 6 (1965) 1839.
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B. Kaufman, Phys. Rev. 76 (1949) 1232.
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T. D. Schultz, D. C. Mattis and E. H. Lieb, Rev. Mod. Phys. 36 (1964) 856.
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L. P. Kadanoff and H. Ceva, Phys. Rev. B3 (1971) 3918.
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B. Schroer, T. T. Truong and P. Weisz, Ann. Phys. 102 (1976)156.
34.
Bo Riemann, Werke, p.379.
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P. Painlev~, Oeuvret.]II, p.187.
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K. Ueno, Monodromy preserving deformation of linear differential equations with irregular singular points, RIMS preprint, Kyoto Univ. 301 (1979).
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G. D. Birkhoff, Trans. Amer. Math. Soc. i0 (1909) 436.
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T. D. Schultz, J. Math. Phys. 4 (1963) 666.
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A. Lenard, J. Math. Phys. 7 (1966) 1268.
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H. G. Vaidya and C. A. Tracy, Phys. Lett. 68A (1978) 378. Phys. Rev. Lett. 42 (1979) 3.
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The authors would like to thank the secretaries in RIMS for carefully typing this manuscript.
INSTABILITY OF PHASE COEXISTENCEAND TRANSLATION INVARIANCE IN TWO DIMENSIONS Michael Aizenman* Physics Department, Princeton U n i v e r s i t y Princeton, N.J. 08540 I.
Description of the main r e s u l t C o l l e c t i v e phenomena which are important features of the physics of bulk systems
are already exhibited by deceptively simple l a t t i c e models. the f i r s t
Among such phenomena are
order phase t r a n s i t i o n s , which are associated with the occurance of several
d i s t i n c t phases of a system in a given temperature, magnetic f i e l d and other intensive parameters. The question I would address is the p o s s i b i l i t y of a stable coexistence of phases at a phase t r a n s i t i o n .
By t h i s I mean the occurance of a state of an extended system
which is in a thermodynamic e q u i l i b r i u m and which in d i f f e r e n t regions of the space
o°c Filg, l,"
Fig. 2:
11] = (U+ +p_)/2
A would-be stable phase coexistence.
*Supported in part by the National Science Foundation grant PHY - 7825390
144 l o c a l l y resembles d i f f e r e n t phases ( f i g . 2).
This should not be confused with the oc-
curance of mixed ensembles of the pure phases ( f i g .
I).
These represent an uncertainty
about the system's phase, rather than stable phase coexistence. S p e c i f i c a l l y , I shall describe a recent proof that f o r the Ising model of f e r r o magnets there is no stable phase coesixtence in two dimensions, in contrast with i t s occurance in three and more dimensions. The Ising model consists of spin variables iE~Zd
o i = ±I, associated with the sites
of a d-dimensional l a t t i c e , which i n t e r a c t by the Hamiltonian H(o) =
-½1i_Jis=
1 a.~. I J - hzoi
For a specified spin configuration on the equilibrium state in A
A' ~ A
(h is the external magnetic f i e l d ) . @A- the boundary of a f i n i t e volume AC~ d,
is given by the p r o b a b i l i t y measure
({o}) : exp.[-6 H(~)] /Norm.
on the space of spin configurations with the specified boundary values
a~6 = ( h T ) - l ) .
More general f i n i t e volume e q u i l i b r i u m states are ensemble averages (~ convex combinations) of ~^ ~ over various boundary conditions. The thermodynamic l i m i t , A + ~ d , is descri bed by' j ~A~l i m i t s of the above states which y i e l d probabil~ty measures on the space of spin configurations,
£ = { { ° i } i E ~ d [~i = ±I} ~ { - I , I } ~
Such l i m i t s describe a bulk system in thermodynamic e q u i l i b r i u m , and are called Gibbs states.
Their c o l l e c t i o n is denoted here by
a(6).
Gibbs states can also be
characterized d i r e c t l y by the Dobrushin-Lanford-Ruelle condition [ 1 , 2 ] . If
d ~ 2, for
6>6c the Ising model e x h i b i t s a phase t r a n s i t i o n at
is associated with the existence of two phases l i m i t s of Gibbs states with boundary conditions For was f i r s t
h+O and with
o = +I,
or
~ = -I,
h = O.
This
p+,~_EA(B) which are the two d i s t i n c t
h+O, and which can also be obtained bv the imposed on
SA, as A+ 77d.
d ~ 3, the system admits stable phase coexistence at low temperatures.
This
proven by Dobrushin [3] (1972), by analyzing the e q u i l i b r i u m states induced
in the volumes [ - n , n ] d = A n ~i = sgn(id + ½)
by the boundary conditions: for
At low temperatures the l i m i t
i = (i I . . . . . id) E ~A n. ~
y i e l d s a Gibbs state which is not i n v a r i a n t under
t r a n s l a t i o n s and which offers an example of a stable phase coexistence.
Additional
l i g h t was shed on t h i s phenomenon by the work of van Bei.ieren [ 4 ] , and the proof of i t s occurance was recently extended, by Bricmont et. al. [5], to the ~!idom-Rowlinson model. The above construction does not produce phase coexistence in two dimensions, as was f i r s t
proven by G a l l a v o t t i [6].
His analysis shows t h a t , at low temperatures,
145 as
n ÷ ~
the long contour, which is a l i n e passing between nearest neighbor pairs of
d i f f e r e n t spins s t a r t i n g and ending between the two such pairs on fluctuations.
~A n , has unbounded
S p e c i f i c a l l y , the contour would t y p i c a l l y be e i t h e r f a r below or f a r
above any given f i n i t e
region
V, provided
n
is large enough.
Consequently, the
l i m i t i n g state (defined on the local observables) is the ensemble average ~ = (~++~/2. G a l l a v o t t i ' s work, and the f u r t h e r results of Abraham and Reed [ 7 ] , Lebowitz [ 8 ] , Messager and Miracle-Sole [ 9 ] , Russo [ I 0 ] , and M e r l i n i [ I I ]
raised and supported the
p o s s i b i l i t y that a stable phase coexistence is not possible in the two dimensional model.
That t h i s indeed is the case is implied by the f o l l o w i n g recent r e s u l t .
Theorem:
Any Gibbs state of the two dimensional ferromagnetic Ising system, ~EA(B),
is of the form:
u = ~ + + (I - ~)u_,
f o r some ~ E [ O , I ] .
Following is an o u t l i n e of the proof, which is given in [12]. also proved, independently, by Higuchi [13].
The theorem was
Both derivations incorporate previous
results of Russo [ I 0 ] who proved a s i m i l a r assertion for states which have one of the main symmetries of the l a t t i c e .
As remarked in [12], the proof given there can be ex-
tended to some other two phase systems at low temperatures. 2.
Steps in the proof The proof can generally be described as stochastic-geometrical.
is the i d e n t i f i c a t i o n of i n f i n i t e
contours as the i n t e r f a c e l i n e s .
I t s s t a r t i n g point This permits the
characterization of any given spin configuration as a patchwork of several pure-phase components.
The fact that l o c a l l y any spin configuration does occur even in the pure
phases has not been overlooked.
On general grounds, any choice of interface which
s a t i s f i e s the f o l l o w i n g three conditions is well j u s t i f i e d . I)
For a specified configuration of the i n t e r f a c e l i n e s , the d i s t r i b u t i o n of
spins in the regions they separate should resemble pure phases. 2)
The absense of i n t e r f a c e lines in the t y p i c a l configurations of a given Gibbs
state should imply that the state is a combination of pure phases. 3) either
No i n t e r f a c e l i n e s should be found in configurations which are t y p i c a l f o r ~+
or
~_.
Of main i n t e r e s t to us i s , of course, property condition. Condition
2), which provides a s u f f i c i e n t
However t h i s c r i t e r i o n would have been useless had
3) not been s a t i s f i e d .
3) would presumably prevent the use of contour surfaces as interfaces in
higher dimensions. The v e r i f i c a t i o n of the above three conditions f o r i n f i n i t e
contours is the f i r s t
step in the proof of the theorem, which is organized as follows: |)
I d e n t i f i c a t i o n of interface lines
2) Proof of the thermodynamic i n s t a b i l i t y of states with a single i n t e r f a c e l i n e , which is assumed to b a s i c a l l y f l a t . 3) Proof that in any Gibbs state the number of interface l i n e s is at most one, and that i f such l i n e e x i s t s , i t has the properties postulated in
2).
146 3.
B r i e f outline of the arguments A very useful tool in the study of support properties of Gibbs states is a gen-
e r a l i z a t i o n of the fact that in a f i n i t e region, extremal Gibbs states correspond to pure boundary conditions.
To formulate i t ,
let
~
be the ~-algebra of subsets of
(events) which are measurable at i n f i n i t y - e.g. the collection of spin configurations with exactly f i v e i n f i n t i e contours. Proposition I : i)
Any Gibbs state,
~(')EA(~),
can be represented as a convex combination of
extremal (non-decomposable) Gibbs states
v (.):
p(.) : ffi(d~)v (') ii) iii)
n EA(B) If
is either 0 or I.
is extremal~=~VA E ~ u ( A )
~ ( A ( B ) , AE ~
then
~ conditioned on A is also a Gibbs state, i . e .
~(" IA) EA(~). The geometrical elements in our analysis consist, in addition to the i n f i n i t e contours, of: + clusters (- clusters) -- the connected components, in the nearest-neighbor sense, of sites of the l a t t i c e on which the spins are
+l(or - l )
+ (or -) * clusters -- same as above with the extension of the notion of connectedness to diagonally neighboring sites.
(These appear naturally as the boundaries of
- clusters). Two key elements in the f i r s t step of the argument are the following support properties. Lemma l (Russo [lO]):
I f in a Gibbs state
i n f i n i t e + clusters then
n = u
~ E A(~) almost surely (a.s.) there are no
(assuming, of course, B > Bc).
Lemma 2 (Implied by a result of Russo[lO]):
In two dimensions, configurations with an
i n f i n i t e contour are a-typical ( i . e . their occurance has probability zero) in the states p+ and p_. Lemma l follows (for any d) from the fact that the absense of i n f i n i t e + contours implies that any f i n i t e volume is surrounded by a - * cluster; and from the Markov property of Gibbs states (some care has to be exercised to insure "non-anticipatory" conditioning).
Lemma l and proposition l imply that condition
interfaces is satisfied by the i n f i n i t e contours.
Condition
2) in our c r i t e r i a for 3) is d i r e c t l y implied by
lemma 2. Lemma 2 is proven by showing that (in d = 2) the i n f i n i t e + clusters of
~+ form a
mesh which completely surrounds any point, preventing the existence of i n f i n i t e -*clusters, and thus of i n f i n i t e contours. The second step in the argument is accomplished by proving the following lemma: Lemma 3:
Let
~l = { ~ ( ~ i ~ has exactly one i n f i n i t e contour,
@, intersections with
{i I = const.} }.Then
~(~), and ¥
~(~l ) = 0 for any Gibbs state
has f i n i t e , u(A(B).
147 Since the level of the lowest i n t e r s e c t i o n of fined variable on
y(a)
with
~I' no p r o b a b i l i t y measure supported on
T2 - - t h e t r a n s l a t i o n in the
i2
direction.
{i I = O}
is a well de-
Q1 can be i n v a r i a n t under
Using the decomposition to extremal states,
one may show that lemma 3 is equivalent to: Lemma 3": that
Let
~ be on extremal Gibbs state of the ferromagnetic Ising system, such
~(QI ) = I .
Then ~
is
T2 - i n v a r i a n t .
To prove lemma 3" one may consider pairs measure
u x T2u.
Since each
a~,
(a,a)E~ 1 x ~ I ' d i s t r i b u t e d by the product
has exactly one i n f i n i t e
contour, everv f i n i t e
volume may be completely surrounded by a connected set on which the spins are of a constant sign on each side of t u a t i o n s , t h e two contours infinitely
often.
y(a).
y(a)
The key observation now is that,due to l i n e f l u c -
and y(a)
would,~ x T2u - almost surely, i n t e r s e c t
Thus each volume may both be completely surrounded bv a set on which
a > a and by a set on which
a ~ a.
This implies that
u = T2~, by an argument s i m i l a r
to the one used in the proof of lemma I . The proof of step 3 would not be discussed here. • 4.
An open problem One aim of t h i s summary has been to expose some stochastic geometrical methods.
I t seems conceivable that a f u r t h e r development of these techniques could help to f i n d out whether in three dimensions the "roughening temperature", at which phase coexistence d e s t a b i l i z e s , is s t r i c t l y
below Tc.
References: I. 2. 3. 4. 5. 6. 7. 8. 9. I0. II. 12. 13.
R. L. Dobrushin: Theor. Prob. Applic. I__3, 197 (1968). O. E. Lanford I I I , D. Ruelle: Commun. Math. Phys. 13, 194 (1969). R. L. Dobrushin: Theor. Prob. Applic. 17, 582 (1972--)-. h. van Beijeren: Commun. Math. Phys., 4--0-, I , (1975). J. Bricmont, J . , L. Lebowitz, C. E. P f i s t e r , E. O l i v i e r i : Commun. Math. Phys. 66, 1 (1979). G__Gallavotti: Commun. Math. Phys. 2__77,103 (1972). D. B. Abraham, P. Reed: Phys. Rev. Lett. 33, 377 (1979), and D. B. Abraham, P. Reed: Commun. Math. Phys. 4__99,35 (1976). J. L. Lebowltz: J. Stat. Phys. I_66, 463 (1977). A. Messager, S. Miracle-Sole: J. Stat. Phys. 17, 245 (1977). L. Russo: Comun. Math. Phys. 67, 251 (1979). D. M e r l i n i : Preprint. M. Aizenman: Phys. Rev. L e t t . , 43, 407 (1979), and M. Aizenman: "Translation Invariance and I n s t a b i l i t y of Phase Coexistence in the Two Dimensional Ising System", to appear in Commun. ~ath. Phvs. Y. Higuchi: "On the Absense of Non-Translationally I n v a r i a n t Gibbs States f o r the Two-Dimensional Ising System". To appear in the Proceedings of the Colloquium on Random Fields (Esztergom, June 1979).
TOPOIfOG~ A~3 DIIHYL~S IN O~D~.ZD ~ D I ~
G. Toulouse Laboratoire de Physigue de l'Ecole Normale Sup~rieure 24 rue I~homond 75231 Paris Cedex 05, France
"~e • topological study of stable configurations is in principal applicable to a vast variety of problems.
So far it has been mostly developed for ordinary statis-
tical mechanics problems (this is the so-called theory of defects in ordered media) and for gauge field problems (menopoles and instantons). An ordered medium is tl_~ically a low t _emperature phase, with a broken si~metry, and characterized by an order parameter (e.g. a ferromagnetic p~hase). Besides uniform configurations in space of the order parameter, which are absolute ground states, there are distorted configurations which may happen to be stable, for continuous deformations, against return to uniform, states.
Among these stable distor-
sions, one makes a distinction between : i) singular configurations, or defects, with a "core" which may be of dL~ensionality 0 (~noints), 1 (lines), 2 (~mlls),... ii) non singular configurations, or textures. There are several metivations for the study of defects.
Their practical im-
portance in governing the rigidity and dissipative properties of ordered media (metallurg]; : plasticity ; magneti~n : hysteresis ; superconductors : high-field magnets, ...). _~ne clue that they provide to the nature of the order (cf the early observations by G. Friedel of defects in liquid crystals and the identification of the smectic order) . The possibility of predicting n ~ search of new materials.
nroperties, guiding thereby the
The role of the defects in the pblase transitions
with the concept of disorder parameter,
(relations
_speculations for new types of phase tran-
sitions,... ). The study of topological excitations can be divided into the following categories : -
one-defect theory_ (classification),
- two-defect t.heo~Z (combinations : coalescence, crossing), - n~ny-defect theory (phase transitions, hydrodyrm~-tics). Before the introduction of homotopic tools, there used to be a theory of defects, called here the traditional theory, which ~,~s an outgrowth of the consideration of crystal dislocations.
In brief, wit/%in the traditional theory,
i) a defect is characterised, fully and ever, by an integer number (or a set of such numbers), ii) to every defect corresponds an antidefect,
149
iii) combinations are ruled by addition of these nt~bers, iv) crossing (say, for lines) is permitted. This traditional theory %.~rks well in cases such as translation dislocation lines in crystals, or point defects in ferromagnets.
The convincing way to prove
that a theory is insufficient, and needs to be inloroved , consists in showing that it leads to paradoxes.
Such a paradox arises here, M%en one tries to extend this trad-
itional approach to point defects in ne~atic liquid c~zstals (I) . The paradox is insoluble wit~hin the traditional theory ; its origin lies in a noncommutative feature and its explanation requires homotopy theory (action of 71 on ~2). The homotopy theory of defects starts with the consideration of the manifold of internal states V = G/H, ~.~ere G is the symaetry group of the high t~.~perature phase and H is the symmetry group of the low temperature phase. r
(V) classifies the defects of di.~-~nsionality d' = d-r-l.
The homotopy group
Thus, in dimension 3,
~0 (V) ÷ walls, 71 (V) ÷ lines, ~2(V) ÷ points ; besides, ~3 (V) classifies the non singular configurations or textures. The Whitehead product (2) Tf
p
x
IT
q
÷
TIT
p+q-I
describes the noncc~nutative product of zl.
features.
If o = q =i, it reduces to t/he cc~mutator
If p = i, c~ arbitrary, it reduces to the action of ~I on ~q.
For
p,q >, 2, it gives to the ens~.~ble of homotopy groups, of a manifold, a graded Lie algebra structure. In the realm of gauge field problems, with G as a gauge group, the homotopy groups of G (or of
C/C, where C is t~he center of G) have been nut to use in order
to classify the topologically stable objects.
Thus in dJ_~ension 3 + i,
~I
+
monoDoles (via t~he flux strings attached to them)
3
÷
instantons.
and
In such simple frameworks,
(homotopy groups of a Lie group), which is all I have
seen so far (in the context of gauge field theories), the ~titehead product is disap[0ointingly vanishing. ~bncommutativity effects for defects in ordered media include coalescence effects (for instance, in 3D uniaxial nematics or in 2D m~ctics) ects (for instance, in 3D biaxial nematics).
Topological obstructions to crossing
bring topological rigidity and entanglement probl~s. confinement of linked rings.
and crossing eff-
Thus also a mec~hanism for
In all this, thec~y is now far ahead of ~ L T e n t s .
Sc~e develor~nents of t~his basic theoretical picture can be listed under the title of double topology.
~TLis includes the solitons of G. Volovik and V.P. 2~ineyev,
for systems with a ~nall perturbing energ!7 term.
%3~e absolute minL~a of the free
150
energy define a manifold V. manifold V'.
~eglecting the ~nall term, the ~minima define a larger
The Volovik-~iineyev solitons, non singular defects, are classified by
the relative homotopy groups ~r (V,V'). ~I (V,V')
÷
Thus in dimension 3,
wall solitons,
etc... Neat exanigles have been described for the superfluid phases of Heli~a 3, where the nuclear dipolar energy provides the small energy perturbation. Double topology includes alsc the study of cholesteric liquid crystals (4) , for which the ~hi.ghenergy cost of singularities in the ~olecular director field leads to a second topological constraint.
This leads to the stability, against un-
linking, of observed optical patterns of linked rings, a stability characterized by a non zero Hopf index. In systems with broken translation invariance, cor0plications arise in the topological classification, because Of the necessity of including metric considerations (V.P. Mineyev, N.D. ~rmin, V. Po~naru). In conclusion of this rapid su~Jey, we give a list of recent review papers (5) -
Z
_.References (i) (2) (3) (4) (5)
G. Toulouse, Proc. of the ~fI Karpacz Winter School (Springer, 1979). V. Po~naru, G. Toulouse, J. Physique 38, 887 (1977) ; J. ~ t h . Phys., 20, 13 (1979). G. Volovik, V.P. T~±neyev, Zh. E.T.F., 73, 768 (1977). Y. Bouligand, B. Derrida, V. Po~naru, Y. Pomeau, G. Toulouse, J. Physique 39, 863 (1978). L. ~[ichel, Proo. of the T[ibingen Conf. on Group Theoretical ~ t h o d s in Physics, 79, (Springer, 1978) ; Rev. Mod. Phys., to appear. N.D. Mermin, Rev. _~d. Phys., 5_~i,591 (1979). V.P. ~'ineyev, Soy. Phys. Uspek/li, to appear. V. Po~naru, Proc. of the 1978 Les Houches Summer School on "Ill-condensed _~atter", Eds. R. Balian, R. Maynard, G. Toulouse (_~brth-Holland, 1979).
DEBYE S C R E E N I N G I N C L A S S I C A L
.MECHANICS
STATISTICAL
David Brydges, U n i v e r s i t y of V i r g i n i a Paul Federbush, U n i v e r s i t y of M i c h i g a n
Two years ago David Brydges,
using techniques d e v e l o p e d
in C o n s t r u c t i v e Q u a n t u m Field Theory,
r i g o r o u s l y proved Debye
s c r e e n i n g for c e r t a i n C o u l o m b lattice systems in the low density region
[i].
The p r e s e n t talk presents
joint w o r k
[2] w i t h
Brydges g e n e r a l i z i n g his results from the lattice case, to continuum statistical mechanics;
from the pure i/r interaction,
to
include the p r e s e n c e of e s s e n t i a l l y a r b i t r a r y short range forces; and from a c h a r g e - s y m m e t r i c species.
situation,
to cover a r b i t r a r y charge
The short range forces are r e q u i r e d to ensure the stability
of the system situation),
(the pure i/r p o t e n t i a l is u n s t a b l e in the c l a s s i c a l
and also to fall off q u i c k l y enough at infinity so
as not to i n t e r f e r e w i t h the shielding.
Rather
d e t a i l these two very p h y s i c a l conditions,
then to here
we present a special
case i n d i c a t i v e of the general result and of interest in its own right. We c o n s i d e r a system of charged, R , so that the p o t e n t i a l
V
I~[ V =
J
e.e. l
1
e = +i
, spheres of radius
is given by
÷
÷ ÷ Ixi-xjl
÷
_> 2R
all i ~ j
i~j 4~,Xi-Xj 1 1 1 (i)
,
otherwise
The grand c a n o n i c a l p a r t i t i o n f u n c t i o n is then Z =
[
1 N 1 ' zN++N-f d3Xl" " .fd3XN++ N e-SV(xi'ei) ÷ N+'
N+,N_
(2)
-"
we are d e a l i n g w i t h the charge symmetric situation,
where
z .
"
The Debye
length,
iD ,
is defined as equal
a l i m i t i n g p r o c e d u r e to be explained,
z
= z_ =
(2z8) -I/~. By
infinite v o l u m e e x p e c t a t i o n
values of p r o d u c t s of smeared density operators are defined. simplicity,
point f u n c t i o n
(rather than an a r b i t r a r y product).
We i n t r o d u c e d i m e n s i o n l e s s p a r a m e t e r s
E 1 = 8/R h(x)
THEOREM.
and a f u n c t i o n
Given
~, 0 < ~ < i, there is
0 ~ ~ < ~, f
~2 < f~(~l )
(~) > 0 , f
c
decreasing,
and
and
A. , i = 1,2 are a r b i t r a r y unit cubes, and 1 functions of a b s o l u t e value less than 1 .
f~(<),
For
we will state our m a i n c l u s t e r i n g t h e o r e m for the two
~2 = zR3"
g(x)
such that for
are
152
I/ d3x I f A1
d3x2g(Xl)h(x2 )
A2 C
d
choice
of
g,h,
or
A1 Ai
the density is low enough, the
limiting
(3)
e
is the d i s t a n c e b e t w e e n
on t h e
< J ( X l ) J ( x 2) > I <
-d (AI,A2) ~/£ D
procedure
and
A 2 , and
Thus
at
any
c ,f
fixed
do not d e p e n d
temperature,
there is Debye screening.
we d e f i n e
VL
if
To specify
as
e.e. -I xi-x j I/hiD) VL : 12 .~. +l 3÷ (l-e 1,3 47[xi-x j 1 l
is a small parameter.
V
4)
is chosen so that S
5)
V = VL + V s We see that i (i _ 1 V L = ~ / J ( x I) _A 0 _ A 0 + i / 1 2 £ 2 ) J ( x 2)
where
A 0... is the free L a p l a c i a n on
define
~.
as
R3 .
6)
For a large box
(6) w i t h the integrals r e s t r i c t e d to
A
A
, we
, and w i t h A
r e p l a c e d by the L a p l a c i a n satisfying D i r i c h l e t data on
~A
A'
limit by
a box c o n t a i n i n g
A , we o b t a i n our infinite volume
With
p e r f o r m i n g the following three steps in order. a)
C a l c u l a t i n g e x p e c t a t i o n values for the grand c a n o n i c a l
ensemble defined in volume b)
Letting
A'
c)
Letting
A
A'
, for p o t e n t i a l
+ Vs
become infinite.
a c o n d i t i o n on the fugacities: Vs
t
become infinite.
If we were dealing w i t h a non c h a r g e - s y m m e t r i c
fugacitie s and
=
¢
alone,
system we w o u l d impose
that the system d e s c r i b e d by the same
should be neutral in the limit.
The m o s t
i m p o r t a n t result of the limiting p r o c e d u r e p r i m a r i l y due to the. "grounding"
on ~A
and the c o n d i t i o n on fugacities,
a layer of surface charge simple p h y s i c a l arguments, volume
limit).
(whose presence,
as one can estimate by
can lead to effects s u r v i v i n g the infinite
We do not here discuss how small
We feel the t r e a t m e n t of Debye screening in but we m e n t i o n three problems l)
is the absence of
left open
l
m u s t be chosen.
[2] is quite complete,
(among m a n y others).
Can one g e n e r a l i z e to the q u a n t u m s t a t i s t i c a l situation?
This c o m p l e t e l y eludes us. Since in
[3] and
[4], Brydges and I have
handled the short d i s t a n c e d i f f i c u l t i e s of the C o u l o m b system, we are p a r t i c u l a r l y frustrated.
153
2)
Can one use a more natural limiting procedure?
particular,
In
can one e f f e c t i v e l y treat situations in w h i c h there is
a surface charge? 3)
In a system w i t h charges and fugacities
(ei,z i) we have
imposed the c o n d i t i o n zi/max z i h c > 0
(7)
1
Our e s t i m a t e s are not u n i f o r m as extremely
c ÷ 0, if one species is of
low density c o m p a r e d to the other species,
charge compared to the other species. to
el,e2,...,e r
if
ef
(ef
and of fractional
is fractional w i t h r e s p e c t
is not a m u l t i p l e of g.c.d. (eI .... ,er).)
have p r o v e n that f r a c t i o n a l charges are shielded! that our limiting c o n d i t i o n
We
This may suggest
(7) is m e r e l y a w e a k n e s s of our procedure,
and not a p h y s i c a l requirement. Two of the basic ingredients of our proof are the same as in
[i],
the use of the Sine G o r d o n t r a n s f o r m a t i o n and the G l i m m - J a f f e - S p e n c e r cluster expansion.
The basic cluster e x p a n s i o n is fully p r e s e n t e d in
[7], and the p a r t i c u l a r a p p l i c a t i o n to situations w h e r e it is combined w i t h a Peierls e x p a n s i o n first d e v e l o p e d in basic i n g r e d i e n t
The third
is the Mayer e x p a n s i o n used to handle the effects of
the short range p o t e n t i a l
V
S
P u r s u i n g the d e c o m p o s i t i o n of ing discussion,
[8].
we w r i t e
e -BV
-~v s e
in
V
given in
(4) and the succeed-
(2) as
-Bv~
(8)
e
and use -BV~ e
iBl/2 [ ei(e)
~(x ) (9)
= /dz0(~)e
_! [¢ [~2~(_A) 2+(_A)]~ d~o(%)
~
d~e
2
lO)
is a n o r m a l i z e d G a u s s i a n m e a s u r e W i t h inverse c o v a r i a n c e a ~0 fourth order d i f f e r e n t i a l o p e r a t o r as given in (i0). A l t h o u g h for this r e a s o n
some of the t e c h n i q u e s of E u c l i d e a n Q u a n t u m Field
T h e o r y are not a v a i l a b l e to us, we have gained the a d v a n t a g e that there are no r e n o r m a l i z a t i o n difficulties;
no expressions
in
~
need
be normal ordered. Inserting
(8) and
in a c o m p a c t notation)
(9) into
(2) we find the following sum
still to be i n t e g r a t e d over
(written
154
N~ zNIe-6Vs
e
i~1/2 ~ei(a){ (xa)
(ii)
This sum may be e x p r e s s e d as the exponential of a Mayer series, M, the Mayer series d e v e l o p e d for p o t e n t i a l s as a p p e a r i n g in w i t h the o n e - b o d y potential,
together s Up to a constant term
-iB -I/2 ei%(x).
V
M is given by the sum
+ "~'.w [
[ fPi(X) si(x) i with
i~i/2 E.(X) 1
Here
Pi,j .... (
the p o t e n t i a l s
+
] Pi,j ( x , Y ) S i ( x ) 8 j ( y )
(12)
....
i,j
=
e i %(x)
e
- 1
(13)
) is a truncated c o r r e l a t i o n function d e v e l o p e d for
in
alone. To control the Mayer series expansion, s fall-off p r o p e r t i e s of the P's are required. These may be o b t a i n e d from results in
V
[6], or alternatively,
tions on the short range potentials, using the f o r m a l i s m of
w i t h slightly d i f f e r e n t condi-
from results in A p p e n d i x 1 of
[2]
[5].
The following a p p r o x i m a t i o n is the m o t i v a t i o n behind m u c h of our procedure
2p~
[COS(6 I/2 %(x) )-l]
e
~ [
~
-
2p6 T u/
({(x) - n W
)2
e
(14)
n=_CO
with
~
a cube,
20{
T = 2~/B I/2.
We write
2p5 2 {(*(x) - h ( x ) ) 2 = ~ e eG (15) h is constant on each cube of a lattice of cubes filling A ,
[COs (~ll2 *(x) - i]
e where
and
h(x)
and assuming values integral m u l t i p l e s of (14) were an equality.
Equation
Y .
G
w o u l d equal
(15) is our Peierls expansion.
0
if Unfor-
tunately we do not have a p h y s i c a l u n d e r s t a n d i n g of the Peierls expansion in (15). C o l l e c t i n g the fruits of our efforts and w i t h some further manipulation we arrive at Z = [ N/dp(~)eE eG e R h 2 (-A)2 D C-A) ave
-i/2I~[12£
d~(~)~ p(~)
+ ~2 +
~]~
D
(17)
is a n o r m a l i z e d Gaussian m e a s u r e whose c o v a r i a n c e may be read
off from with
+
(16)
1
(17).
g(x)
e x p o n e n t of
~(x)
chosen to
is a linear translate of (at least approximately)
(15) linear in
#(x).
ables in the G a u s s i a n integral,
R i/£
~(x),
~(x) = ~(x)-g(x),
cancel the terms in the
arises from the change in vari= 2P6; this term in
(17) arises
155
from the e x p o n e n t i a l terms in
(15) q u a d r a t i c in
term arising from terms in the e x p a n s i o n for is the c o n t r i b u t i o n of all terms in effect is given by less n o r m a l i z a t i o n a function of
h
That d e v i a t i o n s of
(15), and the factor. alone,
In
M
~ is a "small"
quadratic
in
E
E
N and
is a r e l a t i v e l y harmG
were zero, and
R
it w o u l d be easy to show screening took place.
E, G, and
R
from these ideal values are "small"
in a suitable range of parameters,
that screening occurs,
is finally
proven by a p p l y i n g the m a c h i n e r y of the cluster e x p a n s i o n to We feel the inexorable
~ .
except the first term, whose
~ term.
(16) if
#(x).
M
(16).
i n e l u c t a b l e t e c h n i c a l details of this process
are treated quite e f f i c i e n t l y in
[2].
REFERENCES i.
D. Brydges, A r i g o r o u s a p p r o a c h to Debye screening in dilute C o u l o m b systems, Commun. Math. Phys. 58, 313-350 (1978).
2.
D. Brydges and P. Federbush, sity of Michigan.
3.
D. Brydges and P. Federbush, The cluster expansion mechanics, Commun. Math. Phys. 49, 233-246 (1976).
4.
D. Brydges and P. Federbush, The cluster e x p a n s i o n for p o t e n t i a l s w i t h e x p o n e n t i a l fall-off, Commun. Math. Phys. 53, 19-30 (1977).
5.
D. Brydges and P. Federbush, A new form of the M a y e r e x p a n s i o n in classical statistical mechanics, J. Math. Phys. 19, 20642067 (1978).
6.
M. Duncan, D. Iagolnitzer, and B. Souillard, Decay of correlations for i n f i n i t e - r a n g e interactions, J. Math. Phys. 16, 1662165'6 (1975).
7.
J. Glimm, A. Jaffe, and T. Spencer, The p a r t i c l e structure of the w e a k l y coupled P(~)2 model and other appl~ications. In~." Lecture notes in physics, V ~ 2 5 ~ d ~ . G~. ~ Aq. ~-~ghtman), B e r l i n - H e i d e l b e r g - N e w York, S p r i n g e r 1973.
8.
J. Glimm, A. Jaffe, A c o n v e r g e n t e x p a n s i o n about mean field theory I and II,Ann. Phys. i01, 610-630, 631-669 (1975).
D e b y e screening, preprint,
Univer-
in statistical
EQUILIBRIUM
OF
CLASSICAL
LONG
RANGE
FORCES
SYSTI~MS
Ch. Gruber, Ph.A. Martin de Physique Th@orique, Ecole Polytechnique CH-]001 Lausanne, Switzerland
Laboratoire
We discuss acting with Details can noting by q 7vdq... =Tdx
PROPERTIES
WITH
F~d~rale,
general properties of infinite classical systems of particles interlong range forces from the view point of equilibrium equations. be found in [ I, 2, 3, 4]. We consider N types of particles in R ~, de= (x,o) the position x ¢ R v and the charge ~ of a particle with ~ .... VcR~.
The particles interact with a two-body force F(ql,q2)= ~o2F(x2-xl) form background of fixed negative charge F(I) (q) = -pB~. ~ F (x-y)dy,
and a uniPB >~ 0. F(x)
is assumed to be bounded and C I everywhere, except possibly for an integrable singularity at the origin; there is no restriction on the decay of F(x) at infinity. Our discussion includes the Coulomb force, F(x)=~V, x and jellium models with PB ~ 0. We adopt as a definition of equilibrium states of infinite systems the following BBGKY hierarchy for the correlation fhnctions p(n)(q1"'" qn), n = I, 2 ..... p is an equilibrium state with respect to the temperature ~-I, the sequence of finite space regions {V~} and the effective field E if its correlation functions salisfy : (n)
(I) L I -clustering:
~ dxl PT R~
(X1~1
"'" Xn~n) l < oo
(I)
(P (T n) (ql • " • qn ) are the truncated correlation functions) (2) The B B G K Y
(equilibrium) hierarchy:
Vlp(n)(q1"''qn ) = P {(~IEp(X I) + j~2F(ql,qj) }p(n)(q1"''qn ) (2) + p,fdqF(ql, with E~(x)=E+Vk_~Rvlim and
C
q)Ep(n+l)(ql
--. qn q) -P(n)(ql "'" qn )p (1)(q) ]
--$V dy(F(x-y)_F(_y))Cp(y)dy
(x) = E ~ p(1)(x~)-p
B is the charge
density
in the state'
P This
definition
deserves
the following
comments:
(a) The hierarchy applies to extremal states of the infinite system having at least the L I -clustering property (I). (b) E (x) is the effective field in the state p and the parameter ]E =Ep (o) is its v~lue at x = o (in the Coulomb case E$ (x) is the solution of the electrostatic equation v -E (x) = C (x) with the condition E$ (o) = E). (c) If the force is not integrable in R '~, the sequence V X -* R ~ defines Ep(X) as a limit of definite integrals. (d) If the force is integrable in R ~, the BBGKY hierarchy reduces to the usual form it takes for short range forces. We
present
now
the properties
of equilibrium
states
defined
as solutions
of (i)
157
and (2) (assuming their existence). First, of such states in one space dimension.
A.
Fxamples
of
equilibrium
there
are
states
explicitely
known
examples
[I]
(I) The one dimensional Coulomb-jellium has non trivially periodic eq. stateswith E~(x) = 2 -_J'Xcp(y) dy +I~. o (2) The one d i m e n s i o n a l two c o m p o n e n t C o u l o m b gas has an eq. state i n v a r i a n t u n d e r t r a n s l a t i o n s and c h a r g e c o n j u g a t i o n with EO (x) = E = 0 [ 5].
(3) The one d i m e n s i o n a l two c o m p o n e n t C o u l o m b gas has also t r a n s l a t i o n inv a r i a n t eq. s t a t e s , but not i n v a r i a n t u n d e r c h a r g e c o n j u g a t i o n with non v a n i s h i n g effective field Ep (x)= E ~ 0 [ 6 ] .
B.
Shape
dependence
[2,§3]
A state verifying (2) may depend genuinely on the sequence of cut offs V k entering in (2). This cannot occur if the force decreases faster than the Coulomb f o r c e . P r e c i s e l y , let {Vk} = {~Vo} s e q u e n c e of dilated r e g i o n s and PEVo a M s o l u t i o n of (2) with r e s p e c t to E and {kVo}. If F(x) ~ - ~ with v > ~ - ] , then P E V o is i n d e p e n d e n t of Vo .
C.
Local
neutrality
[2, §4]
Let ~ be a discrete subgroup of the translations and 4 o the corresponding fundamental cell. If the force decreases as or slower as the Coulomb force, any T invariant eq. state is locally neutral. That is, if lira XV F(X2)= d (£) ~ 0 x (£=~-~-) with V-< ~J-I, then the average charge of the fundamental cell must
v a n i s h , ,;
Cp (x) dx = 0.
~o
D.
The
canonical
sum
rules
~2, §5]
The canonical sum rules are integral (n+1)-point correlation function: (~ ~i)~ (n)(ql...qn)= i=1
- ~dqc~
relation
that link the n-point
to the
(0(n+l)(ql...qnq)-p(n)(ql...qn)p(1)(q))
One has to note that such relations hold trivially true for a system of N types of particles in a finite volume canonical ensemble, irrespective of the nature of the force. For infinite systems, we have the following theorem: the canonical sum rules hold true if the truncated correlation functions decrease faster than the force, ie if PT =°(F) as I x[-,~ (for a precise statement see prop. 8 of ref. [2] ). We ernph~:size the following points: - ~e sum rules hold for Coulomb of L I- clustering). - Since the sum rules do not hold
systems
when
for infinite
screening
free
(or short
occurs range
(in the force)
sense
158
systems, they have to be seen as an essential characteristic of Gibbs States of long range force system. The validity of the sum rules has been noticed by Dyson and Mekta [ 7, 8] in their analysis of the logarithmic potential.
E.
The
charge
As
a consequence
fluctuations
[3]
of the first sum
rule
one
deduces
that the charge
fluctuations
<'(Q~- are non extensive as A-~ R~, ie Alj~v ~-~ <"(QA-)2> is of the order of the surface IbA of the boundary of A [3, §3]. If the state obeys the sum rules and has suitable clustering properties, one can prove a central limit theorem for the probability distribution ~P~I of the 1 normalized charge Qo= 7-z-7-r-(QA- (QA >) with respect to the surface, in dimension v = 2, 3. In one dimenSiOn°,AI PQI converges to a discrete distribution [ 3, § 4]. F. P a r t i c l e
fluctuations
in one
component
systems
When N = I, QA is identical with the particle number Nh. Thus rule holds, the bulk compressibility X =A_~R~)lim~ <(N A-
[2, § 6]
if the first sum vanishes. From
this, one deduces that the truncated function p{_~ of a compressible fluid J~-]2 = 0) cannot decrease faster than ~ f-o--rce'.l"ff iF did, then the first sum rule' woufd h~ implying X-~. But X~0 in the analyticity region, a contradiction! Under the supplementary assumption I xl~ff.l = O(~.) and with Groeneveld result [ 9] p~ decreases exactly as the potential itself. On the other hand, in ~' T the one component Coulomb plasma, the sum rules hold (because of screening) and thus, the particle fluctuations are not extensive (X = 0) in such systems.
G. Equation Several
of
definitions
state
for
one
of the pressure
component are
a priori
thermal: p(@)= ~lirn ~-I ~-l in Z(V~,~,N, NB)
systems
[4]
possible:
NB= ~BVx
mechanical: p(m)= ,lira ~-I ~ in Z(Vk, ~, IN,PB) ^-, oo o,x
I kinetic: p(k)= ~irn ~] ~-I i vxl
!vx p(]) vx (x) x do
>I 0
It is p(k) w h i c h is e q u a l to the v i r i a l p r e s s u r e and not p(O) a s it a p p e a r s often in the literature. For a fluid (PB- 0J these definitions are equivalent, but they are not if fi~ > 0; it is argued in [ 4J that the correct equation of state is defined by ~p(k) and it is shown that the Coulomb force appears again as the border line where new properties appear.
159
References [1] Ch. Gruber, Ch. Lugrin, Ph.A. Martin, Helv. Phys. Acta 51, 829 (1979) [ 2] Ch. Gruber, Ch. Lugrin, Ph.A. Martin, Equilibrium properties of classical systems with long range forces, to appear in Journ. of Star. Phys. [33 Ph.A. Martin, T. Yalcin, Charge fluctuations in classical Coulomb systems, to appear in Journ. of Star. Phys. [4] Ph. Choquard, P. Favre, Ch. Gruber, On the equation of state of classical systems with long range forces (Preprint) [5] S.F. Edwards, A. Lenard, Journ. of Math. Phys. 3, 778 (1962) [ 6] M. Aizenman, Ph.A. Martin, in preparation [7] F. Dyson, Journ. of Math. Phys. 3, 166 (1962) [8] F. Dyson, M. Mehta, Journ. of Math. Phys. 4, 701 (1963) [ 9] J. (l~oeneveld, in Statistical Mechanics, foundation and application, Ed. A. Bak, Benjamin, N,Y. (1967)
ON THE LACK OF FR~CHET DIFFERENTIABILITY
IN MORE PHASE REGIONS
H.A.M. Dani~is and A.C.D. van Enter Institute for Theoretical Physics, University of Groningen, The Netherlands
Abstract: For the version of the Gibbs phase rule proposed by Ruelle [1,2], Fr~chet differentiability
of the pressure is essential.
pressure is not Fr~chet differentiable sional Ising model below T . c Consider the thermodynamical
pressure as a function on interaction spaces. We
prove that for the 2-d Ising model below T in the space of pair interactions
In more phase regions however the
in general. We illustrate this on the 2-dimen-
the pressure is not Fr~chet differentiable
c
(for the more general case see [3]). This gives a
negative answer to the question raised byRuelle[1,2] + Proof: Let p and p be the extremal pure phases
whether his condition (R) holds. +z and p (s o ) = m > 0 (the magneti-
zation). Divide ~ 2 in layers L Ni of thickness N(L Ni ={(nl,n2)]Ni i n I < N(i+l)}). Define R N by: RN s~3 = s~j
V j 6 LN'2k+I
for any k
R N s~3 = -s~J
V j 6 L Nme
for any k
Define ~N by ~N (A) = 2~ SoL
2N E p+(RN~ i=l
T i A).
is ordered in layers and is also translation invariant. We denote the entropy
per lattice site of the system in a state p by s ( p ) . We prove t h a t s(~ N) = s(p +) [3]. Denote by ~N the antiferromagnetic
interaction of strength one which acts between
two
spins at distance N. Take ~ > 0. From the variational principle P(~Is+e~N) = sup s(0) 0El The fact that O
-
[4] we obtain :
P (A~is+~ N) ~ s(~N ) - ~N(A~I s ) - e~N(A~N ~
(I)
+ . is an e q u i l i b r i u m s t a t e for ~Is implies [4]:
+ P(~Is ) = s(p +) - p (A~
) .
(2)
Is Combining
(I) and (2) we obtain:
P(~Is+ePN ) - P(~is ) + cp+(A~N) g
-gI >-~ --
s(
) -
S(P+)
-
(AOls
) + 0
(A~
Is
) -
161
N + ~ m2 - CPN(Ap N) + cP (A~N) S ~ 2m2 - N ~ ll~IsII > if N large enough. Thus P is not Fr~chet differentiable and also not analytic (compare [5]) in the space of pair interactions.
[]
By making use of a theorem by Israel [6] it can be shown that some interaction in the neighbourhoods proposed by Ruelle has an equilibrium state
which is ordered in lay-
ers. Hence to prove a version of the Gibbs phase rule in infinite dimensional interaction spaces, some conditions on the decrease of the interaction are necessary.
This work is part of the research program
of the "Stichting F.O.M.".
I
D. Ruelle: Commun. Math. Physics 53 (1977) pp. 195-208.
2
D. Ruelle: Theor. and Math. Physics 30 (1977) pp. 24-29.
3
H.A.M. Daniels and A.C.D. van Enter: to appear in Commun. of Math. Physics.
4
D. Ruelle: Statistical Mechanics
5
D, lagolnitzer and B. Souillard: Commun. of Math, Physics 60 (1978) pp. 131-152.
6
R.B. Israel: Commun. Math. Physics 43 (1975) pp. 59-68.
(1969) Benjamin.
LONG TIME TAIL FOR S P A C I A L L Y
Herbert Theor.
Physik,
Universit~t
The Lorentz through rers.
gas c o n s i s t s
in R 2 r a n d o m l y
The Lorentz
bution
correlation gives
of m a g n i t u d e
not to o v e r l a p
function
the d i f f u s i o n
constant
sufficiently
fast decay
indicate
a power
To u n d e r s t a n d an old recipe replace
bility with
Z d there
in the d i r e c t i o n tocorrelation
to be difficult. gin there ting
it came
at the origin,
implies
exponential mensional
performs
Together
with model.
at each site
At a site
continues
This
that at each site except
Since r(t)
= -r(t) return = t -3/2
for this model.
distribution I studied
is c o n t i n u o u s
star-
+ to the for long A sym-
~ -t -(d/2+I)
from a s p a c i a l l y
H. van B e i j e r e n
the ori-
from paths w h e r e
of a first
that
inhomo-
gives
an
the one di-
and the d i s t a n c e s
~. b e t w e e n scatterers are i n d e p e n d e n t l y and i d e n t i c a l l y distributed. 1 ~ Z t < small z we p r o v e o / d t e v(t)'v(O)> = (1/2)<~0> + /z<(~O-<~O>)2>/ /8<£
> + O(zI/2+~),
than°t-3/2.
5>0.
The n e g a t i v e
time tail
au-
seems
for the p a r t i c l e comes
to
with proba-
to prove that the v e l o c i t y
has a long time tail.
A periodic
Time
I use
I/2d in one of
the p a r t i c l e
long time tail
of scatterers.
scatterer
Then
with p r o b a b i l i t y
proves
re-
advices
a r a n d o m walk:
The long time tail o r i g i n a t e s
decay.
one.
which
p a scatterer,
r a n d o m walk.
due to M. A i z e n m a n
random
mechanics
by a stochastic
the e x p e c t e d
distribution
and n u m e r i c a l
at time t, < v ( t + 1 ) . v ( 1 ) >
this
is to
from a
of the long time tail
is the p r o b a b i l i t y
at time t for the simple
follow
v(t)
+ I) in d dimensions.
in one dimension,
times,
geneous
Theoretical
since the only c o n t r i b u t i o n
r(t)
the
One deep p r o b l e m
which would
statistical
Suppose
origin
argument
gas.
is finite,
One w a n t s
Then
is at the o r i g i n
in d dimensions.
process
continues
from.
To simplify:
(r~r) (t) - ..., w h e r e
metry
over
is a scatterer.
the p a r t i c l e
integral velocity
At a site w i t h no scatterer
function
are in e q u i l i b r i -
The time
The p a r t i c l e
the p a r t i c l e
scatte-
a uniform distri-
of the s t a t i o n a r y
the origin
motion
2, FRG
moving
h e a v y hard disk
The hard disks
is with p r o b a b i l i t y
1-p no scatterer.
a scatterer
one.
of .
the d e t e r m i n i s t i c
the 2d directions.
infinitely
8 MOnchen
particle
at the o r i g i n with
law d e c a y as -t -(d/2
qualitatively
37,
classical
of the Lorentz
constant
in n o n - e q u i l i b r i u m
of the lattice
Spohn
the origin.
show that this d i f f u s i o n
sults
starts
R A N D O M WALKS
Theresienstr.
of a single
distributed,
particle
of v e l o c i t i e s
um c o n d i t i o n e d
MHnchen,
INHOMOGENEOUS
cannot
decrease
For
faster
WHY IS THERE A SOLID STATE? Charles ~athematics University Austin, We Consider
78712/USA
(called the "crystal
system of point particles
tial energy when interacting the particles
of Texas
TX
the problem
ing that for a finite
Radin Department
through Lennard-Jones
must lie approximately
the approximation
becoming
joint papers,
exact as the particle
in the Journal
respectively
poten-
of a lattice
with
number diverges.
progress
on this old prob-
of Statistical
with C.S. Gardner,
of show-
type potentials,
on the vertices
We report here the first nontrivial lem -- to be published
problem")
to have minimum
Physics
G.C. Hamrick
in three
and R.C.
Heitmann. In the first dimension, highly
but only for the Lennard-Jones
problem in one space
potential
itself.
In the second paper we demonstrate
that the crystal
sensitive
perturbation
(at least ground
paper we solve the crystal
to physically
in one dimension):
states,
a potential
and a family of perturbations
, such that the perturbed for all
negligible
we exhibit
problem is
of the potential
with only lattice
physically
system has only nonperiodic
small of order ground states
~ ~ O. In the last paper we solve the crystal problem
sions but only for the "sticky disk" delta function geometrical
attraction
one:
pairs of disks, Extension
disks
configurations,
Finally we consider long range orientational
of a finite number
the possibility
order at nonzero
of touching
on a triangular
to longer range potentials that
with
The problem is a
in the plane with the most number
must have the disks centered
of the result
i.e. hard core,
at the hard core radius.
to show that those
of nonoverlapping
potential:
in two dimen-
lattice.
is expected.
sticky disks
temperature.
exhibit
SOME REMARKS ON THE SURFACE TENSION. J.
Bricmont
J.L.
Lebowitz
C.E. Pfister Department Rutgers
* +
** +
of mathematics
University
New Brunswick N.J. 08903
* supported + present
in part by N.S.F.
address:
University,
Math.
Grant Phys77-22302
Dept.
Princeton N.J.
USA
Fine Hall,
Princeton
08544 USA
** supported by the Swiss Foundation
of scientific
research + present
address:
E.P.F.-Lausanne,
We consider
the surface
nearest neighbour [i]
tension
where
B-it
ferromagnetic
Lausanne
for the d-dim.
coupling
Ising model with
J > 0. T is defined
in e.g.
and
+
boundary
sJ, 0 & s & 1 -
(i)
at the origin of height
2L+I. The indices conditions
We do the following
=
ZA'~ 1 lim log ZA ' M+~ +
A is a box centered
with a base of side
iI
CH-IO07
de math~matiques,
:
1 0 & T = - lim - - d _ L÷~ (2L+I)
+
D~partement
+
and
construction:
T(s)
refer to the usual
(b.c.).
we replace
for all nearest neighbour pairs
I- By symmetry
+
2H in the il-direction
is zero for
the couplings
crossing
s = 0
and ~ = T(1)
J by
the plane is equal
to rI
T = kJ 0 where
Ps+
p~
+
ds (Ps (~0~-i)
-
(~0~-i)
)' BJ = k
(Z)
S
(p
±)
is the corresponding
infinite volume
Gibbs state with
s +b.c.
(~b.c.)
(2) is j u s t i f i e d by c o r r e l a t i o n i n e q u a l i t i e s and using
such inequalities
we prove that T = 0
whenever
there
is no spontaneous
165 magnetization. tization
We can also prove
that T = 0
is zero for the semi-infinite
{i e ~ d: il > 0}
with free b.c.
(2) and correlation
T ~ 2k
inequalities
11 P2(O0 ) Psa
at
if the spontaneous
system iI = 0
we obtain
defined
magne-
on
and +b.c.
elsewhere.
From
the lower bound
(a0) ds
0 Using T(k)
duality = m(k*)
correlation where
and correlation where
length
and
in the state with free b.c.
in the Ising gauge model
d=3
we prove that for d=2
k* is the dual temperature
a is the coefficient
For
inequalities
and
k
m
For
is the inverse d=3
~(k)
= ~(k*)
of the area law decay of the Wilson
loop
[2].
large enough we prove
that
x - 2k
and the cor-
+
relation
functions
in the state
more we have the Gibbs
exp(-2k).
Further-
formula
(p+(oi ~]) - p-+(oioj) )
il--~ i2=i3=0
[2]
in
+co
dT dk
see
p- are analytic
and
j : li-j I=l
[3].
References: i)
Gruber math.
2)
C., Hintermann Phys.
Bricmont
J., Lebowitz
for lattice International Annals 3)
A., Messager
A., Miracle
Sole S
Commun.
56 147(1977)
systems.
J.L.,
Pfister
To be published
Conference
C.E. On the surface in the Proceedings
on Collective
of the New York Academy
Bricmont
J., Lebowitz
1 and 2
1979 and to appear
J.L.,
Phenomena
of Sciences
Pfister
tension
C.E.
in Commun.
of the 3 rd
Moscow
1978.
1979.
Commun.
math.
math.
Phys.
Phys.
1979.
66
ANALYTIC STRUCTURE OF GREEN'S FUNCTIONS IN QUANTUM FIELD THEORY by J. Bros
DPhT, CEN Saclay, BP n°2, 91190 Gif-sur-Yvette, France A tentative review of recent results obtained by several authors will be presented. They mainly concern the derivation of anaiyticity properties for the Green's functions and collision amplitudes of the 2 ÷ 2, 2 ÷ 3, 3 ~ 3 particle processes from the general principles of Q.F.T. ; emphasis will be lald on the exploitation of asymptotic completeness which generates the monodromic structure of Green's functions. In this connection, the role played by "generalized Bethe-Salpeter kernels" will be described.
INTRODUCTION In ~he present situation of Theoretical Physics where models of 0uantumFieldsof numerous kinds are under study, bringing with them lots of new ideas and new aspects of 0uantum Field Theory, we believe that understanding the analytic structure in complex momentum space of Green's functions of local fields and of the associated nbody relativistic scattering amplitudes remains an important objective of O.F.T.. At least, this looks justified as far as one believes that fields with asymptotic particles are relevant mathematical concepts and that analyticity remains a basic language for describing certain aspects of the reality of particle physics. In this lecture, we try to make a review of the methods which have been developed and of the results which have been obtained recently in the axiomatic approach of the analytic structure of Green's functions and scattering amplitudes.
From the various recent developments which we shall present here, we wish to emphasize
the following aspects :
i) The use of A6~ptO£ge Compl~ene6s
(in short : A.C.)-considered as an extra=
axiom is a powerful tool for introducing and investigating the monodromic structure of Green's functions around Landau singularities in the general analytic framework implied by locality and spectrum. Ityields a scNemefor anexaet (i.e. nonperturbative) treatment in large complex regions,of the singularity structure of Green's functions (Note that up to now, such a general treatment had only been proposed in the pureS-matrix approach [I], and has only been developed consistently in the neighbourhood of the physical regions [2]). ii) The fact that certain methods 6f the axiomatic
scheme, based on analyticity
properties, may implement important steps in the study of models. As an example the analytic continuation of structural equations of the Bethe-Salpeter type (see Part C II below) is a tool which can be used either in working out the consequences of A.C., considered as an axiom, or in studying the validity of A.C. in a class of models of 0.F.T. (in the case of P(~)2 models, see in this connection [45],[|7]~ As a guide for understanding the results which we describe here, the reader is invited
to refer frequently to Table ! which represents in short the axiomatic
167
scheme of local O.F.T. in complex momentum space. TABLE 1 THE AXIOMATIC SCHEME OF Q.F.T. IN COMPLEX MOMENTUM SPACE i
Axioms for euclidean Green's functions (Extended form)
1
~ ^ ~
i
I
Axioms for relativisAx: tic ti, local quantum fieldsl Extended form) " ! (E:
i
J
The Holomorphic Green's functions
Asymptotic theory S-~natrix reduction formulae
[ ~__
H(n)(kl . . . . . k n)
1
I
Axiom of Asymptotic I completeness
.
Global analyticity properties ) for 2 + 2 scattering amplitudes and partial waves Some analyticity properties for m ÷ n seatterin~ amplitudes.
,
i
Asymptotic completeness c equations for the H (n)
C-I
C-II~ Further analytic continuation of the H (n) -Local properties : meromorphy in "unphysical" sheets - Landau singularities and monodromic structure. - extension of global Properties.
We divide these lecture notes in threeparts of unequal length. Part A is devoted to a work by Eckmann and Epstein [3] on the equivalence of the euclidean formulation of Field Theory in terms of "extended" Schwinger functions and the minkowskian formulation in terms of time-ordered products of fields. This work sets on a firm basis the "bridge" between the euclidean configuration space which is the laboratory of constructive Q.F.T. and the minkowskian momentum space in which the analyticity properties of the scattering amplitudes take place. Part B deals with some progress in the standard derivation of analyticity properties of the scattering amplitudes without using A.C. as a systematic tool. In B-I, improved analyticity domains for the 2 ÷ 2 particle scattering ampli ~ tude and for corresponding partial waves are presented ; these results are due to Roy and Wanders [4 ] .
168
In B-II, some analyticity properties for certain 3 ÷ 3 particle scattering amplitudes, obtained by the Russian school of the Steklov Institute
[5] are described.
For the reader who is unfamiliar with the subject, we found it useful to precede the results of B-I and B-II with a rather long account of standard results which the specialist is invited to skip. A larger extension has been given to Part C which is concerned with the joint exploitation of A.C. toEether with the analyticity properties implied by locality and spectrum. A.C. can be expressed as an (infinite)
set of quadratic integral relations for
the n-point analytic Green's functions H(n)(kl,...,kn ), these relations being offshell extrapolations
of the unitar~yequations
for the scattering amplitudes.
The use of Fredholm theory in complex space allows one to exploit the A.C. equations by two parallel methods ~e~
: the one uses integration on the mass shell (through
application of the A.C. equations) while the other uses off-shell integration
in structural equations of the Bethe-Salpeter
type. These methods and their results
are separately presented in C-I and C-II. In C-I, the former, due to V. Glaser, is presented, and some local results for the 2 + 3 ,
3 + 3 particle scattering amplitudes which have been obtained by this me-
thod [6] are described. In C-II, the latter is presented ;it is the method of the present author [7], which was suggested to him by a pioneer program proposed in 1960 by K. Symanzik in [8a], and
later in [8b] under the name of "Many Particle Structure Analysis"
(M.P.S.A.).
In this program, the notion of "%-particle irreducible"
(in short :
"%-p-i") Green's function plays a crucial role ; the starting point of our method is the systematic study of "~-convolution
associated with a graph ~ " : t h i s
operation
on general n-point functions generalizes the construction of perturbative Green's functions. Some global results for the analytic six-point function H (6) and for the 3 ÷ 3 particle scattering amplitude
(in the equal mass case for an even field theory) which
have been obtained by this method are described ; they can be summarized as follows
:
- a study of the equivalence between 3-particle A.C. and 3-particle irreducibility. - The derivation of a global structural equation for H (6) which exhibits the monodromic structure of the latter in the low energy physical region of the 3 ÷ 3 - a proof of the crossing property 5 6
~ 0 ~ I
2 3
and
~'~'~4 5-~ )--2 6 1 ~ ' ~ "~3
(on the mass-shell)
process.
for the processe~
(~ denoting the antiparticle of particle j) in the equal mass case.
In order to help the reader in understanding where the contents of parts A,B,C are situated in the logical organization of the axiomatic scheme, we have indicated the localizations (A,B,C) on table I.
169
PART
A
THE CONNECTION BETWEEN THE EUCLIDEAN AND HINKOWSKIAN FOR~ULATIO~JS OF QUANTUM FIELD THEORY (O.F.T). The standard formulation of O.F.T.. in Minkowski space* ]DR(d+I)(x) = ]R(x(O))X1~x)
W£g~an a~oms [9] which
(d being the dimension of space) is best expressed by the can be summarized as follows
:
-I In the Hilbert space of states ~ ,
there exists a unitary representation
U(a,A) of the Poincar~ group ("a" denoting a translation in IRd+l , and A ahomogeneous iPP ap Lorentz transformation of 1Rd+l) whose translation group U(a,l) = e - admits the closed light cone
as its spectrum. Moreover there exists a unique state I~> in
called "vacuum", which is invariant under {U(a,A)}
(PNI~>
,
= 0).
In the following, we shall mostly deal with the case of theories having.a minimal mass m > 0 ; in the general developments, spectrum is the set + Hm
(
=
we shall assume for simplicity that the
{0} U H m+ U °+ V2m , where :
p E IR
d+]
; p
2
2 p(O)} + { iRd+] p2 = m , > 0 , V2m = p E ; >
(2m) 2,p (0)> O}
-II There exists a field A(x) (i.e. a set of unbounded operators A(f)
= f A(x)f(x)dx
acting in a common dense domain ~ o
which satisfies the locality condition ~o
: [A(x), A(y)]
can be generated by all vectors of the form
=
of ~
Vf E ~(IRd+l))
0 for (x-y) 2 < O, on ~ o "
A(fl)...A(fm)~
. Moreover the field
A(x) fulfils a covariance condition under the Poinearg group. It turns out that all these properties can be fully reexpressed infinite set of tempered distributions, Wn(X 1 .... ,x n)
=
<~,A(Xl)...
func~o~", namely :
called"Wig~an A(Xn)~>
in terms of an
".
From spectrum and locality, one deduces the important fact that each W n is the boundary value of an analytic functions ~n(Zl,...,Zn)
(zj = xj + iyj) whose analy-
ticity domain contains the whole "euclidean space" :
E (d+l)n
. . {(Zl, . .
,
Zn) E ~(d+l)n
except forthe set (of measure zero) components ~
le~t are equal
o
(Zl,...,Zn)
1,2 ,... ,n
}
,
for which two time-
{S_n} , called "Sehwinger functions" and defined
• the pseudometric being defined by x 2 = { x ; x 2 > O, x (°) > O}
of all points
, j =
(y~O) = n yj(o)) •
The infinite set of functions in E (d+I)n- ~ by : n
V+ =
(o) , +xj)
; zj = ( i yj
x(O) 2
÷2 + - x , and the light cone V by
170
--nS = ~
E (d+l)n-O n
can itself be taken as the starting point of a so-called
"eucgLdean fo~egLon" of
Q.F.T. Such an axiomatic formulation in terms of a given set {Sn} has been proposed by Oster walder-Schrader [I0], and it has been proved in
[10] and [11] that the latter en-
tails the existence of a Wightman field whose associated set of Schwinger functions coincide with the given set {S_n} . This was an important result since it was precisely in euclidean space that the various P(~)2,3 models could be most successfully
studied[12] and since the Schwinger
functions of the latter could be directly constructed and shown to satisfy the O.S. axioms[13]. However a gap had to be filled in order that all the results of the momentum space scheme of Q.F.T. be applicable to euclidean field theory. This gap originates in the following insufficiency of the Wightman axioms general the existence of distributions
sharp
: the latter do
time-ordered or retarded operators
not
imply in
(since multiplying
as A(x) A(y) by the step function @(x (°) - y(O)) is not licit in gene-
ral, due to the possible singular behaviour of A(x) A(y) at coinciding time-components) ; a way of avoiding this difficulty consists tead of tely
sharp)
in working with
regularized
Green's operators. However the complete momentum-space
more tractable in terms of sharp Green's operators,
(ins-
scheme ~s defini-
and in order to set this
scheme on a firm basis it was necessary to produce an extended version of the axioms of local fields. Such an extended version has already been proposed and formulated in terms of retarded operators by O. Steinmann[14]; Stora [15]
a more recent version, due to Epstein, Glaser,
can be set as follows.
One keeps Wightman axioms I and replaces axioms II by the following set of
postalates for (anti)~time-ordered products. T-i) Existence of sets of operators
:
{T(f) = I T(xI ..... Xn) f(x I ..... Xn)dXl...dx n, such that the distributions vectors
T(Xl,...,Xn)
Vn E N, Vf E ~ ( ~ ( d + l ) n ) }
are symmetric with respect to the set of
(Xl,...,Xn) ; in short, one puts : T(Xl,...,x n) = T(Xn) , where X n = ~,2 ..... n}.
- Existence of all products (11,12 (Note that
: T(II).T (12)...T(Ip)
Ip)_ of Xn, in the sense of distributions
T-ii)
for all partitions
on a dense domain
T(Xl) coincides with the basic field A(x I) of Wightman
IAn~l-ca~3~2~ fac/£o*&L687t~on :
for every partition
~o
of
I
axioms II).
(l,J) of any set Xn,
one has : T(X ) = T(1).T(J) in the following region :$ (x ...... x ) ; Vi E I, Vj E J, [ 1 n f f ] n x~ °) < x~°)~ (Note that this implies the locality of the field T(x) = A(x)). ± Jl
171
-T iii) (reap. T iii'))
Time-translation
(reap. Poincar~) covariance of
{T(Xn)} • -T iv) If one puts: T(x) t h e n one has t h e h e r m i t i c i t y
=
~
lip property :
(-1) [X[+p Ix[. T(X) = {(X).
~ T(II)...T(Ip), (I 1 . . . . . Ip)
These axioms implythe Wightman axioms and allow the ~ .
~ .~.
formalism of
scattering theory [16] to be fully justified, as well as the construction of analytic in momentum space with slow increase properties in
Green's functions H(n)(kl...kn) their primitive domains D
n
(see part B and [15]).
A corresponding extension of the O.S. axioms was presented in [ 3 ] ; the axioms can be set as follows.
Axioms for euclidean Green's functions (Eckmann-Epstein). -S i) existence of a sequence of tempered dxstributions Sn(Yl,...,y (o) n) on E(d+1)~ ,
which are symmetric with respect to the n points Yl ..... Yn (Yi = [Yi
'~i ))"
-S ii) Extended O.S. positivity : For every set {!~T ~ ~(]R(d+l)n), N ; 1 (_< m <_ ( fo E~C} such the that : Vm, SUPP fm c , ..,ym); y j°) _> 0 ..... ym°) _> O i , following inequality holds :
'Ym ..... (-Y
'~i )
fn~tYl
'Yl ) ..... (Yn
'Yn )
0 <m,n
x
,
v
v
Sm+n(y 1, . . . . y m , Y l , . . . , Y n ) d Y l . . . d y
-S iii) (reap. siii)') Time-translation
m dy 1 . . . d y n
>_ 0
(reap. euclidean) covariance of {Sn}.
-S iv) Growth condition : n
Vn, ISn(fiO...@f#I_<
K n n! L
FI I[fjIIs j=1
,
for appropriate positive constants K,e,s and a certain Schwartz norm I Ifl Is. We notice that in these axioms, each distribution Sn is defined on the whole space
E (d+l)n, including the set ~
of coinciding points ; thus Sn represents an n (d+1)n extension of the Schwinger function --uS (defined on E - O n ) and (in view of the following theorem) deserves the name of "euclidean Green's function". We also note that in S ii), the original O.S. positivity condition has been extended from E (d+])n- o to E (d+1)n n
Then the following important relationship between the two sets of minkowskian and euclidean axioms has been proved in [ 3 ] :
Theorem: There i s an equivalence between a s e t of ( a ~ )
time-ordered p r o d u ~ s ~ -
lying the postulates T i) . . . T iv) in the framework defined by Wightman axioms I,
172
and a s e t of euclidean Green's function s a t ~ f y i n g S i)
. . . S i v ) . Anyone of these
two set~ can be reconstructed from the other.
More precisely, if one puts ~n(Xl...Xn)
=
<~,T(xl...Xn)~> the following rela-
tions hold for each couple of distributions Sn,~ n) :
f i n f i ( x l , . . . , x n) Sn(Xl,. . ,xn)dx . 1.. .dx n. = f.f ( x 1 . .x n). ~n(Xl . where f denotes any t e s t f u n c t i o n in ~(]R (d+l)n) which is " r a d i a l l y
.x n) dx 1. .dx n, a n a l y t i c " with
r e s p e c t to the time-components of ( X l , . . . , x n) in the f o l l o w i n g sense : the mapping E JR+
÷
f%(xl,...,x n)
=
(o) ÷
f (%x I
,Xl),...,(%x
~o) ÷
,xn)
extends to an analytic mapping in the domain { ~ E ¢ ,
Re ~ > O,
Im ~ > 0 }
Moreover, it has been fully established
in [ 3 ]that the truncated analytic n-
point function'~tr in z-space (z =x+iy), the analytic n-point Green's function H n n in k-space (k =p+iq), the various boundary values and the euclidean restrictions Strn (reap. Sntr ) of ~/tr and H n in their respective spaces are linked together through the following commutative diagram
: this diagram holds for either choice of the basic
quantities, {St} or {Tn}~ taken as the starting point of the theory. Analytic continuation Boundary values in in complex z-space Minkowskian x-space ~tnr ~----~ tr(~r)~ tr mr -tr <~nt>r iWn ~,Tn = , Tn =
Euclidean y-space S S tr n' n
E u c l i d e a n q-space ~tr n' n
~
In this diagram,~ E and ~ M
Analytic continuation in complex k-space H (n) ~
Boundary values in Minkowskian p-space {r(n)},T~tnr,~tr
respectively represent the Fourier transformation in the
euclidean and minkowskian (d+1)-dimensional spaces ;{Wtr(F)} denotes the set of all r(~) (Xln" tr the "permuted" truncated Wightman functions W t ..,xn) = W n (xv1,.-.,xFn) ; {r~n)} denotes the set of all the generalized retarded n-point functions in momentum .... ~tr --~tr space (seepartBand [J5,1~J);T n ,T n (reap. ~tr) n are the ~M-transforms (reap. % transform) of Ttr, ~tr (reap. str).n
*Note however that the facts expressed by this diagram (analyticity properties, alEebra of boundary values etc...) were known before and refer to a long historical development produced by numerous authors in this domain [15]. The main point of [3 ] lies in the rigorous formulation of this rich structure, includin~ the delicate . . . points in x (o) (or y(°))-space. treatment of distributions at .coinciding
173
PART
B
SOME NEW RESULTS FOR THE 2 ÷ 2 AND 3 ÷ 3 PARTICLE SCATTERING AMPLITUDES, OBTAINED BY STANDARD TECHNIQUES OF ANALYTIC COMPLETION. We recall the following facts which concern the so-called p r i m i t i v e a n a l y t i c
structure
of n-point Green's functions in momentum space and are consequences of the
(extended)axioms of local fields. For each integer n, there exists an n-point function H(n)(k) of the set of 4(n-I) independent complex variables k = p + iq -= (k .... 'kn)' ki = Pi + iqi C C 4, k I + ... + k n = O. The analyticity properties of H In) are the following :
I) H (n) lS defined and a n a l y t i c i n t h e following " p r i m i t i v e " domain Dn : Dn
=
where : a) ~ a eo~iste~t
(~
~)
U
= ~4(n-l) m(p)
(
+ i~
u )
, and~
is a cone in IR4(n-l)which is defined by (q)
set of relations
~:~O0= {q E IR4(n-1)
; qj =
here Xn = {l,2,...n} and
~ jEJ
qj E e ( ~ ) ( J ) . V + , VJ E ~ ( X n)} ;
c (7 ) is a sign-valued function defined on
3"
(Xn). In
each tube ¢-~o , H(n) is a function with moderate increase (at infinity and near the boundary
of ~
);
it thus admits a "distribution boundary value"
H~n)(p)
=
lim C ~
H~n)(p + ig) on ~4(n-l)
re(p)
.
s÷O _ (n) The link of tl~
wl"th the bas"lc fie ld of the theory is the following : the tem-
pered distribution ~n)(p) =
Fourier transform of a c e r t a i n n ) % . . . . . xo) =
~4(p I + ... + pn ) H ~(n) (p) (defined on IR,4n. _ )) is the .3
(n~
(P
generalized retarded function r~ ~ ( x l , . . . , x n) suchthat
<, @ ) % , ....
; the
lgebra of the g e n e r a l i z e d retarded
operators R ~ )" can be constructed from the set of time-ordered products s a t i s f y i n g the postulates T i ) . . . T iv)(see part A). Locality (or "causal f a c t o r i z a t i o n " T ' i i i ) ) implies support properties for the tubes~
~(n)
which e n t a i l the a n a l y t i c i t y ofH (n) i n t h e
through the Laplace-transform theorem.
b) For every couple of tubes (~e~, cl~,) w ~ h qI = 0"
(i.e. whose associated sign functions g
on the subsets I and X ~,
7$n)(p)
= f
a r ~ i a d j a c e ~ along t h e face , g
only differ by their values
- I), one defines the corresponding "¢oino~dence region"
CIR4(n-l) n2 # m 2
2 < M2~ , which is the region in which
I p, . ';PI ' PI n)(p)(the l a t t e r is a consequence of the spectral condition).
~
~ / ~ )
represents the i n t e r s e c t i o n of a small complex neighbourhood of gdf~'
with the convex hull of various t u b e s ~
together.
%
U ¢~
Note that the set
{tUo~,,,)}
connects the
174
Remarks about
: i) For any fixed Lorentz frame, let us put :
D
÷ , k (°) = p To) + i q k =(k (°) ,k) IR3Qn-1) (~)
× IR(n-|) (q (o))
(o) , ~ = p÷ + iq ÷ •, then the euclidean space
is contained in D . n ii) Every (n-1)-dimensional section of D
= ÷
p (real) fiYpd is a dense domain in
~(n-I
by k(°)-space, at n
(k(O));
it is the exact generalization of
a cut'plane in which the upper and lower half-planes ( ~ + , ~ _ )
are connected together
through a real interval~. iii) Each manifold k 2 = m 2 (VI C~*(Xn))
is a simple polar
manifold for H (n) .
2) The discontinuity funoJ~ions (or "absorptive p ~ " )
AIH(n) of H(n) :
With an arbitrary partition (or "channel") (l,J) of Xn, let us associate the following momentum variables • kI k I = -kj =
~
ki
,
k(l ) = { ki = k i -]i-- m }
,
i ~I kj k(3 ) = {kj = kj - ~ }
, and rewrite : H (n) (k) = H (n) (kl;k(l) ,k(j))
We define the corresponding discontinuity or "absorptive part" of H (n) : A I H(n)(Pl;k(1),k(j )) = Al H(n)
lim [ H(n)(pl + is ;k(1),k(j )) - H(n)(Pl-ig;k(1),k(j~] ~÷O+ L sEV is : a) a distribution in Pl with support contained in {Pl E H + U VM } m b) an analytic function in (k(l) ,k(j)) in a domain D(1)n(Pl) which de-
pends on Pl and is similar to D n (namely a union of tubes and of neighbourhoods of various "coincidence regions").
p o s ~ v e kernel
c) a
(depending on pl ) in the case Ill = IJ~ (n even)
More precisely, for every suitable test functions
X, ~0 , the following inequality
holds : I
AiH(n)(pl;P(1),_p(j))
X(Pl) ~0(P(1)) ~(p(j)) dPi d_P(l) dp(j)
>
Another basic development in the axiomatic framework of local Q.F.T. is the scattering theory of Haag-Ruelle-Hepp [9b,]6],whichinvolves the construction of asymptotic incoming and outgoing states and the derivation of the f . f . ~ .
reduc-
tion formulae which relate the scattering kernels to n~point Green's functions. In the complex momentum space framework, this can be summarized as follows : l e t c~ ) be the complex mass shell, namely : O]~(c)
=
{ k E ¢4(n-1) ; kl2 = m 2 ... k 2 = m 2 } .
n
The real mass shell
~
~n
= ~(c) O ~4(n-l) n (p)
~n
is composed of the union of all the
0
175
physical regions ~p(IJ) n =
{p 6 I R A ( n - l ) Y;i
6 I , - Pi 6 Hm+ ; Yj 6 J,
of the corresponding reactions { A i } i ~
pj 6 H m+ }
{Aj}j 6 J" (the Ai,A j denoting identical
particles whose states are created by the basic field, as asymptotic states). (n) . . . . Let S~(p) be the connected scatterlng amplltude assoclated wlth the above reaction, s~inj~-Pi}i61, {Pj}j 6 j ) i s a distrlbution" on Q'~n(IJ) which is obtained from H (n) (n =III
+ IJl) through the following procedure : consider the
,,
amputated
n-point function n 2 H (n) H (n) (k) = N (k -m 2) (k), which is analytic in a domain D such n i=l i n that D = D - U {k;k~ = m 2} . Then in D take the "time-ordered boundary value" Sn n " | (p(O) of ~(n) , namely : Tn(p) = lim ~(n~ (i +ie), p). It can be proved that • g > 0 SIj
(ipii
E1 ;{Pj}j 6J]
=
~n (p) ~ I J )
(n)
The derivation of analyticity properties for the scattering amplitudes SIj on the complex manifold ~]~n ¢~c) is based on the following analysis. ^
^
D
N ~ (c) = ~, but D is ;~ot a natural domain of holomorphy, Standard techniques n n n of analytic completion can then be applied to computing parts of the holomorphy envelope of D n. Any such extension D' of D which has a non-empty intersection w i t h ~ (IJ) n n n yields a "maSS shell dom~n" from which Sti~ is the boundary value of an analytic function. S (n) = H(n) I~c), The analytieity
properties
of
the
can
have
either
a local
or
a global
cha-
racter.
By "local ana£y£ic%£y", we mean the fact that in the neighbourhood of a point p, S~)is" the boundary value of an analytic i n ~c ) . - ~
function from a certain "local tube"
It has been proved in [20c], that for every couple (I,J), there exists a
_(n)satisfies the property of local analyticity. subregion o f '~(IJ) ~ in which 3ij Under the name of
"global analytieity",
of a certain analyticity domain i n ~
we mean either the explicit knowledge
(c), or the existence of such a domain which n
connects two distant regions : for example, a"c~os~ing dom~n" on ~(c) links toge~(llJi ) ~ 1 2 J 2 ) n ther a couple of physical regions rJ~n , . Several global analyticity properties have been proved for the two-particle scattering amplitude (IiI = IJI = 2); among them let us quote : i) the "Lehmann- Martin ellipses"[22] which give an explicit size of the analyticity neighbourhood of each physical region in the momentum transfer ~ariable t (at fixed squared total energy s), ii) the crossing domains which exist in the general case of two initial and two final particles with arbitrary masses [20b]~ and become the domains of dispersion relations for certain mass configurations
176
(for example, cut-planes in the variable
s~ at fixed t, with Itl < 4m 2
in the case of identical neutral particles with mass m [22]
).
The "standard techniques of analytic completion" which have been used in obtaining these results can be summarized as follows : i) the ~ocal techniques which are elaborated forms of the edge-of-the-wedge theorem [23] (see in this connection the notions of "essential support of a distribution" [24] or "singular spectrum of a hyperfunction" [25]) ii) the global techniques which essentially reduce to : a) the continuity (or "disc") theorem [26]
: in practice it is often applied under
the form of an explicit continuation of a suitable
Cauchy integral (or "dispersion
relation"), b) the interpolation technique [27] the tube theorem
it often amounts to a suitable application of
(after appropriate conformal mapping).
c) Martin's use of the positivityof the absorptive parts[2]](this special method of analytic continuation can also be given an alternative, more geometrical, form[24a]
B-I. Extension of the analyticity properties of the ~-~ scattering amplitude by using dispersion relations on curved manifolds (Roy and Wanders [4] ). Let ~0~-(c) {k = (k],k2,k3,k 4) k I + k 2 + k 3 + k 4 = O, k 2 = m 2 I < i < 4} be "~4 = ' i ' the complex mass shell in ¢12, by taking into account the (complex) Lorentz inva-
~(c)
riance of this 8-dimensional complex manifold,'ll84
can be parametrized by two inde-
pendent Lorentz invariant variables ; however one usually Considers for convenience the three Mande!stam variables s = (k I +k2)2 , t = (k] +k3 )2, u = (k 2 +k3 )2 which are linked by the relation
4 s +t +u In
=
~ i=]
k~ = 4m 2 . i
~(2). have been taken as a basis"to [4], the following analyticity properties in ~(stu)
further analytic continuation for the mass-shell restriction, S (4) of ~(4)
:
-i) Let ~t = {(s,t) ; Itl < 4m 2, s E ¢}and ~s' ~u the similar domains obtained by permutations of (s,t,u). Then S (4) is analytic in ~t U m s IJ ~ u minus the union of the "cuts"
~s = {(s,t) ; s real
> 4m 2 } , o t and o u
-ii) the absorptive part ASS(4)
-- (c)
(i.e. the restriction to~IID~
analytic in the Lehmann-Martin ellipse
E(s) of the t-plane [22]
^(4))
of A{]2} H
is
, for each s ~ 4m 2 ;
an identical result holds for A t S (4) and A u S (4) The method of [4] consists in expressing S (4)(s,t) througha "dispersion relation" on a complex analytic manifold ~(a,x o) in ¢2(stu)
, and in using the analytic depen-
dence of the latter with respect to the complex parameters a,x ° as a tool of analytic continuation for S (4). The family of manifolds {~(a,Xo);X ° E ¢, a E V(xo)} required to satisfy the following conditions
:
is
177
a) ~(a,x o) is symmetric under the permutations of s,t,u, b) for lal < g(Xo) (g(Xo) being sufficiently small), ~(a,Xo) lies in the domain ~s U ~t 11 ~u' c) Vx ° E ¢, V(Xo) is a star-shaped domain in ¢ such that,
Va E V(Xo) , the
section of ~(a,x o) by every line s = so > 4m 2 is contained in E- . --
S
0
If these conditions are fulfilled, then S(4)(s,t) can be analytically continued in the domain ~
= x
U ~x ' E ~ o
0
o
where :
:'a £ V(x o)
u
The following choice has been made for {~(a,Xo)} : Put x = - l--6l(st +tu +us), y = ~ 2 or in ¢(stu) : (s + 4a)(t + 4a)(u + 4a)
l
stu ; then ~(a,x o) is defined by y=a(x--Xo)
64a[a(a + I) - x o]
This family of cubics has been shown to satisfy conditions a),b),c) for suitable star-shaped sets V(x o) and thus yields a computable analyticity domain previous form. Moreover, i n ~ relation (valid
of the
the function S (4) satisfies the following dispersion
on each curve ~(a,Xo) such that x ° C ¢, a E V(Xo) ) :
S(4)(s,t,u)
=
S(4)(So,O,4m2-So ) + ...
oo
... + - ]
I 4m 2
ds' A s
S(4)(s',T(s',a,Xo )) [ ' I +] , , 4 - -v LS -S S -t S -U
, 1 1 ] ] v , /+s. S -S S S -~m 2 O O
in this formula so denotes an arbitrary parameter such that 0 < so < 4m 2. An investigation of the sections of the d o m a i n s ~ S
at fixed angle e O .
(t(s, cos@)= (2m2 - ~)( | -cose)) has been done. Of speczal importance are the sections at e = ~ which have been computed for a sample of values of x ° and yield a good approximation of the section
of~at
@
~. In the complex s-plane, this
domain is represented on fig.] below ; the two cuts s ~ 4m 2 and s ! 0 are parts of its boundary. 9011ms
~,~60r4"5 z" ~
, -
~
~
~3~
J
' JS~I~
12~31
178
Analyti~ity of the partial waves 1
f~(s)
=
The following property has been proved in
Theorem
P%(cos@)
Io d(cosO) S(4)(s,t(s,cosO)) [4] :
for each value of ~ , the function f~(s] is analytic in the domain ~ ± . ii) f~ ~ a ~ f i e s the following integral representation :
: i)
f~(s)
=
~%° S(4)(4m2'0'O)+
~ I ds' (s s')a~,(s') ~'=O 4m 2 K~, ,
Here
a~(s) denotes the absorptive part of f~(s) (i.e. its discontinuity on the cut 2 s h 4m ),and each kernel K~%,(s,s ) can be explicitly computed, as the ~ntegralon a suitable contour of an algebraic expression which contains a certain product of
Legendre polynomials
P% . P~, .
Rem~ks : i) is essentially due to the following fact : for every s in~i(Xo) , one can find a continuous distortion U cos @£Ys
(s,cos@) c ~ x
Ys of [0,I] in the cos @-plane such that
(this is a simple consequence of the fact that V(x o) is a o
star-shaped set). ii) is a remarkable representation of the analytically continued partial wave f~ in terms of pure physical quantities,
namely the absorptive parts of all
partial waves fL, and the scattering length (through S(4)(4m2,0,O))
; for want of
obtaining a full cut s-plane for f~, this is the best substitute for a dispersion relation that can be found.
B-II Some analyticity properties for the 3 ÷ 3 particle scattering amplitudes : We list very briefly a set of interesting results which have been obtained at the
Steklov Mathematical
Institute
[5] , a part of which have already been presen-
ted or announced in the previous Symposium of Mathemetical wing results of global type have been proved
Physics [28] . The follo-
: the validity of one-dimensional
persion relations for the forward scattering amplitudes of the reactions yyN
÷
yyN
in e4-order [5a] ,
and
yNN
+
yNN
dis-
:
[5b]
for a mass of the virtual photon satisfying k 2 ~ a 2 < O, a certain absoptive part of the latter being relevant for the study of deep inelastic proeesses matter of fact, a "generalized optical theorem" zed retarded operator~
(based on
. As a
the algebra of generali-
mentioned in Part A) has been proved for the 3 ÷ 3 particle
Note that all the works [5] are however presented as developments of the Bogoliubov axiomatic framework [29] which leads to an identical analytic and algebraic structure as that described in [18] (see a recent account of it in [5c]). although its conceptual formulation is quite different.
179
forward scattering amplitude
[5a] ; it expresses the latter as a certain sum of ab-
sorptive parts ; among these, the discontinuities A{iji,} H (6) (represented by the i - - ~ i ' j
diagram
j'
are present and have a physical significance since,
~ - = ~ - - ~ ,
due to A.C. or unitary equations (see Part C) the latter can be expressed as inclusive cross sections of the form ~~ - - ~ - - ~ i ' /---"
9.
.7
J' "k._/-~-
n
I n t h e same s p i r i t the 3 ÷ 3 particle derived
in[5e]
as in
[2~],
some d e t a i l e d
forward scattering
; t h e y a r e b a s e d on a s t r a i g h t f o r w a r d
wedge t h e o r e m , w h i c h l e a d s t o t h e f o l l o w i n g neighbourhood of the physical For H(6)(kl,...,k6 manifold~'
),
region
analyticity
~33
of the forward three-body lies
= (1,2,3)
the physical
Then i f one t a k e s a s v a r i a b l e s region
cubic surface defined in ~ 3
~33 i n ~
for
of the edge-of-the-
of the situation
in the
reaction.
in the linear
d e f i n e d by k 1 + k 4 = O, k 2 + k 5 = O, k 3 + k 6 = O, k i 6 t ~ i "i".
properties
m a s s e s have b e e n
application
description
the set of forward configurations
b e i n g t h e mass o f p a r t i c l e (i,j,£)
local
amplitudes with arbitrary
sub-
(i = 1,2,3),
mi
s i = k~.k£j
is the "interior"
of the sheet of
by :
~s]s2s 3) 22 22 22 223 m] s] + m 2 s2 + m 3 s3 - m] m 2 m 3 - 2sls2s 3 = 0
;
si >- m.m~ j
(i = ],2,3)
~33 is the union of four disjoint parts : i) the three "inclusive regions" 933~(~) (~ = 1,2,3) defined by the conditions _p%)2 _> 2 reglon(Pi +Pj ~ ) s u c h that Mij'%(pi(f°r+pjappr°priate-p%)2 < M2ij£thresh°ldf°r all masses(ij%)Mij'%= (]23),(231),(312).) and the remaining
For p 6 ~33~(%)' H (6) is analytic in a local tube T% {q ; qi+q% 6 V +, qj +q~ 6 V +
'
qi +qj -
with basis q~
6 V + }.
For p 6 ~~(o) 3 ' H (6) is analytic in a(larger) local tube T
with basis
{q; ql + q 2 6 V t , q2 + q 3 6 h V + , q3+ql e 6 V+} . The ~ intersectionso of {To,T~ with ~ 6(c) mass shell O?b c) have been fully described in[Se]; non-empty local tubes on (TIU are thus obtained for large unbounded subregions of a submanifold of ~ )
~33 (for example for p lying in 2 of the form : s3 = 2s], s 2 > M2, s 3 > f(s2) ).
Let us finally mention that a certain crossing symmetry property for H (6) restricted to the forward submanifold~
has been shown in [5f] on the basis of the pre-
vious local results and of the Lorentz invariance properties of H (6).
180
PART
C
ANALYTIC STRUCTURE OF GREEN's FUNCTIONS AND SCATTERING AMPLITUDES IMPLIED BY ASYMPTOTIC COMPLETENESS +
For each local Q.F.T. whose spectrum contains a discrete part Hm (the case of a single mass m is considered for simplicity), the Haag-Ruelle formalism produces two asymptotic subspaces k i n
and ~ o u t of the Hilbert s p a c e r
which are both isomorphic
to the abstract Fock space of relativistic particles with mass m : these two isomorphisms induce i n ~ a (partial) isometry from ~ o u t
to ~'n which is by definition
the scattering operator S. However, it is n0t a consequence of the axioms of local _
Q.F.T. that ~ i n
and
fl
%ut
span the whole space
From now on, we shall consider local field theories in which*~i
n
~ ~ o=
ut
=
"
This property, called "Asymptotic Completeness" is clearly equivalent to the unitarity of the operator S (SS
= S S = | ) .
On the basis of Ruelle's discontinuity formulae R~
-
Ry,
=
i [R~ ,R~
] in the algebra of generalized retarded
operators** (see [18] ), of the reduction ~ormulae and of the analytic structure of the absorptive parts
AIH(n) of the functions ~(n) (see Part B), it can be shown that
the latter satisfy the following (infinite) set ofrelations[3Oa], which we call"A.C. equations" since they are equivalent to the property of asymptotic Vn, and for every partition (l,J) of Xn, with (A.C.)
llI
=
.) = ~| ~| ~(nl+~) ~ (Pl;k(1),
AiH(n)(Pl;k(1),k(j))
completeness :
n I, JJI = n 2 , *(1)H(n2+~)(pl; " % -~ ,k(j))
where the operation .(I) , called in the following "maSs sheg~convolw~Zon o fordeA£
i n the channel [I,7)" is defined by :
~(nl+l),(I)~(n2 +1) ]
2~i ~(p~, m2)O(p~O))~(nl+l)
_~
~
_
V ~ > 2, H(nI+~)*(1)H (n2+~) = ~ ~ -~
...
x
~I i(
H '~"0~'(P I)
(nI+%) ~
^(n2+l) (PI;k(1)) H_~ u f
(Pl;~(1)'~I
(Pi;k(j)) ,r~})
....
...
^(n2+%) d~! d~% H_~ (pl;{-rl...-r~},k(j)) 2~(rl)...2~(r~) .
Such a condition maybe less restrictive than it c a ~ p p e a r at first sight; for instance one can imagine situations in which the s p a c e ~ spanned by the states of a certain local field A(x) is not the space of ~ physical states, but a certain sector of physical states which can have its own interest, independently of other types of fields or observables describing other aspects of reality. these formulae (which hold for every adjacent couple (~,~') are generalizations of the standard discontinuity formula for the 2-point"retarded"and "advanced" operators R± of a local field A(x). R+(x,y) - R-(x,y) = i [A(x),A(y)] .
181
in these formulae, ~(r i) =
{
~(pi
)
=
_i
+ m2,
Z = _(p2 -m 2) H(2)(p)
°-P
2
2
+,p°m
and
}
(rl,r 2 ..... r~); r I +...+r~ = pj
I' ri ~ H m ~ 1 < i < Z . ^tn1+~) denotes the sign of the relevant boundary value of H with respect to qI + (i.e. qI 6 ~V , qI + O) ; the two cases (~ = ±) correspond to either use of "it"states or"out"~tates (
=
projections at the right-hand
=
side of Ruelle's formulae
) in the derivation of the
A.C. equations.
Remarks on A.C. equations and t e c h n i c a l assumption : We first notice that since f f ~ ( p i ) is empty for p~ < (~m) 2 , the term of order -+ £ at the r.h.s, of the A.C. equations has its support contained in {k;Pi real ~ V~m} , so that the summation over % always reduces to a finite number of terms if PI is kept in a bounded region. Strictly speaking, the A.C. equations hold as identities for measures with respect to PI' taking their values in a space of analytic fune ~ tions of k(i),k(j ). The purpose of what follows is to performo some analytic continuation of these equations in the complex variables s I = kS. However this will only be feasible under the following additional assumption~:
Smoothness assumption : t h e A.C. equations hold as p o i n ~ i s e i d e n t ~ between I a n a l y t i c functions i n t h e r e l e v a n t domains Dn[Pi), for each f i x e d vague of PI i n -+ V2m. The analytic continuation procedures which will be used in this Part C are based on the study of the analyticity properties of integrals of analytic functions and specially of the Fredholm resolvent of an analytic kernel. To be more specific, we consider the following general situation in a complex domain ~ mension n
let
=
O k6A
(k,~k
×~),
r k be an n - d i m e n s i o n a l
ly with k (for k in A) and let
: Let F(k,z,z') be defined and analytic Ok,
~
cycle in~) k N
being domains of complex di'
which varies
~k(Z) be an analytic differential
continuous-
form of order n.
We are interested by the analyticity properties of the Fredholm resolvent G(ko,Z,Z';%)
of F corresponding
to a given value k = k ° and to the integration space
F k . The following result has been proved in [33] : o
Lemma : T h e s o l u t i o n G of t h e r ~ o l v e n t F(k°'z'z')
equation :
= G(k°'z'z';%)
+ % IF(k°'z'z]) Fk °
G(k°'z'z';%)
~k ( z ) o
this property can also be considered as a consequence of the "smooth s p e c i a l c o n d ~ 0 n " t h a t has been postulated in [30a].
182
admits a meromorphic continuation of t h e form : G(F) (k,z,s' ;%1
=
N (~) (k, z,z' ;%) D (~1 (k;%)
where N(#)(resp. D (~)) is holomorphic in ~ x
C(%)(resp. A
x C(%)).
For each k i n A,G(r) ( k , z , z ' ;~) i s an a n a l y t i c continuation of t h e . f r e d h o l m r e s o l v e n t of F ( k , z , z ' ] ,
considered as a k e r n ~ on Fk. Moreover, N (~) and D (r) (which can be
defined on r k × r k by t h e standard Fredholm formulae) o ~ y depend on t h e s ~
F of
(~ = {~k~k ~ A}).
I c o ~ n u o ~ l y v~yin~l homoSogy ~ a s s ~ ~k of r k in ~k n ~
This lemma will be applied in the two schemes of analytic continuation which are developed below in C I and C II, the first scheme being purely based on the analytic continuation of the A.C. equations,while the second scheme uses theanalyticity of ~-particle irreducible n-point functions; in the latter, the A.C. equations enter as an element of the proof of the irreducibility properties. In a pedagogical spirit, we shall now illustrate the ideas of these two schemes by treating
the case of the
two-particle structure of H (4) and of the two-body scattering amplitude of an even* Q.F.T. with a single mass m. We consider the channel (l,J) = ({1,2},{3,4}) and the corresponding low energy region : s = kl2 = (k| < (4m) 2 ; the relative four momenta associated with this
+k2)2
channel are : k| - k 2 2
k(1)
. z , k(j)
=
k3 - k4 ~ =
, z
In the considered region, the A.C. equations for ~(4) can be written as follows : IAiH(4)(pl;z,z'
-- H(4)(pl;z,z' ) - H _(4) (Pi;Z, z ,) +
=
(A.C.) /
~. =
,
2w ?
]
~"
i(--~-)- × ]
^
j
H~4)(pl;z,z1(u,s)) H(4) (pl;z| (u,s)z')
u 6 S2
2
d~(u)
/ s ( s - 4 m 2)
To derive this formula, we have assumed that k I = (k~ °) ,O), with
K I(o)
= ~
(this is
not a restriction since ~(4) is Lorentz invariant) ; the mass shell integration manifold ~2(Pl)
=
{Zl E ~ 4
Pl 2 = ; (z] + -~-)
(z I _ ~ ) 2
= m2j
is then a sphere which can be parametrized as follows : z|
=
z](u,s)
=
(0, / s - 42 m 2
u), with
u E S2 ; d~(u) is the
surface element on the unit sphere S 2 Let us now introduce "the following Fredholm resolvent equation, depending on the results which we present have however been proved in tNe general case.
183
the complex parameter s(z(u,s) being the analytic continuation for complex values of s of its definition given for s real) : I
I
~(4) (ki;z(u,s) ~+
~
z'(u' ~ s))- ~(4)(..) •
=
(F) i.2~.2
= ~t~)
I
~(4)' ~(u,s)'zl(ul's) ) ~(4) + (kl; - (kl;Zl(Ul'S)'Z'(u''s))
u I ES 2
where dNs(U )
2
d~s(U)'
d~(u) .
/s(s -4m 2) In this equation, ~(4)_ represents the function ,
°~v4
given in a part of its analyticity domain D_
=
t~k E ~ ) ;
-g
~(4)[~(c) = S (4) considered as
of the following form :
s < 0 ; (2m)2 < Re s < (4m)2; u,u'C $2:c
IIm u I < a, IIm U' I < a
}
,
-
where g and a are certain positive numbers. According to the previous lemma, the Fredholm resolvent ~(4)' of ~(4) + ned as a meromorphic function N(s'u'u')
D(s)
is
identical for s real
tion (A.C.) (they both
in the domain D
is defi-
; but since equation (F)
((2m)2 < s < (4m)2) with the mass shell restriction of equacoincide with the standard "elastic unita~ity equatio~" for
S(4)~ it is a straightforward consequence of the edge-of-the-wedge theorem that~i 4)' IJ J' is the analytic continuation in D -- of ^~ ~ +H~4j the latter being the determination of ~ ~(4) defined in the part D+
=
{k E ~
; O < Im s < ~ ; (2m) 2 < Re s < (4m)2 ;u,u' E S 2 c~
IIm u I < a, lira u' I < a} of its analyticitv domain. So far, a meromorphic continuation of ~(4) through the cut : (2m)2 < s < (4m)2 + has been performed in the complex mass shell ~ q ~ ) ; however it is easy to get rid of the constraints k~ = m 2 (i = 1,2,3,4)
by a simple argument of analytic continuation
for both sides of equation (F). Let us add the following remarks concerning this meromormhic continuation of ~(4) and ~(4) in a second sheet : ~IJ i) the mechanism of analytic continuation of the partial waves across the cut (2m) 2 < s < (4m)2
through using the "elastic unitarity relations", has been explai-
ned long ago in[3]]and refinedin[32]; however the present method is more general and Strictly speaking, the domains D+, D_ are contained in the holomorphy envelope of the primitive domain D 4 (see [2Oa]).
184
more suitable for treating the scattering amplitude S (4) IJ " ii) The pathology
which was discovered by A. Martin [32] in the partial wave
approach and concerns the possible accumulation of poles from the second sheet on the real interval
(2m) 2 < s < (4m) 2, corresponds
in the present approach to the sin-
gular case of Fredholm theory in which the denominator D(s,%) happens to have a i-2~ 2 fixed zero at % = ~(-~) (the "physical value" of the Fredholm parameter % ); this case is here discarded thanks to the "smoothness assumption" which has been postulated above ; the latter may be compared with analogous
regularity assumptions postu-
lated in [32] . iii) A local meromorphic continuation ~(4)'_ of ~(4)_ in D+ can be derived similarly ; it can also be shown that ~(4)' admits an analytic continuation of the form
{Is-4m21
in a domain
< ~ , s # 4m ~ + p} and that it coincides with ~(4)'
in the
intersection of the latter with D+ ; the reason for this "two-sheeted monodromic structure" of ~(4) around the manifold s = 4m 2 lies in the change of sign which is produced in the integral at the r.h.s of equation (F) when s describes a small loop enclosing the point s = 4m 2. iv) Extensions of the second sheet analyticity domain of S (4) outside the local regions described above can be obtained in principle by applying more thoroughly the previous lemma on Fredholm equation. Indeed, for each continuous distortion F(s) of S (2) in the complex sphere S(2)such that the set {z = z(u,s), u E F(s)}iS cone tained in the (first sheet) analyticity domain of ~(4), there exists a second sheet meromorphic
continuation of ~(4)
. We notice for example that such a distortion
been produced in the cos e-plane (see B-I)for the constructionofthe of the partial waves proposed in [4]
has
analyticitydomain
. One can assert that any domain of the partial
waves obtained by such a distortion technique is the projection onto s-plane of a second-sheet domain of S (4). An alternative method
[7]
for deriving these analytic continuation properties
of ~(4) in the two-particle unphysical
sheet consists in using the Bethe-Salpeter
equation : (B.S.) H(4)(kl;Z,Z')in
=
L(4)(kl;Z'Z)in + 2I I r(kl ) H(4)(k in I ;z,z ])
in which ~!4) denotes the "left-amputated"
form of H (4), namely
L!4)(k~;z,,z')d~z in i J ~ | :
in
~!4) in (k)
=
H (4) (k). (k~ - m 2 )(k 22-m2).
Equation
(B.S.) defines
Lln!4)(namely
the "Bethe-Salpeter kernel") as a Fredholm resolvent of 2! 4) , depending analytically in on the complex parameter k I. In (B.S.), r(k I) denotes a four-dimensional contour which coincides with the euclidean space {z I
=
(iyl°),x])} when k I
=
.
(o)
(lql
÷
,pl ) ;
when k I varies in C 4 -{kl;k 2I = 4m2 + P, VO > O} , F(k I) has to be distorted inside the primitive analyticity domain of ~!4). The required conditions for applying in
185
the C.F. lemma are then fulfilled and the following results can be obtained : a) the function
~(4)(k) = ~(4)(k).(k~Lin- m 2 ) ( k ~ - m 2) is analytic in D4' except 2 for possible "C.D.D. polar manifolds" of the form : s = k I = constant, ^ (4) b) the absorptive part A~ ~Lin(4) of Lin satisfies the following relation (A) in the region (2m) 2 < s < (4m2~
A L (4) (~ 1 L(4)) ~(4) I~(4),(I)H(4)~ I~(4)~ (Ll) I in (Pl ;z'z') = - 2 in ~)F+(pl) (Almin - Y m 2 in ~ Gr_(pl)(~ - ~ L i n j
In this relation, the following operator notation has been used : r
(F (Dr(kl)G)(kl;z,z') j?(ki)F(kl;Z,Zl) =
G(kl;Z|,z')d4z I ;
denotes the identity kernel on the space F(kl) ; F+(pl ) and F_(pl) denote limiting positions of
F(k I) from the respective sides
q~O) > 0 and ql(o)< O, obtained when
ql(o) ÷ O. With these notations, equation (B•S.) can be rewritten : (4 + l ~(4)) l ~(4) 2 in (4 - ~ in ) = ~ (for each value of kl) ,
Or(el)
and this shows the "invertibility" of equation (A) under the following form :
1 ~(4),(I)~(4) (A') A I H ~ )
- ~
2
in
(4 =
1 ~(4)~
AI %(4)
+ 2 ~in ) QF+(PI)
in
I ~(4)x G~(pi)(l
+2~in
~
From (A) and (A') one readily concludes the equivalence of the A.C. equation : AI~4
1 ~(4).(I) ~(4~ -~ • 2 = O
with the equation : AI
~4) L.in =
AI
74) L
= O
in the re-
gion (2m) 2 < s < (4m) 2 ,. if the B.S. kernel ~!4) (or %(4)) satisfies this condition, in one says that it is "two-particle irreducible" (see C.II for the perturbative interpretation of this notion). Now, if A I ~(4) = ~, it results from a standard application of the edge-of-thewedge theorem to ~(4), that the latter is ana£y£~c in ~ complex n~ghbo~hood of the
(I,J)
region {k = (kl...k 4) C 'fl~4
2
; s = PI < (4m)2}
. Then by exDloiting this infor-
mation in equation (B.S.), namely by applying the C.F. lemma to ~(4). considered as in the Fredholm resolvent of ~!4), it is possible to recover the analytic continuation of .(4) across the cut (2m) 2 ~n< s < (4m)2 , as well as the two-sheeted monodromic strutin ~(4) ~(4) ture of • around the singularity s = 4m 2 ; the threshold behaviour of . in in in (s - 4m2) 1/2 can also be derived from a general study of the Landau singularities produced in complex Fredholm theory (see [33] theorems 3 and 4) ; let us simply notice here that the singularity s = 4m 2 appears as being produced by the pinch of the two polar singularities (z I ± ~ ) 2
~!4): in this mechanism is a generalization of the production of pinch singularities in the
Feynman integrals•
= m 2 which are present for the analytic function
186
Finally, the geometrical equivalence between the two previous procedures of analytic continuation in the second sheet can be explained as follows ; when Pl is such 2 that (2m) 2 < Pl < (4m) 2' F+(pl) is homologous, to F_(pl) + ~(pl ) in the z|-section of the primitive analyticity domain of ~xn~4)(pl,Z,Zl) : here ~(pi ) denotes a closed four dimensional contour which can always be supposed to be contained in an ~tb/AtkoJty Sma£~ complex neighbourhood in £4 of the mass shell sphere *~IJ)(PI)-
(see [34] ).
Therefore the region of the half-plane Im s j O for which F+(pi) can be distorted, ¢(4) with the constraint of staying inside the domain~of ~. is identical with there,±on ~ e tf~(IJ).(pl)' can be distorted in inside the same domain ; but for which ~(pi ) , and there=or this region of the s-plane is precisely that which can be reached through the first procedure (see remark iv) above). However, differences between the two methods may appear in more general cases at a more global level of analytic continuation (see C II).
C-l.
Local a n a l y t i c continuation of the 2 ÷ 3 and 3 ÷ 3 p ~ l e
sc~t~ng
amplitu-
des obtained by a d i r e c t use of A.C. equations. a) The 2 ÷ 3 case
: The scattering amplitude associated with the reaction chan-
nel (l,J) = ({4,5},{I 2 3}) is a function of the five Lorentz invariants, namely : the three subenergy variables s|2 = (k I +k2)2, s13 = (k1+k3)2, s23 = (k2 +k3 )2 (the -squared total energy s = !k4 +k5)2 = ~ | +k 2 +k3)2 being such that s]2 +s23 +s3|= s+3m 2) and two independent momentum transfer variables such
as
t|4 = (k I +k4) 2~ and
t35 = (k 3 +k5)2. The_ local analyticity of SIN .(5) = H(5)I~Ij)
in t|4 and t35 in a com-
J
plex neighbourhood of all the physical points is an old result [35] which is the andhad logue of the Lehmann ellipses of S (4) • In the variables (s12,s23,s31), _(5) 51J been shown (in [20c]) to be analytic in a certain local tube with conical basis contained
in
{(sij); Im s12 > 0, Im s23 > O, Im s31 > O}, the size of the lat-
ter being very "thin" in the low energy strip (3m) 2 < s < (4m) 2. A much more satisfactory result has been obtained in [6]
by carrying out Glaser's method : in the
low energy strip, the A.C. equations for ~(5) can be written : A
~(5)
~(5) ~!5) + - i- =
{jk} where
1 ~(4) 2 +
,{jk} ~!5) 2 i- -
(i,j,k) = p.c. (1,2,3), and ~5 +
I ~(4) ,{jk} ~(5) 2 2 + '
H(5) denote the boundary values of ~(5) from '
i-
the respective tubes :
{q;ql2 E v+' q13 ~ V+, q23 E V+}
and {q;qij E VE,qik +
V+,- qjk E VE,Vq123 #}+
Then by plugging into the above equation a local decomposition of~Hi 5) involving an auxiliary set of analytic functions F. +
= FI+
F3+ '
l-
= F.l_ +F.j+ +Fk+ with Fi±
'
i-
}
introduced in [20c] (namely : analytic in "wide" local tubes
+
with bases ^ 407i .= {q; ±qjk E V + , q123 E V + ), and by performing the analytic continuation of H~ ~{~k}F i in @i under the form ~4'+ ,{Jk}2 Fi-' one obtains a certain
187
analytic continuation of each Fi+ across the two-particle cut Sol = 4m 2 + 0 (0 > O). ~(5) JK . By iterating twice this procedure and then coming back to n+ , the authors of [6] have obtained the following result :
Theorem : "At almost"every point p = (PI"" P5 ) of the low energy physical region (s = (Pl +P2 +P3 )2 < (4m)2) of t h e 2 + 3 p a r t i c l e scattering oIjO(5) (equal mass case I = {45}, J = {123} ) the l a t t e r i s the bound~y value of an analytic function from a l o c a l tube whose basis i n the space of the i n v a r i a ~ s is Im s
> O.
Remarks : i) the exceptional points at which this result fails are the submanifolds with equations (Pi +Pj -Pk )2 = m 2 , with {i,j,k} = {1,2,3} ; the latter include in particular the two-particle thresholds ~ j study of [6]
= (2m) 2 ({i,j} c {1,2,3}). The microlocal
does not yield the analytic structure of blj^(5)at these exceptional
points, although the latter could in principle be investigated by a more global use of Glaser' method. ii) An aspect of the obtained result is that ~(5) has been analytically blJ continued across the three two-particle cuts s.. = (2m) 2 + 0 (0 < 0 < 5m 2) ; this can 13 be visualized with the help of fig. 2, in which hatchings represent the regions of Im sij-space corresponding to an "unphysical sheet" analytic continuation.
b) The 3 ÷ 3 case : The scattering
~mS=O
amplitude associated with the reaction channel (l,J) = ({456},{123}) is a function of eight Lorentz-invariants, namely : i) the six (incoming and outgoing) subenergy variables s45
Fig.2. The side Im s > O of Im s..space (a section of it by a spherl~ centered at the origin has been drawn).
s12 = (k I +k2 )2, s23 = (k 2 +k3 )2, s3| = (k3+kl) 2,
= (k4 +k5 )2, s56 = (k5 +k6)2 , s64 ~ (k6 +k4)2 which are linked by the relation :
s12 +s23 +s31 = s45 +s56 +s64 = s + 3m ;
s denotes the total energy variable
s = (k! + k 2 +k3)2 = (k4 + k 5 +k6)2 ii) three independent momentum-transfer variables such as : t14
=
(k I +k4 )2 , t25 = (k 2 +k5 )2 , t36 = (k3+k6)2
By using the A.C. equations for the six absorptive parts
^(6) &{jk}H
({j ,k} c{l,2,3}) and A{~,],2,3 } H (6) (% 6 {456}), the authors of [6] have performed a local analytic continuation of ~(6) H(6)~( which is similar to that of ~IJ = I J) I" v 6
S(5) IJ described above. Here again, their starting point is the "weak" local analyticity
188
of ~Ij~(6)derived from locality and spectrum which (in its refined form [20e]) implies a certain decomposability property of bij~(6)in terms of auxiliary analytic functions. From considerations of perturbation theorv and of S-matrix theory, one can expect that the Landau varieties associated with the following tree-graphs and triangle graphs will be obstacles to analytic continuation, since these varieties have real
"~+-branches"
in the low-energy physical region (s < (4m)2) ; these va-
rieties are : ~h~
%h%
=
~
(ki + kj
sij Smr~th%2 m
and The
sij - 4m 2 e+-branch of
+k~)2
-m
2
-
sij +ti~ +tj% -4m 2 = O
i j~--%
for
+ (sij + mn + t %)-2(sij Smn +Smnth% + sij the) = O
h
= O
j
, Sm n - 4m 2
=
O
n
%h~ = O is defined as follows (see fig. 3) :
{(Pl • "'P6) E ~(I,J); 6
lh~
= O,
s ij
+th%
<
4m 2,
s mn +
th£ < 4m 2}
+
Indeed, the varieties
@h% = O, lh%= 0 are involved in the microlocal result of
which can be described as follows :
Theorem : At almost every p o i n t p = (PI""P6) of t h e low energy p h y s i c a l region ( s < (4m) 2} of t h e 3 ÷ 3 p a r ~ c l e s c a t t e r i n g amplitude ~Ij~(6) (equal mass case ; I = {4,5,6}, 7 = { 1 , 2 , 3 } ) , t h e l a t t e r i s t h e boundary value of an a n a l y t i c f u n c t i o n from a l o c a l tube whose b ~ i s i n t h e space of t h e inva~iants i s : i ) Im s> 0 iS p does not belong to any v a r i e t y ii)
@h~ = O,
+
~h~
= O,
Im s > O, Im @h~ > 0 (resp. Im ~h#~ > 0), i f p belongs to a s i n g l e v a r i e t y
+ + 8h~ = 0 (resp. ~h~
=
0),
£ i i ) Im s > O, {Im @h~ > O, Im ~h,~,+> O} , i f p belongs to t h e i n t e r s e c t i o n of t h e co~esponding v ~ e s / m .'~J~/l///////////////////~
5 zl./
{@~ = O, ~h'~' = 0}.
~
V
which this result fails is the union of all the submanifolds
- t~sI X3s(S'"S~s't~)= 0 ~/ .I 0
Remark: the exceptional set at
~
-t35
5m 2
Fig.3. Asectionof the3->3 physical region at t36 fixed ; the latter exhibits the relevant branch %+r_ = O of %~& = O, and a typical local analyticity domaln (Im A ~> O) forpE{)~o~=O} (s1~-si~-4m ; s45 =s45 - ~m ~).
(Pi +pj -Ph )2 =m 2, ({i,j ,h}={4,5,6}) and )2 2 (Pz + P m - P n =m ({£,m,n}={l,2,3}). (see also the remark after the analogous result in 2 + 3 case).
189
C-II. The method of 1-particle ~ e d u c i b l e kernels ; some local and global r e s u ~ for H(6) and for the 3 ÷ 3 particle scattering amplitudes. The £.p.i.
kernels have a "natural definition" in the perturbative expansion
of lagrangian Q.F.T.
: the
1.p.i.-part I - - ~ -
J
of H (n) with r ~ p e ~ to a
given channel If,J} is defined there as the formal series of all Feynman
amplitudes
which correspond to l.p.i, graphs in the following "topological" sense : a graph is /-particle irreducible with respect to (I,J) if (l+ ]) internal lines of ~ at least must be cut in order to yield two disjoint connected graphs, carrying respectively the sets of external lines labelled by I and by J. Equations of the B.S. type were primitively introduced in this context, as identities of formal series which relate the expansions of the H (n) with those of their l.p.i, parts ; an intuitive graphical notation was used for writing the latter; for example for equation (B.S.) :
k c~
•
k ]"~'~
"~" k3
(B.S)
k I"-~k.Q ~ "
k3
=
graph
k2"~'k-'l~'~k'k4
~ k |~x~" ~ +
k2"Y
~
k4
~'k3
1
k2">r"
k~
"~'~k4
where the notation of the last term indicates the operation Jf dkrydkB6(k | ~ as a Feynman z In
'
+k 2 - k
-~k~,
amplitudes ;the link with the previous notation for (B.S.) is : k! -k 2 k~-k$ , k3-k4 2 ' zl 2 , z 2
[8]
, Symanzik proposed to use these equations for defining the l.p.i.
parts ofH (n) in axiomatic Q.F.T. and outlined a program for applving these equations to the Manv-Particle Structure Analysis (M.P.S.A.) of Green's functions. A svstematic and rigorous exploitation of these ideas in the complex momentum space framework was initiated in
[7] and developed in collaboration with M. Lassalle
and more recently A. Katz in [36,37] as a first step, we had to give a mathematical status to the generalized graphical Feynman leads us to introduce and study
notation (as used in (B.S.)graph); this
~-convolution p r o d u ~
;
we now summarize the (old
and recent) results of this study.
_~-convolution: Let us define a
~eneral n - ~ o i ~ f u n c t i o n F(n)(kl,...,kn ) as an analytic function
in the domain D n described in part B, and whose absorptive parts AI-F(n) satisfy the properties a) and b) described there, except that the support of AI F(n) with respect
F (n) iS l.p.i. with respect to the channel (I.J.) i f the support of &iF(n) iS contained in ~+ (l+1)m to PI can be more general and depend on the channel (I,J). We say that
(in the single mass cas~. Let~ {v C ~ } ,
denote a connectedm-looped graph with n external lines,set
set of interhal lines {i C ~ }
vertex v, and k
V
of
vertices
: n v denotes the number of lines incident to
E C 4(nv-|) the momentum variables attached to v, which are linear
190
functions kv(k,kin t) of k E ¢4(n-I) and kin t C ¢4(m-1) (the external and internal momentum variables of ~ ); the momenta k i = ki(k,kin t) E ¢4 are similarly for all internal i
C~. ~
introduced
will be represented graphically with "fat vertices" (or
"bubbles") in place of the pointwise vertices of ~eynman graphs. The following "conservation" theorem has been proved (see [38] for a more precise statement) : (nv) Theorem : B~ng given a set of general nv-point functions {F ;v £ ~ and a s ~ of general 2-point functions {Fi(2).i, E~} which s a t ~ f y appropriate decrease p r o p ~ e s at i n f i n i t y in euclidean space, there exlst~ a g e n i a l n-point function F~n) , tailed " ~ - c o n v o l u t i o n produ~ of the functions F (nv), F~2)'~ whose restriction to the euclidean space is defined by : (nv) F(n) (k)
=
I v~ {kin t fEucl-space}
Note that the
g
(kv(k,kint))
N .[F~2) (ki(k,kint)) ] 1 iE~ J "dkint
analytic function which is integrated in this formula admits
[ki(k,kint)] 2 = m 2 as a
simple polar
manifold (it thereby generalize~ the Feynman
integrands). We call ciated with case III=
Q l-product F 1 G1 F 2 the~-convolution product of* FI,F2,H(2) asso~
= I
IJI = £ ,
J ; £=
ILl. Of special interest is the
the i£~ated ~y-p&oduc£ of a geneBethe-S~pet~ inv~e G (2£) of F (2£) ;
for which one can consider
ral 2 -point function F (2£), as well as the
G (2£) is the solution Of theFredholmresolventequation
: F (2£) -G(2£)+ | F(2~)Q
G (2~),
and on the basis of the C.F. lemma stated above, one can prove :
Theorem : G ( ~ ) ( k i , z , z ' ) (where z = k I ~ , z ' = k ( j ) ) is a meromorphie function of the f~N 21) (ki,z,z')/D(21)(kI) , wh~e~N (21) (resp. D(21)) is a g e n i a l 21-point (reap. 2-point) function which is analytic at least in the somain D21 of F (21) . Note that N and D can be expressed as convergent series which involve all the iterated
~)l-products of F (2£)
The computation of the absorptive parts of%-convolution p r o d u ~
has a crucial
importance for M.P.S.A.; here we shall briefly present some results for the absorptive parts of Q/-products.
a) absorptive part of an i~erated O l - p r o d u ~ i n a crossed chann~. Let IF(2/)]~)/r ted with the graph ~ =
the iterated
Q 1-product of order r of F (2/)
ii__~ f ~ j_ . \i /
. ...... .....
:
it is assoeia-
~F'Y-~----rt___Ij .....~ , with (r+l)"bubbles",
and (l,J) is called the "convolution channel". A "crossed channel" (l',J') is by In certain developments, one prefers to replace H (2) by (p2 _m2)-; which has no C.D.D. zeros.
191
definition such that
I' D I, J' D I, I' D J, J' D J are all # @. The following
general statement has an easy graphical interpretation in perturbation theory :
lemma : IF(21)] ~ r
is r-partite
i r r e d u ~ b l e with respect to a l l the crossed chan~
nels. b) absorptive part of a Q 1-product in the convolution channel (I,J) . We give the expression of the absorptive part of F Q o G
in terms of those of F
and g in the cases £ = ],2,3 (for arbitrary values of llI ~and IJ[ ) : i) ~ = ] [30a]:
AI(F~) ]G )
=
F + .AIGamp" + AIF.Gamp-
ii) ~ = 2 [30b]:
AI(FQ2G )
=
F+ ~) F+ AIG
^
+
^
&IF (~ F_G + F°ut *2 Gin'
2 valid for (2m)2< Pl < (3m)2 ^
iii)
Z = 3 [37] : A I ( F Q 3 +
~ (hij)
G)
IF +
^
F + (~ F+AI G + A I F ( ~ F _ G - + Fout *3 G.In
ij ,h GF+ ij .h ~)Y+ A(iJ} + A{ij} ~)~/-
+
G- ] J
...
'
2 valid for (3m) 2 < Pl < (5m)2' under suitable assumptions of local analyticity for F and G. In i), G amp ~)
= G×[H(2)(kl)] -| (or Gx(k21-m2)).
In ii) and iii) the notations
(inplaceof
~)2, Q ~ refer to appropriate distortions r+,r_ of the integration F J ^ cycle in the llmlts kT = p +- ig, E 6 V , g ÷ 0; F = F ~ [7 (k 2 - m 2) ^ I out ~ i6L .. _n 2 2 ± . k l lJ G. = G. IT (k. -m ) (as in equation (B.S.)). In iii) the slgns G y + * corresin . 6~ I pond to an integratlon prescrlptlon on the momenta k.,k.,k_ ({i,j,h} =-L) such that k.-k. l j n kh m and 6 ¥ + or y_ ( s u i t a b l e o f f - s h e l l c o n t o u r s ) . +
.
•
i
•
oL
•
.
.
These d i s c o n t i n u i t y
+
.
f o r m u l a e ( o b t a i n e d by c o n t o u r d i s t o r t i o n
n i q u e ) a r e b a s i c f o r p r o v i n g t h e e q u i v a l e n c e between ticle
irreducibility
and r e s i d u e t e c h -
, ~ - p a r t i c l e A.C. and t h e ~ - p a r -
of an a p p r o p r i a t e 2 £ - p o i n t B e t h e - S a l p e t e r k e r n e l
( s e e above f o r
the case £ = 2, and below for the case ~ = 3).
A remark concer~ng the applicaZio~ of ~-convolution : Apart from its basic role in M.P.S.A., which will be illustrated below by our analysis of H (6), the formalism of ~ - c o n v o l u t i o n provides a "natural" framework for writing and studying the equations thatGreen's functions of a given lagrangianQ.F.T. must satisfy. However this immediately sets the question of divergences at infinity for Q-convolution products which involve the physical n-point functions
H(n).
192
In M.P.S.A. one gets rid of this difficulty as follows : in the
-convolution
. (n) i s used , products and equations of the B.S. type, a regularized version of the ~ namely, H(n)(k|...kn ) ×o~ (m 2 - O ) ~ ( k ~ - 0 ) - ~ w i t h
O and ~ sufficiently large (note
that H is thus u n m o d i f ~ on the mass shell) ; thereby the analyticity properties n of the H (n) which can be derived in M.P.S.A. are shown to be independent of the asymptotic properties of the latter at infinity and purely determined by the geometry. However, for setting the
~-convolution
it seems necessary to use a renorm~zed
equations of a lagrangian field theory,
form of ~-convolution products. In [39], a
generalization of Zimmermann'srenormalizationprocedure
[40] h a-s been presented; itde-
fines a renorma£ized in£egrand for ~ - c o n v o l u t i o n products involving functions F ~nv) which belong
to Weinberg classes [41]of a special type. The convergence theorem and
the asymptotic properties of the integrals which are stated in [39]
furnish a star-
ting point for developing rigorously the approach of Q.F.T.-models based on the system of equations satisfied by Green's functions in momentum space : this study has been undertaken by M. Grammaticou.
Result~ of Many-Par/~cle S t ~ u ~ e
Analysis :
the main points in this scheme are the following : i) show on the basis of A.C. that appropriate n-point functions, introduced by B.S. inversion and ~ - c o n v o l u t i o n operations involving the physical n-point functions H (n), h a v e
%-particle irreducibility properties,
ii) express each H(n)through R-convolution structural equations which may involve other H (n') with n' < n, together with various
%.p.i. parts of H (n). These
equations should. a) be valid in well-defined regions of complex momentum space, whose size increase with the rank
% of the %-particle structure which is exploited.
b) exhibit for each of their terms an analytic and monodromic interpretation : this amounts to analyze how the Landau singularities attached to various graphs contribute to Green's functions and scattering amplitudes and thus to give a global description of the Riemann surfaces of the latter above the regions mentioned in a). This analysis can be done through a recursion over
The c ~ e £ = I : has been completely treated in[3Oa]:
%. the content of this prelimi-
nary case is that the residue factorization property holds ~ at all the poles of the H (n) which are poles of H (2) (in all channels and for theories with an arbitrary mass spectrum).
a generalization of this property to the case of bound states and unstable particles can also be established [42] .
193
The case I = f : has been partially presented above : the 2-p.i.part L (4) of H (4) in s a given channel (I,J) (k~ = s) has been introduced through equation (B.S.); it is a general 4-point function meromorphic
in D4 ; moreover the two-particle A.C. assump-
tion implies that L (4) is 2.p.i. and this implies in turn the two-sheeted monodromic s
structure of H (4) around s = 4m 2. A similar result has been derived for all the H (n) without new B.S. inversion
[30b].
Moreover, by using the property crossed channels,
(a) of the absorptive parts ofC) 2-Prdducts in
it has been possible to construct and recognize "parts" of H (4)
which are simultaneously
2.p.i. with respect to two channels s and u
(L (4) = L (4) + L (4) - H (4)) or with respect to the t ~ e e channels s t,u su s n (~4) = L(4) + _(4) + L(4) _ 2 H (4)) Such kernels which satisfy crossing symmetry, stu s Lt u " ' may be used for global problems ; in this connection, let us mention Sommer's recent work [43] in which the problem of reconstructing H (4) from L(4),considered as given su in a certain domain, has been solved by a fixed point technique ; this work also contains an interesting off-shell extrapolation of Martin's domain obtained by the exploitation of positivity properties of ASH(4)
(see part B : absorptive parts AIH(n) ,
property c)). The rest of this section is devoted to a further step :
One-two-and t ~ e e - p a r t i c l e structure analysis of H(6) (even Q.F.T. with a single mass) NotationS: We put H(6)(k],...~k6 ) = H(6)(K,Z,Z'),
where K = k ] +k 2 +k 3 =K(k4 + k 5 +k6) ,
Z = (k_4,k5,k6) with : k i =k i +-~ , i =4,5,6 ; Z' = (k],k2,k3) with k. = k i -7' i = ],2,3. •
-
.
.-
(o)
K is kept In the manlfold K : (K varies in a domain
÷
-1
,0), and K (°)
&cut = A M _ { K o > 3m}, where M
.TinK(°|~
A M = {K (°);0 < Re K ° < M, with 3m < M_< 5m}
ReK(°)
Fig. 5. (below) The situation in Re Z'(°)plane ; c0h is the polar manifold ~ =m2,o his the branch manffold sij = (k i +kj) 2 =4m 2 ({i,j,h} = {],2,3}) ; the dark disk at the center of fig.5a) represents the physical region ~ I J ) in projection.
(I = {4,5,6}
, J = {],2,3}),
Fig.4:
The domain AMu t
ko - K •
(K)
~'~,<
Fig.5a. K (°) > 3m
Fig.5b.
K (°) < 3m
kc~~--c
194 For every K (°) in AM, one defines the following domain D(K) in Z-space : DK = {Z = (Z(°)',~); Im Z(°) E IR2 , Re g(°) E ~Q(K), Re Z E IR6 ;Ilm ~ I < ~} where the domain ~(K) is represented in light hatchings on fig.5. ~cut The following domains ~ M , ~ M
are used in the results below :
~ M ( r e s p ' @ M ut) = { (K(°)'Z'Z')' K(°) E AM(reap" ~M'cut'); Z E D(K) ;Z' C D(K)}
One-and-t~-parti~e dressing of H(6) in the channel (I,J). The aim of this step is to express * H(6)
in terms of a suitable irreducible part
G (6) and of auxiliary kernels which only involve H (4) and L (4) . We first define the latter, namely : •
"
T.in is the tree-graph~, product
h-~nnh
Tout
~
which we call respectively U. A =
~
- I h~n
A V = ~
h~n
n
2~
n
and similarly
; they admit B.S. inverses (for Q3-pr0duct) and U
and
h
' ~
. We also introduce the operators V = ~
I + -6
which satisfy : hn
n
! ! - ~ Tin, V A = ~ - ~ Tou t .
The B.S. inversion of Tin, Tou t allows to define the inverse of A as :
! I A -I = V.(I] + ~ Uin) = (I] +~- Uout).V. I
We now introduce the six-point function @(6) through the ~-convolution equation :
11(6) = HI6)
=
At-II6)fi H (6)
_
~T o
(note t h a t A Tou t = Tin fi), where ~
~
is the l-p'i" part °f H(6) ( ~ in~(6) the channel
(I,J). One also introduces two other useful six-point functions ~,6~ in ' Bout are such that : 11(6) = A11(6) = ~!6) A out in
which
The following irreducibility properties can be proved for ~(6), 11!6)- 11(6)- : in ' out a) They are ]-p.i with respect to cb~ channels, b) 11(6)(reap. ~(6)~ is 2-p.i. with respect to the triplet {4,5,6} (reap.{1,2,3}); in ~out ) ~(6) is 2-p.i with respect to both triplets. ("2-p.i. with respect to the triplet {4,5,6}" means "2-p.i. with respect to the three channels ({ij},{h,],2,3}) with {h,i,j} = {4,5,6}"). This implies in particular the following property of $(6)
= 611 ( k ~ - m 2) 11(6) i=l
*Actually, H (6), H ( 4 ) H (2) are replaced here by the corresponding regularized forms (see our remark on ~-convolution products).
195
Lemma : ~(6) is analytic in the domain ~'~ ~ut, such that M = 11m/3. Now the equation which defines G (6) can be inverted and yields the following "two-particle dressing equation" HI 6)
(~
+ 1
G(6)
~ Uout) V
+ l
-6 Uin) + Uout V
V(~
By using the analyticity properties of H (4) and C (6), and by exploiting in complex space some standard combinatorics of Fredholm theory, one can prove
Theorem :
In ~
~(6) ....
: +
(with M = 11m/3) the following decomposition of ~(6) holds •
~-----C )__C ~ + h!n h - ' ~ O l l l ~ h,n
+
~ h,n
+
~ h,n
+ "'"
l~hn
in the l a t t e r , a l l the "bubbles" of the ~ - c o n v o l u t i o n products represent H(4) and each analytic function ~hn of the residual sum following properties :
(h £ {4,5,6}, nC{1,2,3}) has the
(c~) - u i) ~hn ds analytic in a " f i r s t - s h e e t domain" which i s ~ M - (Oh ~n ) ; here the set~ ~h' ~n denote the following "cut~" • ~h = { ( K , Z , Z ' I ; ~ j ( K , Z ) =4m2+ p; p>O}ahd ~n={(K'Z'Z');Slm(K'Z') =4m2+p; p > 0}.
di) ~hn admits a local analytic continuation across the sets oh, ~ and 6n on the Riemann s ~ f a c e asso~ated with the Feynmann graph d ~ ~ h~ ~ n Besides, each ~ - c o n v o l ~ i o n term of t h i s decomposition i s analytic i n ~ cut minus the Landau singular s e t ~ s o c i a t e d with the corresponding graph i t moreover a d m ~ a local analytic continuation on the Riemann surface associated with ~ . Rema~____kk• The o n l y s i n g u l a r i t i e s
of ~(6) which a r e prodnced in ~ut
_
U (~h U ~n ) , hn
considered as a first sheet defined by the extension of the primitive afialyticity domain, are the leading Landau singularities of the 0-loop, l-loop and two-loop "truss-bridge graphs"; the i-loop graphs (~ > 2 ) ~ ~ l ! - - - ' ~ are potentially present in the Neumann expansions of Uin , Uou t
-)~
and illustrate
the two-particle dressing of Hh6)r from both sides of @(6) in the expression given above ; howev,er, it is a consequence of the previous theorem that their Landau singularities are effe~tgve only in other shee£s. This analysis has not yet been done, but our method should make it feasible. qD(e! By taking the restriction of H (6) to the complex mass-shell..~ 6 one obtains the following corollary of the previous theorem :
Corollary : In --t,~ nq~6, ~ e 3 -+ 3 p ~ c l e scattering amplitude admits the following decomposition as a sum of analytic functions :
196
SIJ( 6= ) ~
h,n
-h
n _~EI~
+
~ h,n
~ - ~ ~
n
h cut
+
~ h,n
~hn'
• (c)
wh~e ~ach function ~hn i s analytic i n ~ M n ~ - (S h U 6n); across each cut ~h,~n @hn a d m ~ a l o c ~ analytic c o ~ n u a t i o n w~ch spreads in a two sheeted Riemann s ~ f a c e around each t h r ~ h o l d ~h,~n. ^
Remark
:One reobtains as a subproduct, and for K (°) < M, the result of [6 ] described
in C.I. ; in this region, one obtains actually here a more global description of the analytic structure of ~(6) and ~(6); however our limitation M = I|m/3 should be imbIJ proved (it is due to the apparent obstruction of certain crossed channel singularities in
-convolution
point
deserves
products,
b u t may h o p e f u l l y
a more c a r e f u l
be e x p e c t e d ~ t o
be f i c t i t i o u s
: this
analysis).
The t ~ e e - p a r t i c l e i ~ e d u c i b l e
k ~ n e l L (61 .
Let L(6! (reap. L! 6)) be the B.S. inverse of @(6) (reap. $ ~ ) ) w i t h respect to one in out the~)l-product, and let us put : L (6) = A L (6) (= L (6) A). The following property of (6) 6 out -in L has been proved :
Theorem : There i s an eq~valence between the t h r e e - p a r t i c l e A.C. equation : ~ . ~(6) ,~I) ~(6) and the f a c t t h ~ L (6) is 3 . p . i . , i . e . &i L{6) = O, in the region (pl)2 = (KIOJ)2"" < ~42 (the present l i m ~ a t i o n ~4 = 11m/3 b~ing hopefully provisional). The proof of the latter amounts to showing the following relation for appropriate integration cycles F+(Pl), F_(p I) :
•
in
~+
in fact, as it has been proved in
_
'
[30a], the~racket at the r.h.s, of this relation
vanishes if and only if the A.C. equation for H (6) holds• Thanks to this equivalence pgoperty, the following consequence of three-particle A.C. can be derived : ~(6)
=
i~ I (k~ - m 2) L(6) is analytic in the whole domain ~ M '
and therefore in a complex neighbourhood of the three-particle physical region s =
2 < M 2. Then by plugging this new information into the equation : L(6) + ~(6) A-I = L (6) , one can show that admits £ o c ~ analytic conti-
(6) pI
nuation
across the three-particle cut
~ = {(K,Z,Z');(3m) 2 < s = K 2 < M 2}
from both
sides Im s > O, Im s < O. As a consequence, the same property can be established for ~(6) ; more precisely, this property holds for each term at the r.h.s, formula of the decomposition R~ark
theorem.
:This result contains as a subproduct the local analyticity properties which
are sufficient to ensure maerocausality conditions *in the sense of--Stapp-Iagolnitzer's S.matrix theory
for the 3 + 3 scattering process
197
in the considered region s < M 2. We shall finally present a joint application of the above local analyticity properties based on A.C. and of a typical extension of D6
, obtained by methods of ana-
lytic completion.
A ~ o s s i n ~ domain for t h e 3~ ~ 3~ s c a t t i n g
ampl~de.
We consider the crossing problem for the couple of channels and (I'J') = ({156},{423}),namely S(6) = ~(6)
we seek ananalyticitydomain
~6 •
, >
respectively
0 (s
,
2
= ki,)
IJ
.
from the slde Im s
2
~I
O (s = k I) and ir~6
'J'
from the
will then be
.(6)
the scattering amplitudes .(6) and bI'J'' bIJ
being the following one-dimensional =
>
: the correspondlng boundary values of S
- (c)
We shall define @ as a neighbourhood
~
@ on ~ (c) for
(c)Whose boundary contains parts of both physical regions "~6° IJ and ~I'J',6
and such that it touches ~ 6 side Im s
(I,J) = ({456},{123})
(kl,...,k6)
in 'r~l~6 of a certain domain 0
%c ~ ~ ,
submanifold of "forward configurations"
; kl +k4 = O, k 2 + k 5
kl = (kl°),kll),o,o),
in ~
= O, k 3 + k 6
= O ;kl
= m
:
,
k2 = (mch~,O,m sh~,O), k 3 = ( m c h ~ , O , - m s b ~ , O )
will be chosen sufficiently small• On this manifold,S (6) is singular
}
since the
pole (kl +k 4 +k5)2 = m 2 of H (6) reduces to k 52 = m 2 on the mass shell. Therefore on ~,
it is necessary to consider the crossing problem for the
fom~d p~ot* of
~(6)
which we define by :
F
where the summation
=
hn
I' runs over the set
The following lemma can be proved for
{(h,n); h E {4,5,6}, n E {1,2,3};h # n+3}. ¢(6) ~F"
Lemma : t h e r e ex~st~ a crosnigr domain @ on ~ (0 < ~ < s o) for t h e "forward s c a t t e r i n g amplitude" S~ 6) = ^rH~6) domain, whose shape i s indidated on
(c).~Th/s
~6
f i g . 6 tends t o a cut-plane i n t h e v a r i a b l e s when ~ tends to O. We note that on ~ ,
the physical region of the channels
(I,J), (I~,J') are
symmetric with respect to the origin if one takes k~O)r as the variable, k(O)= = 1
s-m'2(l +4ch2~) 4m ch ~
=
_
and that :
s'-m2(| +4 cb2~) 4m ch
The situation was similar in the works which have been presented or mentioned in Part B II.
198
-"~ a~a'
~t
a a~
rn2
e Fig.6. The crossing domain @~ in the -(O)_plan k! The proof is obtained by performing the analytic completion of the union of the two following regions in the space of two complex variables a) For ~
(2 -ch2~) 2 ch2
real < ~e
m 2, H~ ^ 4)
k o) and ~ = k I .
is analytic in the k. 1(o) cut-plane
(the cuts being given by s ~ 9m 2, s' > 9m2). This results from the Jost-Lehman-Dyson 2 in k]-space. Note that ~ < m 2 and that ~ ÷ m when ~ ÷ O.
completion [44] b) When
kl °)
varies in a neighbourhood
of the low energy physical region consi-
dered above (limited by s < M2), SF t.(4) is analytic in
whose size is independent of kl°)and ~ , when e
~ in a n e i g h b o u r h o o d ~
of ~
tends to mero. This results from the
fact that such a domain is contained in the analyticity domain obtained in the decomposition theorem for ~(6). The standard interpolation technique in two complex variables then yields the crossing domain of fig. 6, if
~
" small compared with the size of - m 2 is
is satisfied if ~ is chosen sufficientltly
: this
small.
To obtain the crossing property for ~(6) itself, it is sufficient to notice that H 6) is analytic in a neighbourhood terms
J
(k i + k j - k n ) 2
0
of @
are analytic = m2 ( t h i s
in
c) and that in @ 6'
e x c e p t on t h e c o r r e s p o n d i n g
f o I l o w s from t h e a n a l y t i e i t y
the tree-graph poles
of the 2 ÷ 2 forward scattering).
REFERENCES [I] G.F. CHEW, The Analytic S-Matrix, W.A. Benjamin, New York (1966) and references quoted therein. [2] D. IAGOLNITZER, The S-matrix, North Holland, Amsterdam (1978) and references quoted therein. [3] J.P. ECKMANN, H. EPSTEIN, Comm. Math. Phys., 64, 95 (1979). [4] S.M. ROY, G. WANDERS, Physics Letters, 74B, 347 (1978) and Nucl. Phys. B]41, 220 (1978). [5a] A.A. LOGUNOV et al., Theor. Math. Phys. 33, 149 (]977).
199
[5b] L.M. MUSAFAROV, Theor. Math. Phys. 38, 36 (1979). [5c] V.P. PAVLOV, Theor. Math. Phys. 37, 154 (1978). [5d] V.P. PAVLOV, Theor. Math. Phys. 35, 3 (1978). [5e] L.M. MUSAFAROV, V.P. PAVLOV, Theor. Math. Phys. 35, 151 (1978). [5f] L.M. MUSAFAROV, Theor. Math. Phys. 35, 291 (19787. [6] H. EPSTEIN, V. GLASER, D. IAGOLNITZER, in preparation. [7] J. BROS in "Analytic Methods in Math. Physics", p.85, Gordon and Breach, New York (1970). [8a] K. SYMANZIK, J. Math. Phys. I, 249 (1960). [8b] K. SYMANZIK, in Symposium on ~heoretical Physics, 3, New York, Plenum Press (1967). [9a] R.F. STREATER and A.S. WIGHTMAN, "PCT, Spin & StaTistics and all that", Benjamin, New York (1964). [9b] R. JOST, "The general theory of quantized fields", Ann. Math. Soc. Providence R.I. (1965). [10] K. OSTERWALDER, R. SCHRADER, Comm. Math. Phys. 33, 83 (1973) and 42, 281 (1975). [11] V. GLASER, Comm. Math. Phys. 37, 257 (1974). [12] J. GLIMM, A. JAFFE, T. SPENCER, Ann. Math. 100, 585 (1974). [13] See [3] and references quoted therein. [14] O. STEINMANN, Comm. Math. Phys. IO, 245 (1968). [15] H. EPSTEIN, V. GLASER, R. STORA, "Structural Analysis of Collision Amplitudes", p.7, North-Holland, Amsterdam (1976) and references quoted therein. [16] K. HEPP in : "Axiomatic Field Theory", Gordon and Breach, New-York (1966). [17] T. SPENCER and F. ZIRILLI, Comm. Math. Phys. 49, I (1975). [18] D. RUELLE, Nuovo Cimento 19, 356 (1961) and Th-esis, Zurich (1959). [19] See R. J O S T ~ and references quoted therein. [20] J. BROS, H. EPSTEIN, V. GLASER, a) Nuovo Cim. 3_!, 1265 (1964), b) Comm. Math. Phys. I, 240 (1965), c) Helv. Phys. Acta. ~5, 149 (1972). [21] A. MARTIN : Nuovo Cimento, 42 A 930 (1966) and 44 1219 (1966). [22] A. MARTIN, "Scattering theory : unitarity analy-flcity and crossing"~ SpringerVerlag (1970). [23] H. EPSTEIN , J. Math. Phys. 1 254 (1960). [24] a) J. BROS, b) D. IAGOLNITZER in Publ. R.I.M.S., Kyoto Univ. 12 Suppl. (1976) . [25] M. SATO, T. KAWAI, M. KASHIWARA, Lect. Notes in Math., Springer Verlag (1972). [26]A.S.WIGHTMAN, in "Relations de dispersion ..." Hermann, Paris (1960), ref.therein. [27] J. BROS, V. GLASER, l'enveloppe d'holomorphie de l'union de deux polyeereles. (1961). [28] M.C. POLIVANOV, "Math.Pb. in Math.Phys.",375 Lect. Notes in Phys. Springer (1977). [29] N.N. BOGOLIUBOV, D.V. SHIRKOV, "Intr, to the ~h@@~Y of Quantized Fields", Moscow [30a] J. BROS, M. LASSALLE, Comm. Math. Phys. __43, 279 (1975). (!957) ~ [30b] J. BROS, M. LASSALLE, Comm. Math. Phys. 54, 33 (1977). [31] W. ZIMMERMANN, Nuovo Cimento, 21 249 (1961) [32] A. MARTIN, in "Problems of Theoretical Physics", Moscow, Bauka (1969). [33] J. BROS and D. PESENTI , "Fredholm theory in complex manifolds" to be published in "Journal de Math. Pures et Appliqu~es". (preprint Orsay 1979). [34] J. BROS, Cours de 3~me cycle, Lausanne, Mai 1979. [ 3 5 ] R. ASCOLI, Nuovo Cimento, 1 8 7 5 4 (1960) [36] J. BROS, M. LASSALLE, "Structural Analysis of Collision Amplitudes", p.97, North-Holland, Amsterdam (1976). [37] A. KATZ, Th~se de 3~me cycle Paris, Juin 1979. [38] M. LASSALLE, Comm. Math. Phys. 34, 185 (1974). [39] J. BROS, M. GRAMMATICOU, "Renorm-alized ~47convolution I", Saclay (1978) to be published in Comm. Math. Phys ; M. GRAMMATICOU, "Renormalized~-convolution II", Ecole Polyteehnique, Paris 1979. [40] W. ZIMMERMANN, Comm. Math. PhyS. 15, 208 (1969). [41] S. WEINBERG, Phys. Rev. 118 (1960)~. [42] J. BROS and D. PESENTI, in preparation. [43] G. SOMMER, in preparation. [44] R. OMNES, in "Relations de dispersion ..." Hermann, Paris (1960), ref. therein. [45] F. DUNLOP, M. COMBESCURE, n-particle irreducible functions in euclidean Q.F.T. Preprint IHES (1978), and H. KOCH, Thesis, Gen6ve (1979).
CONSTRUCTIVE
FIELD THEORY
Arthur Jaffe 1 Harvard University I.
Osterwalder-Schrader
Quantization
The cornerstone of constructive quantum field theory i S the existence of nonlinear quantum fields compatible with both special relativity and quantum mechanics. The solution to this problem requires construction of the Hilbert space H of quantum mechanics, a unitary representation U(a,A) o f the Lorentz group on H and the quantum field }(f) itself, which acts as a linear operator on H There are two standard (and closely related) constructions to solve the problem: The first method is based on Hilbert space methods, approximate Hamiltonians H n defined on Fock space. With this method one constructs the local [ield ¢(f) , satisfying [}(f), ~(g)]
0
when f, g have space-like separated supports. (One can also obtain C*-algebras of bounded functions OL(B) of %(f) for suppt f C B , which satisfy the Haag-Kastler axioms.) Then one constructs the vacuum representation (yielding H and U(a,A)) by taking limits of the ground states Pn of H n The second standard construction method is to verify Euclidean axioms for a field theory at imaginary time, i.e. with time t analytically continued to -it . Thus Euclidean symmetry replaces Lorentz symmetry. In the case of bosons, locality analytically continues to commutativity of ~(~, t) ~ ~(~, -it) The Wightman functions of ¢ analytically continue to moments of a probability measure d~(~) on the space of classical Euclidean field configurations. For convenience, we take this configuration space to be S'(R d) where d is the space-time dimension. (In the case of fermions, the classical fields are elements of a Grassmann algebra, rather than ordinary random variables.) The inverse problem of recovering (~,U,H) from (¢,d~(~)) requires a statement of Euclidean axioms. One such set of ~ axioms for bosons was given by Nelson and involves the assumption of a Markov property for (%,dD) . We present a simple version of the axioms of Osterwalder and Schrader, which we refer to as :'OsterwalderSchrader quantization." The three axioms are O.-S. 1 (Regularity).
The regularity assumption S{f}
=
is a technical restriction
on
[ e i~(f) d~(¢) J
S'
Assume for some
p <2
coost["fE'L,+ IIf" ] P
O.-S. 2 (Invariance). translation, reflection)
For
E , a Euclidean transformation S{Ef}
=
on
Rd
(rotation,
S{f}
O.-S. 3 (Reflection Positivity). Let ~ be a hy~erplane in R d , e the reflection ~n ~ Z ~ t-hee~ e d components of R \ H . For {fj}, any finite sequence of functions supported in ~+ , assume s{f.-eT.} 1
J
=
M..
13
ls a nonnegative matrix.
lRupported
in part by National Science Foundation under ~rant P ~ 7 9 - 1 6 ~ 1 2 .
201
Theorem 1 [i]. If S{f} satisfies O.-S. 1-3, then there exists (~,U,H,~) satisfying the Wightman axioms, with the possible exception of uniqueness of the vacuum g , and such that vacuum expectation values < ~,~(x I) ... ~(Xn) d> continue analytically to I ~(Xl)... ~(Xn) d~(~) = < ~,~(~l,-itl)... ~ ( ~ - i t n ) ~ > Uniqueness of the vacuum ~ is equivalent to ergodicity of d~. Pictorially, the theorem is represented in Figure i, which illustrates the connection between d~ and H • The vertical projection from L 2 to H is given by orthogonal projection of L2 onto L2+ (generated by exp(i~(f)), suppt f ~ . ) followed by identifying L2+ as a subspace of H with the scalar product on L+ 2 given by O.-S. 3. ->
Euclidean translation V(t,x) . ¢(x,t)
L2(d~(~))
>
L2(d~)
O.S°
O.-S.
scalar product
scalar product
\
e
H
The ~3
5H
etH~ (~, 0) e -tH = ~ (~,-it)
Figure i.
2.
~/
-tH + ix •
Commutative diagram describing 0sterwalder-Schrader quantization. The time translation on L2(d~) is mapped into the semigroup e -t/{ on H . The field is mapped into the imaginary time field ~(~,-it).
Model.
To be specific, we present the In particular, let V(~K(X))
=
~4
model in
d= 3
space-time dimensions.
XCK(X) ~ + aKCK(x)2 - De (x)
,
-I where CK(x) denotes a lattice field on a lattice with spacing K denote a ball of radius A , centered at the origin. Let Idx denote define
Also let A C3Elattic e and
A where
d~
is
Gaussian
measure
with
mean
(-A + l)~ -I (x,y) Also, ZK, A
is chosen so that
=
zero
C(x,y)
Id~K,A=I
.
and
covariance
=
i
By taking
the
lattice
version
of
e-lX-yl
< = 2 -n , we can imbed each
lattice in a fixed continuum space of functions. Theorem 2 [2]. (Existence). for ~,o real and for
There exist constants
a<
=
,S> 0
-~%~ + 8 % 2 1 n < + o
such that for all
,
~> 0 ,
202
S{f}
:
~|e i¢(f) d~(¢)
=
K-~lim S{f}<,A
J
=
-~°lim f e i¢(f) d~(¢)K,A r.
exists, satisfies O.-S. 1-3, and is asymptotic to the standard
#4
theory.
Remark [3]. Existence theorems also exist for P(¢)2 models with P bounded from below, for Yukawa 2 (scalar and pseudoscalar) and for the Abelian Higgs 2 model. 3.
The Phase Transition Picture
Theorem 3. Let ~,~ be fixed and a >> 0. Then the measure dD(¢) (~,U,H) satisfies all the Wightman axioms (including unique vacuum). lim S{f} ~+0+
lim S{f} ~÷0-
Theorem 4. Let ~,% be fixed and let a<< 0. If ~=0, then dp(¢) (~,U,H) has a degenerate vacuum. In this case, S{f}+
=
lim S{f} ~+0+
is ergodic and Also
=
lim S{-f} p÷0-
=
is not ergodic and
S{-f}
satisfies all the Wightman axioms and has symmetry breaking 0
=
The idea in establishing Veff(¢ )
is minimized at <¢> .
=
these results is that the effective potential ~¢4 + (~_Oc)~2
_ ~¢ + small higher order remainders
Herewe have the various potentials illustrated in Figure 2.
h/ ~
=
(~-(7
0
>
Figure 2.
Veff(¢)
in the four quadrants of the (~, a-Oe)
space for fixed a< a
and C
~= 0
% .
Here
is clear.
a c = ac(% )
C
k/
parameter
and the symmetry for
203
4.
Renormalization
The divergent constant a< is an infinite mass renormalization. The constant is the finite part of the mass counterterm after performing the cancellation of infinities. To complete the mass renormalization, we wish to choose ~ > o in such c a way that the physical mass m has a predetermined
value.
=
This is justified by
T h e o r e m 5 [4]. For X fixed, o(m) with this mass gap.
aMOc
It is believed ' namely
inf(spectrum H \ { 0 } )
m(o)~ 0
as
(though not proved)
~
that
Hence given
c m(~)
m
there exists
obeys an asymptotic
power law as
- ~c ~ v
m(o)
=
[ ~ j
Here v is the critical index for the mass introduce the dimensionless variable
{
(or inverse
correlation
length).
We
q-~ e c
to w r i t e m(o)
-~ T
,
T ~- O +
In addition to parameterizing H by the physical mass m , w e would like to choose % so that the long range force b e t w e e n two particles is given by a Yukawa potential of the form g 4~r
- -
-mr e
where g is the dimensionless coupling constant. That such a renormalization is in general impossible follows from the nonlinear nature of the theory: In fact, consider the truncated four point function <XlX2X3X4>T = <XlX2X3Xk > - <XlX2><X3X4>
where we use "x" to abbreviate
¢(x).
3~
: Theorem 6 [5].
For
q>Oc(k)
- <XlX4><X2X3>
Define
r
¢
~
2
-m J < x l x 2 x 3 x 4 > ~ x 2 d x 3 d x j [ J < x y > d y
I
, 0
where the constant
- <XlX3><X2X4>
is independent
! of
g
!
m,%
const.
.
As a consequence of Theorem 6, and the fact that g = c o n s t , g , w e see that g is bounded for q > ~c • In particular, g ~ 0 as o ÷ = . The critical behavior of g and of g is described by exponents: For d = 3 , g
T
3v+2T-
(2A+y)
3v -~ y - 2A =
T
204
where y is the exponent for the s u s c e p t i b i l i t y and A is the "gap '~ exponent relating the four point to the two point function. F r o m T h e o r e m 6 w e conclude (i)
3~ + ~ - 2 A
On the other hand, if
g# 0
at
(2)
~ = ~
c
>
, it is n e c e s s a r y that
3~ + y - 2A
The relations (2) is called hyperscaling. and ~ + 2 B + y = 2 , w e rewrite (2) as (3)
0
=
0
By the scaling relations
3~ - y - 2B
0
39 + ~ - 2
0
A=B+y
,
or
(4)
5.
Ising M o d e l
The ¢4 m o d e l is closely related to the Ising model. Let us fix K in the cutoff action V K and take the A - > ~ limit. We can repeat the 0.-S. construction, using i n v a r i a n c e under lattice t r a n s l a t i o n for d~ K . In place of e -H , we obtain a transfer m a t r i x K with eigenvalues 1 Z I 1 Z l 2 Z .... ~ 0 . Define m = i n l 1 as the inverse c o r r e l a t i o n length, m = ~-i Then m = m ( 1 , o , < ) is a function of the remaining p a r a m e t e r s %, a, < . The Ising limit of the ¢24 m o d e l is a limit w i t h I->~ in such a w a y that m remains fixed a n d ¢2 ÷ const. #~0 , i.e. ¢+ ± const. It is clear that for fixed K , w e can choose ~ so that m is fixed and ¢ 2 ÷ const as I~= . Because of the %2 in K term in the mass renormalization, however, it is not clear w h e t h e r the %-~ and ~-~o limits can be interchanged. We can also express the h y p e r s c a l i n g relations (2-4) in the Ising model. 6.
N u m e r i c a l Calculations
There are two frameworks [6] w i t h i n w h i c h ~ and the h y p e r s c a l i n g relations have been calculated: high t e m p e r a t u r e s e r i e ~ and Borel summation. High temperature series have been used by Wortis et al. in the Ising casejand b y Baker and K i n c a i d in the ¢4 case. Borel m e t h o d s have b e e n used by L e G u i l l o u and Z i n n - J u s t i n in the ¢4 case. N e i t h e r m e t h o d has m a t e h m a t i c a l l y j u s t i f i e d error bounds. The results are + .02 .039 _ .03
Wortis et al.
.028 ±
.003
Baker-Kincaid
3~ - y - 2B = 0.000 ±
.003
LeGuillou Zinn-Justin
Ising$ H.T.:
3~ + ~ - 2
=
¢34
H.T.:
3~ + y - 24 =
¢34
Borel:
Thus the high temperature series suggest a b r e a k d o w n of hyperscaling. It is p o s s i b l e to argue that the d i f f e r e n c e in these calculations can be a t t r i b u t e d to the exponent In particular, Ising 3 ¢34
H.T.:
Borel
:
~
=
+ .002 .638 _ .001
~
=
. 6 3 0 0 ± .0008
Thus (5)
3[VH.T.-
~Borel]
=
.024 +.006_.004
205
which appears to account for the discrepancy between Baker-Kincaid and LeGuillo~Zinn-Justin of .028 above. Subtracting (5) from .028 is compatible with zero (hyperscaling). Only time will resolve these discrepancies, when the high temperature series are taken to higher order. We ask: (QI.) Are the high temperature exponents different from the Borel exponents for 43k ? (Q2.) Are the high temperature ~I3 and ~ 4 equal? (Q3.) Do the Ising and 4 ~ scaling limits differ? (Taking the limit 43 o + o c with m fixed.) If this is ture the universality hypothesis breaks down. In that case, ~i3 and2 ~'@3~ could be different. If this is the case, its presumed origin lies in the O(1)mass renormalization term alluded to above. 7.
Statistical Physics Models
Note that the equivalence of field thoery with statistical mechanics has led to new methods to study problems in statistical physics. In particular the equivalence of the grand partition function for the Coulomb gas to the Euclidean Sine-Gordon field theory allowed the application of constructive field theory methods to the Coulomb gas. Brydges and Federbush [7] have established Debye screening, namely a finite inverse correlation length m
=
mD(l+O(z))
onmnrdiS G$ the Debye mass m D = 2/2~ , B = I / k T and z is the activity. The Sinetransformation exNibits this explicitly, namely, using Fourier transformation, ZCoulomb
= I e2Z I:eosIB½4(x)]
:am 64
,
where d 4 is the zero mass free field measure. Then m D arises from the quadratic" term in the cosine, namely -½(2zB:~2:) = -½mD2:~2: , a mass term. In d = 2 dimensions, the question of whether the Coulomb gas has a condensation into a dipole phase has special significance. In fact the xy model (n = 2 nonlinear o-model) has been analyzed by Kosterlitz and Thouless [8] as approximately factorizing Zxy
=
Zspin Wave Zvortex Gas
=
Zspin Wave ZCoulomb Gas
The spin wave (free field) part yields polynomial decay of correlations, while the Coulomb gas has exponential decay (screening) at high temperatures. If the Coulomb gas condenses to a dipole phase, it does not screen. This can be seen from the Sine-Gordon transformation for dipoles: ZDipole
] e 2z I :c°s(B½V#):dx d4
Here the mass in ZCoulom b is replaced by a change in the coefficient of the kinetic energy term in ZDipole , i.e. by a dielectric constant. Thus screening does not occur in the dipole phase and the above approximation suggests that for T < T , c m z 0 in the xy model. This explanation of the line of critical points (m e 0 for T < Tc) supposed to exist in the xy model is not mathematically rigorous at this time. FrShlich and Spencer hope to show the condensation for the d = 2 Coulomb system. This would be a major step in establishing the xy behavior, c.f. Figure 3.
206
m
ing
T
Figure 3.
T
T c
C
The line of critical points for the xy compared w i t h the Ising model (d=2,n=l).
model,
Another fascinating and related problem is the roughening phase transition. Here we consider a d = 3 Ising model w i t h + boundary conditions for x I > 0 and - boundary conditions for x I < 0, c.f. Figure 4.
+
+
+
+
+
+
+
+
+
<0> > 0
+
+ + + + +
xI = 0
< 0
Figure
4.
Phase separation
in Ising 3 , for
T < Tclsing 2
It is known that for T < T c T > Tcising 3
, a sharp interface exists. Also for Ising 2 , we have a translation invariant state (no interface). It is of
interest to understand for melting.
the roughening
(disappearance)
of the interface
as a m o d e l
In a special case, where Jxl ÷ ~ (the solid on solid model), the roughening transition is again described by a d = 2 Coul'omb gas [9]. It is of great interest to study this problem mathematically. Again, the roughening transition is related tO dipole condensation for the Coulomb gas. In addition, we expect the roughening transition does not occur in the continuum ~34 field theory, because of the logarithmically divergent height-height correlations in the s.o.s, model. I have given these examples to show a wealth of m a t h e m a t i c a l - p h y s i c s problems associated w i t h statistical physics and quantum fields . If these questions seem too simple, tell us at the next M O @ conference h o w to construct a non-Abelian (asymptotically free) gauge theory in d = 4 dimensions, or a nonlinear o-model without cutoffs in d = 2 dimensions! References i.
The original Osterwalder-Schrader theorem is in Commun. Math. Phys. 3_ii, 83(1973), 42, 281 (1975). A theorem similar to this can be found in J. G l i m m and A. Jaffe in "New Developments in Quantum Field Theory and Statisticl Mechanics", edited by P. Mitter and M. L~vy, 1976 Carg~se Lectures. The version here is contained in our forthcoming book. The case p = 2 (e.g. free field) can be handled with a separate assumption.
207 2.
The 434 existence theorem combines J. Glimm and A. Jaffe, Fort. d. Physik 21, 327 (1973) with J. Feldman and K. Osterwalder, Ann. Phys. __9780 (1976) and with J. Magnen and R. S~n~or, Ann. de l'Inst. H. Poincar~ 24, 95 (1976). See Commun. Math. Phys. 56, 237 (1977). The strong coupling theory is given in J. Fr~hlich, B. Simon and T. Spencer, Commun. Math. Phys. 50, 79 (1976). Other references can be found in these proceedings, e.g. to the work of Gallavotti et al.
3.
References to the construction of P(~)2 models can be found in "Statistical Mechanics and Quantum Field Theory" 1970 Les Houches Lectures, C. DeWitt and R. Stora Editors, Gordon and Breach; in "Mathematics of Contemporary Physics," R. Streater Editor, 1971 London Mathematical Society Symposium, Academic Press; and in "Constructive Quantum Field Theory" edited by G. Velo and A. Wightman, 1973 Erice S u ~ e r School, Springer Lecture Notes in Physics V25. The construction of the Yukawa 2 model is given in R. Schrader, Ann. Phys. 70, 412 (1972), E. Seller, Conm~un. Math. Phys. 42, 163 (1975), J. Magnen and R. S~n~or, Commun. Math. Phys. 51, 297 (1976) and A. Cooper and L. Rosen, Trans. Am. Math. Soc. 234, i(197~. The recent work of D. Brydges, J. FrShlich and E. Seiler on the Higgs 2 model is in press in Commun. Math. Phys. and in these proceedings. The I/n expansion of Kupiainen is also in these proceedings, as is the work in progress of Balaban. Multiphase ~6 models are studied by K. Gawedzki, Commun. Math. Phys. 59, 117 (1978) and S. Sommers, preprint.
4.
See 1973 Erice and 1976 Carg&se Lectures.
5.
J. Glimm and A. Jaffe, Ann. l'Inst. H. Poincard 22, 97 (1975) and in the 1979 Carg&se Lectures, Plenum Press, in press.
6.
M . A . Moore, D. Jasnow and M. Wortis, Phys. Rev. Lett. 22, 940 (1969); J. Kincaid, G. Baker and W. Fullerton, LA-UR-79-1575; G. Baker and J. Kincaid, LA-UR-79-2655; J. Le Guillou and J. Zinn Justin, Phys. Rev. Lett.
39, 95 (1977). 7.
D. Brydges, Commun. Math. Phys. 58, 313 (1978); D. Brydges and P. Federbush, Commun. Math. Phys., in press; and P. Federbush, these proceedings.
8.
M. Kosterlitz and D. Thouless, J. Phys. C5, 1124 (1972).
9.
See John D. Weeks, "The Roughening Transition" in 1979 Geilo (Norway) School, to appear.
I/N EXPANSION-SOME RIGOROUS RESULTS A. Kupiainen Department of Physics, Harvard University i.
Introduction
It was noted by Stanley [i] in 1967 that certain n-component lattice spin systems exhibit considerable simplification as n becomes large. In fact, formally these models become the exactly soluble spherical model when n --°°. Later Wilson [2] and others developed systematic expansions in powers of i/n. These i/n expansions have several interesting features: they are non-perturbative, supposedly valid near the critical point and for scale invariant theories essentially the only expansions available. The only rigorous result on large n limit was that of Kac and Thompson [3], who proved the convergence of the free energy as n-~o to that of the spherical model for the models considered by Stanley. In this note we present some new results proving that the i/n expansion is asymptotic and establishing a mass gap arbitrary near the n=~ critical temperature for n sufficiently large. 2.
The Model and Results
We consider the n component nonlinear u-model (classical Heisenberg model) on lattice. Let A be a torus obtained from a cube in Zd (d arbitrary) and ~:A÷R n be the field with the probability distribution d~(~) = z-le ½(~'(A-m2)~) ~ ~(~2i-n~)dn@ i i~A where ~ is the inverse temperature and m 2 is a constant to be chosen later [Note that d~ is independent on m 2 since e
-~m2 (~, ~)
=
e-½m2n~].
By inserting the
Fourier expansion of ~-function and performing the gaussian integrals one gets e.g. for the generating functional of D: <e(g,~)> =
<e 2(g, (-A+m2 ~
2ia)-ig)>
(I) a
with <.> a the expectation in the "dual" measure d~(a) = ZI I d e t ( - A + m 2 - 2ia)-n/2e-inBtra ~ da. imA i
(2)
Choosing m 2 now such that a=0 is the saddlepoint of (2) i.e. (-A+m2(B));10
= B
(3)
one can obtain the formal i/n expansion as a loop expansion about a=0. As is well known, (3) gives the mass gap of the spherical model, which is positive above some critical temperature T S. We consider only T > T S. fl
Let now <@~>~ be the correlation function < H }i lEA summarized in the following two theorems: Theorem iall T > T S
>
Our main results are
Let ~ Sk(A,~)n-k be the formal i/n expansion for <~7>.~ Then, for r-I i<*A >-k~ 0Sk(A,~)n-kl < n-rRr(A,~,B)
i~i
The expansion is thus asymptotic. R r has an explicit formula and bound. In particular we get exponential falloff; the following theorem can be stated also for more general correlations.
209
Theorem 2.
There exist constants ~i and e2' only depending on the dimension
d, such that for all T > T S and
n>~im(B) -~2
there is a mass gap:
<91 ~i > < (_j+p(~,n)2)-i 0
X
--
OK
I_1
where (the physical mass) ~>0. ~i and ~2 are explicit and one gets bounds for the critical a
form TC < T S + - - ~
temperature of the
.
n
We have also proved similar results for the i/n expansion for the continuum (~2)2 Quantum Field Theory.
3.
For further details see [4].
Methods
We will now briefly discuss some methods involved. The dual transformation (i) gives i i (_A+m2_2ia) ~ > <~0~x > = <
We start from Theorem 2.
a
(4)
The difficulty with (4) is that d~ is a complex measure and (-A+m2-2ia)~ is non local in a. The second problem is solved by using a convergent Neuman expansion due to Brydges and Federbush [5]:
(-A+m2-2ia);~ =
E m:o+x
~ (2d+m2-2i%) n ( ~ ' k ) ksA ,
(5)
Here ~ are random walks from o to x and n(~,k) the number of times ~ hits k. Although complex, <'> a turns out to be reflection positive. One can apply chessboard estimates to (4) and (5): < i i ~O~x > J ~ k~A
where z(t) = I / I A l . _ pressure estimate
z(t) z(0)
z(n(~,k) z(0)
Theorem (2) is thus a consequence of a
j (2d+~2)-t, VA.
We need sharp upper bounds for z(t)
and lower bound for z(0). The idea is to introduce a cutoff in a-integrals thus confining the integration near the saddle point, which is easier to study. Dependence on the cutoff is shown to be small using chessboard estimates. The expansion itself is generated by integrating by parts with respect to
the quadratic form
~2--F[ 3ai~ajla= 0
where F is defined by dp = ZI-I eF[da].
210 One gets an explicit formula for the reminder, involving expectations of the form
< N a. E (-A+m2-2ia)7~ > K± i 1
(6)
(6) is estimated by inserting (5) and chessboarding the resulting products of local observables thus reducing again to pressure estimates.
References: [i] [2] [3] [4] [5]
Stanley, H.E., Phys. Rev. 176, 718 (1968). Wilson, K.G., Phys. Rev. D7, 2911 (1973). Kac, M.; Thompson, C.J., Phys. Norv. 5, 163 (1971). Kupiainen, A., Thesis, Princeton University (1979). Brydges, D; Federbush, P., Comm. Math. Phys. 62, 5 (1978).
This work is supported in part by the National Science Foundation under Grant No. PHY-77-18762.
THE RENORMALIZATION
GROUp IN TH~ EUCLID~AN
SUPERRENORMALIZABLE
SCALAR
FIELD THeORIeS
+
G, Benfatto Istituto di Matematica Universit~ di Roma 00100 ROMA ~ Italy
I. Introduction.
In this talk I will refer about an approach lity problem
in the scalar euclidean
to the ultraviolet
field theories,which
loped in Roma in the last two years,following
stabi~
has been deve-
some initial
ideas of G,
Gallavotti (1,2,3) The heart of the method me dimensions)
is the rigorous
of a recursion
analysis
relation,which
(in 2 or 3 space-ti~
can be thought
as a repre-
/
sentation
of the renormalization
group
(in the definition
of K,G. Wil-
son(S)). The techniques
which allow to study this recursion
cal of the Statistical
Mechanics
associated
Operators(4),I
to Elliptic
cal aspects
of the method,but
and'the
relation
are typi~
theory of Oaussian Processes
will not refer about these techni~
I will only sketch the main ideas,
2. Notation.
Let
~(~) be
the gaussian random field with covariance
where D is the Laplacian
C~= (i ~ D)
in Rd~d = 2,3.Formaliy
(i)
+ Supported
in part by C,N~R~,G,N~F,M~
212
Then
~(~)
can be thought as a sum of independent random fields
~KI~)
of covariance Ck. Observe the scaling relation
-
(z)
c.
which implies that the fields
¢~KC~,) are
identically distributed up to A
scale factors. ~lq(~)
will be called the field of frequency k and d~ k
will denote its probability distribution. We now define the cut-off field (which is essentially the field regularized ~ la Pauli-Villars)
(3) and the corresponding measure A
A ?. -- K¢O 7[ A PK
(4~
We want to study the measure
(s) where
The counter terms are those needed in the formal perturbation theory (then they are absent for d = 2).
3. The recursion relation.
The ultraviolet
stability problem is essentially the following:to pro-
ve the existence of E + ( ~ ) , E _ ( ~ ) , i n d e p e n d e n t
of N,such that
213 The main idea is to calculate the partition function iteratively. Then, if we define the effective potential of order k , ~ N'k)~ ,by
L : K~.t
we have to study the recursion relation
Eq. (9) can be thought as a representation in the definition of K.G. Wilson(5).Its scaling properties of the field of ~ N , k )
is essentially
Vz
":
where W~ N'k't),which not contained as N--e ~O
group
analysis is made easier by the
~(S~),which
imply that the structure
independent of k.Roughly:
V~:
*
~ A z
is the contribution
~ >
~lO)
of the terms up to order t
in V~ k) in the ~-expansion
and,for some
of the renormalization
of ~ N , k ) , h a s
a finite limit
O
I 0O with
KV,
~K,I=
<
(12)
O0
0
if t is large enough (t >/3 for d = 3 , t ~ l In the proof of'eq. i) The fields
for d=2).
(iO) there are two main technical problems:
~K(~)
are unbounded,so
that it is not possible to _ (,t,k,~)
simply use the Taylor formula in order to control the "remainder" ~ = in eq.
(IO).Furthermore,in order to bound the contribution to A s co.(w,K,,I,~) ming from W:: ,it is necessary to control the HSlder norm o f (~K.,(~) We overcome these difficulties
by using:
a)for the upper bound,the positivity properties V(/~'l K, k')
•
of the interaction and
,
b)both for the upper and lower bound,the fact that the points where the fields
~KC~)
are '¢large" or "rough" give a negligible contribution
to
214 the integral in eq. (9),thanks to the support properties of the free (4) field of covariance C o ( ~ - ~ ) 2) The solution of problem I) could give the rough estimate
A=
',-, I~: I
!13)
To obtain the right dependence on I,we use again the support properties of the gaussian field of covariance Co(~- ~ ) and the exponential decay of Co(~- ~).I will briefly sketch the idea. We divide I in "large" boxes rated by a corridor
r
f'l~ (of side
of width
~
independent of I),sepa-
~ < ~ ~ (see Fig. I for the case d=2).
If_ II II
o,
II
II
iI
By the Markov property of the measure dPk,We can write the integral in eq. (9) (suppressing
?£
where
and ~ a [
all the indices)
I
in the following way
ldenote the restriction
to Y and a ,
of
y
and
P(&?=~! ?~) is the probability distribution of ?=,,conditioned to The approximate local;however
?~.
equality in eq. (14) comes from the fact that V I is not
the terms connecting different regions give a negligible
contribution,as
it can be shown using the exponential decay of C o ( ~ - ~ ) .
Using the results of i) it is now possible to apply the Taylor formula I
to
~C vO~P(~a.~r~
to calculate
'unif°rmly in
~ ~(~)p(~)This
~r
up to a small error.There remains
can be don e by iterating the procedure
described before many times with new grids obtained from the one of Fig. 1 by suitable translations,chosen
so that any point will eventually be
in the interior of some fT,.In this way we construct contributions
successively
to the r.h.s, of eq. (IO) coming from the different
of boxes.The estimate
of
~
will
o b v i o u s l y be p r o p o r t i o n a l
the series
to the
215
number of boxes
I~
contained in I,that is it will be proportional to
Iil I want now to give an idea of the main result,which is of course eq. (lO).Let us define the normalized fields
(15)
<':
(16)
It is easy to verify the following recursion relation
rK-I
~
Q-
By eq. (2) the field in
terms
of
v: <''--
the
.
E")/(~'-,) , ~" ~
~K(1)= Z~(f"i)
c. (i- ~)l~c.(.),independent sion
' ~=~
field
.~(,{-.U~
is a gaussian
of k. Then .vI(k) 2K(~J
as
-
l'
"
(see
(18)
field with covariance eq.
(6))
X~)
proportional
expres-
~
(19)
f ~r/VZ
is "essentially" bounded,then the contri-
bunion of the terms of o~der t in the ~-expansion is
an
° "
+ ) ¢0~.~..l" ~El"mS I If we can show that
has
of
e.l
.(liil.
3~&
#i
to
Z
Izl
Some straightforward algebra shows that this property is not destroied by
vv ~
,thanks to the counter terms. Indeed it is possible to
show that the Taylor series in the variable ~ of the effective potential of order k,~N'k),can" be thought as a power series in the "effective coupling c o n s t a n t " respect to N.Eq.
~ - C k - ~ ) ,with coefficients uniformly bounded with
(20) immediately implies eq. (12).
216
References
i. G. Gallavotti,Memorie dell'Accademia dei Lincei,XV,23,1978 G. Gallavotti,Annali di Matematica Pura e Applicata,lq79o 2. G. Benfatto,M. Cassandro,G. Gallavotti,F. Nicol~,E. Olivieri,E. sutti and E. Scacciatelli,Comm. Math. Phys. 59,143,1978.
Pre-
3. G. Benfatto,M. Cassandro,G. Galiav0tti,F. Nicol~,E. Olivieri,E. Presutti and E. Scacciatelli,"On the ultraviolet stability in the eucledean scalar field theories",Preprint IHES/P/79/251 (in print on Comm. Math. Phys.). 4. G. Benfatto,G. Gallavotti and F. Nicol~,"Elliptic equations and Gaussian processes",Preprint IHES/P/79/251 (in print on J. Funct. Anal.). 5. K.G. Wilson,Phys.
Rev. B4,3174,1971.
EXPANSION AND SUMMABILITY METHODS IN CONSTRUCTIVE
FIELD THEORY
J. Magnen and R. S~n~or Centre de Physique Th~orique Ecole Polytechnique,
91128 Palaiseau
I. Introduction
The following models low), ~
: XP(~)
in D = 2 dimensions
[N,GJI]
sive QED in D = 2 [FS], Higgs model in D = 2 [BFS]... degree of achievement
version of these models consists i) Ultraviolet
field theory.
the construction of the Euclidean
in ~ steps
stability of the free energy per unit volume
This is the removing of the momentum cutoff. More precisely~ interaction with suitable renormalization
eounterterms,
e-diAl S z(K)~ = f e - ~
Z A = lim K ~
~ a~ A > 0
being the cutoff independent
of Ks.t.
~(K)I _< e+Al~l
limit of Schwinger
functions
e
the limit satisfying at least the 0sterwalder
Schrader axioms
Expansion methods are used among other ones (correlation space analysis- renormaiiza±ion In principle~
regions).
~( ~ I
Z ~ K) exists.
2) Thermodynamic
normalizable
Sine Gordon or mas-
have been studied with various
in the framework of constructive
A being a space cutoff and K a momentum cutoff,
and
(P semi bounded from be-
in D = 3 [GJ2], Yukawa in D = 2 [Sc,S] and D = 3 [FMS, MSI]~
group...)
£(K) I
(0 - S).
inequalities- functional
to prove i) and 2).
these combined methods are enough general to control all superre-
local models far from critical regions
Generally expansions
(i.e weak and strong coupling
give detailed information on the physical
structure.
218
Example of expansions 1. Convergent perturbation expansion with respect to the coupling constant Sine Gordon
D= 2
2 . Weak coupling cluster expansion
(GJSI)
It is the quantum field analogous of the high temperature expansion in statistical mechanics.
It solves 2) knowing i). The following expansions are related
expansion (GJS2~ GJ3)
a) n - particle
It is adapted to the study of the Hamiltonian spectrum up to the (n+ i) - particle threshold. Roughly speaking it is as n - successive cluster expansions. a') Spencer's expansion [Sp~ Nice to study n - p a r t i c l e
irreducible parts
(e.g. Bethe- Salpeter kernels)
b) mean field expansion (GJS3) See A. Jaffe contribution. 3) Phase space cells expansion (GJ2) It solves
i) for superrenormalizable theories.
We will now explain how to prove
i) in some simple cases and what are the main ideas
of phase space expansions.
II. Ultraviolet stability
It will be a consequence of superrenormalizability and of the positivity of the effective potential. i) Superrenorma I izabi i ity For each vertex~ using power counting argument~
one gains an extrapower of convergence
P : D = 2n - (n - I)D ~-D 2
0 =
D
for
2n
for Yukawa ~
2
and QED
3 0
~ A vertex~
1
1/~
o
such that one of the momenta of its lines is bigger than ~ is then "conver-
gent like ~-P. This is the asymptotic freedom of superrenormalizable theories
(~>0).
219
2) P o s i t i v i t y
of the e f f e c t i v e
B y this w e m e a n for a scalar
> o
with $ K
and
the cutoff
(x) >
~ [
~
4:
-cst(log K) 2
D = 2
-cst K 2
D = 3
field.
For a theory with o v e r the f e r m i o n
potential
field t h e o r y as e.g.
fermions
leaving
like e.g. Y u k a w a
the b o s o n
or QED,
field constant,
veff(~)"- =
lim-
One can d e f i n e
similarly
an e f f e c t i v e
dary condition
in w h i c h
$ is constant.
1
~,n
potential
we p e r f o r m
the i n t e g r a t i o n
then
I
e
in a "cubic"
box w i t h p e r i o d i c
boun-
O n e gets
- for Y u k a w a a n d ~ large
V elf
~
D=2
~n[$]
2
D=3
Introducing
a fermion upper
c u t o f f K, V telf ~K J > -- _ log K
for D = 2 a n d
>
K 3 for D = 3
- for Q E D V eff is zero by gauge gives
invariance.
Breaking
it w i t h a m o m e n t u m
cutoff o n the fermions
in D = 3
I ct A~ eff
-~--
V(K)~ We now briefly
describe
and 2) in o r d e r to prove
2) The p h a s e Following
Nelson
[A[ << K
[AI >>K
c t A2K
space
the p h a s e
IAI D/D
in space
thus that
e
in D = 3 .
expansion
bigger than
(properly
expansion,
potential
N o t e that ~ has to be s m a l l e r invariance).
where
-V elf is big
each v e r t e x p r o d u c e d
in the sense that at
IAI -I/D
renormalized)
is t h e n as f o l l o w
The idea of using e f f e c t i v e B a l a b a n and Gaw@dzki [BG2
of space w h e r e
< ~.
perturbation
(in cubes A) a n d in m o m e n t u m
of i) e a c h v e r t e x
. The
i)
expansion
of the v e r t e x has a m o m e n t u m
consequence
the case of ~
(N), one wants to s h o w that the m e a s u r e
is v e r y small, proving
is l o c a l i z e d
cell e x p a n s i o n and the use of p o i n t s
I.). F o r s i m p l i c i t y we treat
To do that one p e r f o r m s a t r u n c a t e d
fields
space
(well
has a convergent
(with ~ a unit
in c o n s t r u c t i v e
t h a n p for technical
localized
least one the field).
As a
factor
cube).
field t h e o r y
reasons(in
is due to
particular
translatior
220 Let M > > i, then define ~i "''"~i'
"'" covers of A with cubes of equal sizes. The cubes
of ~i are obtained by subdivision of cubes of ~£i - i ' with
IAi] = IAi_ i I + 6 for some
> O. A perturbation step in A can be symbolized by
z=/e-f~ =/e -lq Lmentumcutoff+ /~ e - ; Z I IA]-I/~ in A =
(I A + PA) z
The expansion is then
z=~AIg2~ Tr I %~ + P A l A aITc A I
(IA2+PA 2 A j c A 2
A2C~ 2
A3C~ 3
T[ %3 + PAj ~. . . . .
Supposing that each produced "vertex PA" gives a convergent factor
~z
IA I~/D (as expec-
ted) and ~ is small enough~ the expansion is convergent and for each term of the sum
-]%IA /~PA zl C o s t
i
~ A+o/D~
one half of the convergent factors being used to hound
-f ~ I~I A e JA
We now show that
-Y
e
-~ £I
IA] flower bound of £I in A I
I I <__e A
I~ xP A e
-; f I
since ~IT cst A
I -~ ost ~ I A!0/D
2 This bound is obtained via gaussian integration positivity of the effective potential Each PA v e r t e x At l e a s t
is roughly
one o f t h e /~ f i e l d s
gration).
I A I -l/D)
In fact
for fieids
,f~ 4
domination) and use of the
(e -x4 domination).
(A 1 / 4 "~A)4
is well localized in A ~'i
(e -x
w i t h ~A =
A-JA •
and can be i n t e g r a t e
well localized
(i.e.
(gaussian
with lower cutoff
inteat
one has
J~i
Ag~i
which is the best bound compatible with power counting. The other bounded using (e -x$ domination)
IAII/$ ~A
are
221
~A
A
~
x e -x
~
cst
A
A
it follows that for each P~ ~ertex one gets IAII/~- I / 2 + V D
= I~I~/D
as convergent
factor. ~) The phase space expansion The Yukawa theory can be treated grated out
(Matthews
for Yukawa and QED. similarly as ~
Salam formulation)~
4
after the fermions have been inte-
the effective potential
V elf being used as
for non gaussian domination. The procedure
is however different
for QED since by gauge invariance
method
is to break the gauge invariance
cutoff
on the
fermion
5) Improvement
fields.
by introducing
As a r e s u l t
of the convergence
V ff.
Oand
can be used
for
vertex is necessary. S u p e r r e n o r m a l i z a b i l i t y insuring
/ I ~I/D I -I/~
IAI ~I per P~
it remains at our
IAI I / D - I / ~ - ~ i. This extra power can be used to define the expansion
to perturb
in smaller
For example~
for ~
lower cutoff at
cubes~
increasing
141 -I/2" but at
III Analyticity
in the coupling
localized not only fields with
IAI -I/~" - E2 for some ~2 > O "
(regardless
of the possibilities
given by domination w i t h V eff (comparable ~
domain is given by the condi-
of extending
of these theories
by
it by complex scaling
are essentially
the ones
to what one gets using Lipatov's
in D = I~ D = 2 or D = 3[GGS~
and p r o b a b l y QED in D = 3 are Borel
determination
(see I 2) and IV).
constant and Borel summability
[EMS2).In the same way, the bounds on the n %h derivatives
As a consequence
This last remark can pro-
expansion
In all the models discussed above the analyticiiy tion Re V e f f > O
i.e.
the localization.
in D = 2~ one can consider as well
bably be used %o cure the defects of the n - particle
[R,MSI2~
domination.
: superlocalization
In the proof of the c o n v e r g e n c e of the e x p a n s i o n in section 3) only
disposal
V e f f = O. The
a (a priori non necessary)
EMS~ MS2~,Yukawa
summable at k = O showing
the p e r t u r b a t i o n
ideas Fp2).
in D = 2 and 3 the complete
series.
IV Questions W e now present a list of open questions On expansions
related to the above
topics.
:
i) In the present that the coupling
status of the n - particle
constant
2) Define an expansion
go to zero as n ~ .
expansion,
its convergence
required
Remove this constraint.
for strictly renormalizable
asymptotically
free theories
222
3) Define a lattice version of the phase space expansion (applicable to gauge
fields),
(see a l s o T. Balaban r e p o r t ) ) .
On analyticity and Borel summability i) If Borel summability holds~ how to extend known properties of the perturbation series to the global Green functions? 2) What is the largest analyticity domain in ~ for ~
4
in D = 2 or 3 ? Can we
see the left singularities of the Borel transform (instanton contributions) ? D
3) Are the perturbation series Borel summable in each pure phase of ~$ in D = 2, as it is expected? $) Is the I/N expansion Borel summable for O(N) symmetric or
(~ 2)2 in D = i, 2
37
5) Using Feynman diagrams
estimates
can we reconstruct the Borel transform~
perform its analytic continuation and define a Global theory (see for ~
4
in D <
I S - R~) ?
'What to think about Khuri result [K] showing a contradiction between Borel summability of perturbation series of $$ in D = $ and the existence of a non trivial ultra violet fixed point?
223
REFERENCES
[BFS2
D.C.
Brydges,
J. FrSlich,
E. Sailer
: Construction
of quantized
IHES preprint [BG 3 T. Balahan,
C. Gaw~dzki
[EMS~
J.P.
[FMS2
J. FrShlich,
Eckmann,
[FS2 J. FrShlich,
: private
J. Magnen, J. Magnen, E. Sailer,
R. S~n~or
Helv.
[GJ 13 J. Glimm~
A. Jaffa,
[GJ 23 J. Glimm,
A. Jaffa
: Fortchr.
[GJ 33 J. Glimm,
A. Jaffa
: Comm.
[GJS
13 J. Glimm~
A. Jaffa,
: Comm.
B. Simon
Acta,
PRL.
in les Houches
T. Spencer
70,
Gordon
21,
in Erice
A. Jaffa,
T. Spencer,
Ann Math.
T. Spencer,
Comm.
D, Vol.
and Breach
327-
376
1973,
A. Jaffa, Review
(1976).
19721-
108.
(1973). (1979).
Springer
Lecture
Notes
in Phys.
25.
[GJS 23 J. Glimm,
Phys.
39 , 251-271(1975)
1976.
6__~7, 267- 293
[GJS 33 J- Glimm, N. Khuri,
Phys.
unpublished
Vol. 49,
Phys.
No
[K~
(1979).
3_~2B 631(1970)
Physlk
Math.
Math.
: CargOs%
Phys.
V. Grecchi,
iOO,
in Math.
i~, No iO, 2665
585-
632
Phys.
(1974).
4_55, 203 - 216
(1975)
(1976)
[MS 13 J. Magnen, R. S~n~or in preparation [MS 25 J. Magnen, R. S6n~or, Comm. Math. Phys. 56, 237 - 276 (1977). IN 3
E. Nelson, in Mathematical theory of elementary particles MIT Press Cambridge (1966).
[P2
G. Parisi P.L. Vol. 66 B No 4 382
JR2
P. Renouard, Analyticit~ et Sommabilit~ de Borel pour Y2
, to be published in
Ann. Inst.' H. Poincar~. [Sch] R. Schrader IS 3
II
communication
R. S~n~or
[GGS 3 S. Graffi,
fields
9auge
Ann. of Phys. 7__O0,412 - 457 (1972).
E. Sailer, Comm. Math. Phys. 4_~2~ 163 - 182 (1975).
[Sp2
T. Spencer, Comm. Math. Phys. 4~, 143 - 164 (1975).
[SR~
4 E. Spear, V. Rivasseau, Borel transfrom in Euclidean ~ ~ ... Preprint Ecole Polytechnique (1979)
THE PHYSICAL-REGION MULTIPARTICLE S MATRIX D. IAGOLNITZER
DPhT, CEN Saclay, BP 2, 91190 Gif-sur-Yvette, France ]. INTRODUCTION. This text is a short guide to several recent works on the physicalregion analytic structure of the S matrix in relativistic quantum theory, for sets of external particles with stric~y
positive masses. In the absence so far, in the
multiparticle case, of a satisfactory understanding and derivation of more general properties,the study of the physical region has in fact appeared to have its own interest
both from a conceptual viewpoint and for applications and further developIll ments. For details on the general physical and mathematical background, see We first recall in Sect.2 the physical region properties that are expected, and are equivalent to basic properties of macroscopic causality and macrocausal factorization (= asymptotic properties of transition amplitudes between sets of initial and final wave functions corresponding respectively to non causal, and causal, configurations of displaced particles). In Sect.3, we briefly explain [2] the particular phenomena that occur in twodimensional space-time, and yield factorization of the multiparticle S matrix itself, for a class of models, into a product of two-body S-matrices.
[3-8]
In Sect.4, we return to four-dimensional space-time and present recent works on the derivation, in S-matrix theory, of the physical-region structure described in Sect.2. The subject was presente
~n the sixties as successfully achieved on the
basis of unitarity and "maximal analyticity". However, the proofs of [9]rely on some assertions which are crucially non correct, even in the simplest cases. On the other hand, the related approa~0~asedr_ ~ on macrocausality and unitarity made use of an
assumption of "separation of singularities" in unitarity equations (see example in Sect.4), which is an ad hoc way of eliminating crucial problems that appear otherwise. The results obtained in [3-8] in the framework of essential support theory, or of hyperfunction theory, will be illustrated here f o r 3 ÷ 3
processes.
Related results have also been obtained in axiomatic field theory, in two recent approaches
. Information was derived previously from microcausality and the
spectral condition. It is, however, much weaker than macrocausality. The new results are based on a further exploitation of the off-shell structure of the theory, together with the axiom of asymptotic completeness and regularity assumptions : see the lecture by Bros. 2. PHYSICAL-REGION STRUCTURE! ] The "scattering function" f of a given process is the connected momentum-space S-matrix kernel, after factorization of its energy-momentum conservation ~-function. Expected physical-region properties of f are :
225
2.1.
Analyticity
outside +a-Landau surfaces L+(G) of connected graphs G, and plus
ig rule = f is, in the neighborhood
of +a-Landau points,
analytic function from the "plus iE" directions to the causal direction(s)
dual, at each point P={Pk } of L+(G),
~+(P;G).
The plus iE rule gives no information of several +a-Landau
the boundary value of an
at points that lie at the intersection
surfaces,
or at the "M points" such that two initial, or two o Property 2.1 is then stated in general in terms of an essen-
final, Pk are colinear.
tial support, or equivalently mlcroanalyticity, property, which is a natural and formulation of a general property of macroscopic causality in
precise mathematical terms of particles. analysis 2.2.
Recent works [5'6] study the situation at M ° points, where the
is more delicate.
Discontinuity
formulae around the +a-Landau
information on the structure at +a-Landau points. in terms of an essential property of
support property,
singularities,
which is now equivalent
surfaces L+(G) encountered.
terms are well defined, modulo analytic backgrounds,
of the +a-Landau
singularities
TWO-DIMENSIONAL
mass particles,
types, and a 3 * 3 process.
and of the final 2,momenta,
one encounters
in two-dimensional
coincide with the set of final 2-momenta.
a number of +a-Landau
line and triangle graphs, which,
mension one surface E defined by requiring
there is no particle production.
space-time,
surfaces in view
all coincide with the codi-
that the set
of initial 2-momenta
should
They are the only ones in models in which
The plus ig directions
associated
subsets of these surfaces are opposite,
2.2 of Sect.2 then entail that the (non connected)
scattering
a sum of two well defined boundary values of analytic It
In each sector of
to a given relative position of the initial 2 -
associated with graphs with one internal
is analytic outside E . In models
this implies consistency
S-
concern the nature
We cQnsider for simplicity a theory with equal-
possibly of different
the physical region corresponding
of E with two different
[4]
in refined situations.
SPACE-TIME [2]
of two-body kinematics
These
in terms of on-mass-shell
matrix elements associated with each vertex of G. Recent works
tions.
to a general
as a sum of terms that generalize Feynman integrals
and are associated with the various +a-Landau
momenta,
is again
macrocausal factorization and yields decompositions of f, in appropriate
parts of the physical region,
3.
which give detailed
Their general statement
at any point P
and properties
2.1,
function is along
functions
from opposite direc-
in which it is strictly zero outside E,
relations between sums of products of two-body functions,
and the two above boundary values add up in a way which exhibits the 3-body S matrix into a product of two-body S-matrices
: see
factorization
[2] ,
of
where the exten-
sion to the m-body S-matrix, m > 3, is also given. 3.
S-MATRIX THEORY.
We shall here consider,
for simplicity,
type of particle and a 3 ÷ 3 process below the 4-particle
a theory with only one
threshold.
In that region,
226 +
and away from Mo points, the only +a-Landau surfaces L (GB) correspond to the 9 graphs
111~4 with one internal line, such as G] = 2 causaT.ity
~
5
, and 9 triangle graphs. Macro-
[1] 3 6 entails that f can be decomposed correspondingly as a sum of 18 terms f~ ,
each of which is analytic outside L+(GB) and is along L+(GB) a plus it boundary value. The aim is then to show that fB is necessarily a generalized Feynman integral associated with G B (see Sect.2.2). We start from the "unitarity equation" that directly follows from SS-I=sst=I and the decomposition of S into connected components. In order to first study fl' we write it, following [8], in the form :
+ ~ where F B = ~x~h(Epi-EPj), terms such as
and 3
~
4
L~# 1
~#1
(1)
~
+~j#4~ + ~ ~+#},j#4 i ~ j ] =0 and ~ are connected kernels of S and S-l, are integrals over on-mass-shell
i s the plus i e - p o l e p a r t of
larity
internal 4-momenta,
(This l a s t term has a s i n g u -
6(k2-II 2), k=Pl+P2-P4 , along L+(GI )) .3
The first three terms, after factorization of their conservation ~-function, +
+
are, like f1' analytic outside L (G I) and plus ig boundary values along L (GI). Moreover, following the idea of "separation of singularities in unitarity equations" [I,41 according to a certain common underlying topological structure, their singularities along L+(GI ) should cancel among themselves(independently of the singularities of the terms in the second bracket of (I)) and their sum a should thus be analytic. If this is established, (see (i)(ii) below), one shows easily, by using two-particle unitarity, that F I =
. Q.E.D. Theanalytieity of ais derived as follows : 3
(i) The detailed study [3] of the terms in the second bracket of (i) shows that +
they do not contain ~+(P;GI) in their ess-support at P, if P E L (G I) lies outside a
certain subset ~+ ~J N of L+(GI ). The equality (I) and previous information an a
then ensure that a is analytic outside this subset. Property 2.1. (= macrocausality) is not sufficient to obtain this result, because one encounters, for a term such as
~
, the so called u=0 problem.
It is a general aspect, in terms of essential supports, of the fact that no information is a priori obtained on a product of boundary values of analytic functions, if they cannot be obtained from common directions. It is solved in [6], in the framework of essential support theory, on the basis of a refined version of macrocausality, which gives information on the way rates of exponential fall-off tend to zero when causal directions are approached. A slightly different, somewhat more refined u=0 conjecture is on the other hand proposed in [5] on the basis of general results
227
on "phase-space
integrals".
It is supported also by recent results [7] on products
of holonomic functions with regular singularities. (ii) Individual terms such as
~
..... do have ~+(P;GI)
in their
ess-support at points of ~+ 0 N . If one cannot show it is absent from the ess+ support of a, this singularity will propagate all along L (G 1) in the final step, when two-particle unitarity is used, and no result on fl will be obtained anywhere. This is solved by making use of a weak "no sprout" assumption on f, which is related to refined macrocausality , and entails a corresponding no sprout property for a itself. Then by Theorem 2 of [3], a , being analytic outside L+(GI ) and a plus ig boundary value along L+(GI ) , cannot be analytic at some points, and singular at other points of L+(G]). For details,
Hence, it is indeed analytic all along L+(GI ).
see [3,8]. The method presented above is that of [8], in which
triangle graphs are also treated and where some extensions are given.
REFERENCES
[1]
-
D. IAGOLNITZER,
The S Matrix, North-Holland
[2]
-
D. IAGOLNITZER,
Phys. Rev. D]8, 1275 (]978), Phys. Lett. 76B, 207 (1978).
[3]
-
D. IAGOLNITZER,
H.P. STAPP, Comm. Math. Phys. i7, l (1977).
[4]
- T. KAWAI, H.P. STAPP in Publ. R.I.M.S. Kyoto Univ. 12, Suppl.,(1977).
[5] - M. KASHIWARA,
(1978) and references therein.
T. KAWAI, H.P. STAPP, Comm. Math. Phys. 6__66,95 (]979).
[6] - D. IAGOLNITZER,
Comm. Math. Phys. 63, 49 (1978).
[7]
-
M. KASHIWARA, T. KAWAI, preprint.
[8]
-
D. IAGOLNITZER,
[9]
-
D. OLIVE, in Hyperfunctions and Theoretical Physics, Lecture Notes in Mathe-
preprint.
matics 449, Springer-Verlag []0]-
(Heidelberg),
1975, p.133, and references
therein.
H.P. STAPP, in St~actural Analysis of Collision Amplitudes, ed. R. Balian, D. lagolnitzer,
North-Holland
(1976), p.191 and references therein.
[i~]-
J. BROS, Cours de Lausanne,
[12]-
H. EPSTEIN, V. GLASER, D. IAGOLNITZER,
in preparation. in preparation.
Q u a n t i z e d Gauge Fields: Results
and P r o b l e m s
Erhard Seller Max-Planck-Institut f~r Physik und A s t r o p h y s i k M~nchen, ? e d . R e p . G e r m a n y
I. I n t r o d u c t i o n
At the previous
M ~
conference in Rome a number of talks d e a l t w i t h the sub-
ject of lattice gauge t h e o r i e s [I]. A t least for me one m a i n m o t i v a t i o n for these investigations was the fact that the lattice p r o v i d e s the o n l y known gauge i n v a r i a n t cutoff for the c o n t i n u u m theory and therefore s h o u l d be a g o o d s t a r t i n g p o i n t for an attempt to construct these c o n t i n u u m gauge theories.
Here I w a n t to r e p o r t some re-
sults that have b e e n o b t a i n e d in that line, in p a r t i c u l a r on w o r k by D. Brydges, J. F r ~ h l i c h and m y s e l f L]2 _r5
in w h i c h we c o n s t r u c t the first
(presumably)
nontrivial
example o f a q u a n t i z e d gauge field theory - the abelian Higgs m o d e l - o b e y i n g all of W i g h t m a n ' s axioms with the p o s s i b l e exception o f clustering.
I also w a n t to draw
your a t t e n t i o n to those aspects of that c o n s t r u c t i o n that are u n s a t i s f a c t o r y in the sense that one cannot hope to g e n e r a l i z e them to more i n t e r e s t i n g a n d realistic models; I think that we are facing not so m u c h a technical p r o b l e m as a c o n c e p t u a l one: we do not really u n d e r s t a n d w h a t phase space l o c a l i z a t i o n m e a n s for i n t r i n s i c a l l y n o n l i n e a r fields.
II.
Some Eacts on Lattice Gauge T h e o r i e s The lattice a p p r o x i m a t i o n
"approximates" E u c l i d e a n space-time ~ 9
b y the simple
cubic lattice S ~ ~ ; a general formalism for lattice gauge theories has been p r o p o s e d by W i l s o n E6] and s t u d i e d b y numerous authors.
F o r m a l i s m a n d m a n y r e f e r e n c e s may be
found for instance in [2] ; here I just want to recall the m o s t b a s i c facts: A lattice gauge field is a map from the links
(nearest n e i g h b o r pairs) o f the
lattice into the gauge group G (typically a compact Lie group): (x, y)L -i such that gyx = gxy
• gxy ~
G
. Wilson d e f i n e d a lattice Yang-Mills action and a correspond-
ing p r a b a b i l i t y measure for these fields.
In our t r e a t m e n t the lattice gauge field
arises f r o m a c o n t i n u u m gauge field A ~ A/~ dx/~ (a l-form with values in the LieY algebra of G): gxy = Pexp [ ~ A w h e r e the integral is along the link (x,y) and P indicates p a t h ordering.
For the abelian case G = U(1) we will consider in p a r t i c u l a r
229
gauge fields A.
that are Gaussian random fields with a covariance < A ~
= D ~ 9 (x-y) whPere D%~
is the Fourier transform of ( ~ 9
t > 0 serves as ultraviolet and / ~ 2 >
- ~ )
(x) A/~ (y)>
i
e_tp2
;
t
0 as infrared cutoff.
By a general result of Garsia [7] these Gaussian random fields may be assumed to be H~ider continuous of any index ~ , i A Bose matter field
so
%A makes
(Higgs field) is a map ~
sense.
from the sites of the lattice into
some finite dimensional Hilbert space carrying a unitary representation U of G. The action for the Higgs field coupled minimally to the gauge field is --
l ~
4-
A
~
(i)
where V is some polynomial that is bounded from below and grows at least as I ~ 1 is
some
reasonable bounded set in
~{%2 .
There is an associated joint probability measure
for the Higgs and gauge
fields
l
- gM,A
-aM'A
~M®(~)
"]]" C/(~(_×)
(2)
(4)
dm (A) denotes the probability measure for the gauge fields. The task is now to send 6, t a n d p
to zero
(which requires of course some renormalization),A to ~9
fy the Osterwalder-Schrader
and veri-
axioms for the correlation functions of gauge invariant
fields in that limit. This has been done in [2-4 ]
(with the exception of clustering).
Among, the many results on lattice gauge theories there are two which play an important r61e there
(see [2] ):
(i) Universal Diamagnetism:
I~'(A) I _< g'(o)
(s)
This is true for arbitrary group G and even for Fermi matter and expresses a physical property clearly related to the diamagnetism in nonrelativistic quantum mechanics that was discussed by Hunziker [8] at this conference. (2) Correlation Inequalities:
-
<~2>
>~ O
(~)
for
G = U(1), if FI, F 2 are functions in the multiplicative cone generated by
I~I
, cos A, cos
(arg~).
These inequalities have a number of interesting phy-
230
excel consequences limits
A/~gand
Is]
; in the constructive program they are used to control the
}~2 ~ o. /
III. Continuum Limit: External Yang-Mills Fields In [3] it is shown that the probability measures
a converge wdaKly as
~ ~
0 provided
independent of the orientation
= 2 and A is HSlder continuous.
The limit is
of the lattice;this crucial fact is responsible for
the Lorentz invariance of the final field theory. The proof of convergence is rather involved;
it requires a lot of subresults
some of which might be of independent interest and not limited to two dimension~ for P example we prove L convergence of the kernels £ v~ ~ )-I
( - A' A for p <
+
9 ~_----~ (
& ~
is the finite difference covariant Laplacean)
of det (( ~£)-i /k~A)-- for normalization,
~ = 2 (the proof could be extended to
~
and convergence
= 3 and, after re-
even V = 4 with some effort). This last result contains the statement
that the sam of the graphs
~
verge individually as ~ 0 .
A similar result for the Fermi case, i.e.
and
det~ ~ ~6+~]'I(~2~]I
~
converges even though they di-
[9]
has been obtained by Challifour and Weingarten
IV. Removal of All Cutoffs for Abelian H i g g s ~ The results mentioned under the joint measure
__~/~£1A,%
as
(III) are sufficient also to obtain a weak limit for ~'-mO as long as t > O; this follows simply from
the diamagnetic bound and the dominated convergence theorem, using the HSlder continuity of the sample fields A ~
for t > O .
To remove the t-cutoff one has to do a "stability expansion ''L4] [ involving cancellation of some divergent graphs against appropriate counterterms. expectation"
(
O
+
~
,
~
+
...~.
)
The stability expansion expresses an "unnormalized
as a telescopic sum:
ZtN
< F>t N
=
~ n=l
(Ztn+l
<
F > tn+l
- {tn < F > t n )
(7)
and bounds each term in the sum by an expression of the form const
n ~ i=l
~ t. l
e c (log t n) 2
which insures convergence of (7) as N ---~ oo appropriately
(8)
(n!) r (log t )n n
(for instance t n = exp (-nY),
if the sequence O <
uses integration by parts to cancel divergencies,
~ <
[ tn~
is chosen
i). The proof of the bound
a procedure to estimate large
Feynman graphs in terms of small ones and a power counting lemma for convergent graphs.
(8)
231
F i n a l l y the limits lities
(6)
A,~
2 and
(which give monotonicity)
e x i s t e n c e of the l i m i t
/~2 ~
0
/-~2~0
are t a k e n u s i n g the c o r r e l a t i o n inequa-
and upper boUnds that are n o t too hard. The
(no i n f r a r e d divergencies)
is a signal o f the Higgs
m e c h a n i s m for d y n a m i c a l mass generation. In the e n d one o b t a i n s a W i g h t m a n field t h e o r y for the fields F ~ , = ~ / , ~ and
: ~
-%~A/~
; e x p e c t a t i o n s of "string" and "loop" o b s e r v a b l e s
-
®
m a y also be c o n s t r u c t e d and fulfill some "axioms" d e s c r i b e d in [I0] and to be discusn e d in more d e t a i l elsewhere.
V. P r o b l e m s The a p p l i c a t i o n o f the techniques o f constructive q u a n t u m field theory to gauge fields turns out to be r a t h e r tricky.
I think t h a t the technical d i f f i c u l t i e s that
are coming up here are really the top o f an iceberg c o n s i s t i n g o f U n r e s o l v e d conceptual problems. The first one is the following: Gauge fields are r e a l l y i n t r i n s i c a l l y nonlinear: One can c o n s i d e r them as fields taking values in an infinite d i m e n s i o n a l fibre b u n d l e w i t h the gauge group as structure g r o u p []l lr, 1 2
or, alternatively,
as m a p s f r o m a
space o f loops into the Lie g r o u p G. So far constructive m e t h o d s have n o t even b e e n s u c c e s s f u l l y a p p l i e d to the m u c h s i m p l e r n o n l i n e a r n o n l i n e a r i t y w i t h gauge theories.
~ -models that share intrinsic
In o u r t r e a t m e n t we simply force a linear structure
on these fields in o r d e r to be able to apply the m a t h e m a t i c s o f r a n d o m fields. It is p e r h a p s i n t e r e s t i n g to note that one meets a r e l a t e d d i f f i c u l t y in trying to p r o v e e x i s t e n c e theorems for the classical Y a n g - M i l l s equations, both in the e l l i p t i c and h y p e r b o l i c cases. There one w o u l d like to d o m i n a t e some suitable n o r m b y the classical action resp. energy, b u t o f course the c o n c e p t of n o r m r e q u i r e s a linear structure. It is also obvious that in our c o n s t r u c t i o n o f H i g g s
we did n o t p a y m u c h atten-
tion to the g e o m e t r i c a l m e a n i n g of gauge fields w h i c h s h o u l d b e i n t e r p r e t e d as conn e c t i o n s in some fibre bundle. O f course we know that will be d i s t r i b u t i o n s ,
(euclidean) q u a n t i z e d fields
so one w o u l d have to d e v e l o p a c o n c e p t of " d i s t r i b u t i o n v a l u e d
connections". Furthermore,
if we d r e a m e d of ever c o n s t r u c t i n g a theory like Q C D w we w o u l d have
to be able to see and e x p l o i t " a s y m p t o t i c freedom" in the construction. The central p r o b l e m b e h i n d all this is the following:
B e c a u s e of the lack of a
linear s t r u c t u r e we do not really know w h a t it m e a n s to d e c o m p o s e the fields into p a r t s l i v i n g in d i f f e r e n t m o m e n t u m ranges
(or b e l o n g i n g to d i f f e r e n t scale sizes).
This m o m e n t u m or p h a s e space l o c a l i z a t i o n p l a y s an e s s e n t i a l rSle in c o n s t r u c t i v e
[13]
field theory b o t h for s t a b i l i t y e x p a n s i o n s and for m e a n field (= semiclassical) [14] expansions ; this r61e b e c o m e s p a r t i c u l a r l y obvious in the a p p r o a c h of G a l l a v o t t i and c o w o r k e r s
(cf.
[15].) that is inspired b y W i l s o n ' s r e n o r m a l i z a t i o n group.
232
VI. How to Proceed Of course I don't know the final answer. What I outlined in this talk is a rather pragmatic approach: It fixes a gauge and proceeds as if the gauge fields were not fundamentally different from other kinds of fields; the penalty seem to be the tremendous technical complications that discourage us from attacking higher dimensional and n o n a b e l i a n m o d e l s
with these methods.
It should also not be forgotten that gauge fixing will run into the Gribov ambigui~
7
ties [II] that will certainly make things more difficult even though they do not seem to force any formal modifications of the Faddeev-Popov prescription [16] . A more gauge invariant way to decompose fields might try to imitate polyhedral approximations to a curved surface or the finite element method of the numerical treatment of PDE's. A polyhedron is piecewise flat; so one might try to define "slowly varying" gauge fields to be piecewise "as flat as possible", i.e. one might require them to obey the classical equations over the elementary cells of a certain scale size. The matter fields one would require to be piecewise covariantly harmonic
(an
analogous approach to ordinary scalar fields is equivalent to the standard lattice approximation).
Obviously such an approach would require a lot of knowledge of the
classical boundary value problems,
than we know at the moment.
more
F
I
A different approach will be discussed by T. Ba~aban [17] at this conference; it 4 is inspired by the work of Gallavotti et al. on ~ 3 and according to the announced results deserves great attention even though
(or because ?) it avoids dealing with
the conceptual problems mentioned here. References [i] Mathematical Problems in Theor. Phys., Proc. Rome 1977, G. dell'Arltonio et al.r Springer-Verlag Berlin, Heidelberg, New York 1978. [2] Brydges, D., Fr~hlich, J. and Seller, E., Construction of Quantized Gauge Fields I, General Results, to appear in Ann. Phys. [3] ... II, Convergence of the Lattice Approximation,
to appear in Conam.Math. Phys.
[4] ... III, The Two-Dimensional Abelian Higgs Model Without Cutoffs, preprint in prep. [5] .... Nucl.Phys.
B152
[6] wilson, K., Phys.Rev.
(1979) 521. DIO (1975) 2445.
[7] Garsia, A.M., Proc. of the 6th Berkeley Symp. on Math. Statistics 2, p. 369. [8] Hunziker, W., these proceedings. [9] Challifour, J. and Weingarten,
D., Indiana preprint 1979.
[IO] Fr~hlich, J., Lecture given at the Coll. on "Random Fields", Esztergom 1979. Eli] Singer, I.M., Commun. Math.Phys.
60 (1978) 7.
[12] Narasimhan, M.S., Ramadas, T.R., Commun.Math. Phys. 67 [13] Glimm, J., Jaffe, A., Fortschr.Phys.
21 (1973) 327.
[14] Glimm, J., Jaffe, A., Spencer, T., Ann. Phys. [15] Benfatto, G., these proceedings. [16] Hirschfeld, P., Nucl.Phys.
B157
[17] Ba~aban, T., these proceedings.
(1979)
(1979) 37.
iO1 (1976) 631.
121.
CLUSTERING,
C H A R G E - S C R E E N I N G AND THE M A S S - S P E C T R U M
IN L O C A L Q U A N T U M F I E L D T H E O R Y
Detlev Buchholz
and
Klaus F r e d e n h a g e n
II. I n s t i t u t fur T h e o r e t i s c h e Physik, U n i v e r s i t ~ t H a m b u r g F.R.G.
M a n y f u n d a m e n t a l results c o l l i s i o n theory,
in q u a n t u m field theory
the d i s p e r s i o n relations,
s t a t i s t i c s theorem)
(such as the H a a g - R u e l l e
the T C P - t h e o r e m and the spin-
are based on the a s s u m p t i o n that p h y s i c a l
states are
local e x c i t a t i o n s of some v a c u u m state. The c o n v e n t i o n a l way of expressing this h y p o t h e s i s
is to assume that there exist s u f f i c i e n t l y m a n y lo-
cal field o p e r a t o r s w h i c h g e n e r a t e the p h y s i c a l [ I ] . A less technical cannot be d i s t i n g u i s h e d
states from the v a c u u m
input is the a s s u m p t i o n that p h y s i c a l
states
from the v a c u u m by m e a s u r e m e n t s m a d e in the cau-
sal c o m p l e m e n t of some s u f f i c i e n t l y large, b o u n d e d region [ 2 ] . It is well known that n e i t h e r one of these c o n c e p t i o n s q u a n t u m field theory. ged ~ h y s i c a l
is a d e q u a t e in gauge
There one is faced w i t h the p r o b l e m that the char-
states cannot be d e s c r i b e d in terms of local fields
[3 ]
or
as s t r i c t l y l o c a l i z e d e x c i t a t i o n s of the v a c u u m [ 2 ] , and it is not quite clear w h a t the actual
l o c a l i s a t i o n p r o p e r t i e s of such states are. There-
fore the above m e n t i o n e d canon of f u n d a m e n t a l results applicable
is not d i r e c t l y
in gauge field theory, w h i c h is quite annoying,
b e c a u s e the
only m o d e l s w h i c h seem to be r e l e v a n t in physics belong to this category.
In v i e w of this situation it is of c o n s i d e r a b l e
interest to derive
c a l i s a t i o n p r o p e r t i e s of p h y s i c a l states from other assumptions,
lo-
which
are general enough to cover gauge field theories and w h i c h clearly display the p h y s i c a l
situation under consideration.
for such an a n a l y s i s basic o b j e c t is a net
~--~@)
of ~ -
b o u n d e d regions ~ of M i n k o w s k i b e i n g g e n e r a t e d by the local m e a s u r e d in ~ , a n d
The a d e q u a t e f r a m e w o r k
is the t h e o r y of local o b s e r v a b l e s [2]
a l g e b r a s a s s o c i a t e d w i t h the
space. The a l g e b r a s ~ ( ~ )
(gauge invariant)
they are t h e r e f o r e
lity and t r a n s l a t i o n a l covariance.
.There the
are r e g a r d e d as
o b s e r v a b l e s w h i c h can be
subject to the p r i n c i p l e s of loca-
One also c o n s i d e r s the a l g e b r a
w h i c h is g e n e r a t e d by all local o b s e r v a b l e s .
The p h y s i c a l
states of a g i v e n charge are r e p r e s e n t e d by v e c t o r s in some
irreducible representation
(~,~)
of ~ ,
lations are u n i t a r i l y i m p l e m e n t e d o n ~
and it is assumed that the transwith generators
P~ (energy-momen-
234
mentum)
satisfying the spectrum condition.
This general characterisation
of physical states should apply to all field theoretic models of elementary particles [4 ] . In this note we present the results of an analysis of the localisation properties of physical states for the restricted class of models with a complete particle
interpretation and a minimal mass
disregard models like quantum electrodynamics,
(mass gap). So we
but our starting point
is general enough to cover massive gauge field theories like the Higgs model and quantum chromodynamics.
The above restrictions on the models
can be easily expressed in terms of the mass-operator P ~ .
we assume
that to each particle type there corresponds an irreducible representation (TCp}~p~ ~
of
~
, called a particle representation,
0
has an isolated eigenvalue
represent the one-particle
in which
. The corresponding eigenvectors
states and we assume that the energy-momen-
tum spectrum of these states has finite
multiplicity.
The fact that
q~% is isolated guarantees that there are no massless excitations
in the
theory. If the algebra of local observables ~ has a particle representation, turns out that it has also a vacuum representation ducible representation,
(~0~0),
it
i.e. an irre-
in which the energy-momentum spectrum contains
the isolated point 0; the corresponding
(unique)
eigenvector ~
is cal-
led the vacuum. Moreover it follows that for a dense set of vectors @~ in ~p,arbitrary
local operators
A~ 5,
and arbitrary m e n
[5 ]
I(~, ~'~(~CA))~ ) - (@,¢Z).(~ ~-o(A)C~ J~ ~' (~* '~I)-~ (Here K ~
(~)
is the automorphism implementing the spacelike translation ~.)
This estimate, which resembles the well known clustering theorem in conventional field theory [ 6 ] , shows that the vectors in ~ p cannot be distinguished from the v a c u u m ~
by local measurements in the causal comple-
ment of some sufficiently large, bounded region. However, exclude the existence of sequences of observables
~
complement of an increasing set of bounded regions ~ , discriminate these vectors. The limit points ~
~-"~ ,~@,which
of these observables
commute with all local observables and therefore also with momentum
energy and
[ 7 ] . So they may be interpreted as charges.
Such charges, which can be measured at spacelike infinity, ristic for gauge field theories, charge.
this does not
in the spacelike
are characte-
the standard example being the electric
It follows immediately from (I) that in a massive theory e l ! p a r -
ticles carry zero electric charge, because the electric field
(which is
235
a local observable) decreases rapidly at infinity [5 ]. The first general proof of this charge-screening phenomenon
in massive abelian gauge field
theories was given by Swieca [8 ] . It is an open and difficult question whether a generalisation of this result holds also for non-abelian gauge field theories. So far we did not need to specify whether ~ or unbounded operators.
is an algebra of hounded
We restrict now our attention to bounded observa-
bles(which can be obtained by spectral resolution of unbounded oberservables)
and assume that the local algebras ~ ) a r e
C ~ -algebras.
We
must also consider algebras associated with certain unbounded regions of Minkowski
space.
If % is any pointed,
small aperture and ~t
spacelike cone with arbitrarily
its causal complement we denote by ~ ' )
algebra which is generated by all local algebras ~ @ )
with
the C e
~c 3'
main result is then that the restrictions of the representations ~p)
and
( ~ o )
to the algebra ~ ' ~
there exists a unitary operator
V~pCA) : R~(A)V This relation in ~ p quence,
and
.Our (~p,
are unitarily equivalent. That is,
Vfrom ~p
onto~
such that
, A~ ( ~ ' ) .
(2)
shows that it is not possible to distinguish the states
Z0 by
measurements
in the unbounded regions
the above mentioned charges ~
. As a conse-
, which can be determined
causal complement of an arbitrarily large bounded region ~ , infinite sphere,
in the
i.e. on the
cannot be approximated by observables on segments of
this sphere. Because
of the topological differences between a sphere,
and a sphere with a point removed one may reasonably argue that in a massive t h e o r y ~
can only be a topological charge.
In conclusion we want to stress that relation
(2) can also be used to
establish in our general framework some of the results mentioned at the beginning.
The reasoning is very similar to that expounded in
particular we found that if ~
has particle representations
exist also representations carrying the composite charges.
~].
In
then there If these re-
presentations are irreducible then the particles can only be Bosons or Fermions,
otherwise they must obey para-statistics.
Moreover,
to each
particle there corresponds an antiparticle with the same mass and one can construct incoming and outgoing collision states for these particles. So starting only from assumptions on the mass-spectrum and the causality-requirement
for local observables we get a very satisfactory account
of qualitative
features found in relativistic particle physics. A de-
tailed elaboration on these results will be published elswhere.
236
References:
1.
R.F.Streater and A.S.Wightman: that
2.
(Benjamin; New York,
S.Doplicher, 199; 35
3.
PCT, Spin and statistics and all
1964)
R.Haagland J.E.Roberts:
Commun.Math.Phys.
23
(1971)
(1974) 49
R.Ferrari,
L.E.Picasso and F.Strocchi:
Commun.Math.Phys.
25 4.
H.J.Borchers:
Commun.Math.Phys.
5.
D.Buchholz and K.Fredenhagen:
6.
D.Ruelle:
7.
H.Araki:
8.
J.A.Swieca:
i,
(1965), 281
Nucl.Phys.
Helv.Physica Acta 35 (1962) Progr.Theor.Physics Phys.Rev.
32 (1964)
D 13 (1976)
312
B 154
147 844
(1979)
26
35 (1974)
THE ULTRAVIOLET
STABILITY BOUNDS FOR SO,rE LATTICE
6 - I~0DELS AND LATTICE HIGGS-KIBBLE
NODELS
T. Ba~aban Warsaw University, 00-901 Warszawa,
A.)
Inst. of iv~th.
PKiN
IX p.
Let us consider a 6 -model with an action
U(x)6G,
G = U(N), SU(N)
is a parallelepiped
~
and periodic boundary conditions are assumed.
This model is considered than Higgs-Kibble
(or any Lie subgroup of U(N)), / k C 6 ~ for methodological
reasons - it is easier
model and the methods were worked out for it at first.
It is also much simpler to describe them in this case. A result:
if E(A, go,6) is property chosen
normalization expansion),
(it is defined as a sum of
constants and a counterterm computed from perturbation
d = 2,3, then there exist E+__independent
exp(-
z(^,go,~)
z(A, go,6) =
of A , g
such that
<e~p(~+I^l),
dU(x) sxp(-s(A, g o , ~ ) ) ,
dU is a Haar measure on the group G ( I~I = £ & (ntumber of sites i n A ) ) . A method:
iteration of renormalization
noff-Wilson
group transformations
in Kada-
form (3), (4)° Domain /k is divided into blocks B(y), y -
a site of a block, each block B(y) contains L d sites
(L = 2,3,°..,
the estimates are the best for L = 2). Let ~ : #k' 9 Y
> V ( y ) 6 G denote a configuration
of sites of the blocks° Ta[F]--
defined on the set
We define integral transformation
~]---[dU(x) t a ( ~ ,
~
Ta :
)F( Z ) ,
~-QA
a~-4 Q ~<'t)
where
'.IA"12 = tr(A*A).
Of course a n o r m a l i z a t i o n
c o n d i t i o n hol
s.
238
The use of renermalization group transformation T a is based on a crucial observation that it provides a natural "slicing" of momentum range, an effective integration in Ta~eXp(-S)1___ is over momenta ~T~.~ < ~ ~ 7_gTa. " Thus it implies a kind of "phase space cell •
11
expanslon as in (2). This is done by method of Benfatto, Cassandro, Gallavott± et alo (1). Let us now describe the essential points of the method. I.) The integration over each U(x), x m A , grations over U "small" : and U "large", g(a) = g o ~ ~ domains
is divided into subinte-
g(a)-111 - U(x)l ~ 0 ( 1 ) ( 1 + l o g g(8)-1) p, and /\is next divided into two sub-
A = M u M c, where ~ is contained in the set of sites x for
which the corresponding U(x) are "small". Ta[eXp(-S)] is estimated by different w&ys in ~c and M. 2.) In ~c the integral is estimated using positivity properties of the action. It is a difficult part of the proof because the action is changed after each application of renormalization group transformation and the positivity properties are not evident. 3.) In M it is calculated and estimated by cumulant expansion. To do this we expand each U(x), x~ M, in power series U(x)
= exp(i~g o
~
(x))
= I + ieg o
~=~a(x)
+ ...
,
the integral is expressed in terms of ~ -variables and after rescaling ~(a~) = a - & - ~ ~i(~) written in a form
and other transformations it can be
Now cumulant expansion method of Benfatto et al. is used for this integralo It is a laborious part of the proof. 4.) The very important ingredients in the considerations of the points 1
, ,I
~/~
2) and 3) are the properties of the quadratic part ~ Q ~ ) ~ g ~ . M -i of the action (exponential decay of covariances -~i )
C
and so on). In fact a lot of uniform estimates are needed for a sequence ~ ~ n m(k) ~ l obtained by iteration of renormalization group t tans format ions o The procedure described in the points I.) - 4.) is iterated until LkE = 0(I), and then the required bounds are obtained. B.)
A really interesting model is Higgs-Kibble model on a domain with action
239
& s(m,~t, go, 8)
+
~ , P ~A, P-plaquet t e s
~'[I
- Re trU(P)] + p,(A,X,go,&)
and partition function z(A'~t'g°'8) = l ~ d ~ ( x ) I -
IdU(x,x')exp(-S(A,#t,go,~))
,
(x,x'>c/~
where U ( x , x ' ) E C (only the cases G = U(1), SU(2) were considered), @(x) ~ corresponding vector space (= @I,~2). A result: if ~m2(~,go,£) and E(A,~t,go,E) are properly chosen, d=2,3, then there exist E~independent o f A ,~ such that
exp(-~_IAl) ~( z(A, ~,go,~) ~exp(z+IAl) . A method: renormalization group transformations in a Wilson form are used again. We define
(4)
,
ta(
where ~ y , x
v ,+,
"
~
' ~,y~
"
)= T--[const.exp(_a y~K'~ A
H
~
a.~l~(y) y,x I
are properly chosen contours connecting points
y,x or y,y~, and for a contour ~
passing through the points Xl,X2,...
• .,x n u ( p ) = U ( x ~ , x ~ ) U ( x 2 , x 3) . o . u(Xk_~,Xk). The above transformation T a transforms a gauge-invariant function into a gauge-invariant one°
Proof of stability bounds proceeds in the same manner as the proof in A.), with one additional step: a gauge fixing procedure is necessary in order to reduce the gauge invariance and to make possible the introduction of "large" and 'small' fields and the estimations of points 2.) and 3.). It is the author "s duty to mention here, that for non-abslian case the point 3°) is not completed yet.
I
240
References. (I)
(2) (3)
G. Benfatto, Mo Cassandro, Go Gallavotti, F. Nicolo, Eo Olivieri, Eo Presutti and E. Scacciatelli "Some Probabilistic Techniques in Field Theory" Communo Math° Pnyso 59, 143-166 (1978). Jo Glimm, Ao Jaffe Posltlvlty of the Hamiltonian" Fortschritte der Physik 21, 327 (1973). L. Kadanoff "The application of renormalization group techniques to quarks and strings" ReVo ~od. Physo, VOlo49, Noo2, 267-296
(1977). (4) K . Wilson "Quantum Chromodynamics on a Lattice" in "New Developments in Quantum Field Theory and Statistical ~echa~ics',' Cargese 1976, ed. by ~° Levy and Po ~itter.
MATHEMATICAL ASPECTS OF QUANTUMFIELD THEORY IN CURVEDSPACE-TIME Bernard S. Kay Blackett Laboratory, Imperial College, London SW7 2BZ, England I n s t i t u t e for Theoretical Physics, University of Berne, Switzerland In t h i s short paper, I discuss the q u e s t i o n : To what extent does the basic structure of f l a t space-time quantum f i e l d theory generalize when we curve the space-time? The discussion is meant to introduce and complement that of two recent papers ,2] to which the reader is referred for f u r t h e r d e t a i l s , motivation and references. For completeness, mention should also be made of closely related work by Ashtekar and Magnon ~3], Hajicek ~4], Isham~] and Moreno~6].
There has also been interesting
work by Dimock ~ ] and Wald[8] on the existence of an S-matrix for space-times which are Minkowskian o f f a compact set. One may divide the basic structure of f l a t space-time quantum f i e l d theory into: A)
What might be called the algebraic structure - i . e . local algebras, states, e t c . , and
B)
What might be called the vacuum structure - i . e . the whole complex of ideas arising from Poincar~ (actually time-translation) invariance:
The
existence of a preferred vacuum state, i t s associated H i l b e r t space representation, the existence of a positive Hamiltonian generating the dynamics etc. I shall i l l u s t r a t e the problem by considering the covariant generalization of the Klein Gordon equation ( g ~ v ~ + m2)~ = 0 Such a simple l i n e a r system can be understood quite completely.
(See
especially the work of Segal and co-workers e.g. [9]] and thus one can answer our question quite e x p l i c i t l y . I)
I shall outline t h i s answer in four p a r t s : -
The algebraic structure A generalizes for a class of space-times known as
g l o b a l l y hyperbolic space-times.
These are space-times (M,g) where M has the form
II~ x C where I~ constitutes a global time - coordinate on M in such a way that the equal-time surfaces ~ } x C; t ~II~ are Cauchy surfaces.
For such a space-time, we
have Leray's theorem that C~ Cauchy data onanyCauchy surface define a unique solution ~ C~(M) which has compact support on a l l other Cauchy surfaces.
Given a classical
l i n e a r system with such a well-posed Cauchy problem, i t is then straightforward to construct the algebraic aspects of quantization [2].
One equips the space of Cauchy
data C~(C) • C~(C) (or equivalently the space of solutions ~) with the natural
242
symplectic form arising from the conserved current 41 ~ 2 ' algebra over the resulting symplectic space.
and constructs the Weyl
Thus one obtains a complete theory in
the sense of the C algebra approach to quantum f i e l d theory. 2)
However, there i s , in general, nothing that could be called a vacuum state and no
natural H i l b e r t space-based theory. with a simple analogy:
I shall i l l u s t r a t e the physical meaning of t h i s
namely, a time-dependent system in flat-space-time.
f l a t space-time Klein Gordon equation with a variable mass:
m2(t) takes some constant value ml 2 before some "early" time t l ;
a d i f f e r e n t constant
value m22 a f t e r some " l a t e " time t 2 and varies smoothly in between. "vacuum representation" for mass mI
Take the
(I~ + m2(t))@ = 0 where Clearly, the
w i l l be appropriate for t < t I and that for a
mass m2 w i l l describe appropriately dressed p a r t i c l e s at late times.
But these two
representations are u n i t a r i l y inequivalent, and, in general, the dynamics between times t I and t 2 w i l l not be implemented.
So i t makes neither mathematical nor physical
sense to have a representation that w i l l do for a l l times. 3)
I f one specializes f u r t h e r to (globally hyperbolic) stationary space-times, one
has a time-translation invariance and thus one expects to have a vacuum structure B. Standard theorems ~ 0 , I I , 12] e s s e n t i a l l y reduce the problem to the construction of a " f i r s t quantized system":
One seeks a symplectic map K from C~(C) + C~(C) with dense
range in some complex H i l b e r t space H which intertwines the classical dynamics with a positive energy unitary group on H:
One says that K simultaneously Hilbertizes the
classical Cauchy data, and unitarizes the dynamics.
The construction of the vacuum
then proceeds via second quantization. We have the: Theorem [I S (ii)
I f , in a stationary space-time ( i ) there is a Cauchy surface C
In stationary coordinate systems
goo,~°°> ¢ > O.
suitably adapted to C the metric s a t i s f i e s Then a f i r s t quantized system exists, has a mass gap and the
complexified range of K is a core for the one-particle Hamiltonian. The proof depends on results on the causal structure of space-times and results relating the Cauchy problem in n + l dimensions to essential self-adjointness properties in n dimensions.
Conditions ( i ) and ( i i ) essentially specify what
constitutes "good behaviour at i n f i n i t y " .
In fact, they are always satisfied
when C is compact. 4)
Finally, I shall i l l u s t r a t e further the physical meaning of the conditions of the
theorem by giving an example where they f a i l :
Take the so-called "Rindler wedge".
That is one of the wedges in two-dimensional Minkowski space-time which is space-like separated from some origin.
T h i s is a globally hyperbolic stationary space-time,
the time translation symmetry being generated by Lorentz boosts. Now the space-time contains a Cauchy surface, but the p o s i t i v i t y condition ( i i ) is violated - because the
243 generator of the symmetry gets more and more l i g h t - l i k e as we approach the null lines bounding our space-time.
This is known to relativists as an ergosurface.
The
result of this is that, even though we're quantizing a massive quantum f i e l d , we end up with a quantum theory with no mass gap ~3].
o
•
\
FIG 1 : THE RINDLER WEDGE REFERENCES
I)
B.S. Kay: Linear Spin-Zero Quantum Fields in External Gravitational and Scalar Fields I. A One Particle Structure for the Stationary Case, Commun. math. Phys. 6__22,55 (1978)
2)
B.S. Kay: Generally Covariant Perturbation Theory (Linear Spin-Zero Quantum Fields in External Gravitational and Scalar Fields I I ) . Commun. math. Phys. (in press).
3)
A. Ashtekar, A. Magnon: Proc. Roy. Soc. Lond. A346, 375 (1975).
4)
P. Hajicek: In Differential geometric methods in mathematical physics II. Proceedings Bonn, 1977. Ed. K. Bleuler, H.RLIPetry, Ao Reetz: Berlin: Springer 1978.
5)
C.J. Isham: In Bonn Proceedings (as Ref. 4).
6)
C. Moreno: J. Math. Phys. 18, 2153 (1977), J. Math. Phys. I_9, 92 (1978)
7)
J. Dimock: Scalar Quantum Field in an External Gravitational Field, SUNY Buffalo preprint (1979)
8)
R.M. Wald: Annals of Physics 118, 490 (1979).
9)
I.E. Segal: In: Cargese lectures on theoretical physics, New York: Gordon and Breach 1967.
I0)
I.E. Segal:
I I I . J. Math. 6, 506 (1962)
II)
Mo Weinless:
12)
B.S. Kay: A Uniqueness Result in the Segal-Weinless Approach to Linear Bose Fields, J. Math. Phys. (in press).
13)
S.A. Fulling: Phys. Rev. D__7,2850 (1973)
J. Funct. Anal. 4, 350 (1969).
CLASSICAL AND QUANTUM ASPECTS OF TI-~ INVEP~SE SCATTERING METHOD l V.S. Gerdjikov * ", P.P. Kulish** ) and M.I. Ivanov *)
*) Joint Institute for Nuclear Research, Dubna, USSR **)
I.
Leningrad Department of the Steklov Mathematical Institute, Leningrad D-II, USSR
The inverse scattering method (ISM) [see the review paper of Ablowitzl)]
enabled us to solve exactly a number of physically important non-linear evolution equations (NLEE). Starting from an arbitrary linear operator (or an operator bundle), it allows us to construct the exact solutions of a whole class of NLEE;
this class
is generated by a certain integro-differential operator A, closely related to the operator L i-~). For the particular case s,7,8)
L =
[i
d
o3 ~ +
~
Ip 0 (x,t)
q(o't)l
_ ~2
]
~(x,X,t)
= 0
(1)
oo
A=7
~
,
I =
(x,t)
dy (p,-q) (y,t)
(2)
X
it is Possible to Prove s) $ that/(~.*-~ the vectors .~P) [q] and (1 + i I ] $ \-P~I / qtl may be expanded over the eigenfunctions ~- = (~f\~%~] of the~ operator A, ~- being the Jost solutions of L defined by their asymptotics at x + . The coefficients in the expansion for (q) are determined by the scattering data of L, and those for (i + il)(~l
by the
t-derivatives of the scattering data of L. Thus one may consider the ISM as a + generalized Fourier transform2,S,9, ~°) where instead of e iX2x we use ~-(x,X), and instead of (I/i)(d/dx) there naturally appears the integro-differential operator A. Using the uniqueness and invertibility of the expansions over {~+-}, we obtain the most general form of the NLEE related to L [Eq. (I)]:
f(A) (1 + iI)
-Pt ~
245
where f and g are arbitrary functions.
For example, if f(z) = l/z, g(z) = 4iz, and
q = -p*, we obtain the modified non-linear Schr6dinger equatio ns, 8), iqt + qxx + i(q2q*)x = 0. The spectral theory of the operator A s,11) allows us to derive, in a uniform manner, the main properties of these NLEE s,9,10) , and in oarticular their complete integrabilityg, 12-18) and the explicit form of the action-angle variables,
lqe also
see that to each NLEE we can relate an infinite number (a hierarchy) of symplectic structuresS,9,xg), generated by the symplectic two-forms: co
if
flk = 2-
dx (6p,6q) ^
Ak(1
+ iI)
-6p
-co
Using the important notion of gauge equivalence z° ,21) , we obtain that the class of NLEE (3) contains the massive Thirring model and other physically interesting equations.
2.
Now let us briefly discuss the quite recent development of the quant~n version
of the ISMG,22-26).
This method has been used to quantize the sine-Gordon equation z@,
different versions of the non-linear SchrSdinger equation6,23), and some spin models on a chain z4) (for a review, see Ref. 26). Here we consider the vector non-linear Schrgdinger equation:
i . . . . ~t Xz in the case n = 2;
2e
q~qB
qa (x't) '
s = +I
,
(4)
=I
the generalization to n > 2 is trivial.
The operator-valued
functions qa(x,t) satisfy the following commutation relations:
[qc~( x ) ' q~(Y)] = ~c~f~6(x-y) • As usual, when we start from a given classical theory to construct the appropriate quantum one, there arises the infinitesimal transition (monodromy) matrix Zn(~ );
in our case 6):
xn
o
if x n-1
Tn(X) =
x
n
X n-1
dy (s)
246
where the points x n divide the interval (-L,L) into K = 2L/A equal parts. The main relation in the quantum ISM is
(6)
R ( t - ~ ) TL(t ) ® TL(U) = TL(P) (9 TL(X) R(X-!J) , where R(I->) is a 9 x 9 (c-number:) matrix
R(t) =
be
ce
~o
~e'
0
0
r(1) =
° ° i]
~o
c
b
[i °° !] ~ c 0
(7)
0
K
TL(I ) =
Tn(l ) = n= i
B
B
,
b(k)
I + i '
b+c=l
.
~ Ca
From (5) we obtain the commutation relations between the matrix elements of TL(X); in particular we obtain that [tr TL(I), tr TL(~)] = 0, i.e. tr TL(1), is the generating functional of commuting quantum integrals of motion.
K Now we construct the states in the Hilbert space ae = ~={ ae~n) , on which tr TL(1) is diagonal.
First we note that the vacuum I0) e ae, for the operators
qa (x), satisfies
(Vl)lo> = o] ~(1)lo> = %
e-ilL
Io), D(1)lo> =
eikL
Io>
The eigenstates of tr TL(I ) in ag are generated from the vacuum I0) by Ca(l ) . Using the commutation relations (6) we see ~) that these states are labelled by two sets of quasi-momenta 11, ..., IN, ansatz 2 7) : A
l k
At, ..., AM, satisfying the generalized Bethe
]~__M A k
A. - i
-
J
j=~
j#k
N
M
o~J~i ~-~ ~ ~ :,:~,• k=-l
k#j
J
247
The corresponding eigenvalue of tr TL(X ) equals N
e
-ikL
i 1=1
c(h_hi )
1
+
1
C(Ak_ )
+ eiXL 1~__1~ - X
)
=.
where c(h-~) = (X - ~)/(X - ~ + i).
REFERENCES I) M.J. Ablowitz, Studies in Appl. Math. 58, No. 1 (1978). 2) M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Studies in Appl. Math. 5_33, 249 (1974). 3)
F. Calogero and A. Degasperis, Nuovo Cimento 32B, 201 (1976).
4) A.C. Newell, in Non-linear evolution equations solvable by the spectral transform (ed. F. Calogero), Research Notes in Mathematics 26, 127 (1978). 5) V.S. Gerdjikov, M.I. Ivanov and P.P. Kulish, JINRpreprint E2-12590 (1979). 6)
P.P. Kulish, LOMI preprint R-3-79 (1979), Leningrad.
7) E.A. Kuznetzov and A.V. Mikhailov, Teor. Mat. Fiz. 30, 303 (1977). 8)
D.J. Kaup and A.C. Newell, J. Math. Phys. 19, 798 (1978).
9) V.S. Gerdjikov and E.Kh. Khristov, JINR preprint E2-12742 (1979). I0)
D.J. Kaup and A.C. Newell, Advances in Math. 311, 67 (1979).
II)
V.S. Gerdjikov and E.Kh. Khristov, JINR preprints E5-I1668 and E2-12731 (1979).
12)
L.D. Faddeev, in Solitons (ed. R. Bullough) (Springer, 1977).
13)
H. Flashka and A.C. Newell, Integrable systems of non-linear evolution equations, preprint (1974).
14) M. Adler, Inventiones Math. 50, 219 (1979). 15)
B. Kostant, Inventiones Math. 48, i01 (1978).
16)
A.G. Reiman and M. Semenov Tjan'-Shan'skii, Funktzional'nii analiz i ego prilozhenia (in press).
17)
Yu.I. Manin and D.R. Lebedev, Preprint ITEP-155 (1978).
18)
I.M. Gel'fand and L.A. Dikii, Institute of Applied Math., Preprint 136/78, Moscow (1978).
248
19)
P.P. Kulish and A.G. Reiman, Zapiski nauchnikh seminarov LOM177, 134 (1978).
20) V.E. Zakharov and L.A. Takhtadjan, Teor. Mat. Fiz. 38, 26 (1979). 21) V.E. Zakharov and A.V. Mikhailov, JETP 7__4,1953 (1978). 22)
E.K. Skljanin and L.D. Faddeev, Dokl. Akad. Nauk USSR 243, 1430 (1978).
23)
E.K. Skljanin, Dokl. Akad. Nauk USSR 244, 1338 (1979).
24)
P.P. Kulish and E.K. Skljanin, Phys. Lett. 70A, 461 (1979).
25)
E.K. Skljanin, L.A. Takhtadjan and L.D. Faddeev, Teor. Mat. Fiz. 400, No. 1 (1979).
26)
L.D. Faddeev, LOMI preprint (1979).
27)
C.N. Yang, Phys. Rev. Lett. I_99,1312 (1967).
MAGNETIC MONOPOLESAND NON-ABELIAN GAUGETHEORIES
D.I. Olive Blackett Laboratory Imperial College, London
Magnetic monopoles arise naturally as soliton solutions in spontaneously broken gauge theories in D = 3+I space time. The possible values of the magnetic charge are restricted by a generalized Dirac quantization condition, and s t i l l further by s t a b i l i t y requirements.
Key questions concern the quantum field theory
of these objects and the possibility of a dual symmetry between electric and magnetic charges. Thesecould have a bearing on quark confinement. In one class o f model a universal mass formula hints at a role played by hidden extra dimensions of space time.
250 MAGNETIC MONOPOLESAND NON~ABELIAN GAUGE THEORIES Experiments in p a r t i c l e physics have revealed that apparently disparate phenomena can a l l be explained in terms of gauge theories.
Mathematically these are
principal bundles over a base manifold which, in physics, is Minkowski space time with three space and one time dimension (D=3+I).
This base leads to special features,
the dimensionlessness of the gauge coupling constant and the possible existence of isolated sources of magnetic f i e l d , namely magnetic monopoles.
These features could
bear upon the s t i l l unsolved problems of p a r t i c l e physics, namely quark confinement and the u n i f i c a t i o n of a l l the p a r t i c l e interactions. There has therefore been a renewed i n t e r e s t in the theoretical study of magnetic monopoles.
The fact that we can pose unanswered questions concerning them
c e r t a i n l y means we have a l o t to learn about gauge theories.
In any case recent
progress in understanding monopoles has synthesized many hitherto unrelated ideas, some of them important in other branches of theoretical physics; s t a t i s t i c a l physics, general r e l a t i v i t y etc.
solid state physics,
Thus we may well be on the way to
seeing a grander mathematical structure, underlying much diverse physics. In this t a l k I shall attempt to explain some of t h i s . Gauge theories are specified by the relevant Lie group and the simplest choice, namely U(1), governs Maxwell theory as we shall see. Maxwell's equations in vacuo e x h i b i t a peculiar d u a l i t y invariance with respect to the substitution 1 + i B ÷ e i@ (E + iB) ~) rotating between the e l e c t r i c
and magnetic f i e l d s ( i f D = 3+I~).
To maintain this
symmetry in the presence of matter one must introduce magnetic as well as e l e c t r i c charges. Classical equations of motion involving E, B and the p a r t i c l e coordinates can e a s i l y be written down with the desired symmetry (for a review giving more details of t h i s and many other points see 2.
But i f a p a r t i c l e carrying electronic charge q
is to be described by a quantum mechanical wave function there is a problem since the minimal coupling p r i n c i p l e ~÷p-qA introduces the gauge potential A and so the operator -ihV-qA= -i~(?-iqA/h) must enter the Schrodinger equation. so
Now B=V~A, and
dS.B = 0 by Stokes theorem i f A is defined g l o b a l l y , and so magnetic charge
cannot e x i s t . But A is not uniquely defined:
we have the gauge ambiguity A~÷A~+@~× and
¢ ( x ) + exp (iq×(x)/b)~(x),where X(x) is usually a single valued function of position. Notice that the phase factor exp(iq×/~) l i e s in a U(1).
This is the gauge group.
Surround a magnetic charge g by a sphere s p l i t into northern and southern hemispheres N and S, with gauge potentials AN and AS defined in each hemisphere but not i n d e f i n i t e l y extendable into the opposite hemisphere. For consistency we demand the existence of a gauge transformation connecting AN and As on the equator where the hemispheres overlap. So AN = AS + V× and ~N(x) = exp(iqx(x)/~)~S(x).
251
Since #N and ~S are each single valued~ and we suppose~ non zero, exp Ciqx(x)/~) must be single valued as the equator is encircled (but not necessarily X i t s e l f ) . The change in phase factor (which should be I) is formally exp(iq~ dI%FX/~) = exp. (iq~____~l (AN-AS)/I~ = exp ~ q .~d__SS,B(x)/~) , using Stokes theorem in the appropriate hemispheres for AN and _
The surface integral ~ d__SSB(x) extendsover the whole
sphere and therefore equals the magnetic charge g enclosed.
So
exp (i q g/h) = 1 This is Dirac's argument
(2) 3 , as refined by Wu and Yang 4
I t is close to the
mathematical treatment in f i b r e bundle theory. I t follows that for any electron charge q and magnetic charge g in nature qg = 2 n ~ , n = 0,
±I,
±2 ...
{3)
and that i f there is a g # O, q = nqo as observed in nature. Notice also that as the equator is encircled, the phase factor exp. (i q× /~) provides a map from the c i r c l e S' (the equator) to the c i r c l e S' (the gauge group U I ) ) , whose "winding number" is g/2q~, i . e . the n in equation C3). Thus g is a "topological quantum number", and we shall see further examples yet i t is quantized in the same way as the e l e c t r i c a l
charge q which could not be said to be topological.
Thus there are two ways of looking at a similar object, a quantized charge. This is connected to the fact that although quantum f i e l d theory is believed to provide the correct theoretical framework for elementary p a r t i c l e physics there are two d i s t i n c t ways of associating particles with f i e l d s : I)
As a quantum e x c i t a t i o n ; particles are created by the f o u r i e r components
of an___yf i e l d satisfying canonical commutation relations (x,t),
=
(4)
irrespective of the f i e l d equations. 2)
As sol±tons;
particles with structure occur as smooth localized f i n i t e
energy classical solutions to special non-linear equations - loosely speaking as "waves at rest". The simplest example of a sol±ton occurs in Sine Gordon theory in one space and one time dimension, (D=I+I), with Lagrangian densityj~'=½ (@2_~,2 -V(@)) and s e l f interaction V(@) = ~ (l-cos6@). ,~ has a symmetry @+#+2~/6 which generates the group of integers• The vacuum corresponds to ~ = 0, ±2~/B, ±4~/6 . . . . etc, the minima of V(#), which are equivalent because of the symmetry. The sol±ton solution arises when ~ changes by 2~/6 between adjacent minima as x runs from -~ to =. To see that there is such a solution consider the energy:
252
-co
fd+
I
(5)
This lower bound Is] on H is saturated exactly i f ~ = 0 which means the soliton is stationary, and i f @' = ± W F(@), for s o l i t o n / a n t i s o l i t o n .
So the soliton
s a t i s f i e s f i r s t rather than second order equations and i t s mass is given by a simple integral.
Time dependent m u l t i - s o l i t o n solutions also e x i s t .
possess a "quantum number", proportional to @(x:~) -
All these solutions
@(x = - ~), related to the
topological way the scalar f i e l d @ realizes i t s boundary condition at large distances. This charge corresponds to the current ~v@v@ which is conserved independently of the equations of motion. The most remarkable feature of the Sine Gordon model is that i t is possible to construct a quantum f i e l d operator [6] ~ {x) which creates the soliton in the sense ( I ) above, and s a t i s f i e s the equations of motion of the massive Thirring'model [?]
This
model has a phase convariance ~ + exp.(iqx) ~ whose Noether current Tzy~ coincides up to a factor with the ~ v ~ ~ already mentioned. At this Stage one may say that two space time dimensions are very special, but there are also other special cases where "miracles" may happen, amongst them our Minkowski space time D = 3 +I, when the aforementioned d u a l i t y symmetry [1] can occur, and in which there i s the special t w i s t o r structure [8] r e l a t i n g i t to CP3. When one t r i e s to generalize the concept of a soliton with a topological quantum number supplied by non t r i v i a l
boundary conditions upon a scalar f i e l d at large
distances, to D = 3 + 1 one is lead to consider a very interesting type of theory namely a Yang-Mills gauge theory [9] based on a "big" gauge group G (the symmetry ofj~) together with a Higgs f i e l d @(x)[Z°].
Like the scalar f i e l d above, this has
a set of non vanishing vacuum expectation values, but now this set is a continuous manifold, the coset space G/H of points related by the action of G on @.
H is the
subgroup of G which leaves invariant a @(x) of the vacuum set and is observed physically as the exact gauge symmetry group whose f i e l d s persist at long range. The reason for a l l this is that without the gauge f i e l d one nf the positive contributions to the energy is ½
]d3x (v@)2,which,
since V@ = O(I/x), diverges
unless @ is constant at large distances, which is uninteresting. Vi@ is replaced by the covariant algebra
d e r i v a t i v e ~ i @ = Vi@ + ie Wi@.
valued gauge or connection f i e l d for G.
I f we can gauge G, Wi is the Lie
Now the energy integral can mnverge
i f Wi has a term inversely proportional to e x, and there is a delicate cancellation between the two terms in~),-@.
Then the space-space (or magnetic) component of the
gauge f i e l d strength (or curvature) Bi : ½ E i j k Fjk, is inversely proportional to ex2. This inverse square law magnetic f i e l d suggests that the soliton is a magnetic monopole with magnetic charge inversely proportional to e in agreement with [3] ' t Hooft and Polyakov were the f i r s t to fiealize this [11]. They considered the simplest example in which G : S0(3) and the Higgs f i e l d formed an i s o t r i p l e t
253 (@i,~2, @~) so that the vacuum expectation values mo,i.e.the minima of the S0(3) i n v a r i a n t s e l f interaction, formed a sphere $2.
The special point about spacetime
D : 3 + 1 is that the set pf points at large distances also form a sphere S2 so that the boundary condition on the Higgs f i e l d provides a map from one S2 ( f a r ~ a y ) to another S2 (mo in isospace), i f we have f i n i t e energy.
This map can again be
c l a s s i f i e d by an integer winding number l a b e l l i n g homotopic equivalence classes [12] By theorems on exact homotopy sequences [13]this is
~2 (mo) : ~2(S 2) : ~2
(SO(3)/U(1))
~ ~1(U(1))
II
II
Z
Z
showing the r e l a t i o n to the previous topological way of classifying monopoles. For a general (simply connected) G the same theorem y i e l d s
(mo) z ~(G/H) ~
~(H)
(6)
I
Physically the l e f t hand side is defined by the boundary conditions on the scalar Higgs f i e l d while the right hand side concerns the long range magnetic fields [14] The soliton monopoles look like Dirac monopoles at long distances but now have a detailed f i n i t e internal structure and mass, and so provide an interesting model for elementary particles whose structure usually results from renormalized quantum corrections. The mass of the solitonmonopole can be estimated, rather as in the bound (5), whenever the Higgs f i e l d @lies in the adjoint representation of G, irrespective of G.
Define the Lie algebra generator dim G Q = eh E ~i Ti/a i=l
(7)
In the vacuum @ is covariantly constant ~)~@ = (V ~- i e W~iTi) dp = 0 and so has constant length, a say.
In a representation corresponding to a f i e l d ~,
Q has constant eigenvalues q which can be thought of as the Q charges carried by ~. The generators of the exact symmetry group H are precisely those generators of G which commute with Q. I t follows that Q generates an i n v a r i a n t U(1) subgroup of H, and that H has the structure [15] H =" U(1) x K" at least l o c a l l y near the unit element.
So Q is K s i n g l e t and so i f G = SU(4),
K = SU(3) t h i s would be a good model of QCD + QED (quantum chromodynamics and Maxwell theory).
Other currently fashionable gauge theories with a major Higgs f i e l d in the
adjointmepresentation are the"grand unified theories" of strong and electroweak i n t e r actions, based on G = SU(5) [16], SO(IO) [17] or E6 [18] Then Q is the weak hypercharge.
254 The Higgs f i e l d kinetic energy ½ ( ~ ) ~ ) 2 has a term quadratic in the gauge particles which can be interpreted as a mass term [10] for the gauge f i e l d s . The mass matrix M is then, by equation (7). (M2)~ = e2h2 ~o T~ TB ~o = a2(Q2)~B remembering that when @ is in the adjoint
representation, the T's are t o t a l l y
antisymmetric structure constants. Diagonalizing and equating have for gauge p a r t i c l e masses [9' 10] M=a
eigenvalues, we
Iql
To evaluate the mass of themonopo]esoliton, consider the classical Hamiltonian
H = ½~d3x [~2 +~2 + (~i@)2 + ~o@)2 + V(@) ] writing(~)2 + ~)i@)2 : ((Bi$~i#i2 ± 2~)i@~i
usingthe Bianchi i d e n t i t y ~ ) i ~ i
= O.
we see
Now far~ay,~)i@ = 0 and @ ~ i / a
magnetic f i e l d associated with the invariant U(1) subgroup. g is the U(1) magnetic charge of the monopole soliton [s'21]
is the
So H > algl where The bound is
saturated i f the terms dropped vanish. V(@) = 0 means the s e l f interaction coupling constant must vanish, even though @ "remembers" Lts length a in vacua!22]This leaves ~ i =~bi ~ = o , ~ i = ± ~ i @ , which mean that the soliton is at rest and s a t i s f i e s f i r s t order " s e l f duality" equations. Similar arguments applied to a dyon soliton [23] carrying both U ( 1 ) e l e c t r i c chargeqandmagnetic charge g yields. IS'21] M = a ~2
+ g2
(8)
When V(~) = 0 the Higgs particles are massless (and chargeless) so the remarkable conclusion is that all the particles of the theory, whether gauge, Higgs o r s o l i t o n i c s a t i s f y
the universal mass formu~ (8), irrespective of t h e i r
o r i g i n , or the gauge group G [19,20] The formula (8) manifests the electromagnetic dual symmetry (I) with respect to q + ig ÷ eie(q + ig). ' t Hooft [24] has found another manifestation of this symmetry ( I ) . When G = SU(2), so H = U(1), t h e ' s e l f dual' equations can be solved a n a l y t i c a l l y ,[22]
ra
~a ( r )
=
Wia ( r )
= -~ .. alj
er
2
(~ c o t h ~ - I) r__~j (I - ~/sinh~) er 2
= aer
255 This "Bogomolny-Prasad-Sommerfield" monopole is not a Dirac monopole since the Higgs f i e l d contributes equally with the magnetic f i e l d to the long range energy density.
T h i s also has the effect [2s] that forces between like charged BPS
monopoles vanish while forces betweenunlike onesare doubled, at least to an accuracy of (separation) -N . This raises the question as to whether l i k e - l i k e forces vanish altogether, or equivalently whetherthere exist time independent solutions describing N(>l) like BPS monopoles (as happens[26] for vortices in superconductors at the transition point between~pes land If), Nah~27] and Weinber~ 28]have shown, using index theorems, that i f such a solution exists i t lies on a manifold of 4N dimensions. displacements but the physical meaning of the
3N are spatial
remaining N dimension is as yet unclear.
The charge quantization condition satisfied by two dyons with charges ( q i ' gi )' i=1,2 is qlg2-q2gI : 2~ hn ~2 where ~2 is an integer [ 29] I f we accept the existence of an electron (q,o) this oDndition implies thatd~nswithg~-~haveq~q(n~) where n is an integer and ~ is undetermined, but the same for all dyons with the mmeg. CPinvariance implies that i f (q,g) exists so does (-q,g).
T h i s then implies
= 0 or½ so that we see that the departure of ~ from these values is a measure of CP non invariance,
Anothermeasure of this is the e angle of instanton theory [ 3 ° ]
The action A of a CP conserving non-Abelian gauge theory is replaced by Afh Ot where t is the topological (Pontryagin) integer, associated with the gauge f i e l d .
Witten[31]
has shown that these two parameters ~ and9 , measuring CP violation are related, = - 0/2~ in such a way that their periods coincide. Now l e t us return to the mass formula (8) and notice that i f we l e t aq and ag constitute two extra components of momentum so that we have a six momentum ~)~= (po, pz, p2, p3, aq, ag)
~nen (8) states that for a particle state~ 2= O.
Where dot!~se extra dimensions come from?
First we shall j u s t i f y p5= aq by noting
that since the Higgs f i e l d lies in the adjoint representation, like the gauge potential W~, i t can be thought of as a f i f t h component of the l a t t e r : [32] @= W 5, providing we agree that a l l fields are x 5 independent; a5 = O.
Since~)~@=F~5 the gauge +
Higgs Lagrangian density in D = 3+ l forms a gauge Lagrangian density in D = 4+I. Now the ordinary, gauge invariant, momentum operator Pu must generate covariant
displacements [Pu '@ ( x ) ] : i T ~
- iI~ ( ~
- i e W!Ji Ti) @
Hence i f the f i v e dimensional theory is g(x 5) i n v a r i a n t ,
while a5 = 0
[Ps' ~ ( x)] = e~W~ Ti~ = aQ~ by (7).
T h i s shows that P5 acts as the charge operator [20]
I f ~ couples
minimally with no e x p l i c i t mass term, the corresponding particle must be five dimensional l i g h t - l i k e , so satisfying (8). At the price of an x 5 dependent gauge transformation we can gauge the Higgs
256
f i e l d ¢ to zero sincethis is an "axial gauge" via [20]"
g (×5, x~) = T exp i e
Wi5Ti dx5 = exp i(a x5 Q/h) 0
Gauge non invariant fields now depend on x5, periodically i f Q has quantized eigenvalues, and so now the theory is very reminiscent of the Kaluza-Klein theory [33]. The clue to understanding P6 = ag is the question as to how to ~uantize the theory in D = 3+ I. There is an ambiguity which we shall seek to f i x by demanding as much symmetry as possible, by f i t t i n g ¢ and W~, which carry d i f f e r e n t D = 3 +I spins
into a single symmetry m u l t i p l e t [34]
The only known symmetry generator with
spin is a supercharge, in fact an anticommuting generator of SO(2) or S0(4) extended supersymmetry.
These symmetric theories are nice in D = 5 + 1 or 9 + 1 spacetime
dimensions respectively and consist of a fermion in the adjaint representation of the gauge group
G coupling minimally.
9 + 1Weyl and Majorana [3S]
In D = 5 + 1 the fermion is Weyl and in
In D = 5 + 1 the supercharge Qm carries an 8
component Dirac index and s a t i s f i e s the fundamental anticommutator 0
{q~ , Q }: [ ( I + 2~) r ~ r ]~B
(9)
Taking ~=6 and summing, we find pO = H = ~ {Q~, Q~ } )
0 so that the
Hamiltonian H is guaranteed to be positive, and to vanish only for supersyrrnetric states with Q~ = 0 [36]
In particular i f the theory is quantized in a supersymmetric
way, there are no quantum corrections to the energy of the vacuum.
Now equation (9)
also implies Ipofp2 : ~ {( r,~Q)~ (r.~ Q)~ } ) 0 s o ~ 2 ~ o and vanishes i f and only i f F.~Q = 0 . Now when we put B5 = B6 = 0 we find [37] that indeed[i~5 = aq, [ ~ = ag and that r ~ Q does vanish, at least c l a s s i c a l l y f o r the solitons. condition r~Q
Hence i f
(g) and the vanishing of the supersymmetry
holds in the quantum theory, so doest2= 0 and hence the mass formula
(8). The analogous statements for the supersYmmetric Sine-gordon theory in D = I+I have been checked p a r t i a l l y , [38]although there is some disagreement as yet [.39] The charges q and g we have discussed so f a r are Abelian, but non- Abelian gauge groups H seem to occur as exact symmetries in nature. compact with generator Ti s a t i s f y i n g
Let H be simple and
IT i, Tj] = i f i j k Tk where f i j k are totally antisymmetric structure constants. The field strength tensor F 6 = dimz H Fi6Ti can be divided into "electric" components~i = Fio (i=1,2,3) and i=l "magnetic" components H i = ½ EijkFik .
A reasonable way of recognizing generalized
257 e l e c t r i c and magnetic charges is to look at the components of Ei and Bi at large distances and seek a generalized inverse square law form ~-i
(r) = ( ~ ) i (r__) = ( ~ )
(r i /~3) Q ( r ) ; ~ . Q
= 0
(r i A~3) G ( r ) ; ~ i G = 0
) )) r large )
This means we are considering configurations with no "radiation". This is important since the d i s t i n c t i v e feature of a non Abelian theory is that gauge particles do carry charge which can therefore be radiated. The Q's and G's are gauge covariant objects and t h e i r equivalence classes modulo gauge transformations w i l l y i e l d the macroscopic structure of these configurations. Let us examine Q in quantum f i e l d theory, assuming for the time being that G = O, so that there exists a non-singular gauge transformation putting Wi= 0 (for large r ) so that Q = const = F d S ~ = ~ d 3 x V ~ I d 3 x jO = ~, where j~ is the noncovariant current
weakly equal to @~F~
by the Gauss law constraint equations, and
including contributions of the form i e Zi ~ D (Ti)@ Ti from all the fields including the gauge f i e l d i t s e l f . By the canonical commutation relations (4), we find
[Qi'
Qj] = i e ~ f i j k
Qk
We "measure" the Q's via the eigenvalues of a maximum mutually commuting set called a Cartan subalgebra, I say. Then by the theory of Lie a l g e b r a s , i f l A > denotes a simultaneous eigenstates of the Qi' i ~ I < A ] Qi I A > = e ~ { Ai 0
i ~ I i ¢ I
where Ai c A (H), is a "weight" of the group H,i.e.a set of simultaneous eigenvalues of Ti , i E I, in a single valued representation of H. Since l A > is constructed from the Fock space of matter f i e l d s , i t is t h e i r transformation laws which determine the set of weights and hence the global structure of H. Here i t is important to distinguish the d i f f e r e n t groups with the same Lie algebra but d i f f e r e n t global properties e.g. SU(2) with weights (0,_+½,_÷I,_÷ _.3 ,-+ 2 ... , and S0(3) with weights 2
(0, ±I, ±2 . . . .
).
In general i f H is simply connected (~,(H) = 1),
2 ~ /~2 = integer, any root ~ } w h i l e instead i f H has a t r i v i a l
A(H) = { ~ i '
centre (Z(H) = I ) ,
A(H) = Aroot (H) ={~ ,~=Zn i ~ i ' ni integers, ~ root}, There are intermediate cases we shall not discuss here. I f IA> is a physical state, the expectation values of Q and Q coincide, and we have found a specific quantum structure for Q, rather analogous to the Noether charge in the massive Thirr~ng model. Now l e t us turn to the magnetic charge G, treating i t by analogy with the Sine Gordon theory, purely c l a s s i c a l l y . Let h(r) be the ordered integral of the
258 exponentiated gauge potential around a curve ? far from the monopole at O, and subtending solid angle ~ at i t . Then h(?) = exp (i e G (P) ~ / 4 ~ , where P is the starting and finishing point of F. As ~ increases to 411, ~ shrinks to a point and so h(r) = I 40. This is the generalized Dirac quantization condition 1 = exp( i e G ( P ) ) .
As ~ increases from 0 to 4~ h provides a closed path in H,
thereby defining a homotopy class
~,(H) which is the topological quantum number.
I f G(P) = dim H gi T i ' a general theomm t e l l s us that there exists a gauge
i{l rotation sending gi ÷ O, i ¢ I . The remaining coefficients g i , i ~ I are not quite uniquely determined since there is an ambiguity due to the Weyl group, the f i n i t e group generated by reflections in the hyperplanes perpendicular to the roots (for SU(2) this is a sign ambiguity) Modulo this the g i m f l e c t the long range gauge invariant structure, and because of the Dirac quantization condition s a t i s f y 2e T~
Z gi Ti = integer i~l
But the Ti can be simultaneously diagonalized and so replaced by all possible 'weights of H. So 2e g i / 4 ~ must l i e on the l a t t i c e reciprocal to the l a t t i c e A(H). ~ This new l a t t i c e is also the weight l a t t i c e of a group, H~ say, in general d i f f e r e n t from H 40
For example i f Z(H) = I, A(H) =Aroot
(H) so the gi satisfy e_ ~ 2gi~ i
= integer. Put ~i ~ = ~ i / ( ~ i ) 2 so ~i = ~ /(~i ~)~ These "dual roots" 4 ~ i v , are by general theorems also the roots of a Lie algebra, Hv with the same rank and dimension as H. Now, we see comparing with the formula above for the weight system • 4~ ~ , of a slmply connected group, that gi = ~ w h e r e ~ ~ A4(H ), ~CHv ) = l . In • ~" • V • , ~ general i t always works that ~,(H ) = Z(H), Z(H ) = ~,(H), ( H ) = H. e.g.
so(3)
~ su(2)
SU(N)~ N ,-~ SU(N) SU(6)/Z2 ,-~ SU(6)/Z3 Notice that the fundamental representation of H and H~ cannot both occur for SU(3) say.
This is a possible explanation of quark confinement advocated by t ' H o o f t 42 43 and Englert and Windey The conclusion is that the gihave a very similar structure to the < A IQil A >, but on the basis of quite d i f f e r e n t argumenis, namely a classical f i e l d version of the Dirac condition, rather than a canonical quantization < AIQil A> = {e~ A(N) o gi
i ~ l (CSA) i ¢I
(10)
= { ~ -A(H~) i ~ l o i¢I
This formal s i m i l a r i t y
lead Goddard Nuyts and Olive 40 to conjecture that i f
these monopoles were solitons, and the quantum f i e l d theory of the monopoles were
259 constructed, then i)
Hv would be a symmetry of that theory;
ii)
the gi would label
the irreducible H~ representations of the f i e l d creating that monopole, i i i )
H~ is a
gauge symmetry (because of the long range forces). I
Then the ambiguity in addingmonopoles,g + ~ ( g ) ,
~ ~ Weyl group, would
correspond to the Clebsch Gordon series ambiguity in adding i rreps, but there is a problem in understanding the missing weight of t h e i r r e p . n o t i n theWeyl group o r b i t of gi" The basis for this was analogy with the Sine Gordon-Thirring system mentioned above, t'Hooft has made related arguments, apparently using as basis for his analogy the KramersW~nier duality of the Ising model on a square l a t t i c e 42,23 This relates two d i f f e r e n t phases of the same system, and t ' H o o f t talks of confined and Higgs phases. He uses as order disorder parameters, loop operato~ creating e l e c t r i c and magnetic f l u x lines. Mandelstam 44 has tried to construct the l a t t e r e x p l i c i t l y , and hence what might be the gauge potential of H~. One would only expect a subset of the possible magnetic charges (I0) to correspond to stable monopoles, with the specification depending on the detailed internal structure, supplied by the Higgs f i e l d s , but this is not so as Brandt and Neri 45 recently showed. An i n s t a b i l i t y of the Coulomb t a i l can occur into the non Abelian directions i ~ I , unless gi = ~ - ~' 2~ ~/~2 = O, ±I. The p o s s i b i l i t i e s ~i0)~ ±2,±3...then are all unstable. For example i f Hv = SU(2), only the 2 is stable, i f H~
= SU(3) only the 3 and 3. Then I) the problem of "missing weights" mentioned above then disappears since the Weyl group acts t r a n s i t i v e l y on the weights of the stableirreps.46 2)
stable weights, modulo the Weyl group, i . e . HVirreps., correspond in
a one to one way with the non t r i v i a l the topological quantum number 47
elements of X, (H)z ZCH~)which labels
The second point means that, to any unstable monopole of weight ~u ' there corresponds a stable monopole of weight ~s such that ~u - ~s = Z n i mi
qi
integer
Since the roots arethe weight of magnetic Hv gluons, this is the group theoretical
condition that permits the unstable monopole to decay into the stable one
emitting magnetic gluons, and maybe this is the i n s t a b i l i t y the GNO conjecture. 40
mechanism in terms of
I wish to apologise to all those to whose important contributions I have failed to do j u s t i c e . I should l i k e to close by repeating that the subject poses many interesting unanswered questions covering a very wide scope of mathematics and physics.
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E. Schr~dinger:
2)
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3)
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4)
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5)
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6)
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R. Streater and I.F. Wilde:
7)
Nucl. Phys. B24, 561 (1970).
S. Coleman: Phys. Rev. DII 2088 (1975). S. Mandelstam: Phys. Rev. DII 3026 (1975).
8)
R. Penrose and M.A.H. MacCallum: Phys, Rep. 6C, 241 (1972).
9)
C.N. Yang and R.L. Mills: Phys. Rev. 96, 191 (1954) R. Shaw: Ph.D Thesis, Cambridge University (1955).
lO)
P.W. Higgs:
Phys. Rev. Lett. 12, 132 (1964);
Phys. Rev. Lett. 13, 508, (1964)
Phys. Rev. 145, 1156 (1966). F. Englert and R. Brout: T.W.B. Kibble:
Phys. Rev. Lett. 13, 321 (1964)
Phys. Rev. 155, 1557 (1967).
ll)
G. ' t Hooft: Nucl. Phys. B79, 276 (1974) A.M. Polyakov: J.E.T.P. Lett. 20, 194 (1974).
12)
J. Arafune, P.G.O. Freund and C.J. Goebel.
13)
N.E. Steenrod:
J. Math. Phys. 16, 433 (1975).
The Topology of Fibre Bundles (Princeton, N.J.;
Princeton
University Press, 1951). 14)
M.l.Monastyrskii
and A.M.Perelomov: JETP Lett. 21, 43 (1975).
Yu. S. Tyupkin, V.A. Fateev and A.S. Schwar~, JETP Lett. 2__II, 41 (1975). S. Coleman: Proc. 1975 Int. School of Physics "Ettore Majorana", ed. A. Zichichi (New York Plenum, 1975), 297.
15)
E. Corrigan, D. Olive, D.B. Fairlie and J. Nuyts:
16)
H. Georgi and S.L. Glashow: Phys. Rev. Lett. 32, 438 (1974).
17)
N.P. Chang and J. Perez-Mercader:
Nucl. Phys. BI06, 475 (1976).
CCNY-HEP-78-15 (1978)
261
18)
Y. Achiman and B. Stech:
Phys. Lett. 77B, 389 (1978).
19)
C. Montonen and D. Olive:
Phys. Lett. 72B, 117 (1977)
D. Olive:
Phys. Rep. 49, 165 (1979)
20)
D. Olive:
Nucl. Phys. B153, 1 (1979).
21)
S. Coleman, S. Parke, A. Neveu and C.Mo Sommerfield:
Phys. Rev. DI5, 554
(1977). 22)
M.K. Prasad and C.M. Sommerfield:
23)
B. Julia andA. Zee: Phys. Rev. DII, 2227 (1975).
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G. ' t Hooft:
25)
Phys. Rev. Lett. 35, 760 (1975).
Nucl. Phys. B153, 141 (1979).
N. Manton: Nucl. Phys. B126, 525 (1977) J. Goldberg, J.S. Jang, S.Y. Park and K. Wali: S. Magruder: Phys. Rev. 17, 3257 (1978)
Phys. Rev. 18542 (1978)
W. Nahm: Phys. Lett. 79B, 426 (1978). 26)
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E.J. Weinberg: CU-TP-157 (Columbia preprint).
29)
J. Schwinger:
Phys. Rev. DI2, 3105 (1975).
A.M. Polyakov:
Phys. Lett. 59B, 82 (1975)
3O)
Phys. Rev. 19, 3008 (1979).
A.A. Belavin, A.M. Polyakov, A. Schwartz and Y. Tyupkin; Phys. Lett. 59___BB85 , 1975 VoN. Gribov:
unpublished
G. ' t Hooft:
Phys. Rev. Lett. 3_77, 8 (1976)
C. Callan, R. Dashen and D.J. Gross: Phys. Lett. 63B, 334 (1976) R. Jackiw and C. Rebbi: Phys. Rev. Lett. 37, 172 (1976). 31)
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M. L~he: Phys. Lett. 70B, 325 (1977).
33)
T. Kaluza: Sitzungber Preuss. Akad. Wiss. Berlin, Math. Phys. KA 966 (1921) O. Klein: Z. Phys. 37_7,895 (1926) W. Thirring:
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Phys. Lett. 76B 298 (1978)
262
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F.
Gliozzi, J. Scherk and D. Olive:
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H. Osborn: Phys. Lett. 83B, 321 (1979). 36)
B. Zumino: Nucl. Phys. B89, 635 (1975).
37)
E.Witten and D. Olive:
38)
A. D'Adda and P. Di Vecchia: R. Horsley:
Phys. Lett. 78B, 97 (1978). Phys. Lett. 73B, 162 (1978)
Nucl. Phys. BI51, 399 (1979).
39)
J.F. Schonfeld:
4O)
P. Goddard, J. Nuyts and D. Olive:
41)
F. Englert and P. Windey:
42)
G. ' t Hooft:
43)
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Phys. Rev. DI4, 2728 (1976).
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44)
S. Mandelstam: Phys. Rev. DI9, 2391 (1979).
45)
R. Brandt and F. Neri:
46)
W. Nahm (private communication).
47)
S. Coleman (private communication).
NYU preprint.
INTRODUCTION
TO PAPERS
PRESENTED:AT
THE GAUGE T H E O R Y
Andrzej Institute
Hoza
Most ment
the
of the papers invited
00-681 Warszawa,
Poland
at the gauge
by D.I.
by B. Schroer,
Olive
theory
and I.}%
these papers
session
Sinner.
deal with
comple-
Except
for
classical
as-
of the theory.
W. T h i r r i n g tion.
Following
tute the gauge tials
are
describes Kibble
identified
as the basic
group.
Nieuwenhuizen), the r e m a i n i n g connection
with
This
by U t i y a m a
translations
bundle
point
to be the s t r u c t u r e
dle of affine
frames
situation
mal...)
whereas
of linear
[4].
frames;
(Rarita-Schwinger)
are d e s c r i b e d
it is thus
the Lorentz
which
by c o n n e c t i o n s
group p r o v i d e d
defines
analogous
maps
linear
I favour on p r i n c i p a l
between
that
space-
in the bundle,
The P o i n c a r 6
group m a y
one considers the bun-
in a t h e o r y
by a c o n n e c t i o n
group
(P. Van
group y i e l d the
fields,
group.
of view,
field.
"vertical"
structure
of tetrads
point
a sharp d i s t i n c t i o n
and
to consti-
to give the tetrads,
P o i nc a r 6
to gauge
to aravita-
gauge poten-
to s u p e r g r a v i t y
"gauged"
In any case,
is d e s c r i b e d a choice
An a l t e r n a t i v e
approach
approach
of the
translations
~he c o r r e s p o n d i n g
are also
in the base
approach
[i] is to co n s i d e r
o f view allows
by the action
be taken
considers
of the graded
configurations
time t r a n s f o r m a t i o n s induced
on the gauge
the tetrads.
In the gauge
generators
fibre
gauge
bundles.
and Cho, h e
and the g r a v i t i n o
In the [2,3],
his views
g r o u p of gravitation;
w h i c h was o r i g i n a t e d
gauge
Physics
University
presented
lectures
the c o n t r i b u t i o n pects
69,
Trautman
of T h e o r e t i c a l
Warsaw
SESSION
of gravitation,
(linear, a section
affine,
a
confor-
of the b u n d l e
to fixin~ the gauge
in electro-
264
dynamics.
Incidentally,
an essential
-and often o v e r l o o k e d - d i f f e n c e
between g r a v i t a t i o n and gauge theories of the Y a n g - M i l l s type is due to s o l d e r i n g of the bundle of frames to the base m a n i f o l d geometric view of gauge theories of M.E. Mayer and L. Vinet. gravitational
[5,6]. The
is r e p r e s e n t e d in the c o n t r i b u t i o n s
Gary Gibbons describes large classes of
instantons which are important since they d o m i n a t e the
functional integral o c c u r r i n g in the E u c l i d e a n approach to q u a n t u m gravity.
The c o n t r i b u t i o n by M. Mulase extends the d e s c r i p t i o n of the
Y a n g - M i l l s instantons given by Sinoer.
Some of the simplest g r a v i t a t i o n a l not admit any spin structure.
instantons -such as
CP 2-
do
P.G.O. Freund d i s c u s s e s a e n e r a l i z e d
spin structures o b t a i n e d by e x t e n d i n a the Spin(4)
group so as to over-
come the t o p o l o g i c a l obstruction. A n o t h e r type of c o h o m o l o g y o b s t r u c tion, namely the one to l i n e a r i z a b i l i t y of n o n l i n e a r d i f f e r e n t i a l equations,
is d e s c r i b e d by D. Sternheimer.
J.S.R. C h r i s h o l m shows that
e l e c t r o m a g n e t i s m w i t h two vector p o t e n t i a l s may be used to d e s c r i b e m a g n e t i c m o n o p o l e s of a r b i t r a r y strength.
In a b e a u t i f u l
lecture, W. F r i t z s c h reviews the recent attempts
to c o n s t r u c t a unified, weak interactions.
gauge theory of strong,
SU(5)
electromagnetic
and
is amon~ the groups given the most serious
a t t e n t i o n in this connection.
I gratefully acknowledge
financial support from the U n i v e r s i t y
of Geneva w h i c h enabled me to p a r t i c i p a t e in the IAMP C o n f e r e n c e at Lausanne.
REFERENCES [i]
R. Utiyama,
Phys.
Rev.
i01
(1956) 1597.
[2]
A. Trautman,
Rep. Math. Phys.
(Torun) ~(1970)
[3]
A. Trautman,
Czech. J. Phys.
[4]
A. Trautman,
Symp. Math.
12
[5]
A. Trautman,
Bull. Acad.
Polon.
27, No 1 (1979) [6]
A. Trautman,
B29
29; 10(1976)
297.
(1979) 107.
(1973)
139.
Sci., S~r. Sci. Phys.
et Astron.
7.
A r t i c l e in E i n s t e i n C o m m e m o r a t i v e Volume,
A. Held et al., P l e n u m Press, New York 1979-80.
Ed. by
THE UNIFICATION OF PHYSICS
Harald Fritzsch I n s t i t u t e of Theoretical Physics U n i v e r s i t y of Bern and CERN, Geneva
Abstract The strong, electromagnetic, and weak i n t e r a c t i o n s are described as d i f f e r e n t manifestations of a grand u n i f i e d gauge theory. Candidates of u n i f i e d theories based e.g. on the groups SU(5) and SO(IO) are discussed.
At the present time high energy physics explores the structure of the elementary p a r t i c l e s at distances of the order of 10-15 o.. 10-16 cm. I t has turned out that we can describe the world at those distances or l a r g e r rather well
by three types of
gauge i n t e r a c t i o n s : I . Gravity (gauge group ~ Poincar~ group P) I I . Flavor Interactions (gauge group SU(2)xU(1)) III.
Strong Interactions (gauge group ~ SU(3)c).
The three d i f f e r e n t gauge groups involved are independent of each other, i . e . they commute. The t o t a l gauge group G is given by the d i r e c t product G = SU(3)Cxsu(2)xU(1)xP. As f a r as the elementary fermions are concerned, they come in two d i f f e r e n t varieties: a) SU(3) c s i n g l e t s (leptons) b) SU(3) c t r i p l e t s
(quarks).
A l l lefthanded quarks and leptons are doublets under the weak isospin SU(2), and a l l righthanded quarks and leptons are s i n g l e t s .
266 Quarks and leptons can be grouped together in f a m i l i e s according to t h e i r mass. The l i g h t e s t leptons and quarks c o n s t i t u t e the f o l l o w i n g e i g h t f o l d family:
e
i dr dg d b
(the three colors are denoted by red ( r ) , green (g), and blue (b)). The f u r t h e r , yet
CrCC)
heavier fermions c o n s t i t u t e t h e family
s r Sg s b
.
Furthermore there e x i s t s evidence f o r a t h i r d family:
I b r bg b b !
.
Thus f a r one has only evidence f o r the e x i s t e n c e of the T - l e p t o n , i t s n eu t r i n o , .and the b-quark. The t-quark has been searched f o r a t PETRA, thus f a r without success. The present lower l i m i t on the t-quark mass is of the order of 14 GeV. There seems to be no d i f f e r e n c e between the various lepton-quark f a m i l i e s , except the masses. Furthermore the dominant weak t r a n s i t i o n s occur within the same f a m i l y , i.e.
u++d, c + + s , t + * b ,
and not u + + s e t c . This suggests t h a t t h er e must e x i s t some
connection between the fermion masses and the weak i n t e r a c t i o n . The world of s t a b l e p a r t i c l e s which we observe in everyday l i f e c o n s i s t s of the members of the f i r s t family. We do not know how many lepton-quark f a m i l i e s e x i s t , and nobody knows why nature p r e f e r s to have more than one. I f someone is s a t i s f i e d to d e s c r i b e the world at d i s t a n c e s of the order of ~ 10 -16 cm by the theory described above, he can do so. However he i s f a c i n g a s e r i e s of important problems:
a) The number of free parameters is rather large. F i r s t of a l l the three coupling constants of QFD and QCD are e s s e n t i a l l y free parameters. Furthermore the lepton and quark masses are free parameters, and moreoever a l l elements of the lepton-quark mass matrix (e.g. the weak mixing angles). In case of the three family scheme denoted above one counts more than twenty parameters. b) Since the generator of the e l e c t r i c charge contains an Abelean U(I) f a c t o r , the e l e c t r i c charges of the quarks and lePtons are not quantised, i . e . the e l e c t r i c charge of the d-quark need not be I / 3 of the electron charge. c) The gauge group PxSU(3)Cxsu(2)xU(1) contains four factors. This is unsatisf a c t o r y , and one should aim to reduce the number of factors. The most ideal s i t u a t i o n would be to deal only with one gauge group. I am r e s t r i c t i n g myself to a more modest
267 approach, namely the construction of models which u n i f y the f l a v o r and color i n t e r actions. Gravity is l e f t out. The idea is to i n t e r p r e t the gauge group SU(3)CxSU(2)xU(1)
as a subgroup of a
larger gauge group, which is e i t h e r simple or semisimple such that the various subgroups are related by a discrete symmetry (e.g. p a r i t y ) I )
I t is clear that in such
a theory one needs to incorporate i n t e r a c t i o n s which are not observed, i . e . the associated gauge bosons must be much heavier than the c a r r i e r s of the conventional weak i n t e r a c t i o n s , whose masses are believed to be of the order of 102 GeV. Thus a u n i f i e d gauge theory of the color and f l a v o r i n t e r a c t i o n s must e x h i b i t at least two stages of symmetry breaking as shown below: G +I. U(1)xSU(2)xSU(3) + II. U(I)xSU(3). At the f i r s t
stage I . a l l gauge bosons which are not gluons, W, Z or y, acquire
a large mass, say of the order of M, while at the second stage I I . the masses of the W and Z (of the order of m) are generated. One has to have M >> m, in order to understand why the subgroup U(1)xSU(2)xSU(3) plays such an important r61e at r e l a t i v e l y low energies. At energies much below ~I the u n i f i e d i n t e r a c t i o n s can be neglected f o r many purposes, and we are l e f t with an e f f e c t i v e gauge theory, based on the group U(1)xSU(2)~$U(3). Within such an approach the fermions (leptons, quarks) form representations of the large group G, and one may have the hope to learn something about t h e i r mass spectrum. -
Let us consider the members of the f i r s t
+
-
_
family: (v e, e ; u, d; e , u, d). I f we
construct a u n i f i e d theory incorporating these fermions one finds two predictions: sin2e W = 3/8 (SW: SU(2)xU(1) mixing angle) g~ as = ~ =
8 e2 ~ ~ • ~ ~0.02
(g3: SU(3)c coupling constant). On the other hand the experimental values of ms and sin20w measured at energies of a few GeV are ~s~0.3 sin20 W = 0.23 L 0.02, i . e . there e x i s t s a large gap between the predicted and observed values.
268 One possible way out is to assume that the u n i f i c a t i o n mass H is a v e ~ large mass, and there e x i s t large renormalization effects. Indeed the observed value of
s
is obtained f o r M ~ 1015 GeV. One predicts in this case sin28 W ~ 0.20, which is not in disagreement with the observed value. The reader may f i n d i t u n s a t i s f a c t o ~ that in t h i s case the u n i f i c a t i o n of the i n t e r a c t i o n s sets in only at such gigantic energies l i k e 1015 GeV. However on the other hand one gains an understanding of the fact that the strong i n t e r a c t i o n s are much stronger than the f l a v o r i n t e r a c t i o n s . At distances corresponding to the unif i c a t i o n energy of the order of 1015 GeV a l l i n t e r a c t i o n s are u n i f i e d , and a l l coupling constants are of the same order, namely of the order of e. A t " n o r m a l " energies, say at a few GeV, the strong interactions are much stronger than the f l a v o r i n t e r a c t i o n s , since the gauge group SU(3) c is l a r g e r than the f l a v o r group SU(2)xU(1), hence the renormalization effects f o r the strong interactions are much larger than those f o r the f l a v o r i n t e r a c t i o n s . At energies of the order of 1 GeV the strong coupling constant a s becomes of order one, while the f l a v o r coupling constants are much smaller than one, namely of the order of 1/137. The group U(1)xSU(2)xSU(3) has rank 4. For t h i s reason the u n i t i n g
group G must
have a rank larger or equal 4. The fermions transform under SU(2)xSU(3) as follows: f =
, e-
L
, e , UL' dL + other generations d
L
= (2,1)+ (2,3)+ (I,I)+
( 1 , 3 ) + (1,3)+ other generations.
This representation is complex, and each generation is composed of 15 fermions. The smallest group which can serve as a u n i t i n g
group is SU(5) 2 )
which has
rank 4. The ( 5 ) - r e p r e s e n t a t i o n o f SU(5) decomposes under SU(2)xSU(3) as (5) = ( 2 , 1 )
+ (1,3).
The lO-representation of SU(5) is obtained as the antisymmetrized product (5X5)a. I t is easy t o w o r k out i t s SU(2)xSU(3) content: I0 = (5 x 5)a = ( I , I )
+ (2,3) + (1,3).
Thus the fermion representation is obtained as the reducible representation + I0. We make the f o l l o w i n g comments about this scheme: a) The basic set of fermions including the electron and i t s neutrino as well as the l i g h t quarks u and d appea~in a reducible representation:
g=
e~a e
!
269
10 =
(u, ) d
I! U~ e ÷ i
b) The o t h e r generations o f fermions (u, T . . . . r e p l i c a t i o n s of the f i r s t
) are simply i n t e r p r e t e d as
one.
c) In the 1 0 - r e p r e s e n t a t i o n of SU(5) appear both quarks and antiquarks. Theref o r e baryon number could not be a conserved q u a n t i t y . d) In SU(5) there is no room f o r a righthanded counterpart of the lefthanded neutrino. e) The gauge bosons ( 2 4 - r e p r e s e n t a t i o n of SU(5)) transform under SU(2)xSU(3) as: 24 : (1,8 c) + (3,1 c) + ( I , I c) + (2,3 c) + (2,3c). Besides the gauge bosons belonging to the U(1)xSU(2)sSU(3) c subtheory (~, Z, W; gluons) one has 12 a d d i t i o n a l gauge bosons which are c o l o r ( a n t i ) t r i p l e t s
as well as
weak doublets, We shall denote them by the doublet: yC
+ antiparticles .
The a d j o i n t r e p r e s e n t a t i o n of SU(5) can be obtained by m u l t i p l y i n g the 5 - r e p r e s e n t a t i o n w i t h i t s complex conjugate: 5 x 5 = 24 + I . Using the decomposition of the 5 r e p r e s e n t a t i o n in terms of quarks and leptons i t the couplings of X and Y.
is easy to work out the
They couple to a d i - a n t i q u a r k
configuration
(qq)
and to a leptoquark c o n f i g u r a t i o n ' ( q T ) . The X and Y i n t e r a c t i o n s lead to the decay of the proton i n t o leptons and mesons in the second order o f the gauge coupliog. As a consequence the masses of the X and Y bosons must be very large (not less than 1015 GeV). f ) The decay o f the proton is induced by the i n t e r a c t i o n s due to X/Y exchange. 6 0 0
The leading decay modes are: p ÷ e
~ , p , ~ .....
p ÷ ~
+ +
e
~ , p .....
The l i f e
time
of the proton can be c a l c u l a t e d in terms o f MX. One obtains: • (proton) = 103 . . .
104 • [I]/[I~ .
Since the mass of the X-bosons is estimated to be about 1015 GeV, the proton l i f e time is expected to be of the order of 1030 yrs. However the c a l c u l a t e d l i f e r a t h e r s e n s i t i v e to [IX , and i t
time is
is not possible to improve our knowledge about MX such
as to determine i t to b e t t e r than a f a c t o r o f ten. Thus we can say t h a t w i t h i n the SU(5) scheme the proton l i f e
time is expected to be ~ 1028 . . .
1032 yrs.
On the other hand the present experimental l i m i t on the proton l i f e 1029 y r s , i . e .
very close to the t h e o r e t i c a l value. For t h i s reason i t
to improve the present experimental l i m i t s on the proton s t a b i l i t y
time is is important
by 2 - 3 orders of
magnitude. I f the decay o f the proton is found at a l e v e l o f e.g. T(proton) = 1031yrs,
270 i t would, of course, not establish that the SU(5) scheme is correct, but i t would support the ideas about a unified theory of a l l interactions. A more extended scheme involves the gauge group SO(IO) 3). The simplest way to a r r i v e at the group SO(IO) is to consider the group SU(4)xSU(2)LXSU(2)R, where SU(4) is an extension of the color group SU(3) incorporating the quarks and leptons (the lepton number acts as the fourth color) 4):
dr
dg db 1
R.
su(4) The group SU(2)LXSU(2)R acts on the f l a v o r indices of the lefthanded or r i g h t handed fermions. All generators of SU(4)xSU(2)LXSU(2)R which do not belong to the subgroup SU(3)xSU(2)xU(1) are supposed to be broken such that the associated gauge bosons are much heavier than e.g. the masses of the W and Z bosons (of order lO 2 GeV). The group SU(4)xSU(2)LXSU(2)R is isomorphic to SO(6)xSO(4) which can be viewed as a subgroup of SO(lO). The fermions transform as a (16)-representation, which is the spinor representation of SO(lO). Under SU(4)xSU(2)LXSU(2)R one has (16) : (4, 2, I ) + (4, I , 2). Under SU(5), the (16)plet transforms l i k e I0 + 5 + I , i . e . i t contains the SU(5) family of lefthanded f i e l d s discussed above plus a s i n g l e t (a neutral lepton). The l a t t e r can be interpreted as the righthanded counterpart of the conventional l e f t handed neutrino. In order to break the SO(IO) gauge symmetry down to SU(3)xSU(2)xU(1) one needs several (I0) representations of scalars and, eventually, the (126) representation of SO(IO) ( f o r d e t a i l s see e.g. r e f . (5)). In any case the symmetry breaking in SO(IO) is more complicated than in the SU(5) scheme. The interesting new feature of the SO(IO) scheme compared to the SU(5) scheme is that i t allows the incorporation of the f i r s t family of fermions in one irreducible representation. Furthermore one can obtain, by choosing a suitable framework in order to carry out the symmetry breaking, relations between the fermion masses and the weak mixing angles 5), 6) which are in good agreement with the experimental informations available at present. Besides the schemes mentioned above one has investigated theories based on more extended gauge groups, e.g.
SU(4) 4 (see ref. (7)) and the exceptional gauge groups
E(6), E(7), and E(8) 8). For details we refer the reader to the original l i t e r a t u r e .
271 I f one accepts the idea of a unified theory of the strong and f l a v o r interactions, there are s t i l l
many unsolved problems, for example the problem of incorporating the
many flavours observed in nature, or the problem of what determines the huge gap between the grand unified mass scale of the order of 1015 GeV and the "normal" mass scale of the order of a few GeV. F i n a l l y the u n i f i c a t i o n of the strong and f l a v o r interactions and g r a v i t y must be regarded as an unsolved problem. I t is obvious that we are s t i l l
f a r away from constructing a unified theory of
a l l interactions, whose completion can be considered as the ultimate goal in physics and would imply the end of the development in fundamental theoretical physics. Nevertheless i t is interesting to note that at the present time we have arrived at a stage, at which the construction of an ultimate theory of a l l interactions has become thinkable.
REFERENCES I ) For reviews see e.g.: J. E l l i s , Proceedings of the Int. Universit~tswochen, Schladming 1979).
Austria,
H. Fritzsch, Proceedings of the Int. G i f t School, Jaca (Spain, 1979). 2) H. Georgi and S.L. Glashow, Phys.Rev.Lett. 32, 438 (1974). 3) H. Fritzsch and P. Minkowski, Ann.Phys.N.Y. 93, 193 (1975). H. Georgi, in: Particles and Fields, AIP, N.Y. 1975, p. 575. 4) J.C. Pati and A. Salam, Phys.Rev. DIO, 275 (1974). 5) H. Georgi and D.V. Nanopoulos, Harvard preprint HUTP-79/AO0). 6) H. Fritzsch, CERN preprint (March 1979), to appear in Nucl.Phys. B. 7) See e.g.: J.C. Pati, Proceedings of the 19th I n t . Conference on High Energy Physics, Tokyo 1978, p. 624. 8) See e.g.: P. Ramond, Caltech preprint 1979.
GAUGE T H E O R I E S OF G R A V I T A T I O N
W. T h i r r i n g I n s t i t u t f~r T h e o r e t i s c h e P h y s i k Universit~t Wien
W h e n E i n s t e i n f o r m u l a t e d general r e l a t i v i t y he took as L a g r a n g i a n the b e s t piece w h i c h the differential g e o m e t r y o f that time o f f e r e d a n d it w o r k e d m i r a c u l o u s l y well. E v e r since p h y s i c s and m a t h e m a t i c s have e v o l v e d a g r e a t deal and today we think that gauge theories describe the fundamental interactions. H e n c e one m i g h t study the similarities and d i f f e r e n c e s b e t w e e n gauge theories and general r e l a t i v i t y to see w h a t the chances o f further unifications are. For that I shall emphasize the p h y s i c a l rather than the g e o m e t r i c a l aspect. All classical field theories can be e x p r e s s e d by means of p-forms. L e t E this linear space. One needs the e x t e r i o r p r o d u c t A:
(Ep,Eq) ÷ Ep+q,
denote P the e x t e r i o r
derivative d: Ep ÷ E p+l and in a p s e u d o - R i e m a n n i a n m a n i f o l d the i n n e r p r o d u c t i: (Ep,Eq) ÷ E p _ q and the duality o p e r a t i o n ::: E p ÷ E m _ p (m = 4 is the dimension of space-time M, see
[i] for d e f i n i t i o n s and calculational rules).
The L a g r a n g i a n k ~ E 4 will be c o n s t r u c t e d from gauge p o t e n t i a l s A s e E l and their e x t e r i o r d e r i v a t i v e s d A
d : " F = _Xj ~
u s i n g A and x. T h e n the E u l e r e q u a t i o n s
aL
,
~:F~ := ~dA
8L
,
;"J~ := -~A -
(i)
'
have the form o f the i n h o m o g e n e o u s M a x w e l l equations. T h e y do n o t o n l y i m p l y charge c o n s e r v a t i o n dXJ
= 0 b u t also total charge zero in any c o m p a c t 3-dimensional sub-
manifold N without boundary
¥
i
P
Xj
N
=
-
{
XF
=
O
if
~N
(2)
=
~N
Examples i. Electrodynamics:
1 T h e r e is just one v e c t o r p o t e n t i a l A, L = ~ dA A XdA + [matter'
F = dA, and we have the usual M a x w e l l equations dF = O, dXF = -::J, ;:J = ~Lmatter/~A. 2. Chromodynamics:
Here the gluon p o t e n t i a l s A
are one-forms w i t h values in the Lie-
algebra SU(3), e = 1.... 8 and L is c o n s t r u c t e d from the c o v a r i a n t e x t e r i o r deriva1 B A A c By, c ~y b e i n g the structure constants in a b a s i s tire D A = d A + ~ A _ w h e r e the C a r t a n - K i l l i n g form is ~ ~. T h e n i L = ~ DA e A :"DA
+ [matter
,
1 F e = dA ~ + ~ A B A A 7 cesy
,
273
~[matter :~J = c A A :~F + - e pay y p ~A
Thus 3. T h e
the c o n s e r v e d simplest
colour c h a r g e J
has also a c o n t r i b u t i o n
t h e o r y of gravitation:
e x p r e s s e d b y an o r t h o g o n a l gauge p o t e n t i a l s .
The
The p s e u d o
f r o m the gluons.
m e t r i c o f M is m o s t c o n v e n i e n t l y
frame e e, i
currents
J
e 8 = e 8 , w h i c h p l a y the role o f the e~ are then the currents o f e n e r g y and momentum.
e
F o r instance,
1
I
(~ dA A : ~ d A ) =
is i n o b v i o u s components shall
analogy with
T 8 = ie8 J
consider
although
dA A i
the H a m i l t o n [ a n
are just the M a x w e l l
is not perfect;
are an a b e l [ a n
A ::dee +
seems
Again
the f i e l d s t r e n g t h s
corresponding
it is f u r t h e r p u r s u e d
mechanics.
tensor.
Its
Hence we
to the translations, in
and exterior
[2]. Since derivative
the coincide
to be
= i [matter
- L o f classical
energy momentum
group the c o v a r i a n t
a n d the s i m p l e s t L a g r a n g i a n
I [ = ~ de
H = ~
the e e as the gauge p o t e n t i a l s
the a n a l o g y
translations
:~F - i e (~ 2 d A A ~:dA) e e
+
;
grav
e
[matter
~
are F e = de s and there
- e
for e = O
+ ee
for e = 1,2,3
=
is a g r a v i t a t i o n a l
.
contribution
~L gray ~e ~ to the e n e r g y m o m e n t u m theory.
Thus
de ~ A i
XF 8 - i e~
currents J
. It has
Hence,
the same s t r u c t u r e
f r o m the M i n k o w s k i
metric by
cally since i t gives o n l y h a l f theory:
Here
i
g
and e~(x)
i n the linear a p p r o x i m a t i o n
~ [ m a t t e r / ~ e e we g e t for a p o i n t mass
4. E i n s t e i n ' s
grav
the theory is just 4 x e l e c t r o d y n a m i c s
+ I Dret(x-x')J~(x')d4x'"
differing
L ee
as in M a x w e l l ' s = const + w h e r e J~ =
at r e s t o n l y the O - O - c o m p o n e n t
mK/r.
Thus
o f the m e t r i c
this t h e o r y is r u l e d o u t e m p i r i -
the l i g h t deflection.
is r e q u i r e d
to be i n v a r i a n t
u n d e r local L o r e n t z
trans-
formations
e
which
change
÷ Le 8 (x) e 8
the L a g r a n g i a n
3). One
(3)
finds t h a t
Lg = l ( d e ~ A e ~) A :¢(de8 A e e) - l ( d e ~ A e ) A ~:(de8 A es)
changes
under
(3) o n l y b y an e x t e r i o r
derivative
since
it differs
only by
274
1 d(e e A ::dee) f r o m E i n s t e i n ' s L a g r a n g i a n ~ R X ~
w h i c h is invariant. In this theory
the field s t r e n g t h b e c o m e s a little more c o m p l i c a t e d
::Fe = e 8 A ~'¢(de8 A e e) - ~1 e e A X(de 8 A e 8)
the g r a v i t a t i o n a l e n e r g y momentum
,
k e e p i n g its form
~L t e
:=
g = de S A i ~e e e
;"F8 - i e
L e
e
g
Thus we see that E i n s t e i n ' s theory has the structure o f a gauge theory of translation.
Since L depends on the e e x p l i c i t l y and through the X - o p e r a t i o n there is g e a contribution ~ to the source from the field, t e is the s o - c a l l e d Landau-
L i f s c h i t z p s e u d o - t e n s o r w r i t t e n in an o r t h o g o n a l basis. In c o n t r a d i s t i n c t i o n to the M a x w e l l energy m o m e n t u m forms they do n o t t r a n s f o r m h o m o g e n e o u s l y u n d e r
(3)
w h i c h leads to all s o r t of unusual features. G a u g i n g the translations only there is no reason for r e q u i r i n g i n v a r i a n c e under local L o r e n t z transformations
(3). This r e q u i r e m e n t does n o t o n l y single o u t E i n s t e i n ' s
L a g r a n g i a n b u t also is essential for a c o n s i s t e n t e o u p i i n g to a Dirac field. O n l y if the Dirac L a g r a n g i a n L D is made locally L o r e n t z - i n v a r i a n t the ~UD/6e e gives the s y m m e t r i z e d energy m o m e n t u m tensor and n o t the canonical one. T h i s suggests that also the L o r e n t z group s h o u l d be gauged. Thus we introduce 6 more gauge fields e ~
e El
and the c o v a r i a n t e x t e r i o r d e r i v a t i v e s c a l c u l a t e d w i t h the structure constants of the P o i n c a r ~ group
De s = de s + e
D~e8 = dm B + e y
A e 8 = torsion
A ~YB = curvature
N o w the q u e s t i o n is h o w to construct a L a g r a n g i a n w i t h this material. One p o s s i b i l i t y is to take E i n s t e i n ' s L a g r a n g i a n :¢ee~D~eS b u t not to impose De s = O. T h i s yields t h e E i n s t e i n - C a r t a n theory
[4]. A n o t h e r Lagrangian, w h i c h is more s i m i l a r to the Y a n g -
M i l l s - L a g r a n g i a n of example 2) is [5] o b t a i n e d b y r e p l a c i n g de s b y De s in ig and adding the square o f the curvature:
L = Lg (e,De)A~ - f-ID~e~ A ::D~6e
If one uses in L Hehl e t al.
o n l y the f i r s t term one o b t a i n s i n this w a y a theory p r o p o s e d b y g [6]. W h e r e a s torsion is d e t e r m i n e d b y the spin d e n s i t y l o c a l l y i n the
Einstein-Caftan
theory it can p r o p a g a t e in these gauge theories and the a d d i t i o n a l
degrees o f f r e e d o m c o u l d give o b s e r v a b l e effects.
275
Re ferences
[i] W. Thirring, A Course in Mathematical Physics, Vol. II, Classical Field Theory, Springer, New York-Wien 1979 [2] Y. Ne'eman, T. Regge, Gauge Theory of Gravity and Supergravity on a Group Manifold, Rev. Nuovo Cim. 5, 1978 [3] F.G. Basombrio, A Comparative Review of Certain Gauge Theories of the Gravitational Field, preprint, Centro At~mico de Bariloche, [4] A. Trautman,
Symposia Mathematica 12,
1978
139 (1973)
[5] R. Wallner, Notes on Recent U 4 Theories of Gravitation, Vienna preprint, UWThPh-79-12 [6] F. Hehl, J. Nitsch, P.v.d.
Heyde, Gravitation and Poincar~ Gauge Field Theory
with Quadratic Lagrangian, Einstein Commemorative Volume, Plenum Press (1979-80)
(A. Hel et al. eds.)
SUPERGRAVITY AS A GAUGE THEORY DERIVED FROM MATTER COUPLING
Peter van Nieuwenhuizen Institute for Theoretical Physics State University of New York Stony Brook, L.I., N. Y. 11794
Supergravity is the gauge theory of supersymmetry (Fermi-Bose symmetry).
It is
also the unique theory of interacting spin 3/2 fields, or general relativity with a symmetry between fermions and bosons.
These definitions are all equivalent since
the gauge field of supersymmetry is the real massless spin 3/2 field ~ a (the gravitino), and because local supersymmetry can only be implemented in Lagrangian field theory if one is in curved spacetime, as we shall see. Historically, s u p e r g r a v i t y w m derived in refs. (1,2) by first constructing the gauge action, and only afterwards was the problem of matter coupling solved 3. the discovery of auxiliary fields
Also
which close the gauge algebra 4 and the establish-
ment of
a tensor calculus for supergravity 5 was done without considering matter
fields.
Today we will rewrite history and start at the matter end.
will rederive all results of simple (N=I) supergravity.
In this way we
Besides being a paedagogical
exercise, it is hoped that these new considerations presented here may be useful for extended (N>I) supergravity where only partial results are known. The simplest matter system in flat spacetime with a global supersymmetry, is the Wess-Zumino model 6. which are needed
To begin with we omit the two matter auxiliary fields (F and G)
if one wants a closed global algebra
(i) W~use the conventions of ref (3). For constant sa (a=l,4), the Lagrangian varies into a total derivative, and thus the action I ° is invariant and leads to a Noether current.
Replacing in (i) E
"P We cancel ~I ° by local E(x), 6I ° is equal to~ ~(x) times the Noether currents iN" u by adding to I % c o u p l i n g IN between JN and the gauge field of supersymmetry. S i n c e a c is a real anticommuting spinor, the iaule field of supersymmetry is a real massa
less veetof-spinor @~¢alled gravitino , and we must require that ~ (the factor 2 is only a convention).
= 2~ ga + more
The Noether coupling reads
(2)
277
Since the dimension of A is [A] = i, while [%] = 3/2, it follows that [e] = -1/2 so that [4 ] = 3/2.
Hence a
now on order by order
dimensional coupling appears in (2).
We will work from
in K.
The only terms left to order K in 6(I°+I N) come from 6(A,B,%) in (2).
Varia-
tion of % yields the AA and BB variations
where T (A) is the energy momentum tensor ~ A~ A-½6 v need for curved space, since only by coupling minimally
~ ~
A.
Here we discover the
to gravity and requiring
do the terms in (3) cancel. Starting all over again, but now in curved space with ~ XOab%
with e a
(4), (3) is
+
D°%
~
+ i ~ ab(e)
the tetrad field, one finds the same Noether coupling, and again with
cancelled.
The AB terms in ~IN are given by
.
~,
(s)
~
This term is cancel~d by partially integrating ~oA (and 3 B, one takes half of both). The terms with ~ ~ are cancelled by adding a term to the action of the form P 2 K 2 ~4ASB (since ~4 = -- 3 E + ...) but the remainder can only be cancelled if one identifies R ~ = c priate term to 6~ .
y5y ~p~o with the gravitino field equation and adds an approH e n e ~ the gravitino action is deduced to be
im y5 E) ( A ~ B ) and, indeed, 64~ = (-~
cancels (5).
It is encouraging that 13/2, when
linearized and in flat spacetime, is indeed invariant under ~
~ ~E
and that 13/2
is the unique action with positive energy. In curved spacetime (6) is not invariant under 640
= ~2
D ° os, but since
[D~,D~]~ is proportional to the Riemann curvature and since there are not enough indices in the
order K=0 terms in 613/2 to saturate the Riemann ~ensor, only its
contractions survive. 7 Hence ( ~ )
only the Einstein tensor appears in 613/2 .
Thus
one can cancel 613/2 by adding the Hilbert action 12 = -½eK-2R, since its variation is also proportional to the Einstein tensor.
One only has to define
(7)
278
Fortunately, this is in agreement with (4).
Without proof we state that the complete
gauge action is simple I2+I 3/2 and is invariant under (7) and ~ one use
Palati~
formalism:
= ~2 D e, provided
replace every where m ab(e ) by an independent field
then solve ~(12+I3/2)/~m ab = 0 to obtain ~ a b m~ab' way torsion induced by gravitinos.
= ~ a b (e'~)"
One finds in this
Returning to (2) and varying A,B (and using also (7) in (i)) all variations with %% can
cancel again if one
adds either term to action and transformation laws.
be found in S. Ferrara, F. Gliozzi, J. Scherk and P. van Nieuwenhuizen, Nucl.
Phys. BII7, 333, 1976).
and
(Details
if
one
adds
extra
All order K terms in ~(I°+I N) cancel, if
order
2 K terms
of an axial current squared.
to
the
action
which
happen
to
be
all
of
the
form
The supercovariant derivative DC°VA has the property
that ~(D~°VA) contains no ~ s(x) terms and is equal to D~A = 3~A - ~K ~ %.Not all is well.
Matter terms in the variation laws of gauge fields preclude
the possibility to add invariant actions and still to obtain an invariant action. Repeating the Palatini formalism for I2+I 3/2 + I°+I N eliminates only the %% terms in (8) but not the AB terms. Lagrange multipliers.
Thus one is forced to introduce auxiliary fields, i.e.,
Since there are
two •auxiliary axial-vector fields.
two spinor structures in (8), one would need
There exists such
and our procedure might lead to them.
a set of auxiliarly fields, 8
We prefer here, however, to restrict our
attention to the minimal set of ref (4). In order to Obtain only one spinor structure in (8), we use the freedom there always is in
the Noether current, and add an extra term j~(imp) so that
~ (j~+j~(imp)) = 0
on-shell.
Thus we replace (2) by the improved Noether current
coupling
a
where e = det e
.
Since also D j (imp) = 0 on-shell, we only need to consider to
order m the variations of A,B,%in this new term in the action. D
= ~
~ where R ~ is the gravitino equation defined before.
Note now that o ~ The %~ and A+~B
terms obtained from (9), are thus removed by adding extra terms to ~ V " now only one spinor structure in the gravitino law
One finds
279
Hence there is an axial vector auxiliary field A
and we consider A
in (i0) as an
independent field, no longer given by (ii), but add terms to the action which lead
to (ii)
_ C
D
Indeed, varying A
_ArA
B
(12)
~L
one finds back (11).
The normalization factor 1/3 is arbitrary,
and from (i0) it follows that if one defines
than all order
K terms with A ~ B
the terms linear in A
and XX coming from (9) cancel, while automatically
from 6(13/2 + I(A )) cancel, too.
These are indeed the results
of ref. (4). Note that D ~voPv
is proportional t o ~ ~.
term in (9), one finds terms ~ ~ (A)2~R .
Thus varying 6X = ~As in the last
These one cannot eliminate by a suitable
~@~, since this would violate the one-spinor-structure of (i0). 6(eR) contains a total derivative ~ D p ~ m a b ( e , ~ ) ~
ab(e,~)_ one finds terms of the form ~R.
K~
by adding a t e r m ~
Note, however, that
(which we usually discard) while in
One can indeed cancell the 8 (A2)~R H term
(A2+B2)eR to the action.
This is the onset of nonpolynomiality. 9 To see how S and P come in, we prefer to first quote the full answer , and then
to see where one would have made further (re)discoveries.
If one couples the Wess-
Zumino model with its auxiliary fields F and G to the gauge action with its auxiliary fields S , P , A (i)
and requires
that the total action is invariant under the standard transformation rules of ref (4).
Then the action is necessarily nonpolynomial
(eq.(30) of ref (9))
~ii) that after a~Weyl rescaling, the scalar kinemetic terms are as in (i) and the R 1 term is the standard - ~eR(eq(55) of ref(9)) (ii@ that afther this Weyl rescaling the action is polynomial (eq(59) of ref (9)) then the action is unique and given by I0
77 2 ~o where x = ~-- (A2+B 2) and
N and ~ i m p
(14)
contains torsion-free derivatives D °. P
Furthermore,
J
=
_
)[
A
]
(15)
280
Note that to lowest order in X,_aXx makes all derivatives in ~ chirally covariant. A ~ + C°VB.
Also
the second and
+ ~'" + ~'." mmp
third terms combine to
Finally
(16) We have omitted four-fermion terms in (14).
To pick up the discussion where we broke i 2 2 it off, we solve the S and P equation and replace in (16) the S,P terms by ~X(F +G ). Suppose we would have started with
(i) plus a term ~2(F2+G 2) in the action.The
flat space laws, gravitationally covariantized, read
Repeating all previous steps, the Noether current and improvement addition remains the same. D°
Invariance of I °
by D c°v
in (17).
and (2) under (17,7) to order
is obtained by replacing
In particular
(18) Consider now the extra Noether terms in (9).
The only new variations
comes from ~% = __(F+iYsG)e and lead to a new term in
to order
6~!J (19)
Hence,
there are a scalar S and a psendoscalar P auxiliary field and one must add
the S and P dependent terms in (16) to the action and finds, in agreement with ref(4)
Complete invariance of the gauge a c t i o n ~ 2 + ~ / 2 simply by replacing R
by RC °v in (20) and (13)
-
I 2+P 2-A 2 ) is then obtained ~(S
( except that in (13) one must
replace A
by A = A e ~). ~ m ~m These steps are completely analogous to the steps we did already for A .
recapitulate. structure.
We found A
This forced
from the requirement that in ~
Let us
there be only one spinor .
the improved Noether coupling (and brings supergravity near
281
to conformal supergravity).
Had we allowed
jectured that another tensor calculus might and
two spinor structures in ~ , have emerged.
Gin the flat-space globally supersymmetric matter
structures in 6 ~ ,
we con-
The auxiliary fields F
action lead to two new spinor
which led to two new auxiliary fields S and P.
There really is no
choice, the auxiliary fields follow directly from our analysis. We tions of
will stop here. (14) cancell.
The
reader may enjoy the puzzles of how all further varia-
It is also remarkable that the gauge algebra closes.
cancellation pattern is certainly quite
The
subtle sometimes, but what we have shown is
that from the simplest matter model, all features of supergravity can be deduced. The simplest set of auxiliary fields needed the improved Noether coupling, which is not surprising,
since the tensor calculus 5
and auxiliary field structure 4 have ii been reobtained previously from conformed supergravity
REFERENCES i.
D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. DI3, 3214 (1976).
2.
S. Deser and B. Zumino, Phys. Lett. 62___B, 335 (1976).
3.
S. Ferrara, J. Scherk and P. van Nieuwenhuizen, idem with P. Breitenlohner,
Phys. Rev. Lett. 37, 1035 (1976),
D.Z. Freedman and F. Gliozzi, Phys. Rev. DI5, 1013
(1977). 4.
S. Ferrara and P. van Nieuwenhuizen,
Phys.Lett.
74___B, 333 (1978); K.S. Stelle and
P.C. West, Phys. Lett. 74__B, 330 (1978). 5.
S. Ferrara and P. van Nieuwenhuizen,
Phys. Lett. 76__B, 404 (1978) and Phys. Lett.
78___B, 573 (1978); K.S. Stelle and P.C. West, Phys. Lett. 77___B,376 (1978) and Nucl. Phys. B145, 175 (1978). 6.
J. Wess and B. Zumino, Phys. Lett. 49___B, 52 (1974) and Nucl. Phys. BTO, 39 (1974).
7.
For spin 5/2 one finds, however, uncontracted Riemann curvatures and one cannot couple consistently spin 5/2 to gravity. See F.A. Berends, B. de Wit, J.W. J.W. van Holten and P. van Nieuwenhuizen.
8.
P. Breitenlohner,
9.
E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello and P. van Nieuwenhui-
Nucl. Phys. B124, 500 (1977) and Phys. Lett. 80___B, 217 (1979).
zen, Nucl. Phys. B147, 105 (1979). i0. Put g=0 and ~=-3 exp(-x) in ref(9). Replace in eq.(30) multiply in eqs.(63,67)
the term
abcp
~---aYb~c e -I by
~2
A2
½.
by
1-~--~A2P and
Then one reobtains
the results of ref.(3). ii. M. Kaku and P.K. Townsend, Phys. Lett. 76__B, 54 (1978); A. Das, M. Kaku and P.K. Townsend, Phys. Rev. Lett. 40, 1215 (1978).
GRAVITATIONAL
INSTANTONS: A
SURVEY
G. W. Gibbons D.A.M.T.P. Silver Street Cambridge U.K. A Review talk given at the "International Conference on Mathematical Physics", Lausanne, August, 1979.
Defn (I)
A Gravitational Instanton is a complete, non singular, 4-dimensional
Riemannian manifold (signature ++++) which satisfies Einstein's Equations
Gravitational Instantons are believed to dominate the path integral for Euclidean Quantum Gravity [I, 2, 3, 4, 5, 6 ] .
One considers expressions of the form
(1)
is the Euclidean action of some manifold M, boundary JM, metric inducing a metric ~
on ~ ~ • ~-- ~
fundamental form of
~M.
~#
,h= ~ ' ~ n d
Cj~/3
K is the trace of the rsecond
E is the Riccl scalar of ~ #
.C
denotes the boundary
conditions that the metric satisfies - corresponding to the quantum mechanical state or density matrix and O (g) is a functional of the metric whose expectation value or matrix element is given by (I).
In what follows I shall regard non compact
manifolds as the limit of compact manifolds with boundary as the boundary recedes to infinity. ~=
O
(If
~
= ~ ~
and ~
~ there can be no boundary ~31~ . If
there can be at most one "end" or "infinity" (N. Hitchin, private
communication).)
The "integral" in (I) is over all possible manifolds, with all
possible topologies subject to ~
.
~6
is a correction term designed to render
zero the action of any flat metric satisfying
Three types of boundary
conditions ~ , are important in physical applications. I)
A~ymptotically Euclidean (A.E) together with a weaker, local version -
Asymptotically Local Euclidean (ALE).
This corresponds to vacuum or zero
temperature physics [5, 7, 8 1 . II Asymptotcally Flat (A.F) together with a weaker, local version - Asymptotically Locally Flat (ALF). III
This corresponds to finite temperature physics [i, 9 ~.
Compact without boundary.
This is used in the discussion of "spacetime foam"
283
3, 4, IO] . Defn (2) ~j~ on E /[ .
is ALE if outside a compact set the metric approaches the flat metric E
zs flat Euclldean space, ~ a discrete subgroup of SO (4) with free
action on S3. ~i# is A.E if ~
is the identity.
Defn (3) ~ is A.F if outside a compact set the metric approaches the standard flat metric on S I x R 3. Defn (4) ~fi
is ALF if outside a compact set the metric approaches -.
0-~
4-
v(G
Crd)
where [(~, } are left invariant one forms on S3/ SU (2) x U (I) invariant metric.
and
is an isometry of this
AE Instantons Thm (i) (Shoen-Yau
[ii, cf. 8] ):
There are no AE instantons other than E 4.
The Positive Action Theorem [I17 shows that the conformally invariant part of the action (in the sense of
[5 1 ) has an absolute minimum at E4.
ALP Instantons Thm (2) (Hitehin, private communication) then q~l {~) is finite.
Let ( ~i ~d~ ) be ALE and Ricci flat,
Thm (3)
Let ( ~, ~¢~ ) be ALE and half fiat
(Hitchin, private communication)
( ~:~6~>~R
~-~ ~ =
0 ) then
-n;,(~)
=~
or ~ @ Z "
and
the universal~ cover___has a discrete# subgroup of SU (2) whose action on S3 is given (for ~ # # ~ = - 4 - ~ Aft ~ Af~v ) by right actions of SU (2) on itself. The discrete subgroups of SU (2) are Zk
cyclic order k
D*k T* O*
binary dihedral order 4K binary tetrahedral binary octahedral
I*
binary icosohedral
Almost certainly there are half flat metrics for all these groups. metrics were given in the Zk case by Gibbons and Hawking
[7].
Explicit
Later H i t ch i n [13~
used "Twistor" methods to give the Zk case and he now has the D~ (private communication). The Gibbons-Hawking metrics may be written in coordinates (-£,~) as
284
(s) k = I is flat space, E 4 k = 2 is the Eguchi Hanson metric [14, 15, 8 7 which had been discovered earlier.
The Zk ALE's depend on 3K-6 essential parameters which
correspond exactly to the possible infinitesimal variations. the possible half flat ALE metrics with p = ZR.
They probably exhaust
He has also constructed scalar
Green's function explicitly for these spaces [ 16 ] .
Atlyah (private co~unicatlon)
subsequently gave a Twistor construction of these Green's functions using sheaf cohomology methods. Any half flat metric is a local minimum of the action amongst metrics with R = 0 suggesting the following analogue of Theorem I. Conjecture has
~
(I)
The only ALE instantons are half flat and their universal cover
SU (2).
AF Instantons The known A.F instantons are the flat metric on S I x R 3, the Euclidean Scwarzschild and Euclidean Kerr with imaginary angular momentum F23] . Thin (4)
(modified Israel [18~
)
Let (~j~#) be A.F. and Ricci Flat on R 2 x S2
with a hypersurface orthogonal killing vector, then ~g~
is the Euclidean
Schwarzschild. A. Lapedes (private communication) points out that the analogous modification [19l for axisymmetric metrics on R 2 x S2 with another non
of Robinson's Theorem
hypersurface orthogonal killing vector does not go through.
Nevertheless on the
basis of the Black Hole No Hair Theorems (see [17~ for a review) one might Conjectur~
(2)
The only A.F instantons are flat space, Euclidean Schwarzschild
and Euclidean Kerr. Schwarzschild and Kerr are not local minima of the action amongst A.F metrics with vanishing Ricci scalar [32] .
ALF Instantons
t=~ =
I --
'
}
(6)
in equations (3) and (4) one obtains the "multi-Taub-NUT" ALF metrics of Hawking ~27. The boundary at infinity is a cyclic lens space.
Presumably these metrics (and
those corresponding to D~) can be constructed using Twistor methods. groups cannot occur [9] .
The other
Page's Green's function construction also works in the
multi-Taub-NUT case. The metric (3) and (4) exhaust the class of metrics with a self-dual killing vector ~ 7 has ~X#~
self-dual.)
This is also the class with a killing vector in
which the connection forms in the "obvious" basis are self-dual. The remaining, explicitly known ALF solutions are not half-flat.
There are
285 the solutions of Page 1211 and its spinning generalization whether more complicated ALF metrics are possible. the language
of
Schwarzschild.
I " 1
1221.
One might enquire
The answer seems to be no.
In
[237 such metrics would be multi "bolt" metrics - e.g. multi It turns out that these may be rdled out by similar arguments used
in black hole physics.
Thus it seems that the ALF class of metrics is a little
richer than the ALE class but not very much richer.
Compact Instantons
A~O
If these are to admit killing vectors then
The only known examples
are I.
S4
2.
CP 2
[24, 25 ]
3.
S2 x S2
[251
4.
cp2#
[26]
Example 4 is not homogeneous.
The manifold is the topological sum of 2 copies of
CP 2 with opposite opposite orientations. If A = 0 the only known examples are half flat I. 2.
S1 x S 1 x S1 x S1
this is flat
~ 3 with the Yau metric.
The Einstein Kahler metric on
K3
is known only implicitly from a theorem of Yau ~27 ]
or approximately by gluing togetherfl6
the Eguchi Hanson solutions [25, 28].
exhausts the possible compact half-flat metrics. Conjecture (3)
~<3
One is tempted to
The only compact, non flat Ricci flat metric is the Yau metric on
K3. If
~ < O there are many implicit examples given by Yau's theorem ~27] . None
of these is known explicitly.
The only semi explicitly known instantons with ~ < O
are
I. 2.
Spaces of constant curvature ("anti de Sitter space/discrete group) Spaces of constant holanosphere sectional curvature (~.e. set i
negative in
the CP 2 metric and factor by a discrete group) 3,
Metric products of 2-2 spaces of constant negative curvature. An interesting class of Yau examples are complex hypersurfaces of degree ~ in
CP 3, with k ~ 4 ~ 3 0 ~
. k = 4 corresponds to ~ 3 .
Taking more complicated algebraic subvarieties of CP class of examples. however.
provides an enormous
There exist important inequalities limiting their behaviour
Define for each compact Einstein space the number f by
The classical action'S( is
The fact that the Weyl tensor satisfies
yields y29]
Ci~J
C ~'J
C ~:'g
286
~2
~. (9)
X
is the Euler number and "~the signature with equality if the Weyl tensor is
self-dual.
This implies
(lO) with equality for constant curvature.
If $ ~
is Kahler we have (ii)
where C~
and
G
are the first and second Chern numbers.
So if /j/ is Einstein
Kahler
(12) and
(13) with equality if Weyl tensor is self-dual (i.e. constant holomorphic sectional curvature, e.g. Cp2). For algebraicsubvarieties of CPn we have (N. Hitchin private co~unication)
equality on the left is for hypersurfaces ( ~ =
~ (~-~
+~
T=~{
~){~-~))
and on the right for products of 2-spaces of constant negative curvature genus gl and g2 ( ~ = 4(l-gl)(l-g2 )'~ = 0). These results and some qualitative arguments have encouraged Hawking Conjecture
As Z
increases most compact instantons have A ~ O
and
~oC
~4, lOTto ~
Friedan (private communication) has shown that amongst metrics with constant Ricci scalar all known compact instantons except S2 x S2 are local minima of the action.
For S2 x S2 there is just one direction in which the action decreases.
287
REFERENCES I. 2. 3. 4. 5.
6. 7. 8. 9. I0. II. 12. 13. 14. 15. 16. 17. 18. 19. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
S.W. Hawking, G. W. Gibbons, Phys.Rev. 13 2752 (1977). S.W. Hawking, Phys.Letts. 60A 81 (1977)7S . W . Hawking, article in "General Relativity" edited by S. W. Hawking and W. Israel, C.U.P. 1979. S.W. Hawking "Euclidean Quantum Gravity" lecture held at the NATO Summer School at Cargese 1978, to be published by Plenum Press, S. Deser editor. G. W. Gibbons, M. J. Perry and S. W. Hawking, Nucl.Phys. B 138 141 (1978). G. W. Gibbons and M. J. Perry, Nucl.Phys. B 146 90 (1978). G. W. Gibbons and S. W. Hawking, Phys.Letts. 78B 430 (1978). G. W. Gibbons and C. N. Pope, Comm.MAth.Phys. 6"-~267 (1979). G. W. Gibbons, H. Romer and C. N. Pope, Nucl.Phy-'-s. in Press. S. W. Hawking, Nucl.Phys. B 144 349 (1978). R. Shoen and T. S. Yau, Phys.Re----v.Letts.42 547 (1979). G. W. Gibbons and S. W. Hawking, unpublis-hed. N. Hitchin, Math.Proc.Camb.Phil.Soc. T. Eguchi and A. Hanson, Phys.Letts. 74B 249 (1978). V. Belinsky, G. W. Gibbons, D. N. Page and C. N. Pope, Phys.Letts. 76B 433 (1978) D. N. Page, Phys.Letts. in Press. B. Carter, article in same volume as ref. 3 above, W. Israel, Phys.Rev. 164 1776 (1967). D. C. Robinson, Gen.Rel.'---'Grav.~, 695 (1977). D. C. Robinson, Phys.Rev.Letts. P. Tod and R. Ward, to appear. D. N. Page, Phys.Letts. 78B 249 (1978). G. W. Gibbons, to be publ-~hed. G. W. Gibbons and S. W. Hawking, Commun.Math.Phys. P, G. O. Freund and T. Eguchi, Phys.Rev.Letts. 37 1251 (1976). G. W. Gibbons and C. N. Pope, Commun.MAth.Phys. 61 239 (1978). D. N. Page, Phys.Letts. 80B 55 (1978). S. T. Yau, Proc.Natt.Acad--.'.Sci.74 1798 (1971). See also Asterique. 58 (1978). D. N. Page, Phys.Letts. 79B 235-~1978). N. Hitchin, J.Diff.Geom. 9 435 (1974). A. Back, M. Forger and P.--Freund, Phys.Letts. 77B 181 (1978). J. Milnor, "Morse Theory" Princeton University Press. D. N. Page, Phys.Rev. DIS 2733 (1978).
INSTANTONS
AND
LINE
GEOMETRY
M. MULASE Research Institute for Mathematical Sciences Kyoto University KYOTO 606, JAPAN
Let M be a conformally fiat 4-dimensional oriented Riemannian manifold, and let A ~ be the bundle of 2-forms on M. Since the Hodge star operator ~ acts on ~ as an involution, ~ splits into a direct sum of the ~ I eigenspaces of ~ ; A
-
It is known that the sphere bundle S(A~) of A ~ has a complex structure defined by the Riemannian metric of M ([I]). Put Y = S(A~), and let ~ : Y---)M denote the natural projection. This sphere bundle Y is a 3-dimensional complex manifold of Kodaira dimention - ~ . Each fibre of "g is a complex submanifold of Y and is isomorphic to ~4, the 1-dimenslonal complex projective space, which we call a line. The normal bundle of a fibre in Y is isomorphic to
the direct sum of two hyperplane section bundles on IP~ . Since we have H~( Y
P|,
OEt(1) ~ 0 ~ ( I ) )
= O
and
dim H°(~I,O[l(1) ~ ) O ~ I ) )
= 4,
contains a 4-dimensional complex family of l~nes by Kodaira's deformation theory.
So let X denote the connected component of the Douady space of Y which contains the points corresponding to the fibres of ~ . Roughly speaking, X is a parametrizing space of all lines in Y. It is a 4-dimensional complex analytic space. We define an injection L: M--~X by sending a point m in M to the corresponding point in X of the fibre ~ ( m ) , and this map gives a complexification of M. A generic point in X Corresponds to a line in Y. Let F o X • Y denote the graph of this correspondence, and let ~ : ~---~X and ~:r---~Y be the natural projections. The correspondence is given by % = ~ o ~ I This map ~ is proper and a projection map of a IP1-bundle on some Zariski open subset of X. Now we have the whole stage of instantons: Diasraml.
X <
This diagram is c o ~ u t a t i v e
P---~M
be an
~ Y
in the sense that for every point ~o~
Next let
~C
m
in
M
we have
o~(m) = ~-1(m).
SU(2)_principal fibre bundle defined on
M~ and let
A
289
denote a real analytic connection on P. A connection on P is given by a l-form on M with values in the associated ~64A,(2)-bundle ~ of P which is defined by the adjoint representation of SU(2). The 2nd Chern class c~(P) • H4(M,Z) ~ • can be regarded as an integer. We deal only with anti-self-dual connections, which are connections satisfying the equation (dA + defined on a bundle
P
I/2[A,A])
=
-(dA + I/2[A,A]),
with the positive 2nd Chern class. We call them instantons.
An antl-self-dual connection A corresponds injectively to a rank-2 complex algebraic vector bundle E(A) on Y. Since ~ is a proper map,
~A = ~i~~E(A) is a coherent analytic sheaf on X. The restriction of E(A) to a fibre of ~ is isomorphic to the trivial bundle, hence the restriction of E(A) to a general line in Y must be trivial. Because of this fact ~ A has rank 2 and is actually a vector bundle on some neighborhood of M in X. From the commutativlty of Diagram I, the restricted bundle ~A i M is isomorphic to the rank-2 vector bundle associated with P given by the usual 2-dimensional representation of SU(2). Then the connection A can be exteded as a complex analytic connection ~ of ~A defined on a neiborhood U of M
in
X.
The anti-self-dual equation is interpreted as a partial integrability condition on X. To explain this, we need certain 2-dimensional subspace in X. For a point y in Y, let T(y) denote the parametrizlng space of all lines passing through the point y. Set theoretically this is given by ~(y) = ~Q~l(y) and in our case it is a 2-dimensional subspace of X. Then we have Theorem. A connection
A
is
inte~rable for every
We can reconstruct ~f(U) fibre
antl-self-dual if and only if the restriction y
in
E(A)
has a connection ~-l(y)~-i(U) of
fibre at
y ~ Y
by using
is
EA. The pull-back bundle
~J(
U)
defined on
~. Since this connection ~ is integrable along each ~ : ~'I(U)--->Y, we can define a bundle Eflat(A) on Y whose
is a vector space of all flat sections of the bundle
~*(~AIU)l['(y)n~"(U) This bundle
A q[(y)~ U
Y.
Eflat (A)
w.r.t, the connection
is isomorphic to
~
~-l(y)a~-l(U ).
E(A).
Now let us consider the case (I) M is S~, the real 4-sphere with the canonical metric, and (2) M is T 4, the real 4-torus with the flat metric. If M = S 4, then Y = ~ , X = Gr(l,3) (the complex Grassmann manifold which parametrizes lines in ~ ) and F = FI(O,I,3) (the flag manifold). If M = T 4, then Y is the total space of certain complex analytic deformation family of complex 2-tori whose parametrizing space is ~(, X = (C$) 4 and F = (C~)4xIP I • In this case, Y is not a K~hler manifold because the Douady space X = (C*)~ is not compact. In the both cases, ~ - - ~ X is a ~[ bundle on X. For a general instanton
A
on
M = S4
or
T +, the coherent sheaf
EA
=
E(A)
is locally free everywhere on X, and A is continued analytically onto X as a meromorphic connection K of this bundle EA . The singularities of ~ form an analytic subset J(A), which is a non-slngular divisor of X, and A has simple poles along J(A).
290
Diagram 2. ~E(A) c($~E(A)
A
1- ^
E(A) E: the rank-2 vector bundle associated with P
This divisor J(A) has a geometrical meaning• Take a point x E X. The corresponding line T'l(x) in Y of x is called a jumping line of the bundle E(A) iff the restricted bundle of E(A) to T-l(x) is not trivial. The divisor J(A) of poles of an instanton A coincides with the set of all jumping lines of E(A). It does not meet M in X. In the case of (I), we have degJ(A) - c~(E(A)),~= c~(P) = the instanton number (note that degJ(A) E Ha(Gr(l,3),~) = Z )? and cI(EA) = -c=(E(A)). Moreover E(A) is uniquely determined by the divisor J(A). It meahs that an instanton solution of the (anti-) self-dual Yang-Mills equation is uniquely determined by its poles on the complexified domain
X = Gr(l,3).
A self-dual connection
B
defined on
S4
corresponds to a vector bundle over the
!!a~ d~i!!i~ ~!~ ii!ii! n~Br~m;i!! zi!mlP!~ei~ iii~i~d~aTXI[!~ i i~ a&n]p$ s o~ipnd!a~ eii°ai~!nilnm;;stanton B is extended onto Gr(l,3) as a meromorphic connection B. This connection is integrable along each 2-dimensional subspace of Gr(l,3) which parametrizes all lines in a hyperplane of ]pJ. Finally let us consider the case that M is pg~ with the canonical K~hler metric. As a Riemannian manifold, 1~ is not conformally flat, however, we have Y, X and r ([I]). They are given by Y = FI(O,I,2) (the 3-dimensiona flag manifold contained in ~x]P z) and X = ~ . The injection £: ]P ~ - - ~ x • is defined by ~(z) = (~,z) e ~,]pl for z £ ipz. A generic fibre of the projection map ~ is ]l~t but the fibre on a point contained in the degree (I,I) divisor of I~xlPXwhich coincides with FI(O,I,2) w.r.t, the natural inclusion is isomorphic to l~|v~|, the one-polnt union of two lines. Let A be a real analytic connection defined on an SU(2)-principal bundle over M = ]PX with positive 2nd Chern class c~, and ~ the analytic extension of A. If satisfies the partial integrability condition (~)
q~(~)is integrable for all ~E FI(O,I,2),
then A is anti-self-dual. The converse is not true in general. So let us suppose that A +satisfies the condition (A~). Then we have a vector bundle "EA = ~(~E(A) on X = ~x]P ~ as well aS the former cases, and ~ is a meromorphic connection on this bundle with simple poles along some divisor J(A) of degree (c~,c~). This divisor J(A) coincides with the set of all jumping lines of E(A).
Reference . [I] M.F. Atiyah, N.J. Hitchin and I.M. Singer: Self-duality in four-dimensional Riemannian geometry, Proc. R. Lond• A. 36__2, 425-461 (1978).
GEOMETRIC ASPECTS OF SYMMETRY BREAKING
IN GAUGE THEORIES
(Higgs Models W i t h o u t Higgs Bosons) M e i n h a r d E. Mayer University
of California,
Irvine,
CA 92717 USA
/
1. INTRODUCTION.
for spontaneous mental
The B r o u t - E n g l e r t - H i g g s - K i b b l e
symmetry breaking
in gauge theories has been instru-
in the success of the electroweak
models of "grand unification" duction of the Higgs bosons with p r e d i c t e d masses
surprising
(sometimes
fundamental
by experiments)
and makes many of us uncomfortable.
that alternatives
exploiting
values,
and predic-
is neither unique It is therefore
the nonlinearities
the Higgs bosons as excitations
(much as their prototype,
field, was related to electron pairs If we take the fiber-bundle symmetry breaking
The intro-
not
to the BEHK model are actively pursued,
mechanisms,
or c o n s i d e r i n g fermions
and the various
in quite large numbers,
rapidly approaching m a c r o s c o p i c
either as "dynamical" the problem,
unification
which are so popular today.
tions which are hardly refutable nor elegant,
(BEHK) m e c h a n i s m
of
of more
the G i n z b u r g - L a n d a u
in BCS theory).
approach to gauge theories
should be interpreted
seriously,
as the reduction of the struc-
ture group G of the principal bundle P (the hidden symmetry group) a subgroup H
(the manifest,
nomy group of the vacuum), connection
and curvature
variant under H. Mills
field
surviving,
and at the same time,
corresponding
and a part c o r r e s p o n d i n g
of H in riant
model
a reduction of the
form)
the action of the Yang-
breaks up into a part inva-
its conformal
invariance
(massless-
to the c o m p l e m e n t of the Lie algebra h
( a classical m a n i f e s t a t i o n bosons,
inva-
of the "masses" of the appropriate
such as the W ± and the Z 0 in the W e i n b e r g - S a l a m
If one wishes
to preserve
(renormalizability)
the other
Lagrangian
of that model,
being played by certain extra components With this in mind,
advantages
of the BEHK
one should look for a reduction m e c h a n i s m
which in a certain gauge and a p p r o x i m a t i o n Ginzburg-Landau
to
or the holo-
~ (the Lie algebra of G) which is no longer conformally
intermediate model).
subgroup,
to the gauge field to one in-
In the reduction process
(norm of the curvature
riant under H which also maintains ness)
symmetry
I first analyze
reproduces
the quartic
the role of the Higgs bosons of the connection.
the conditions
under which a
292
G - p r i n c i p a l bundle P reduces to an H - p r i n c i p a l bundle Q over the same (4-dimensional)
base space B. It is well known that a n e c e s s a r y and
sufficient c o n d i t i o n for this is the e x i s t e n c e of a cross section of the a s s o c i a t e d bundle E = P/H w i t h fiber G/H~ T h e s e facts and the n o t a t i o n are s u m m a r i z e d b r i e f l y in Section 2. In Sec.
3 I p r o p o s e a model in w h i c h the section ~ is i n t e r p r e t e d
as a n o n l i n e a r field, derived from an action p r i n c i p l e w h o s e Lagrangian has b e e n i n v e s t i g a t e d in c o n n e c t i o n w i t h h a r m o n i c maps. In Sec.4 I discuss a new v a r i a n t of g e o m e t r i c symmetry breaking, in w h i c h the original b u n d l e
(with group G) is r e i n t e r p r e t e d as
a
p r i n c i p a l bundle w i t h structure group H over the base space E = P/H ("fiber-flipping"), always possible),
then e x t e n d i n g this bundle to a G - b u n d l e
(which is
and finally a c h i e v i n g the symmetry r e d u c t i o n by an
i n t e g r a t i o n over the symmetric space G/H.
The latter step,
together
w i t h a p a r t i c u l a r choice of gauge, reduces the Y a n g - M i l l s action to an e x p r e s s i o n e q u i v a l e n t to the Higgs Lagrangian,
the role of the Higgs
particles being p l a y e d by the dim@ - dimh c o m p o n e n t s of the c o n n e c t i o n in the e x t e n d e d bundle over E.
It is g r a t i f y i n g to see that this re-
p r o d u c e s the B E H K m o d e l in all its features,
the only d r a w b a c k b e i n g
that the "Higgs" p a r t i c l e s are in the adjoint,
rather than fundamental,
r e p r e s e n t a t i o n of G. 2. R E D U C T I O N OF BUNDLES AND CONNECTIONS.
Lack of space forces me
to summarize briefly the facts about r e d u c t i o n of s t r u c t u r e groups of p r i n c i p a l bundles and connections. in ref.
The reader can find all the details
2, the n o t a t i o n of w h i c h I will follow as far as possible.
Given two p r i n c i p a l bundles P(B, G) and Q(B, H) w i t h the same b a s e space B and H a c l o s e d subgroup of G,and a m o r p h i s m f:Q ÷ P such that f(qh) = f(q)h
(h ~ H c G, q E Q, f(q)
E P) we say that Q
(or f) is a
r e d u c t i o n of P and P is an e x t e n s i o n of Q. Fact i. E x t e n s i o n of the structure group is always possible. Fact 2. The bundle dle E(B~G/E,G,P)
= P/H
P has a r e d u c t i o n to Q iff the a s s o c i a t e d bun(with fiber G/H)
has a cross section ~:B ÷ E.
In the c o n t e x t of gauge theory B is either M i n k o w s k i space, or ]R
4
(or its c o m p a c t i f i c a t i o n S 4) and G is a compact Lie group. The YangMills p o t e n t i a l is the p u l l b a c k Y of the ~ - v a l u e d c o n n e c t i o n 1-form on P to the base s p a c e B, and the field strength M is the p u l l b a c k to B of the c u r v a t u r e 2-form ~, both being p u l l e d back by the locally triv i a l i z i n g section s: Y = s*~, M = s*~. Under a change of t r i v i a l i z a t i o n (gauge transformation)
s' = gs we have Y' = A d ( g - l ) Y + g-ldg, M' =
Ad(g-l)M, w h e r e g is a G - v a l u e d f u n c t i o n on B. The d e f i n i t i o n M = DY = dY + ½[Y,Y]
implies the Bianchi i d e n t i t y D M = dM + [Y,M] = 0, and
293
the Yang-Mills
equation D*M = *J follows
Here *J is the current
3-form
from the action principle.
(dual of the usual current vector)
the particle
fields - not a closed form by itself.
are sections
of associated vector bundles or, equivalently,
ant functions
The particle
of fields
G-equivari-
on P with values
in the vector space V of the representai~3 the associated vector bundle; cf.,e.g.,
tion defining
The subgroup H describing group of the vacuum)
may be considered
group of the connection. (the set of points
the residual
symmetry
as the
(the isotropy
(restricted)
sub-
holonomy
The reduced bundle is then the holonomy bundle
in P which can be reached
from a given point by hori-
zontal curves. Considering
the associated bundle E = P/H with fiber G/H and a sec-
tion ~:B ÷ E which gives rise to a reduction H, a connection
~ in P is reducible
@ = h + m, H - e q u i v a r i a n t l y respect to the transport
to a subbundle Q with group
to Q (i. e., will split according
for the K-component)
iff ~ is parallel with
induced by the connection
One of the models discussed below simplifies curvature in
splits
in E.
considerably
if the
in such a way that the bracket of two m-components
h , i. e., if G/H is a symmetric
space
is
(this is the case for the
SU(2)/U(1), SU(N)/SU(n)U(N-n),
models based on
to
but is not true in some
popular grand unifications). 3. SYMMETRY BREAKING BY HARMONIC of spontaneous
SECTIONS.
In the BEHK m e c h a n i s m
symmetry breaking of a gauge theory with the action S = ~IM ^*M + I ~ , d ( v o l ) , B
where ~ denotes rator,
the Dirac
field(s)
a bundle associated
one adds to the action the G i n z b u r g - L a n d a u
field ~
a representation
and ~ the covariant Dirac-Weyl
being sections of
SGL = ID~A*D~ + V [ ~ ] , where the Higgs
(i)
B
the Dirac field(s)
by a representation,
V[~]
= -~II~II2 +
tial" V has a nontrivial
representation).
gauge-fixing
is to consider
(2) to P by
total action is
of Y b e l o n g i n g
the sections ~ : B ~ E, required
functions
the general principles
term
The "poten-
fields".
on P, equivariant
one must introduce
to m =
(cf. ref.4).
the scalar fields ~ "by hand",one
duction of P to Q as new "nonlinear also as G / H - v a l u e d
the
as do the surviving Higgs particles
Instead of introducing possibility
opeto P
extremum ~0 if ~>0, assumed invariant under
then rewritten in a form where the components "acquire masS",
klI~l[4,
is a section of a bundle associated
(usually the fundamental
the subgroup H of G. By a p p r o p r i a t e @ - h
4
obvious
for the re-
They may be c o n s i d e r e d under G. In line with
an action term for
o, p o s -
294
tulating that the extrema of the action determine There is essentially
the particular
~.
only one invariant one can form out of the exte-
rior covariant differential
of ~ (this is required
in order to couple
it invariantly
to the connection Y), namely the "harmonic map"
nal considered
in another context by Eells and Sampson £
functio-
5:
SHM = ]DOA*D~.
(3)
B
Specializing
to a basis shows that this action is identical to the
action of a 4-dimensional independently
by Misner
sigma model.
6 and myself
Fixing the section o produces
Such a model has been proposed
7 and is related to earlier work.8
the r~quired
"mass-terms"
in S, but it
is not clear in what sense such a theory is renormalizable
- the main
attraction of the BEHK model. 4. SYMMETRY BREAKING BY FIBER-FLIPPING. of symmetry breaking,
A more attractive method
which I call the "fiber-flipping
method"
for
reasons which become clear from the figure below, makes use of the details of the reduction process described low are represented for graphical
simplicity
by line segments literally,
the various bundles all manifolds
or rectangles
U x G, U x H, and E by
view the lines as representing m, corresponding
its curvature). (U x G/H) H.
the figure more
open subset U, P and Q res-
U x G/H;
alternatively,
to G/H, when considering
In particular,
where
and groups have been represented
the Lie algebras
you may
g, h, and the vector the connection
and
over U, we may consider U x G = PIU as
x H, i. e., by "flipping"
the principal
2. In the figure bein the process,
(one may also consider
replacing B by a trivializing
pectively by
space
in Sec. involved
part of the fiber
it is locally
bundle over the base space B x G/H with structure group
The second drawing
shows clearly how a choice of section o in the
"new" base space picks out in the bundle space P a m a n i f o l d w h i c h becomes the reduced bundle Q in the third frame of the figure. picture represents
an extension of the bundle P ÷ E
The fourth
(second picture)
~ r~:~~
JH
I I
' " :"
I1~. , , ' , II "
i
"1-
j
,,I
0.. W.
~.:..~j u.,., ,_,.j I ~ L__!l'fi',h',l, E - PIH
B
I I
f.9
Q
o-
c 9 ~ E
B
IH -
H
295
to a principal bundle with structure group G over E (achieved by another "flipping" of the portion G/H into the vertical direction).
We
denote this new bundle by R and extend the original connection in P to a connection e with curvature ~ on R. The pullbacks to E will be denoted by
(the script letters)
Y, M.
The latter are forms on the
4 + dim m -dimensional base s~ace E, and the appropriate action is
SyM = %JM ^ *M A ~,
(4)
E
where ~ is the a symmetric
volume m-form of G/H
(the measure of G/H). If G/H is
space the connection and curvature forms Y, M can be split
along the Lie algebra h and the vector space m , with the brackets of the m-components
ending up in h. Then by an appropriate gauge-fixing
one can reduce the action
(4)
in such a way that
ponds to a reduced Yang-Mills action which involves
the m-part corres-
(invariant under H) and a part
the covariant differentials of the m-components of Y.
The final reduction is achieved by carrying out an averaging over the homogeneous
space G/H, made possible by the existence of a finite
invariant measure on this space. One ends up with an action which, one identifies
the G/H components of Y in m
if
with fields which are
"scalars" in B, looks exactly like the Higgs action
(i) + (2), with<
the quartic functional coming from the corresponding terms in (4). The details are too lengthy and will be published elsewhere;
one should
note only that the "Higgs" components end up belonging to the adjoint representation of G, which is a disadvantage 5. DISCUSSION.
which is both geometrical and nology,
for many physical models.
Although I have not succeeded in producing a model satisfying the requirements of phenome-
the model described in Sec. 4 (which is related to earlier
work by Witten, Forgacs,
and Manton 9, and to the "dimensional reduc-
tion in supergauge theories discussed in the talks of Scherk and Julia 10) is attractive,
since it does not require too many ad hoc assumptions.
One might be able to relax the requirement that G/H be symmetric, placing it with the milder reductivity condition.
re-
It is not clear how
to take the new "Higgs" states out of the adjoint representation. BRIEF BIBLIOGRAPHY i. 2. 3. 4. 5. 6. 7. 8.
(for further references cf. 1,4)
M. E. Mayer, Lect. Notes in Physics, Vol. 67, Springer, 1977. S.Kobayasi and K. Nomizu, Found. of Differ. Geometry, voi. I,1963. A. Trautman, Bull. Acad. Pol. Sci. 27,7 (1979) and earlier work. cf.,e, g., L. O'Raifeartaigh, Rep. Pro----g.Phys. 42, 159-223 (1979). J. Eells Jr. and J.H.Sampson, Amer. J. Math. 8--6,--[09 (1964). C. W. Misner, Phys. Rev. DIS, 4510 (1978). M. E. Mayer, Proc. VIII Intern. Symp. Group Theor. Meth. to appear. J. Madore, CMP 56,115 (1977). J. Harnad, S. Shnider, L. Vinet, this volume, p, 295 9. E. Witten, PRL 38,121 (1977). P. Forgacs and N. S. Manton, preprints. 10. J. Scherk and B. Julia, this volume, pp. 342, 367.
INVARIANCE CONDITIONS FOR GAUGE FIELDS by John HARNAD , Steven SHNIDER
and Luc VINET
W
C.R.M.A. Universit~ de Montr6al, Montreal, Quebec, Canada.
WW
Dept. of Mathematics, McGill University, Montreal, Quebec, Canada
Invariance considerations have proved very useful in the context of gauge theories especially in the search for classical solutions to field equations. example [I]
and
(See for
[2].) The problem arises in this context of how to give a
general definition for invarianc¢ of gauge fields under space-time transformation groups and a way to construct the most general symmetric field configurations. Let
H
be the gauge group with Lie algebra
(e.g. space-time) and presented by an
an open set in
h valued i- form
differentiably on invariant.
U
M,
fg:
~.
M
Let
M + M, gEG;
h ,M
a differentiable manifold
on which the gauge potential is reG
be a transformation group acting
and suppose for the moment that
The local condition of invariance under
G
U
is
is:
fg(m)x = Adp-l(g'x)(~)x + p-l(g,x)dxP ( g , x ) ~ where
p:
G×U ÷ H
d e f i n e s a gauge t r a n s f o r m a t i o n
(I)
and s a t i s f i e s
the composition
rule:
(2)
p(gg',x) = p(g',x) p(g,g'x)
(If
U
is not G-invariant, the domain of definition of
neighborhood of the identity in A change of gauge replaces
G p
p
must be restricted to a
and the composition rule restricted accordingly). and
~
by equivalent
p'
and
~'.
We are
interested in determining a characterization of the invariance condition which is To do so, the problem must be formulated in-
independent of this choice of gauge.
In this language, the transformation
variantly in terms of fibre bundles. (locally) a lift of the G-action on U
U
to that on a principal H-bundle
EIU
p
defines over
by: fga (x) = ~ (fgX) p-i (g,x)
where
is the local section defining the gauge.
0: U ÷ EIU
and
properties of
p
on a covering of
the global action on a brindle E connection form
(3)
~
on
M
over all of
The transformation
by overlapping open sets M,
E (whose local pull-back by
U
guarantee
The gauge field is given by a ~
is
~)
and the invariance
297
conditions (i) become simply
fg
~ = ~
(4)
The problem of finding all classes of inequivalent
p
is equivalent to that of
classifying all principal H-bundles with G-actions as automorphisms projecting to the given G-action on M. Solving the invariance conditions (i) amounts to determining all invariant connections on the given bundles.
Formulated thus, these
problems turn out to have known solutions for the case of homogeneous G-spaces and the results may be extended straightforwardly to more general G-spaces provided the orbit structure is uniform and the isotropy and gauge groups are co~oact.
The main
theorems together with some simple corollaries clarifying the results are presented below. Let
x~M
and
G x be the isotropy group at
~
the structure of
E
over
G/Gx
is determined by: Proposition I. [5] a)
There is ~ one-to-one correspondance beh~een
equivalence classes of principal H-bundles
E
action projecting to left multiplication on
over G/G X
G/Gx. admitting a (u~ to G-equivalent
isomorphism) and b)
conjugacy classes of homomorphisms
Three corollaries follow.
~: G ÷ H. x
~le first states when the bundle is trivial over the
entire orbit and hence allows a global definition of Corollary 1
The bundle
E
extends to a smooth function
is trivial over A: G ÷ H
p.
GZG x. iff the homomorphism
~: Gx ÷ H
such that
A(gg I) = A(g)X(gl)
for
g~G, gl~Gx
The two other corollaries analyze the possible reductions of
(5) p
in terms of intrinsic
properties of the bundles.Corollary 2 establish when the transformation function can be chosen independent of the point in the orbit so that the inhomogeneous term in condition (I) disappears.
Corollary 3 gives the criterion for
p
to be the
zdentity, obviously the simplest transformation function possible. Corollary 2 aJ
The bundle point
b)
The following two conditions are equivalent: E
is trivial with gauge function .p(g,kGx)
independent of the
kG x,
The homomorphism
~: Gx ÷ H
extends smoothly to a homomorphism
A: G ÷ H.
p
298
Corollary 3 ~e
The transformation function p(g,kGx)
reduces to the trivial function
iff it is trivial when restricted to the isotropy group
Gx,
When corollary 3 applies, the invariance condition (I) simplifies to the ordinary invariance condition
fg~ = ~
for 1-forms on
occurs is when the G-action on
M
M.
One case in which this always
is free, i.e.
Gx = e. We would like to point out
that the constructive nature of the proofs of prop. 1 and its corollaries permits the explicit construction of expressions for the °transformation functions
p.
(See
[3] for details.) We now come to the generalization of these results when the base space is not a homogeneous space. open set ~(S)
Suppose that for all
xcM,
S ~ R k :'{k= dim M - dim G/G0)
into
~(S)
is the same.
Theorem 1 groups. M.
Let Let
M
E
Assume
and
~(0) = x and
We then have:
be a manifold with simple
be a principal
H
bundle with
is compact.
Let
~: S ÷ M
H
U = G.~(S) ~ M.
G
action and compact isotropy G
action projecting on the G-action
be a special cross-section through
Then there is an isomorphism
EIU = EX x S Ek
with
We call such a situation a simple G-action and
such an imbedding a special cross-section.
xcM
~: S + M
intersecting each orbit in a unique point and further that the isotropy group
of all the points
on
there is a smooth embedding of an M,
for some
X: Gx ÷ H .
is the bundle determined by the homomorphism
~.
This theorem together with prop.l and its corollaries completely analyzes the structure of bundles with G-action over a neighborhood of an orbit in any space with a simple G action.
On these bundles the G-invariant connections are characterized by
a generalization of Wang's theorem (see [4]): Theorem 2 h
Let ~ be the Lie algebra of
the Lie algebra of
H.
G,~ 0
the Lie algebra of
The G invariant connections on
E~ × S
GO c G
and
are determined
by (i)
A family of linear maps
As:~+
h
depending smoothly on
~
(the cross-
Section variable) and satisfying As(~) = ~.(~) the homomorphism
~*:~0 ÷ h
Ad X(g)-l(As(g))
for
(ii)
A one form
~
SE~ on
~0
and
~.
determined by the differential of
and S
for
and As (Ad~-l~)
geG 0.
with values in the subalgebra of
invariant under the adjoint action of
X(G0).
h
of elements
Again, the theorem is constructive in
299
nature and its application allows the explicit determination of all invariant gauge fields for group actions satisfying the requirements of theorem i.
Cases with a more
complicated group orbit structure may also be treated, by using continuity arguments, but no general theorem is known.
Detailed examples illustrating these results and
their applicability in solving field equations are given in 1,2,3].
References
1,1]
R. Jackiw, C. Nohl and C. Rebbi, "Classical and Semi-Classical Solutions of Yang-Mills Theory"
in D. Boal and A. Kamal Particles and Fields,
Plenum Press 1978.
[2]
J] Harnad, S. Shnider and L. Vinet, J. Math. Phys. 2_~0,931 (1979).
1,3]
J. Harnad, S. Shnider and L. Vinet "Group Actions on Principal Bundles and Invariance Conditions for Gauge Fields" McGill preprint (1979).
1,4]
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry
i5]
Interscience, 1969. G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, 1972.
Wiley
NONLINEAR AND
GROUP
THE LINEARIZABILITY
REPRESENTATIONS OF N O N L I N E A R
EQUATIONS
Daniel STERNHEIMER C.N.R.S. Paris and Physique Math4matique, Universit4 de Dijon, FRANCE
Basic definitions and some results in the recently developed theory of nonlinear Lie group representations in Banach and Fr4chet spaces are presented. Using cohomological methods, this framework permits a study of the linearizability of
covariant
nonlinear evolution equations. Formal linearizability i5 proved under some conditions on the linear part of the representation, for massive and for massless Poincar~ covariant equations. In particular, pure Yang-Mills equations supplemented with a relativistic gauge condition are formally linearizable.
The relation between linear differential equations, covariant under the action of' a Lie group G, and the (linear) representations of this group is rather well-known. In the nonlinear case such a connection has not been developed --as a matter of fact, even a systematic theory of nonlinear group representations in linear spaces was lacking. The basic ingredients for such a theory have been developed recently by Flato, Pinczon and Simon hi, relying in part on earlier works on extensions of linear group representations and l-cohomology [2], and using the notions of analytic functions and formal series on Banach or Fr~chet spaces (see e.g. ref. 3). This permits to exploit sl~aetries of nonlinear equations and transform them into a simpler form (e.g. linear) instead of a direct study for each equation. A similar procedure is used in classical mechanics : the Hamilton-Jacobi method. (For generalization to field theory, it seems that a reduction to a cc~pletely integrable system such as the harr~nic oscillator would be preferred). If we r~member that quantum mechanics can be treated in an autonomous manner
[4] using a deformed product
of functions over classical phase space and that quantum fields can be viewed as functionals over classical fields (initial data), the advantages of performing well-chosen transformations on the Initial data space, which will induce transformations of the quantum fields (in the same way as general coordinate transformations induce linear transformations on function spaces), beccmes evident. Due to lack of space, we shall only indicate some essential defir~tions and results, referring to the quoted literature for details.
301
1 - Nonlinear Lie 9roup and al~ebra representations in Banach and Fr~chet spaces.
If E is a Fr~chet space, L (E) the space of syrsnetric multilinear maps from E n to n
/~
E (also considered as linear maps ~ ~
÷ E, where ~ denotes the projective tensor pro-
duct topology), then to each fn e L n (E) one can associate a monamial ~n on E by ~n(#) = ~(~,...,~),# 6 E . Thus one defines the space F(E) of formal series (with fixed ^n Z f , with the usual ca~position law, and formal representations of a ~0up n>l G as homcmorphisrmS : G ÷ F(E) such that the maps g ÷ sn(#) are measurable. A one-tog one map A between F(E) and the linear operators L(~) on the "Fock" space point) f =
= ~) ( ~i n ~ E) can be defined, the cc~position of formal series being mapped into n>~l p= the product of operators. Its differential dA defines a Lie bracket on F (E), which corresponds to the cc[snutator in L (~), whence the definition of a fcrmal representation of a Lie algebra g as a homcmorphisrn g ÷ F (E). In a natural way one then defines anal~tic functions and representations. Two (formal or analytic) representations of G are said equivalent if there exists a (formal or analytic) series S'
g
= A S
(S,E) and (S',E) A such that
A -I for all g 6 G.
g A useful notion is that of a s~nooth anal~tic representation, for which the terms
of degree >i do not bring more differentiability requirements than the linear term, that is the function g÷S
1 S g-i g
frcm G to a space of analytic functions on the Banach space E is differentiable in sane neighbourhood of the identity. Since the successive "layers" in S are built by successive extensions of the linear representation S 1 , and extension l-cocycles can be taken differen~able up to a l-cohoundary, one can show that there always exist an equivalent smooth representation
(S' ,E) to an analytic one.
Then a kind of "Lie theory" is developed : passage from (nonlinear) smooth Lie group to Lie algebra representations, and vice-versa when the linear part is the differential of a (linear) group representation. This theory provides an easy proof of the linearizability
[5] of the action of semi-simple Lie group around a fixed point in
a finite-dimensional manifold. Several other cases have been studied, including inhomogeneous classical groups in finite-dimensional space [6] where an interesting phenomenon of quantization of coupling constant (between two nonlinear terms) appears.
2 - Applications to covariant nonlinear evolution equations
A nonlinear evolution equation ~t ~t = Po(~t ) is said covariant under a Lie algebra g if there exists a representation
(T,HJ of ..g such that T 1 = dU, U being a linear
representation of the corresponding Lie group G on the Banach space H and H
its Fr~-
g (the time-transchet space of differentiable vectors, and if Po = T (Xo) for some X ° e _ lationsgenerator /~i the case of the Poincar~ group).
302
If (S,E) is a formal representation of a Lie group G in a Fr@chet space E of differentiable vectors, we let G act on ~ E
via ~n S 1 and define in this way the cohano-
logy spaces HI 'n = HI(G,L(~ n E,E)) If H l'n = O for all n>~2 there exists AEF(E) such tb~t S is formally linearizable, i.e. Sg = A Sg1 A ~I ; the equatio D (d ~__dt) = Po(~t ) is then formally linearizable, with solution #t = SexptXo~' The vanishing of these cohomology spaces is easier to prove using infinitesimal methods, for which the "Lie theory" mentioned above is essential. When G is the Poin_ l,n = 0 ~f S 1 is unitary irreducible with mass m 2 >O. Hence equations car~ group [7] , n~ Of the type (~/~t) = (-A+ ~ "mZ)i/2# + f(%) are formally linearizable. When both energy signs are present, linearizability can be shown [8] for initial data with either positive or negative energy ; this is the case of
([3+ m2)~ = f(#) for instance. The pro-
ble~ of convergence of the linearization operator A, which is obtained algorithmically in the form A = n~2 (i - ~n) , Bn~Ln(E) is mere delicate since it requires in particular an implicit function theorem in Fr~chet spaces. In the massless case, mere can be said on less (i.e. on the time-translations only) : H I ~ Po,Ln(E)) = O when I~ Po acts on E by £ -~uL(exp tXo), U L being a unitary induced representations of the Poincar~ group on E = S[M,F) or C~(M,F) , where M is the vertexless future Cone C+ or past cone C_, or C + U C _ , and F is a finite-dimensional space (namely, any finite number of massless particles is allowed, with both e~gy
signs). Therefore (~/~t) %t = Po(~t ) is formall[ linearizable when Plo generates
U L (exp tXo). This is in particular the case of Yan~-Mills equation~ in Minkowski space \
supplemented with a relativistic gauge condition such as 3 A ~ = O or 3 [] A ~ = 0 (conformally covariant), etc.. This is also the case of Einstein equations (without matter) and others. RI~ERI~CES [i] • [2] [3] [4]
M. G. L. F.
[5] [6] [7] [8] [9]
V. J. M. M. M.
FIATO, G. PINCZON and J. SIMON : Ann. Scient. Ec. Norm. Sup. i_~O,405 (1977). PINCZON and J. SIMON : Reports On Math. Phys. (1979). NACHBIN : Topology on Spaces of Holcmorphic Mappings. Springer Verlag (1969). BAYEN, M. FLATO, C. FRONSDAL, A. LICHNEROWICZ and D. STERNHEIMER : Ann. Phys. (NY) iii, 61, iii (1978). GUTLT.~4IN and S. STERNBERG : Trans. Amer. Math. Soc. 130, IiO (1968). SIMON, G. PINCZON : Lett. Math. Phys. 2, 499 (1978). FLATO, J. SIMON : Lett. Math. Phys. 2, 155 (1977). FIATO, J. SIMON : J. Math. Phys. 21, (1980). FLATO, J. SIMON : Lett. Math. Phys. 3, 279 (1979).
Determinants, Green Functions and Induced Action B. Sehroer Institut f~r Theoretische Physik, Freie Universit~t Berlin In quantum field theories with Lagrangians which are bilinear in the matter fields, relevant physical properties often only become exposed after integration over the matter fields. The rules for obtaining a functional representation for correlation functions in terms of integration over gluon fields only are well known I), however there are some subtle modifications 2)" if topology and zero eigenvalues for the matter field equations came into play e.i. in gauge theories. The functional integral representation for quarks field correlation functions is of the form
<~(~) ..... 5 "-- O/~independent factor )
] ~ £1~& .....)
where the induced interaction consists of three parts.
....
)
:
s . + r
÷
....
The first two parts are the Yang Mills or Maxwell action SO as well as the modified (by removal of zero modes) determinant of the quark field; they are usually referred to as the effective action. Apart from the zero mode modification of QCD model :
this part is universal, i.e. does not depend on the correlation function. The originates from the (exponentially written) external field dependence ource of the quark Green Functions and therefore depends on the specific correlation
C
function under consideration. A powerful Seeley heat equation - or f-function
mathematical formalism (the De Witt
formalism) was used to explicitly compute
Sin d in abelian and nonabelian two-dimensional gauge theories. For (generalized) abelian theories with vanishing quark mass the complicated nonquadratic Am. 1 terms which appear as an effect of nontrivial topology in C cancel against similar terms in /Win d and the remaining quadratic functional can be integrated reproducing the e -vacuum -expectation values of the (generalized) Sehwinger model. In QCD 2 the Sind remains complicated (nonpolynomial) and was only studied by quaslclassical approximations.
These saddle point method applied to correlation
functions of gauge invariant composite fields becomes exact in the limit of large euclidean seperations; they asymptotically approach the correlation functions of the corresponding abelians "torus" gauge theory. A detailed account
304
of the methods and results has been given elsewhere 3". ~ The reader who is interested in details and furthergoing investigation is referred to the 4)5)6)7) literature
References i) P.T.Mathews and A.Salam, Phys.Rev. 90,690 (1953). 2) N.K.Nielsen and B.Schroer, Nucl.Phys. B120,62, (1977) and Phys.Lett.66B,373(1977). 3) "Green Functions, Determinants and Induced Actions in Gauge Theories", K.D.Rothe and B.Sehroer, FU preprint, Lectures given at the International Summer Institute in Kaiserslautern August 13-24,1979, to be published in the proceedings of this Summer Instituts. 4) M.Hortagsu, K.D.Rothe and B.Schroer, FUB/HEP March 1979, to be published in Phys.Rev. D (Nov.79). 5) N.K.Nielsen, K.D.Rothe and B.Schroer, FUB/HEP May 1979, to be published in Nuel.Phys.B. 6) "Do Quark Correlation Functions exist in Confining Gauge Theories" K.D.Rothe and B.Schroer FUB/Oet.1979. 7) "Zero Energy Eigenstates for the Dirac Boundary Problem" M.Hortagsu, K.D.Rothe and B.Schroer, FUB/HEP 0ct.1979.
SPIN GAUGE THEORY OF ELECTRIC AND MA(INETIC SPINORS by J.S.R. Chisholm and Ruth S. Farwell University of Kent, Canterbury, Kent, England.
The work is in two sections: (a)
A theory of four-component "generalised" electrons, with the Dirac electron as
a special case; (b)
An eight-component dual theory of modified generalised electrons and monopoles;
it was not possible to include monopoles in a four-component theory.
Our aim has been to formulate a dual gauge theory of electric and magnetic sources~ without unphysical singularities such as strings and patching.
The gauge theory has
some definite differences from standard gauge theories, and we call it a "spin gauge theory".
The basic physical problem is that magnetic sources have d i v H ~ O, so that
the relation H = curl A, where A is the magnetic vector potential, must fail somewhere. To meet this difficulty with purely classical sources, Cabibbo and Ferrari I introduced two potentials, the usual A
D
and the dual '~agnetoelectric potential" B ; the cor-
responding fields are
F v
~ A
=
- ~A,
G
=
~ B
- ~ B .
Their work was further elucidated by Hart and Biedenharn~
The total electro-
magneti£ field K B~ = F ~
-G ~
~F ~
+ ~,q ~ p o b.Do
is invariant under the 8-parameter set of transformations A
÷A
+ a~ o + c~aA
(la)
B
÷ B
+ b~ ~ + c ~ A
(ib)
the A terms defining "mixing transformations"; A~v is an anti-sy~netric tensor with DA yv = O. We use transformations of type (i) as local spin space invariance transformations
306
of a generalised electron Lagrangian.
We define covariant derivatives as spin space
operators by y~D @ ~ y~(~ @) - (a-iB)(iA-Y5B)Y~¢
(2a)
and its conjugate
where e,B are constants and ~ = ~%F~ with F~ a constant matrix. e
. -
L I = ~l[~(y~D 9) - ( ~ yn)~] - M ~
~ Leo + ~ 9
Then
+ ~ Y 5 Bg'
where L e is the symmetrised free fermion Lagrangian, is invariant under the infinio tesimal local spin space transformations
, + s,, ~ ÷ ~s -~,
(3)
where S = I - iA
(x)o pv - i@(x)I - ys¢(X),
(4)
together with v~ ÷ SvPS -l
(5)
and (with fixed y~) A y p + A y ~ + (B2+~2)-! { ~ A a u Y 5 YU + B y~ ÷ B y~ + (~2+~2)-i { _ ~ A u Y 5 7 B The transformations
B~aiapy p + B~pT
- a 3 @ y p + B3%75yP}
(6a)
~ _ B~peT5 YU _ R~@y~}.
(6b)
(6) have several important properties:
(i) they satisfy duality,
(ii) they preserve parity, unlike the well-known transformations of E + i_H, (iii) they preserve the conjugacy of D
p
and a ,
(iv) since they are of form (I), K v remains in-
variant and a spin scalar. We have ensured invariance under an eight-parameter set of transformations by introducing only two gauge fields, with eight components.
The reason for this is
that the spin differential operator ~ converts (S,T,P) type terms in (4) to (V,A) type terms.
This economy in numbers of gauge fields can only happen for Clifford algebras
in 2n dimensions.
Equations (2) and the transformations
(6) deal essentially with
spin space operators ~, ~( and ~, and we must regard these operators as being defined within x-space and spin space simultaneously.
It seems to us that we should define
"spin dynamics", in which ~ is the single dynamical group generator, within a Clifford algebra formalism.
In this spirit, we can combine (5) and (6) to give ~( ÷ S~S -i + derivative terms
307
÷
so that ~, ~ transform as A ,
SgS -~ + derivative terms
B~ do in standard gauge theories.
The transformations
are very sinrple in terms of Hertz tensors~ '3
The invariant Lagrangian Le = Leo + a ~ ¢ + 6~¥5~* + 41KpvKpv - 21Kpv(aPAV-aVA~) + ~Kpv (a~Bv-avBp)
(7)
gives the generalised electron equation
(8)
(i~-Y0@ + ~4~ + BYsg~ = 0 and its conjugate, the definition of K~v, and the generalised Maxwell equations: quantisation, (7) needs minor modification.
for
If a = e, g = O, (8) is the Dirac equation;
so changing the local spin frame introduces a B-field into the Dirac equation; and equations (3) - (6) define an eight-parameter invariance aigebra of the equation. The fact that ¥P is transformed by (5) requires modification of Noether's theorem to the form ~
[~L__. 6@ + 6~ ~L + ~L 6A + ~L 6B I = ~Av,p v ~Bv,p v ~,p ~0,p --L-
- -
- -
t he last term implying replacement of y~ by ~xp. tion, we have only the one usual conservation law.
~L ~Yp ~Yp'
- - -
Since6xp = O only for the @-variaThe other seven variations give
"partial conservation laws". It proved impossible to define monopoles, with g and ~ interchanged, in the fourcomponent theory.
We define
*_-= ~ , w h e r e ~ = CT~* is the charge conjugate of a generalised electron ~.
Introducing a
set { pr } of Pauli matrices acting on ~_ allows us to define a square root of the charge conjugation operator; this acts on ~ to give the eight-component monopole spinor x (four independent components), for which B
is the Coulomb-Ampere potential.
The
Lagrangian exhibits dual syn~netry, but this symmetry is broken by the gauge transformations of ~, ~.
Also, the theory becomes singular when a = B, imposing a second
asymmetry.
References i.
N. Cabibbo and E. Ferrari, Nuovo Cim. 23, 1147 (1962)
2.
M.Y. Hart and L.C. Biedenharn, Nuovo Cim. 2A, 544 (1971)
3.
J.S.R. Chisholm and Ruth S. Farwell, Kent Preprint, April 1979
SPIN STRUCTURESAND GAUGETHEORY Peter G. O. Freund University of Chicago, Chicago, l l l i n o i s
60637
The electroweak gauge theory and QCD, or the more ambitious "grand unified" gauge theories I do not provide theoretical c r i t e r i a for the selection of the gauge group i t s e l f or for the symmetry m u l t i p l e t assignments of spinor matter f i e l d s .
Here
I would l i k e to review b r i e f l y a body of recent work 2-8 that sheds some l i g h t on these problems from a quite unexpected source:
the global consistency requirements that
arise when coupling spinor f i e l d s to g r a v i t y . Given a vierbein we can always l o c a l l y define spinors on a Riemann 4-manifold. A global d e f i n i t i o n on the whole manifold is possible only i f i t s second S t i e f e l Whitney class w2 vanishes.
This is connected with the properties of closed curves
in the orthonormal frame bundle that l i e e n t i r e l y in one f i b e r and correspond to vierbein rotations of 27 so that spinors change sign. tracted to a point while staying in the f i b e r .
Such curves cannot be con-
But by deforming them through distant
regions of the bundle t h e i r contraction to a poi;nt can s t i l l
be achieved i f w2 ~ O.
The instructions for spinor signs are then s e l f - c o n t r a d i c t o r y , and no global d e f i n i tion of spinors is possible. The simplest procedure for avoiding this problem is to construct "generalized spin structures" on the manifold. 4 ' 5 ' I 0 Essentially, one assigns spinors to multiplets of some gauged internal symmetry and absorbs the undetermined sign in the gauge indeterminacy. I I The problem is thus to devise a procedure that w i l l upgrade the orthonormal frame bundle (an SO(4) bundle) to a bundle that allows the global d e f i n i t i o n of spinors.
I f w2 i 0 then the SO(4) bundle cannot j u s t be upgraded to a Spin(4)
SU(2)XSU(2) bundle and the question is what group to use instead of Spin(4). simplest p o s s i b i l i t y 5 turns out to be Spin(4) = Spin(4)X z SU(2).
The
Indeed, there
exists a homomorphism. ~: Spin(4) ÷ Spin(4)XSU(2) define~ by the i d e n t i t y map to Spin(4) and projection to one of i t s SU(2) factors as the map to SU(2). specify Spin(4) by i d e n t i f y i n g (g,h)~(-g,-h) for g~Spin(4), h~SD(2).
We now
The groups
SO(4) and Spin(4) d i f f e r from Spin(4) and Spin(4)XSU(2) each by a Z2 factor. j u s t described
~-homomorphism then induces an S0(4) ÷ ~ ( 4 )
The
homomorphism that can
be used to upgrade the SO(4)-bundle into a Spin(4) bundle and thus to consistently define spinors on any Riemann 4-manifold once one has gauged SU(2). work there is one more requirement.
But for this to
Any representation A of Spin(4) on a vector
space can be pulled back to a representation of Spin(4)XSU(2) ~ (SU(2)) 3.
But the
element ( - I , - I , - I ) of (SU(2)) 3 must map to the i d e n t i t y in A. Label A by three spins ( J l , J2, J3)- This then requires ei~(2Jl+2J2+2J3)=l, or jl+J2+J3 = integer. But the ordinary s p i n - s t a t i s t i c s connection t e l l s us that jl+J2 = h a l f odd integer (integer) for fermions (bosons). Thus we find that the "internal spin" J3 must be
309
half-odd integer (integer) for fermions (bosons) and thus establish an " i n t e r n a l spin" - s t a t i s t i c s connection. This construction can be r e a d i l y generalized to other nonabelian Lie groups 5-8 that contain a Z2 in t h e i r center.
For any such group G replace Spin(4) in the above
argument by SpinG(4) = Spin(4)X Z G. (the construction above had G = SU(2)). G = SU(2)XSU(2) gives an amusing ~henomenology 5 i f i d e n t i f i e d as the SU(2)LXSU(2) R of electroweak gauge theory. theories.
We shall rather go on d i r e c t l y to "grand" u n i f i c a t i o n
Such theories as a rule postulate some "very" large simple group that en-
compasses SU(3)colorX(SU(2)XU(1))electrowea k. T y p i c a l l y , SU(N) and SO(N) groups have been considered. Since the "grand" u n i f i c a t i o n group is the maximal gauged i n t e r n a l symmetry group, i t is the only candidate for G in SpinG(4).
Now f o r SU(N)
groups only those with even N have a Z2 in t h e i r center (those with N = odd have ZN but no Z2 in t h e i r center). Glashow. 12
This rules out the famous SU(5) Of Georgi and
(.For SO(N) groups rather than a selection in N, we are instructed to
take the twofold coverings Spin(N) which is possible for a l l N > 3). the t h e o r e t i c a l group selection c r i t e r i o n at work. out the much discussed SU(5)-theory.
We thus see
Weak as i t i s , i t manages to rule
Recently, based on maximally extended super-
g r a v i t y , 13 we have studied 14 the phenomenology of a grand SU(8) u n i f i c a t i o n group. Remarkably, the favored a l t e r n a t i v e has the spin I / 2 fermions (leptons and quarks) in 56 representations of SU(8) which s a t i s f y the " i n t e r n a l spin" - s t a t i s t i c s connection as do a l l the other Bose and Fermi m u l t i p l e t s in that case. (less favored) assignments of spin I/2 fermions to 56, 28, 8 out by the " i n t e r n a l - s p i n " - s t a t i s t i c s connection.
The a l t e r n a t i v e
m u l t i p l e t s is ruled
We thus see that t h i s general
c r i t e r i o n again has some b i t e . I t is remarkable that considerations of world topology thus lead us I I to statements about the nature and m u l t i p l e t structure of gauged i n t e r n a l symmetries. This work was supported in part by the United States National Science Foundation. REFERENCES I.
See, e . g . , A. Salam, Proc. 19th I n t . Conf. High Energy Physics, Tokyo, 1978, S. Homma, M. Kawaguchi and M. Miyazawa eds, Phys. Soc. of Japan, p. 933.
2. 3.
T. Eguchi and P.G.O. Freund, Phys. Rev. Lett. 37, 1251 (1976). S. Hawking, Phys. Lett. 60A, 84 (1977).
4.
S. Hawking and C.N. Pope, Phys. Lett. 73B, 42 (1978).
5. 6.
A. Back, P.G.O. Freund and M. Forger, Phys. L e t t . 77B, 181 (1978). M. Forger and M. Hess, Comm. Math. Phys. 64, 269 (1979).
7.
S.P. Avis and C.J. Isham, p r e p r i n t ICTP/78-79/21.
8.
S. Hawking and M. Ro~ek, private communication.
9.
R. Geroch, J. Math. Phys. 9, 1739 (1968).
310 I0.
For compact Riemann 4-manifolds SpinC(4) = Spin(4)X Z U(1) structures can a l ways be defined, F. Hirzebruch and H. Hopf, Math. An~. 1361, 156 (1958); for the case of P2(C) the relevant U(1) gauge f i e l d is that considered by A. Trautman, Int. J.-Theor. Phys. 16, 561 (1977).
II.
The obvious a l t e r n a t i v e is to l i m i t the functional integration that defines the quantum theory to spin-manifolds. Here we explore mainly the consequences of not making t h i s very r e s t r i c t i v e assumption.
12.
H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974).
13.
E. Cremmer, B. J u l i a and J. Scherk, Phys. Lett. 76B, 409 (1978); E. Cremmer and B. J u l i a , Phys. Lett. 80B, 48 (1978-~-and preprint LPTENS 79/6.
14.
T. Curtright and P.G.O. Freund, preprint EFI 79/25 and unpublished.
RECENT RESULTS ON DIFFERENTIABLE
DYNAMICAL
SYSTEMS
David RUELLE IHES.
Differentiable
dynamical
91440 Bures-sur-Yvette.
France
systems.
We may think of dynamical
systems as describing
plest case the time will be discrete
(a natural
time evolution.
integer
In the sim-
n ) and the evolution given
by Xn+ 1 = f(Xn ) where
x
is a
n
m-dimensional
For instance
if
vector
m = 1
and
f
a differentiable
function on
]Rm .
we may take
Xn+ I = axn(l-Xn ) this quadratic m = 2
and
x
transformation
of the line will be discussed by J.-P.
has components
n
x' x" n' n
Eckmann.
If
we may take
x' = x"+l - ax '2 n+l n n x" = bx' n+l n This is the H~non model, w h i c h has been much studied numerically, a = 1.4
and
in particular
for
b = .3 . J. Curry will speak about it.
A continuous
time dynamical
system is described by a differential
equation
d d--~ xt = X(xt) Instead of etc.),
]Rm
w e may work on a general differentiable
or perhaps
a Hilbert
we are interested
manifold
space, but the main features
already occur for ~ m
(sphere,
of the problems
torus, in w h i c h
.
Ergodic theory. Let
p
be a probability
measure with compact support in
ty) and invariant under
the map
that for
x , and every continuous
p-almost
all
f . If
p
is ergodic,
the ergodic
function
~ ,
~m
(for simplici-
theorem asserts
312
1 n~l lim
~
n--~.~o
~(fkx)
There exists a multiplicative (or rather
= f p(dy)~(y)
k=o
ergodic theorem corresponding
exp ~ ) is replaced by a m a t r i x - v a l u e d
T n = T(fn-lx) . . . . . x Then for
p-almost
all
x , and all
exists.
For fixed
x ,
teristic exponents. almost everywhere.
X(X,U)
T(fx).T(x)
m
different values
P , the characteristic
The upper characteristic +
.
= X(X,U)
may take at most
For ergodic
T . Write
u E IRm
nI log llT:ul[
lim
to the situation where
function
exponent
exponents
called charac-
are constant
p-
is
i nll = lim --n log lITx n-+~o
X
We shall now apply these notions to the case where T(x) is the matrix of partial ~f T n derivatives ~ . Then x is the matrix of partial derivatives ~fn/~x .
If
X + > 0 , a small error in
ponentially w i t h
x
x
will yield an error on
o
n
growing ex-
n : ~fn 6x
This w e call sensitive
n ="~'~-x
+ 6x
dependence
~ exp n
o
X
on initial
condition.
+ It is important not to confuse exponential definition
rate of mixing determined by of
h(p)
(the characteristic
with the entropy f
positive
h(p)
char.
[i]
when
p
is smooth,
the
p . For a
. One has
exponents w.r.t.p
exponents have to be counted with multiplicity).
(unpublished)
, measuring
w i t h respect to the measure
see for instance Billingsley
h(p) ! E
by Margulis
X
This was noted
then proved in general by Ruelle
[6] *). In particular, if h(p) > 0 , there is always one strictly positive + : X ~h(p)/(m-l) .
characte-
ristic exponent
If
f
has a differentiable
the same as with respect
*) Also S. Katok
to
inverse the entropy with respect to
f , and the characteristic
(unpublished)
when
f
is invertible.
exponents
f-I
is
have the opposite
313
sign. Thus if
h(p)
X- ~ - h ( p ) / ( m - l )
> O
there is a strictly negative
characteristic
exponent
.
Suppose t h a t
the derivatives
of
f
a r e H~lder c o n t i n u o u s , and t h a t
p
is
ergodic. - If all characteristic periodic
orbit
exponents
(Ruelle
- If all characteristic support of
p
are
< 0 , then
p
is carried by an attracting
are
# O , the periodic points
[5]). exponents
are dense in the
(Katok [2]).
Stable manifolds. Consider
the vectors
u
such that
×(x,u)
< O , they form a linear space
V
. One may consider this fact as the infinitesimal v e r s i o n of a stable m a n i f o l d x theorem which is most easily forr~alated if we assume that IRm is replaced by a com-
pact m a n i f o l d
M , that -i f
f : M ~ M
inverse
(i.e.
is a set
A = M , with
fA = A,p(A)
and if
x 6 A
y
= 1
manifold
and a diffe-
f-invariant
probability
there
measure,
around so as to be dense in parts of exponent,
< O}
n
contained
for some Riemann metric,
characteristic
for every
derivatives,
Under these conditions
log d(fnx,fny)
= {y 6 M : lim s u p n->~o
is a differentiable and
is a diffeomorphism).
the set
7~x
x
f
has H~ider continuous
rentiable
in
~x
A
~).
(Here
is tangent
to
d(x,y) Vx
at
is the distance
M ). If there is at least one strictly negative
the manifold
~
is not reduced to a point.
~
X
a stable manifold.
of
x , but may turn
The stable manifolds
is called X
for
f-i
are called unstable manifolds.
Pesin theory.
The greatest progress
in the study of differentiable
recent times was made by Ia. B. Pesin with a d i f f e o m o r p h i s m sure
f
p . He first proves
the existence
respect to
p
ted above).
The stable manifolds
mily.
However,
(actually,
his results
what is m e a s u r a b l e
subset of measure
arbitrarily
by two transversal
manifolds,
~) For a proof
see Ruelle
[4]. Pesin considers
w h i c h is twice differentiable,
[5].
becomes
M
a smooth mea-
p
as we indica-
but only a measurable
continuous w h e n restricted
close to i. Intersecting
in
almost everywhere with
to non smooth
do not form a continuous,
systems
a compact manifold
and preserves
of stable manifolds
can he extended
dynamical
fa-
to a suitable
the bunch of stable manifolds
one gets a natural map from a subset of one transversal
314
to a subset of the other transversal. This map is absolutely continuous i.e. it sends sets of measure
0
to sets of measure
striking fact. The set
M~
ferent from zero is (up to a set of ponents.
0 . From this Pesin derives the following
of points at which all characteristic exponents are difp-measure
In other words the decomposition of
O ) a countable union of ergodic comp
into ergodic components is discrete
rather than continuous where the characteristic exponents do not vanish. This applies for instance in the situation of Moser's twist theorem. "hyperbolic points"
If one could prove that the
(i.e. those with nonzero characteristic exponents)
form a set o f
positive measure ~)
one would Nave a decomposition of this set into countably many
ergodic components.
For a number of further results in the line of Pesin theory (but
not assuming a smooth invariant measure)
see Katok [2] .
Asymptotic measures. In Hamiltonian mechanics there is a natural measure invariant under time evolution,
and it is smooth so that Pesin theory applies. For dissipative systems
instance in hydrodynamics)
(there are in general uncountably many different ergodic measures).
One finds however
often that one invariant measure is stable under small random perturbations, the other measures are simply not seen. The random perturbations
and that
could for instance
be the roundoff errors in the case of a dynamical system generated by a computer. some cases the measure which is stable under small random fluctuations matically determined
(for Axiom A diffeomorphisms,
this asymptotic measure
p
(for
one does not know a priori what invariant measure to use
In
can be mathe-
see Kifer [3]). One finds that
is selected by the condition that its conditional measu-
res on unstable manifolds are absolutely continuous with respect to the measure defined by the Riemann metrical). h(p) =
This implies
(Waiters and Katok, unpublished),
that
E positive char. exponents
The difference between the entropy and the sum of the positive characteristic exponents reaches therefore here its maximum value, which is zero.
Hydrodynamic turbulence. My personal interest in differentiable dynamical systems with sensitive dependence on initial condition comes from the belief that they are needed in the description of hydrodynamic turbulence. This requires however going over from finite dimensional manifolds to infite dimensional functional spaces. I have recently been able to extend the multiplicative ergodie theorem and the stable (unstable) manifold theorem
to Hilbert spaces
(R. Ma~e - private communication - appears to have similar
~)
Unfortunately no such proof exists
!
~)
This condition is unfortunately not universal,
there are counterexamples.
315
extensions to Banach spaces). It seems therefore that one is slowly getting closer to the measures which describe turbulence in the realistic setting of partial differential evolution equations of the Navier-Stokes type.
References i
P. Billingsley. Ergodic theory and information, John Wiley, New York, 1965.
2
A. Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Preprint.
3
lu. I. Kifer. On the limiting behavior of invariant measures of small random perturbations of some smooth dynamical systems. Dokl. Akad. Nauk SSSR 216 N°5, 979-981 (1974). English translation. Soviet Math. Dokl. 15, 918-921 (1974). On small random perturbations of some smooth dynamical systems. Izv. Akad. Nauk SSR. Ser. Mat. 3 8 N°5, 1091-1115 (1974). English translation. Math. USSR Izvestija 8, 1083-1107 (1974).
4
la. B. Pesin. Lyapunov characteristic exponents and ergodic properties of smooth dynamical systems with an invariant measure. Dokl. Akad. Nauk SSSR 226 N°4, 774-777 (1976). English translation Soviet Math. Dokl. 17 N°I, 196-199 (1976). Invariant manifold families which correspond to nonvanishing characteristic exponents. Izv. Akad. Nauk SSSR, Ser. Mat. 4 O N ° 6 , 1332-1379 (1976). English translation Math. USSR Izvestija IO N=6, 1261-1305 (1976). Lyapunov characteristic exponents and smooth ergodic theory. Uspehi Mat. Nauk 32 N°4 (196), 55-112 (1977). English translation Russian Math. Surveys 3 2 N ° 4 , 55-114 (1977).
5
D. Ruelle. Ergodie theory of differentiable dynamical systems. Publ. Math. IHES. 50.
6
D. Ruelle. An inequality for the entropy of differentiable maps. Bol. Soc. Bras. Mat. 9, 83-87 (1978). A number of further references can be found in my review at the Rome confe-
rence in 1977 (Springer Lecture Notes in Physics N ° 80). See also Bifurcation Theory and Applications in Scientific Disciplines, Ann. N.Y. Acad. Sci. 316 (1979).
ON SOI~ SYSTEMS }~TIVATED BY THE LORENZ EQUATIONS: NUMERICAL RESULTS J. H. Curry National Center for Atmospheric Research Boulder, CO 80307/USA and Department of Meteorology Massachusetts Institute of Technology
I.
Introduction
All of the work reported in this article is motivated by a system of three ordinary differential equations which was introduced i n t o t h e literature by Lorenz [i] in his study of atmospheric prediction and convection.
The three-variable model is of in-
terest not only because it provides a simple example of a strange attractor but also because the careful analysis of Lorenz suggested several areas of further study.
Among
other things, Lorenz reduced (heuristically) the study of a complicated attracting set to the simpler problem of studying a transformation which maps the unit interval into itself; we refer the reader to the articles of Eckmann and Lanford in this volume for more details concerning maps of the unit interval. Motivated by work on the Lorenz equations, H~non [2] discovered a polynomial transformation which maps the plane into itself and has some of the properties of the Poincare map of Lorenz's system.
In Section 2 a description of H~non's transformation is given
along with a discussion of its dynamical behavior for several parameter values. principle tool which we shall use is the frequency spectrum.
A
In Section 3 we describe
a 14-variable generalization to Lorenz's equation which has a completely different sequence of bifurcations leading to chaotic behavior.
In the final section we mention
some recent experimental work and its connection to the 14-variable model of Section 3.
317
2.
H~nonVs
transformation
Given a system of differential its dynamics by constructing
equations
it is possible to study
a surface of section transversal
ering the resulting transformation is called a Poincar~ map.
in three variables,
to the flow and consid-
which carries the section into itself; such a map
(Lanford
[3] has carried out this analysis
for Lorenz's
system.) The H~non transformation
is an invertible polynomial mapping of the plane into itself
which shares some of the properties
of the Poincar~ map of Lorenz's
system and is de-
fined as follows: T(x,y)
=
(i + y - ax2,bx>
(i)
T has two fixed points whose coordinates
x
=
(b - i) ±~(b - 1) 2 + 4a 2a
These stationary
'
are given by
Y = bx
solutions are real provided
(2) a > (i-b)2/4,
in which case the stationary
solution in the left half plane is always unstable while that in the right is unstable when a > 3(i-b)2/4. In [2] a and b are 1.40 and 0.3, respectively. two-dimensional
attractor.
id We see successive
Fig. la is a picture of the H~non attractor.
enlargements
observed on magnification
What H~non found is apparently a strange
of the region indicated.
In Figs. ib-
The structure which is
is a family of lines which suggest that locally the attractor
is the product of a Cantor set and interval.
It is possible to prove using the com-
puter, modulo certain technical
facts, that there is indeed a Cantor set in the neigh-
borRood of the H~non attractor,
but such Cantor sets are not attracting.
H~non was also able to show that for the parameter values he considered
there is a
compact region M which is mapped into itself under the action of T, M contains object pictured in Fig. la.
the
In [5] Feit proves that for a large range of parameter
values such an M exists. Numerical experiments
indicate that if several thousand points are placed on the eigen-
space associated with the expanding direction of the fixed point and then iterated by T, one recovers
the object pictured in Fig. la.
This suggests
that the H~non attractor
is the closure of the unstable manifold of the fixed point in M. tractor is the closure of its unstable manifold, (determinant = -b), Figs. la and ib suggest
If the H~non at-
then, since T contracts volumes
that the unstable manifold may have arbi-
trarily sharp bends and therefore may not satisfy SmaleWs Axiom A [6]. Non Axiom A systems are not yet well-understood, cerning such systems.
Newhouse,
H~non may have discovered
for example,
an extremely
of errors inherent in computing, in Figs. ib-ld.
however,
there are some results con-
has proven results which suggest that
long, stable periodic orbit
the process never exactly repeats.
[7] and because Hence the graphics
318
HENON M A P 0.4
B = Q30
A = 1.40
i L , i i i i i , i i i i i i i+ i i i I , i i i i i i
,.,, " . : . ~ 0.2
-
-
o,-
....
--
O3 X i
I
>-
-0.1 -0.2 -03
/
,j'i
/.~
--0.4
I I I I 1"~11
I I I I I
I i
"1.2
-0.4
0
-0.8
I I I I
I I I I I I
0.4
0.8
I I
1.2
X-AXIS Figure la,
Henonls attractor
HENON MAP 0.110
i,\l
i
•. 0.108
~ \
'"
A = 140
p.~l :1 i ..~. i I ,~ 'i ~\ •. .i'~3 ~ ,%. !,. "%
\
"
"t
\\
,... ",,
0.106
a = 1.40+ b = 0.3
~,
i
I
(/') x
""~ '..
...
'\~%
:... ~
\~
\
% %.
Q094
i
'.,
,..~,,
"~. '
.....
""
".
"~,
\
""
0.092
~.~:~,
,:..,
\
0.096
I
,.,.,.
~,.
\,
O.098
i
~.,, ",
•
',:
>-
I
,:,;~..i,t
',.,,
.,
0.100
i
,.
',
..
".,
I
"-~. ""!~i.'.:
".~.. •. \
0.102
'
.'...
'.
%
0.104-
B = 030
\
"%.
";t:,
.
'-%. .
v~.~q~
1.00
I
i
I
1.02
*
I
I
I
I "'1
1.04
I
1.06
I
I',
"~,.
I +,
1.08
I
r,
1.10
X-AXIS Figure lb.
A n e n l a r g e m e n t of the b o x e d s e c t i o n f r o m la
319
HENON MAP 00940
i
I
I
A -- 1.40
I \~.
I
w
I
l
I
I
I
B = 030 '
Ik "I
I
I
00938 "~',.
0.0936 0.09:34 - '~:::
'.,, ..
{~
_-
,
:.
-_
',
00932 0.0930
~:
00928
v,
..,
-
0.0926
,',,,,,
~: 0.0924 >I0.0922 " 0.0920 00918 0.0916 0.0914
I
_-
:,
V,\-.,
'--
'i",,..&..",:, \-~
-
0.0912 0.09 I 0
,
\~,,
-
'k~
\
l'tl
,
I
I
,
I
,
I
i
1.074
1.070
-
l
I
I
1.078
I
1.082
I
l
I
I
1.086
1.090
X-AXIS Figure ic.
An enlargement of the boxed section from ib
HENON MAP 009170 0.09168 0.09166 009164 0109162
A = 1.40
.:. I "'-,1'-.. % 1
-",..
I
I
'
''
I'
B = 030 l ~' ~il\ ' \l
"V,,\
-
'
_
~ ~~
,
-,,-~
t ! \
\
009160 (/)
0.09158 0.09156
I
>-
-
"",.
~
t.-
0.09154 0.09152 0.09150 0.09148
"')"""
"'" ""-
i
0.09146 0.09144
X
-
0.09142 0.09140
"",..,.
%
\ "\
I
1.0790
I
i
\.
I ;,,= ".1
1.0794
i
I
i
1.0798
I
i
I
i
1.0802
I
i
I
I
1.0804
X-AXIS Figure id.
An enlargement of the boxed section from Ic
320
In order to demonstrate the significance of machine error when iterating T, consider the following experiment.
Form Tn(O,0) using two different machines;
no reason why, after 60 iterations,
there is ~ p r i o r i
the outputs from the separate machines should agree.
Indeed, as was reported in [4], there is no agreement.
This experiment provides sup-
port for the suggestion of Newhouse. There are, however,
computations which can be done to determine whether a process is
random or chaotic.
The characteristic exponent provides a method for determining sen-
sitivity to initial conditions.
If it is positive,
then neighboring trajectories expo-
nentially diverge with exponent the characteristic exponent.
For a definition of the
characteristic exponent, we refer the reader to the paper of Ruelle
[8] in this volume;
we remark that from its definition it is clearly a statistical quantity. In [5] and [4] characteristic exponents for a wide range of parameter values of T have been calculated.
In ~he specific case studied by Henon it was found that the charac-
teristic exponent is positive and equal to approximately 0.419.
Therefore what H~non
found has sensitive dependence on initial conditions. Another statistical measure of randomness is the frequency spectra of a process.
Re-
call that if the spectra consist of solitary narrow spikes the process is (multiply) periodic, while if there is a broad band of frequencies present the process has continuous spectra and is not periodic. Fig. ~ is the frequency spectra of the H~non transformation [4] frequency spectra were also computed; however, lution of the spectra presented there. using two different machines.
Further,
(a = 1.40, b = 0.3).
In
in Fig. 2 we have twice the reso-
in [4] the spectra were also computed
It was found that the results were virtually identical.
This latter observation indicates that even though it is difficult to follow individual trajectories on two different machines,
the statistics are machine independent.
We
remark that the statistics of Axiom A systems are known to be stable under small stochastic perturbations
[9].
Finally, in Figs. 3a and 4a we present the phase space plots for two additional parameter values for the H~non transformation a = 1.15 and 1.16 (b = 0.3), in Figs. 3b and 4b their associated frequency spectra can be seen.
Note that in going from Fig. 3 to
Fig. 4 the power in certain frequencies jumps by an order of magnitude. A careful look at the dynamics for a = 1.15 indicates that points are mapped from the upper branch to the lower branch and vice versa. process is noisily periodic with period two. in Fig. 3b.
Hence for this parameter value the
We see little if any evidence of this
For a = 1.16 the dynamics is very similar to that for a = 1.15.
However,
there is now a region of overlap; once in the new region, a phase point remains there for several iterations of T and upon leaving executes behavior similar to that for a = 1.15. clear.
Why there is a strong period two for a = 1.16 and not for a = 1.15 is not
We remark that there is an a
c
(1.15 < a
c
< 1.16) for which the stable and
321 unstable manifolds
o f the fixed point cross; C. Bimo [23] has calculated a
fixed point and other periodic points as well. new orbits to come into existence
[6] or [7].
for the c When this happens we expect complicated
This may explain the sudden increase
in power for certain frequencies.
1.400 0.30 0.0060
rl
O.0055 0.O050 0.004.5 0.0040 0.0035 0.OO30 0.0025 0.0020 0.0015 0.0010 O.O005 0
0
0.05 0.10 0.15 0.20 Q25 0.30 0.35 0.40 Q45 0.50
LINEAR-LINEAR
Figure 2.
Frequency spectra of H~non's map b = 0.3 linear-linear
a = 1.40,
322
HENON
MAP
A = 1.15
B = 030
0.40 __~-L~,.,, i. I ' I i I i I ' I l I l I i I i I l I i __0.:35 030
_
0.25
.......,,,.,.
0.20 015 u')
O.lO
_-
0.05 >-i
0
',!)
-O05 -Q
IO
-0.
15
-Q20 -Q25 I I I I I I i I t I I I It''~l
-OSO
-I.0-0,8-0.6-0.4-02
O
0.2 0.4
I t I I I I I I 0.6
0.8
1.0
1.2
1.4
X-AXIS
Figure 3a.
Noisily periodic attractor
a = 1.15, b = 0.3
1.150 Q30 I0 -z 1 4 i i i l l i l l l l l i l l l = i i l i i i l l i l l i l l l ,
iliilllll
ililll
10-:3
E
8 J 10-4
10-5
IIIIIlilliitllillllitilllllZltlltltiitltlil[tll 0.10 0.15 O,200.25Q30Q~50.400.450.50
0 005
LINEAR-LOG
Figure 3b.
Frequency spectra of Fig. 3a (linear-log)
323
HENON MAP 0.40 ~ i, i, .... ...-... .... 0.35 -0.30
--
A = 1.16 ,,
i,
......
,,
B = 030
I,
i,
i,
i,
i
:
~',~
0.25 0.20 0.15 ¢n
0.10
0.05
;-
o -0.05 -0. I0
-
J
-0. 15 -O20 -"
.~j
-0.25 -030
I
I
i
I
i
I
i
I
i
-1.0-0.8-0.6-Q4-0.2
I
i
0
I.'?"i"
I
J
I
I
i
I
i
I
Q2 0.4 0.6 0.8 1.0 1.2 1.4
X-AXIS Figure 4a.
A t t r a c t o r after h o m o c l i n i c i n t e r s e c t i o n a = 1.16, b = 0.3 1.160
io-i
030
Ilillllillllllllllllllllllllllllillllillllllll~
10-2
L.I_
n. 10-3 t~ 0 J
10-4
IIIIIIIIIIlilllllllilllllllllllllllllllilllllll
10-5 0
Q05 0.10 0.15 Q20 O?_5 0.30 Q35 O40 Q45 Q50 LINEAR-LOG
Figure 4b.
Frequency spectra of Figures 3 and 4a (linear-log)
324 3.
Higher dimensional systems
The equations that eventually lead to the three-variable model of Lorenz are due to Saltzman.
In [i0] Saltzman studies a seven-variable system derived from the following
system of partial differential equations: =
-
(A¢) t
8(~,A~)
+ ~ A 2 V~ +
d~e
~(x,z)
~x (3)
(e)t
=
S(x,z) + R ~ +
-
Ae
where ~(x,z,t) and 8(x,z,t) denote the stream function and departure of temperature from its linear profile while d and R denote the Prandtl and Rayleigh numbers respectively.
These equations describe convective motion in a fluid layer which is uniformly
heated from below and cooled at the top.
For a discussion of the equations in the
above form see [ii]. Saltzman's seven-variable model was derived from Eq. (3) by assuming that ~(x,z,t) and @(x,z,t) are expressible in the following form ~(x,z,t)
=
~ ~mn(t) sin (amx) sin (nz) (m,n)eA m # 0
e(X,Z,t)
=
(4)
~ @ran(t) cos(amx) sin(nz) (m, n) eA
A denotes a subset of the nonnegative plane integer lattice and a is a geometric constant. Saltzman noted that for some initial conditions all but three of the seven variables in his model tended to zero, and the nonzero variables executed aperiodic behavior. This three-variable model is the Lorenz system. In our previous studies we have concentrated on a 14-variable system of ordinary differential equations with A = {(0,2),(1,1),(2,2),(1,3),(3,1),(3,3),(2,4),(0,4)}.
Since
A contains the subset {(0,2), (i,i)} which are the same modes (up to geometry) which are included in Lorenz's three-variable model, we call such a system a generalized Lorenz system.
For a more complete treatment of these equations, we refer the reader
to [12]. In treating the 14-variable model we are particularly interested in the sequence of bifurcations which ultimately lead to an attractor which has some characteristics of the Lorenz attractor. It is convenient when describing the numerical result to introduce the parameter R
r
6.75
where the value in the denominator is the critical Rayleigh number below which
(5)
325
convection is impossible,
r is the bifurcation
parameter and ~ is fixed and equal to
ten.
Numerical Results In reporting the results of our numerical parameter range into several intervals. more complex than the preceding one.
experiments
it is convenient
In each succeeding
interval the dynamics
1 ~ r ~ 44.3
Inverval II
44.3 < r < 45.19
Interval III
r > 45.19
As r exceeds i, the origin in the 14-dimensional
stable and a symmetric pair of stable fixed points bifurcate and C2.
is
The intervals are:
Interval I
Interval I.
to divide the
phase space becomes unoff, we label them C 1
As r crosses the sequence of values defined by 2(0.5 m 2 + n2) 3 r
=
(6) (6.75)(m) 2
for (m,n) g A new symmetric pairs of unstable stationary
solution bifurcate
from the
origin. For r g (1,43.48) stationary
the model exhibits very simple dynamics.
solutions,
two of which are stable,
condition in which the Lorenz components
C 1 and C 2.
are nonzero,
There are at least seven Further,
given any initial
the solution curve will converge
to either C 1 or C 2. When r passes 43.48 the Com become unstable by having a pair of complex conjugate eigenvalues cross into the open right half plane and the stability of each C. is transferred i to an attracting closed cycle. For r = 43.48 the imaginary part of the eigenvalues of interest
is ±35.672.
From the Hopf Bifurcation
be 2~/135.6721 m 0,17613. the predicted value.
Theorem
As r is increased
to 44.3, the Nopf orbit undergoes
monic bifurcation which causes its period to double. remarking that for r ~ 44.2 we find the unexpected closed orbit pair.
This pair of hyperbolic
and is not associated with the bifurcation points.
[13] we expect the period to
The computed period is 0.174, which is less than 2% from
We conclude
appearance
a subhar-
this subsection by
of a stable-unstable
orbits suddenly appears in phase space of any of the seven unstable
critical
The orbits persist until around r = 45.10, when they coalesce.
Intei-val II. transverse
We denote by y a closed orbit, by ~ a 13-dimensional
to ~, x g y, and by Pr(X)
It is not difficult
to define a Poincar~ map using the computer.
the section defined by 804 = constant, an accuracy of I0 -I0.
local section
the associated Poincar~ map. E was chosen to be
the solution was required to intersect
We shall concentrate
E to
on the behavior of the flow in a neighborhood
326
of C I.
Recall that t h e
stability of y can be determined by studying the spectrum of
DP the derivative of the Poincar~ map.
If the spectrum is contained inside the unit
circle y is stable, while if a portion lies outside y is unstable. In [14] we have described a method for finding numerically The method has been applied to the 14-variable model. bifurcates
a point on a closed orbit.
For the subharmonic
orbit which
from the Hopf orbit, we find that when r = 44.87 a complex conjugate pair
of eigenvalues
cross out of the unit circle for the spectrum of the associated Poincar~
map. Fig. 5 is the projection provides
of E onto the (~ll,ell) plane for r = 44.96.
evidence that the y mentioned
ferred its stability
to an invariant
Ruelle, Takens bifurcation
theorem
above has undergone
two-dimensional
This figure
a bifurcation
torus.
and trans-
The Naimark-Sacker,
[13] does not provide any information
concerning
the nature of the flow on the torus in this situation.
I0.00
:.'303.50 -129
'
I
t
I
'
I
t
f"r'-.
I
I
I
t
I
t
-t3o l "',,,
-131
"-"
'..,.
\
-132 -133
[" ,.,"
-134 l (/) x
>I-
-- 155
--
-156
~-
-137
-
-
--140
-
,,,o'"
:"
y ....'"
-
- 142
~-
143 -39.4
I
, ...""
--141
-
/
/
,.,.'
- - 1:38 -- 139
./
.,.." =
...."
..
--~-~-'"T'"'"l
I
I
I
t
I
i
I
t
I
t
I
I
I
I
'
-39.0
-38.6
-37.8
-38.2
-37.4
X-AXIS
Figure 5.
Projection
of E onto the (~ll,ell) plane
(r = 44.96)
the motion is on a two torus
In Fig. 6 (r = 45.148) we have a projection tional loop in the torus should be noted.
of E onto the (~ll,Oll) plane. Recall that a two-torus
than three does not have an inside and outside and more loops appear inside the projected
[7].
torus.
The addi-
in dimension
greater
As r is increased to 45.19, more
327 304Z5 -118
III
IIIII
Kill
I0.00 i I
III
i
iiilii
Ji
llll
-120 -122 -124 -126 -128 (1)
-130
-132 <[ -134 )-I -136 -138
-140 -142 -144
~...o "
-146
-°
--148
o ° "
~ li',l",,,,I,l,,lllll
-40.0
-39.0
I
-38.0
-37.0
I
-36.0
X-AXIS Figure 6-
Interval III. behavior
Projection of ~ onto the (4 ,8 ) plane . . . . ii ii the torus has aeve±opea loops
(r = 45.15)
As r exceeds 45.19 the flow on the torus becomes unstable and we find
similar to that which was observed by Lorenz in his three-variable
We remark, however,
that the dynamics
not at all well-understood. provides evidence
model.
in this regime are extremely complicated and
We refer the reader to [12] for more detail.
that the flow on the torus in unstable,
r = 45.19.
Fig. 7
328
I0.00
305.00 -120
.11
II
~ III
I
III
I
III
~111
li
I
Iil
| I-
-125
-130 -I 35 • ...:
. .
-140 ~'"
(f.) -- -145 x
-
., "'.
•
. ,
.iii~:.'~
~1 -150 -I 55
--160 -165 -170 lilt i lilt I--175 - I ' t - I , t J L I I . I . L ; . I , t I I I t I t I I l l l l -41.0 -40.5 - 4 0 . 0 -39.5 -39.0 -38.5 -38.0 -37.5 -37.0
X-AXIS Figure 7.
Projection of ~ onto (Vii,ell) plane (r = 45.19) flow on the torus is unstable
329
4.
Periodic forcin$
In the previous
section we noted that Eq.
(3) provides
a description
of convective
motion in a fluid layer which is uniformly heated from below and cooled at the top. The first quantitative
experiments
on this problem were done by B~nard in 1900 [17].
For a review of the state-of-the-art
prior to 1960 we refer to [18], for a recent
review see [19]. Recently
Gollub and Benson
[20] have redone the transition
of B~nard.
By computing the power spectra of the velocity
experiments
indicate that the transition
to turbulence
to turbulence experiments field of the fluid,
occurs in the following way.
There is first a laminar state which, as the Rayleigh number is increased, stability.
their
The spectra in [20] then indicate sharp peaks corresponding
loses its
to a single
frequency and its harmonics. As R was increased
further,
of the original frequency.
they saw a subharmonic bifurcation which caused a halving As R crossed another critical value, they found a range
for which the motion was quasiperiodic with two frequencies. regime they found complicated qualitatively
turbulent behavior.
Beyond the quasiperiodic
This sequence of bifurcations
similar to those which we noted in the previous
is
section for our 14-
variable model. Gollub and Benson then introduced perature
at the lower boundary,
a small amplitude periodic modulation
i.e., they modulated
ported that when the flow is in the quasiperiodic
in the tem-
the Rayleigh number.
It is re-
regime a small perturbation
of the
type just described does induce turbulence well below the threshold for the unforced p r o b l e m provided the forcing amplitude
is sufficiently
large.
Rayleigh number never exceeds the value which is necessary to transition
the 14-variable
for the unperturbed
model
I have introduced
[21].
flow
a small amplitude periodic modulation
The modulating
in the autonomous
frequency in [20] was chosen from a small neighborhood
of the two naturally occurring But our numerical
frequencies;
calculations
both resonance and nonresonance become chaotic.
into
There it is also found that the flow for the nonautonomous
system became unstable well below the threshold which occurred
hence,
the behavior m y
case.
of the difference
be a simple reso-
indicate that there is a large frequency range
for which the flow for the nonautonomous
system will
Just as in [20] the effective Rayleigh number never exceeds the value
which is necessary
for the unperturbed
The recent work of Newhouse,
flow to become chaotic.
Ruelle and Takens
behavior observed
in [20] and [21].
any quasiperiodic
motion with three frequencies
By adding a perturbing certainly
the effective
to turbulence.
Motivated by their work,
nan6e.
Further,
frequency
in a situation
[22] may provide an explanation
for the
In [22] it is proved that in the neighborhood
of
there is a motion which will be chaotic.
in the parameter
mange where there is a torus we are
for which the results of [22] is suggestive
at the very least.
330
Acknowledgments This work was done while the author was a visitor at the National Center for Atmospheric Research, which we thank for their hospitality and for the use of their computation center.
(The National Center for Atmospheric Research is sponsored by the National
Science Foundation.)
J.H.
Curry is on leave from the Department of Mathematics,
University of Colorado, Boulder, CO 80307/USA. drafts of this manuscript.
Thanks to H. Howard for typing several
This work was supported by the Air Force Geophysics Lab-
oratory under contract #AF F19628-78-C-0032.
REFERENCES i. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci_____t , 20, 130. H~non, M., 1976: A two-dimensional mapping with a strange attractor. Comm. Math. Phys., 50, 69. Lanford, O. E., 1979: Statistical and qualitative theory of dissipative systems. CIME Lecture, 1977. Curry, J. H., 1979: On Henon's transformation. Comm. Math. Phys., (to appear). Feit, S. D., 1978: Characteristic exponents and strange attractors. Comm. Math. Phys., 61, 249. Smale, S., 1967: Differentiable dynamical systems. BAMS, 73, 747. Newhouse, S., 1979: Dynamical systems. C I ~ Lecture, 1978. Ruelle, D., 1979: Lecture in this volume. Ruelle, D., 1977: Lecture presented to New York Academy of Sciences. Saltzman, B., 1963: Finite amplitude free convection as an initial value problemI. J. Atmos. Sei., 19, 329. Curry, J. H., 1978: Finite dimensional approximation to the Boussinesq equations. SIAM J. Math. Analysis, i0, 71. Curry, J. H., 1978: A generalized Lorenz system. Comm. Math. Phys. , 60, 193. Marsden, J., and M. McCracken, 1976: The Hopf bifurcation and its application. In: Lecture notes in applied mathematical sciences, Vol. 19, Berlin-HeidelbergNew York, Springer. Curry, J. H., 1979: An algorithm for finding closed orbits. In Proc. Int'l. Conf. Global Theory o fDyn, Systems. Ruelle, D., and F. Takens, 1971: On the nature of turbulence. Comm. Math. Phys., 20, 167. Lanford, O. E., 1973: In Springer lecture notes in Math., vol. 322. B~nard, H., 1900: Les Tourbillions dans une nappe liquide. Revue general des Sciences pures e_~tappliqu~es, ii, 1261 and 1309. Saltzman, B., 1977: Thermal Convection, Dover Publishing. Busse, F. H., 1978: Nonlinear properties of thermal convection. Reports Pro S. Phys., 41, 1930. Gol!ub , J., and S. Benson, 1979: Phys. Rev. Lett., 41, 948. Curry, J. H., 1979: Chaotic response to periodic modulation in a model of a convecting fluid. (submitted). Newhouse, S., D. Ruelle, and F. Takens, 1978: Comm. Math. Phys., 64, 35. Sim6, C., 1979: On the Henon-Pomeau attractor. To appear in J. Stat. Physics.
PROPERTIES OF THE
INTERVAL
P. C o l l e t , J.-P.
Continuous dynamical which
maps
systems
seem most
function
f
x ° ~ [-l,l], interested these
we
maps
study.
into
We
[-i,i]
into
on c a n r e c u r
to t h e
are a m o n g
here
some
of n a t u r a l
itself,
xn = f(x n 1 ) thus
of Geneva
itself
review
context
set o f p o i n t s
*
University
in t h e
can define
in t h e
iterates
can
MAPS
ITSELF
University
of an i n t e r v a l
one
TO
Harvard
Eckmann,
relevant
which
OF CONTINUOUS
and
for
of the properties
sciences.
Given
an i n i t i a l
point
In o r d e r
method
i. T h e
x~+
f(x).
of the
As
fixed
*Supported
f(x)
that
is w e l l point,
by NSF
We are
shown
in Fig.
i.
I X
function
It is c l e a r
a
to v i s u a l i z e
-I
Fi~.
simplest
n = 1,2 . . . . .
obtained.
graphical
the
the p o i n t known,
one
Grant
= 1 - 1.4 x
2
.
marked
in o r d e r
analyzes PHY-77-18762
~
is a f i x e d p o i n t m f
to examine
f'(x).
If
the
local
If' (x) I < 1
and by the Fonds
the map
stability then
National
the
Suisse,
332
fixed
point
n ÷ ~.
is stable;
If
point
If' (x) I > i,
x
of
Ifp' (Xp) pI ~ tant,
fP
1
is u s e d
,
f(0)
on
role
1.
periodic
This
will
that
the
fn(0)
is the
will
A fixed and
A n easy,
but
impor-
xo
is p e -
is
this
point
in the
on
This
important We
assume e~uiv-
[-i,0)
and
for o u r p u r p o s e ) .
if
to it as
f
has a s t a b l e
n ÷ ~.
following
review
SHOULD ALREADY
An
sequel.
assume = 0.
f • C3[-I,I]
is c o n v e x
for the
f'(0)
below.
is for
this
if
henceforth and
assumptions,
throughout
0
x,
is s u f f i c i e n t
attracted
starting
guideline
and
shall
be m a d e
above
is
0
in m i n d
If' 1-½
condition
THE STUDY OF THE ORBIT OF OF
f
condition
we
be b o r n e
derivative;
Under
equals
f' (x) = - s i g n
should
about
then
be our
p,
as
.
discussion,
sign
0
latter
[Singer].
orbit,
÷ x
unstable.
of p e r i o d
stability.
= j=o~ f' (xj)
the
Schwarzian
observation
which
by
assumption
(This
fn(x)
of d i f f e r e n t i a t i o n
x 3 it is stable.
then
and that
played
(0,i].
on its
rule
then
is c a l l e d
point
of the p o i n t s
p,
to the w e a k e r
THEOREM
chain
to s i m p l i f y
has n e g a t i v e
alent
point
x,
p-i
if one
= i,
simplifying f
fixed
to d e c i d e
of the
of p e r i o d
special
is n e a r
a periodic
= j=o~ f' (fj (x))
In o r d e r that
the
x
p-1 (x°)
In p a r t i c u l a r , riodic
if
is c a l l e d
consequence
fP
i.e.
strategy,
:
REVEAL ~ N Y
PROPERTIES
f.
If we do this, = 1 - 1.5436
e.g.
x2
for
f(x)
(Fi~.
2b)/
(with
200
= 1 - x2
x
=
0
O
for i0000
points
intervals
(Fig.
2a)
w e get the '
and
for
f(x)
followin~ -"
on
[-i,i]).
=
histograms
333
~00-5000 --
4000
-
400 3000 -
2000-
a00--
1000"
0
0
!
0 Fig.
2a,b.
Two
typical
So a p p a r e n t l y , orbit Fig. we
of p e r i o d 2b.
The
shall
THEOREM
I.
following
very
theorem
I_~f { f n ( 0 ) } n = l , 2 , . . an i n v a r i a n t
are
faced with
drastic
covers
if
the
f(x)
density
(cf
and
a stable
irregular
in fact
the
case
periodic
happens of Fig.
in 2b as
avoids
measure
no s t a b l e
a neighborhood
Which
periodic
o f zero,
is a b s o l u t e l y
orbit.
then
continuous
f
with
re-
measure.*
"
consists
of the
so that
has
above
of f i n i t e l y measure
then"
a combination
many
different
is piecewise
f3(0)
= f4 (0),
of T h e o r e m
points,
convex.
and
1 and
2.~
explains
[Ruelle]).
It s e e m s be s a t i s f i e d . know whether A measure
"
= 1 - 1.544. .. x 2 ,
If' (f3(0)) I > i, 2b
f
Assume
to the L e b e s g u e
I_ff { f n ( 0 ) } n = l , 2 , . . then
Fig.
2a we
2, s o m e t h i n g
[Misiurewicz].
2.
spect
Now
in Fig.
see below.
has
II.
while
histograms.
~
that
the
Therefore the
condition
it is a l e g i t i m a t e
conclusion
is c a l l e d
of T h e o r e m
of T h e o r e m
invariant,
if
2.I and
is r e l a t i v e l y important
2.I h o l d s
"often".
9 = 9 o f-l.
hard
question We n o w
to to
334
a~alyze f
this
question.
(x) = 1 - ~ - x 2 / ~
esting
because
through
the
THEQREM
3.
bi£}.
Let
~
for
it r e s e m b l e s
change
the
L(M)
> 0
This
means
that
M =
lim L ( M ~ ~÷0
~easure.
aperiodicity
for
family
1 >
Ixl
> ~
of f u n c t i o n s
obtained
from
and
is i n t e r -
(4 - ~ 2 ) y ( l - y )
y=sin2(~x/2).
Let
and
This
functions
of c o o r d i n a t e s
[Collet-Eckmann].
Then
(x) = 1 - 21x I
Ixl w ~.
{~]f
has no s t a b l e
[0,~])/~
is q u i t e
= i,
a common
where
thing,
periodic
or-
L is L e b e s g u e
when
the
A
graph
of
f
we ~ analyze for
some
is v e r y in q r e a t
~',
W~ d i s c a r d T~e
~'
Volume
to m e a s u r e
Orbits
of
M'
fast
is then the
tion,
due
impOrtant
which
to
bounded
notion
if t h e r e x ~I
Again,
is an
is a
of the
y E U
dependence orbit
Thus we obtain
studying
of
~
of d y n a m i c a l
will
sensitive
and
n
such
We g i v e
M~.
that
the
the
by
formulaon i n i t i a l
I c[0,1]
x
is
separate
dependence
of
that
says
appealing
an i n t e r v a l U
and
d~(0)/d~
systems
notion
intuitively
has
orbit
are t r a v e r s e d .
eventually
on i n i t i a l
If
by
~'
points
an
4. [ G u c k e n h e i m e r ] .
it).
This
An
~n, (0) = 0
around
neighborhood
0.
If
conditions.
is : f
of
0.
involved,
periodic
study
below.
and
is o u i t e
of a s t a b l e
from below
~ > 0
and e v e r y
sensitive
the b a s i s
close
which
of the p o i n t
values
in the
on i n i t i a l
[Guckenheimer]
for e v e r y
orbit
interval
is b o u n d e d
conditions
: There
the
critical
of t w o a r b i t r a r i l y
amount
In the proof,
in the p r e s e n c e
(and a small
dependence
so~e
true
detail
a~e
how
A Very s~nsitive
we
pointed.
such
following
that is
Ifn(x) ~- fn(y) I > c.
conditions
can be d e c i d e d
a simplified
version
on
of the
result.
THEOREM
tained in
[-x,x]
Period
V
to
(where
(i.e.
~ p ~m
there
f(x)
f(x)
= x, L
Re~.~!k f~
P
then
:x ~
set
the
with
that
0
If
conjugate
of
returns
fmP(0) ~ V ) ,
conditions.
V
V =
then
0
with f
[-x,x],
to a p i e c e w i s e
has
cona sen-
where linear
2 % < p < 2.
M I in T h e o r e m
c0~clusions
such
that
i s topologically
1 - QIxl,
: The
,~E M'
f
= x)
such
si£ivity with respect t9 i n i t i a l
is no n e i g h b o r h o o d
3 is c o n s t r u c t e d
of T h e o r e m
4 h o l d for
in such ~ II .
a way
that
335
We next turn our a t t e n t i o n to some d i s c r e t e aspects of maps on the interval, w h i c h reflect the o r d e r i n g of points on the line. A v e r y useful concept is the s o c a l l e d k n e a d i n ~ sequence , i n t r o d u c e d in [Milnor-Thurston].
Let
x n = fn(0)
be the successive
images of the
critical point. A s s o c i a t e to it the sequence of symbols a c c o r d i n g to w h e t h e r right).
I
~{L,C,R} n (=left, center,
x
is <0, = 0, >0, respectively n this is almost a t o p o l o g i c a l invariant. We give
Surprisingly,
the easiest v e r s i o n of a m u c h more complete result.
T H E O R E M 5. sequence
[Guckenheimer].
(i.e.
orbit,
then
phism
h
In(f)
f
If
f
and
= In(g) Vn),
g
and if
~ + f
f
is t o p o l o g i c a l l y conjugate to
such that
has no s t a b l e p e r i o d i c g
(i.e.
~
homeomor-
h - l o f o h = g).
We now turn again to p a r a m e t r i z e d that
have the same k n e a d i n g
is c o n t i n u o u s from
~
families of maps,
to
C I.
and assume
C o n s i d e r now the set of
all p e r i o d i c k n e a d i n g sequences, w h i c h may occur as k n e a d i n g sequences of a map and w h i c h are of the form
S = ~Ii,...,Ip_l, with
Ij ~{L,R},
sequences,
j = l,...,p-l.
d e f i n e d as follows
If
Ij(S I) = Ij(S 2)
if
In+l(Sl)
then we say of
C, I1,...,Ip_l,
R's,
< In+l(S2)
S1 < S 2
and
for
There is a linear o r d e r i n g of such
:
j = l,...,n, (where
if the set
$2 < S 1
C,...}
and
L < C < R)
{Ij (S I) }n j=l
contains an even number
otherwise.
THEOREM 6. [ M e t r o p o l i s - S t e i n - S t e i n , Guckenheimer, Lanford]. Suppose ~i < ~2 with
and suppose the k n e a d i n @ sequence of
S 1 < S 2.
~i < ~3 < ~2
If
S1 < S3 < S2
such that the k n e a d i n ~ sequence of
In the above ordering,
S =
f~i
then there is a
the sequences
(Ii,...,Ik_l, C,...)
S, = (Ii,...,Ik_l,
and
X, Xl,...,Ik_ I, C,...),
is ~3 f~3
Si,i = 1,2 satisfying is
S3 .
336
where
X = R
contiguous;
if
{Ij} k-I j=l
that
p e r i o d of l e n g t h
k
k, 2k,
4k,
8k,...
values
of
p
FACT
7.
[Feigenbaum].
and of the
T his
In the p a p e r
--i
~(0)
= i,
linearizing
of the f o r m ~
g
k
of
on a
Let us
to the r e n o r m a l i z a k = i.
N o t e that
is a s t a b l e
fixed
"~o:~ ÷ ~ e~.
it is m o r e instead This
Since we
appropriate
is a c o m p l i c a t e d by first
~,
with
~(x)
in a p p l i c a t i o n s
of s t a t i s t i c a l
e
is
We have then
F o r suffilciently small
~e (x) = fe (Ixll+C)'
We shall
~ = i,
Our v a r y i n g
mechanics.
non-
= f(Ixl l+a) is
g > 0,
W i~h
(i + I )t + 0 ( s 2 1 o g s ) , 2 + 0(~ ).
-slogs
The derivative
D~
i__nn Ixl l+S = t
fg~
a~t
~c
(on a B a n a c h
Itl < 2) ,
s pace of f u n c t i o n s
h a s one s i m p l e e i ~ e n v a l u e
a
looking
a r o u n d the f i x e d point. =
to
(this only c h a n g e s
in t h e s e p r o c e e d i n g s ) .
s-expansion
~e'
is true.
and it is s i m p l i f i e d
[Collet-Eckmann-Lanford].
= 1 -
for
stable periods).
contribution
by the
2n
(the case of i n t e r e s t
has a f i x e d p o i n t
1£
of
aspects
we h a v e e x p l a i n e d
are r e l a t e d
a = -~(i),
space,
and t h e n
s > 0
see L a n f o r d ' s
fs(t)
is
(See at the end a b o u t the
for the case
length
~ e~(-ax),
l o o k for f i x e d p o i n t s
~
aspects.
[Collet-Eckmann],
the c o n d i t i o n
= -a
for a f i x e d p o i n t
8.
IP n - P~I
(independently
w h y the a b o v e r e s u l t
(shortest)
l i n e a r map on a f u n c t i o n
THEOREM
and
thus the d i s c r e t e
of the p r o o f w h i c h
s c a l e and does n o t a f f e c t
motivated
exists
We thus w i s h to s t u d y the m a p
w a n t to p r e s e r v e
and small
be the
2k + 4k,...
the f o l l o w i n g .
relates
to a n a l y t i c
aspects
g 2n.
~(x)
k ÷ 2k,
[Collet-Eckmann-Lanford]
a s t a b l e p e r i o d of
study
a
n ÷ ~.
sound basis
tion group analysis
for
are
of this r e g u l a r i t y ) .
mathematically
point
as
otherwise,
~i' ~ 2 ' ' ' ' ' ~ j '''"
from
(4.66920...) -n
observation
m a p s on the i n t e r v a l
Let
limPn = ~ to
R's,X = L
is no o t h e r p e r i o d b e t w e e n
and h e n c e by i n d u c t i o n ,
T h e n we h a v e
The
fp),
striking
sketch those
2k,
a transition
proportional
family
consequences
there
are c o n t i g u o u s .
t a k e s place.
asymptotically
p
and length
for w h i c h
2J - l k ÷ 2Jk
has an even no. of
is, as w e v a r y
analytic ~(g)
> 2,
337
and the r e m a i n d e r of the s p e c t r u m in a small disk around
Note
:
T H E O R E M 9.
~(£ = i) = 4.66920 . . . .
[C-E-L]. The Fig.
3 b e l o w holds for any fixed small
and allows a proof of F e i g e n b a u m ' s o b s e r v a t i o n s p l a c e d by f (t)
~(e))
for a family of functions
a n a l y t i c in
Explanation m a n i f o l d for 4(0)
= 1
:
G.Z 1
Similarly,
has p e r i o d ÷ ~ 6(e)
on
Then for
Wu ZM
~-i(~i)
; and that
~'J(ZM
(x) = f ([xll+e),
x = 0
is a set of ~-n(Zl)
has p e r i o d 2. Now it
4
for w h i c h
is a set of
4
0
has p e r i o d
for w h i c h
0
The t h e o r e m follows then by i n t e r s e c t i n g the curve
, the set of ~
re-
is the u n s t a b l e manifold, W the stable u s is the m a n i f o l d of 45 w i t h 4(1) = 0; since
w i t h these manifolds,
define
e > 0
~(E = i)
W
the m a n i f o l d
2 n+l.
4
(with
It[ < 2.
these are the maps for w h i c h
is easy to see that 4.
0.
~ 4
and due to the fact that
can be l i n e a r i z e d for w h i c h
), j = 0 , i , 2 , . . . ; ~
tinuous i n v a r i a n t measurep
(cf [C-E-L]
~3(0)
~
expands by
[C-E-L]. We can s i m i l a r l y = ~4(O),
cf. T h e o r e m 2 . ~ .
will have an a b s o l u t e l y confor more details).
We end this talk by the o b s e r v a t i o n that the p h e n o m e n o n of F e i g e n b a u m is not r e s t r i c t e d to maps on on
~n
[0,i],
and for flows. A p r o o f of the case for
[Collet-Eckmann-Koch]p
cf Fig.
4.
it occurs for maps ~n
is in p r e p a r a t i o n
338
I l-I
)
~LooO
/,,ix Cl - x ~ )
fi
- .z ~ " )
~o0
Navier
o~
- -~-o~e ~
S~S,
'10
kor~
2
Fi~.
3
~
5
6
7
~
4. E x p e r i m e n t s
(Calculations for N a v i e r - S t o k e s
and Lorenz done by
[Franceschini]
et
al.) Note also the following p r e d i c t i v e c h a r a c t e r of the geometrical picture Fig.
3. If some system
(physical or numerical)
cessive p e r i o d d o u b l i n @ b i f u r c a t i o n s
for values
~o
Shows two suc-
and
~i
of a
339
parameter,
then
one m a y
~j = ~ o ( 6 1 - J - i ) / ( 6 iour"
near
~'3 ~•
and p e r i o d
3-23
•0,803/(6 above),
expect
(~i
(this
~i ~ is not
bifurcations
- ~l-J)/(6
- ~o )/(~
near
- i)
other
- i) + ~i(6
near
- i),
and
"chaotic
behav-
- i) - ~l-J. (~o - ~ ] ) . 0•, 3 3 2 4 1 / ( ~
(6~i - ~o)/(6
- i)
the
3, 2.3,
sequence
- i)
'
~ i - 3 ( ~ ° - ~i )4.3,...
described
etc.
REFERENCES D.
Sin@er.
SIAM
Stable
J. Appl.
Math.
M. M i s i u r e w i c z . interval. D.
~
the
Preprint, J.
IHES,
sur
J.-P.
interval.
J. M i l n o r ,
W.
Eckmann.
dimensional
Commun. Abundance
University
Thurston.
On i t e r a t e d
on the
IHES,
Stein,
unit
P.R.
interval.
J.
165
39,
Quantitative J.
P. C o l l e t ,
Eckmann,
on an i n t e r v a l , P. C o l l e t , archical V.
Franceschini,
turbulence Stat.
V.
Franceschini.
model.
Stat.
Eckmann.
Phys.,
Preprint.
maps
55,
47
par
(1977).
behaviour
for m a p s
on
(1979).
of the
interval
to i n i t i a l
I, ~ .
conditions
finite
limit
Comb.
Theory
(A)
of m a p s
for one
(1979).
On
universality Phys. O.E.
19,
25
of the
sets
for t r a n s -
1-5, 25
(1973).
interVal.
Inven-
for a c l a s s
of n o n - l i n e a r
(1978).
Lanford.
Universal
A renormalization
in s t a t i s t i c a l C.
Tebaldi.
in a 5 - m o d e s
J.
Phys.
continue
properties
of m a p s
to appear.
J.-P.
model
of an
communication.
transformations. J.-P.
maps
(1977).
]3I. P r i v a t e
M. F e i g e n b a u m .
absolument
Bures-sur-Yvette
tiones
Math.
for c e r t a i n
of c h a o t i c
Stein.
On the b i f u r c a t i o n
Lanford
Math.
of G e n e v a
J. G u c k e n h e i m e r .
O.E.
interval.
(1979).
une m e s u r e
dependence
Preprint
M.L.
on the
(1977).
Sensitive
maps.
N. M e t r o p o l i s ,
measures
Bures-sur-Yvette conservant
[0,!].
Princeton
of m a p s
(1978). continuous
Preprint.
Guckenheimer.
formations
260
Applications dx
P. C o l l e t ,
35,
and bifurcations
Absolutely
Preprint
Ruelle.
rapport
orbits
mechanics. Sequences
truncation
group
Lecture
analysis Notes
of i n f i n i t e
of the h i e r -
in Phys.
bifurcations
of the N a v i e r - S t o k e s
Vol. and
equations.
to appear. A Feigenbaum University
sequence of M o d e n a
of b i f u r c a t i o n s (1979).
in the L o r e n z
74.
Remarks
on the
accumulation Oscar
of p e r i o d - d o u b l i n g
E.
Lanford
bifurcations
llI
D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of C a l i f o r n i a B e r k e l e y , C a l i f o r n i a 94720
This
note
tribution,
be r e g a r d e d
giving
conjectures itself,
may
for
a brief
status
problem
on the
~ = i . For a p p r o p r i a t e
Eekmann's
proof
mappings
is to show
= - ~a 9 o g ( - a x )
that
~
has
~ ~
and that except
report
to J.-P.
of the
~ of
con-
Feigenbaum
[-I,I]
into
we d e f i n e
~(x)
The
as an a p p e n d i x
the
derivative
for a s i n g l e
We w i l l
look
of
i-
~
at
positive
for a f i x e d
a fixed
1.4 x
%
has
eigenvalue point
,
%
a
=
-9(1)
point
%
,
2
spectrum larger
which
inside
than
is e v e n
the u n i t
circle
one. and
analytic
so
we w r i t e
%(x)
We t h e n w a n t
to s o l v e
6"7 f = f
~f)(t)
We w i l l and
apply
analytic
convenience;
Newton's in some a disk
The a r g u m e n t i.
Find that
goes
= f(x 2)
method
domain
(by n u m e r i c a l
where
= - ~ f ( [ f ( a 2 t ) ] 2) a
of r a d i u s as
.
in the ~
Banaeh
. The d o m a i n 3 centered
; a : - f(1)
space
of f u n c t i o n s
is c h o s e n
at 1 seems
bounded
on the b a s i s
to be
follows: computation)
an e x p l i c i t
polynomial
of
satisfactory.
fo such
341
n'Jfo is very small. ly speaking,
- fJ
The
fo
I have been using is of degree 20 . Strict-
the c o n s t r u c t i o n of
does not need to be justified;
fo
is not part of the proof and
it is simply an educated guess of
a good a p p r o x i m a t i o n to a fixed point. Show that
2.
by 3.
D~(f o) - ~
is invertible;
denote
~ ( D ~ ( f o) - ~ ) - ~
~ .
Let
r
o
: 2~E
and let
sup {ID25(f)g : gf-fol[ < ro } An e l e m e n t a r y c o m p u t a t i o n shows that if m e t h o d with
if
f o
then Newton's
as a first a p p r o x i m a t i o n converges to a solution
9f:
f ,and
Thus,
if we can find
~
m < i/2~2s
f
r°
is not too big
f
o (< 20)
such that
g
is very small
(< i0 -I0) and
, the existence of a fixed point will
follow from very crude estimates on D ~ ( f ) for f near f . The o latter estimates are s t r a i g h t f o r w a r d to obtain using the explicit formula for
D~(f)
. As a by-product,
this argument
shows that
fo
is an
excellent a p p r o x i m a t i o n to a true fixed point. How do we bound
s
and
~ ? The idea is to construct
strict bounds
which can be e v a l u a t e d by f i n i t e l y m a n y a r i t h m e t i c operations and to do the a r i t h m e t i c on a computer. is more i n t e r e s t i n g than ~hat for
The c o n s t r u c t i o n of the bound for E , but we will discuss only the
latter since it illustrates c l e a n l y the issues Since
involved.
f
is a p o l y n o m i a l of degree 20, ~ ( f ) - f is a polyo o o nomial of degree 800 . We need to estimate its s u p r e m u m over a disk. It can easily be shown that the s u p r e m u m of a p o l y n o m i a l of degree over a disk is m a j o r i z e d by its s u p r e m u m over any number
n
m > 2n points
d i s t r i b u t e d u n i f o r m l y around a circle c o n c e n t r i c with the disk and of a slightly larger radius.
Thus to estimate
g
we have only to evaluate
l~f ° f [ at 1601 p r o p e r l y chosen points in the complex plane (and o this number can be r e d u c e d to 801 by e x p l o i t i n g the r e a l i t y of f ). o A p r o b l e m w h i c h arises i m m e d i a t e l y
in making such estimates air-
tight is that f l o a t i n g point a r i t h m e t i c
on a c o m p u t e r is inherently
342
imprecise because of r o u n d o f f error.
It is possible, however,
to write
programs which a u t o m a t i c a l l y generate strict error bounds on all arithmetic operations as they are done
(so-called interval arithmetic).
This
in fact defines the status of my proof of the existence of a fixed point. All the n e c e s s a r y bounds have been checked ignoring r o u n d o f f error, and the process of w r i t i n g programs which generate error bounds is, I estimate, half finished. of D~(f)
V e r i f i c a t i o n of the spectral properties
should be feasible by extensions of these methods.
AN O V E R V I E W OF S U P E R S Y M M E T R Y AND S U P E R G R A V I T Y J. Scherk
L a b o r a t o i r e de P h y s i q u e Th6orique,
Ecole Normale Sup6rieure
24 rue Lhomond 75231 Parix Cedex 05 FRANCE
ABSTRACT:
Supersymmetry
is defined in terms of t r a n s f o r m a t i o n s which
leave an action invariant and t r a n s f o r m fermions versa.
Graded Lie algebras
used in s u p e r s y m m e t r y models
GLA, the
and vice-
(GLA's) are the basic m a t h e m a t i c a l structures
/ supergravity
(SG). Several GLA's used in SS/SG
as well as their r e p r e s e n t a t i o n s ,
Poincar6
into bosons
are discussed:
(extended) De Sitter GLA, the
GLA. The N = 8 model and its r e l e v a n c e
the
(extended)
(or irrelevance)
(extended) conformal
to physics
are d i s c u s s e d in great detail.
i. S U P E R W O R D D I C T I O N A R Y Let us c o n s i d e r a small and incomplete dictionary of the w o r d "super" as many people ask:
"Why do you call y o u r subject "supersymmetry"?".
If you read the Dictionary, (I) you will find a long list of words starting w i t h super-, many of which are scientific,, some others being of other origins. definitions
A list of some of these words, , is given in Table I.
t o g e t h e r with their
* Les Editions Larousse decline all r e s p o n s i b i l i t i e s for these definitions, except the last one, the others being purely a figment of the author's imagination.
344
TABLE I
Super
: Word of A m e r i c a n origin.
"super"
is a small additional investment
liter versus
~
compared to " r e g u l a r " ( ~ l . 0 8
SP/
:
-nova, -star, -aerodynamics,
Political
In Switzerland
1.04 SF/liter)
Scientific words
-helix
(A.D.N.),
-phosphate,
-conductivity,
-polyamide,
-fluidity,
-sonic, -structure.
:
-powers
(in 1979
: Monaco,
bird unknown to the Antiquity,
Touristic
Liechtenstein),
-phenix
(mythological
of Gallic origin)
:
-Tignes Amer.
Opposite of "regular".
Colloq.
(where is the CERN staff today?)
:
"Gee, it's super Pop Music
!", "Super - Duper"
:
Supertramp Daily life
:
-markets
(Migros,
Coop),
-man
(comic strip by Friedrich
Nietzsche) Poetic touch
:
The French poet Jules SUPERvielIe
(Montevideo 1884 - Paris
1960)
wrote in 1925 a c o l l e c t i o n of poems called "Gravitations" Supersymmetry,
s u p e r g r a v i t y and their offsprings
words to be added to this list. by the words
Supersymmetry
contained in the next table
:
are just a few new
(flat space)
is d e s c r i b e d
345
TABLE II Supersymmetric
t r a n s f o r m a t i o n laws
(SSTL)
formations w h i c h t r a n s f o r m classical,
: a set of c o n t i n u o u s trans-
commuting,
Bose real or complex
integer spin fields into c l a s s i c a l a n t i c o m m u t i n g G r a s s m a n n Fermi variables half integer spin fields and vice versa. S u p e r s y m m e t r i c model
(SM)
: a c l a s s i c a l L a g r a n g i a n field theory,
in flat
space time, w h o s e action is invariant under N = 1 global SSTL Extended supersymmetric model Supersymmetry
(SS)
(ESM)
: the field of
: same as before w i t h N > M ~ ~
i.
which studies SSTL, SM's
and ESM's Superspace zA
=
: a set of c o o r d i n a t e s
(X~
~.) ~
Superfield
X~ : :
(~
: a function
Supermultiplet or e x t e n d e d N >
~MAX
g e n e r a l i z i n g o r d i n a r y space-time
~,...
Bose c o m m u t i n g c o o r d i n a t e s N
~(zA~
) Fermi,
;
anticommuting coordinates
with or w i t h o u t indices.
: an i r r e d u c i b l e r e p r e s e n t a t i o n of S.S.
(simple N = 1
i) in terms of fields.
Matter supermultiplet (
~
~)
: a supermultiplet with
= 1 : vector s u p e r m u l t i p l e t s
~-MAX
; ~sax = 1/2
=i, or 1/2.
: scalar supermulti-
plats) S u p e r s y m m e t r i c Y a n g - M i l l s theories
(SSYM)
: the self i n t e r a c t i n g
(~
0)
field theory based on a v e c t o r supermultiplet. Goldstino
: The~
= 0, ~
= 1/2 fermions a s s o c i a t e d with the spontaneous
b r e a k i n g of S.S. Gluinos
: SU(3)c octets w i t h J = i ~
Superalgebras
: c u s t o m a r i l y called by M 6
~
addicts graded Lie algebras
(GLA's)
A simple example of a s u p e r s y m m e t r i c
field theory is given in the
next table (2) : TABLE III N = i, S.S.Y.M.
A4 ~. a. Spectru/n :
~
b. C o u p l i n g c o n s t a n t
~ ~
= i, real, = 1/2, M a j o r a n a
:
c. I n f i n i t e s i m a l S.S.T.L. :
e.
Lagrangian
parameter
:~
J
= 1/2, constant,
:
:
~=-
~
~p~"
~v£
÷
X K
Majorana
346
If we now go to curved space-time, we have the words r e f e r r i n g to supergravity, w h i c h are defined in Table IV. Simple supergravity, L a g r a n g i a n and t r a n s f o r m a t i o n s
its
laws, are d e s c r i b e d in Table V :(3 ) TABLE IV
S u p e r g r a v i t y models
(S.S.M.)
: S.S.M's in curved space-time,
invariant
under N = 1 local S.S.T.L. Extended supergravity models Supergravity
(E.S.G.M.)
: the field of M ~
: same as before with N >
which studies local,
1
S.S.T.L.'s,
S.G.M.'s and E.S.G.M.'s. Pure s u p e r g r a v i t y multiplets, Gravitino simple
: S.G.M.
or E.S.G.M.
w i t h a self c o u p l i n g
)~
u n c o u p l e d to any m a t t e r super, of d i m e n s i o n M -I.
: the gauge particles of spin 3/2, a s s o c i a t e d with local,
(or extended)
Super Hiqgs effect w h e r e b y w h e n S.S.
S.S.T.L.
: the analog of the Higgs e f f e c t for s u p e r s y m m e t r y is s p o n t a n e o u s l y broken,
"eatsup" a g o l d s t i n o
(or several)
a gravitino
(or several)
and becomesmassive.
TABLE V N = 1 Supergravity a. S p e c t r u m
:
V~
= 2 real field
% b. C o u p l i n g c o n s t a n t
= 3/2 M a j o r a n a field :
c. I n f i n i t e s i m a l S.S.T.L. ~
~
d.S.S.T.L.:
e. L a g r a n g i a n f. D e f i n i t i o n s
~
:
:
= 1/2 ; A - d e p e n d e n t
; Majorana
V~.
~-:
parameter
~ 6
-- ~ % 4 "-
V ~
V ~ ~v~
V'b ~v~ * ~a
-
-~ ~ ~ ~ w f ' - ' ~ ~S ~ w ~ f ~
~w&&
- 6~4-~u~
NOW, we go into the technical facts about SS/SG, crop up. The basic facts of life about Dirac, Weyl, spinors,
are s u m m a r i z e d in Table VI
see ref.
( 4 ).
and m o r e words
and M a j o r a n a
: (For the theorems of this section
347
TABLE VI Clifford
algebra
Irreducible Majorana matrices Theorem
a M.R. - ~
configuration spinor
such that
spinors spinor
Majorana-Weyl in
of the
~
matrices
: 2LD.~j £ /.~ dimensional
(M.R.) : a representation
in which
the
exists =
in even
0,2
matrix
if and only if
Mod
8
~
:
:
:
exist only if a M.R.
exist
= ~
are ~ X real matrices. :
Majorana
a Weyl
~
representation
A spinor
Weyl
{ ~
representation
$
Charge
:
~
~ ~
exists.
~o ~
if
In the M.R.
: Set for e v e n D : ~ D~i~ is such that spinors
:
~*'~ ~
= i. Majorana ~
and
~ ~ o ~ , .. ~D-I
~:
~
spinors ~ ~(~
(~D~)Z= ~
-_ ~_
is both Majorana
~) even if a n d only if
~
repr. C =
S- ~
= O
and Weyl. mod 8
M+W spinors
348
2. S U P E P ~ L G E B R A S The
USED BY SUPERSYMMETRISTS
superalgebras
in s u p e r s y m m e t r y 15 + N 2 B o s e
a r e of
super
S.D.S.A. The are
field
given
S.C.A.
algebr~
theories VII
VII
:
: the
Bose
which
of
supersymmetry.
superconformal
generators)
are
Sitter
generators)
of g e n e r a t o r s
;
as t h e
charges).
the o f f s p r i n g s
of these
superalgebras
:
SUPERALGEBRAS
AND
THEIR
OFFSPRINGS
: Flat
space
Curved S.D.S.A.
: S.S. ~ S.S.Y.M.
space
: S.C.
theories t h e o r i e s (N = 1,2,4)
supergravity
:
C u r v e d s p a c e : e x t e n d e d (N > s u p e r g r a v i t y , w i t h SO(N) g a u g e g r o u p S.P.A.
i) or s i m p l e N = 1 D e
Sitter
:
in D = 6,
Flat space : all supersymmetric i0 : S . S . Y . M . t h e o r y
gauging.
C u r v e d s p a c e : D = 4 : S.G. a n d E . S . G . D = ii • N = 1 supergravity
The most
interesting
supergravity symmetric
models
theory which
Yang Mills
known
to h a v e
is u n s u r p r i s i n g ,
remarkable
fully
l-loop
will
to t h e p r o p a g a t o r finally
for
1 and
one
for any M n ~
coupled
physicist
to N = 4 s u p e r g r a v i t y breaking
of
is t h a t
Nature.
is t h e
the W e y l action,
the g a u g e
Another
superconformal
theory which
group
model
may
is a r b i t r a r y
to d i p o l e
- ~ ghosts
infinite identically
if w h e n
exhibits
to
this
to t h e SO(8)
time
Bose
for
in c u r v e d
sector
4 derivatives
is u s u a l l y
no
supergravity
a n d c a n be f i t t e d
algebra
it is
attractive
of
model
compared
and has and
and
a very
coupled
model
is
one-loop
vanishes
be a w a r e
the G . S . O . while
S.C.A.
gauge The
(finite
s u p e r g r a v i t y ( 8 ), w h o s e
of g r a v i t y
leads
: indeed
of t h e
last model
It is r e n o r m a l i z a b l e ,
~[~]function
of t h e G . S . O ;
application
or t h e N = 4 s u p e r -
This
identically
supersymmetry,
it d o e s ( 7 ) . A n a d v a n t a g e
later
is n e e d e d ) .
Nature
without. SO(N)
s u c h as t h e N = 8
the G . S . O .
, while
(N = I)
cases,
the Fermi-Feynman
the G e l l - M a n n - L o w
2 loops ( 6 ) . This makes
spontaneous
theory
vanish
space
theories
properties.
(no r e n o r m a l i z a t i o n
parts),
in f l a t
model) (~).
convergence
corrections and
be d i s c u s s e d
(G.S.O.
but further,in
finite (5)
models
are the extreme
theory
which
space
used
(S.C.A.:
super-De
; 4N F e r m i
: same number
of c e n t r a l
Those
algebra
; the
generators
(S.P.A.
in the p r e s e n c e
in T a b l e TABLE
skeleton
8N F e r m i
i0 + N(N-I)
Poincar~ except
the
3 types
generators,
algebra(S.D.S.A.: the
are
rejected
includes in the
on these
349
grounds,
as well
The S.C.A.
as on the c l a s s i c a l has no d i m e n s i o n a l
field
theories
based
(S.C.
gravity,
S.S.Y.M.).
ground
of one word.
constant
entering
on it have d i m e n s i o n l e s s The S.D.S.A.
it and thus the
coupling
contains
c o n s t a n t m of the d i m e n s i o n s of a mass. T r a n s l a t i o n s 2 and give a r o t a t i o n times m . The U n i v e r s e d e s c r i b e d thus not a flat universe, universe.
The radius
= ~ units).
The c o n s t a n t m plays
for the g r a v i t i n o the Lagrangian, N ~
field,
where
2, the v e c t o r
minimal
but a De Sitter
of the De Sitter
coupling
and a c o s m o l o g i c a l
K is N e w t o n ' s
fields
such as
constant
~
later) ( 9 )o The SO(3,2)
symmetric
solution puts
a stringent another
Actually, is a c c e p t a b l e
on ~ :
of v i e w
very
space
time
strongly
occurs
the d i m e n s i o n l e s s see m o r e
is the simple,
~ ~ 10 -120
in
with
of such
maximally
and the actual s i z e
is that
of the
!
~ ~
foam p i c t u r e
out,
is flat or n e a r l y
curved
I0-33cm. e n semble
~
=
~0-~'~w
1 and
_2~
,~
of Wheeler,
(101~GeV) 4
Hawking
~
and T o w n s e n d ~I0) . As they point w here
~
(we will
universe
: it is the s p a c e - t i m e f
~
equations,
bound
point
term
In the theories
couple with
~"~
is
than SO(4,1))
N o ~'n~-I (in ~
constant
constant.
~p
and
of the field
by the S.D.S.A.
rather
is
coupling
do not c o m m u t e
the role of a "mass"
relations
universe
(S0(3,2)
universe
constants
one d i m e n s i o n a l
the De Sitter
at d i s t a n c e s
of the order
The p h y s i c a l
space-time
of f l u c t u a t i n g
space
time
theory
has
flat at large d i s t a n c e s
foams
solutions
(icm)
of P l a n c k ' s
but
length
m a y well be a s t a t i s t i c a l
of a r b i t r a r y
sizes
and
topologies. FinallY, have
we have
the d i m e n s i o n
central
charges
the N = 2,
occur
~ =
the N = 2,4, In these
the S.P.A.,
of a mass,
for m a s s i v e
supersymmetric theories,
M~,~ ~n~ space
As O l i v e and
supermultiplets
Yang-Mills
we have
pointed
only
These
with
(for instance solutions
of
theories. 2 and
6 central
are the electric
(Q~
charges.
of the S.P.A.
or for c l a s s i c a l
respectively
out (12),
central
charges.
and m a g n e t i c
charges
G ~)= 0 this
is s u g g e s t i v e
of i d e n t i f y i n g
w i t h P5' P6 in a M i n k o w s k i a n (S = 5, 6 = i) 6 d i m e n s i o n a l Z time w h e r e M 6 = 0. Similarly, in the N 4 m o d e l we have 6
central
~G
charges
~
through holds
or w i t h o u t
1/2 supermultiplet)
In the N = 2 model, the 2 charges (ii and the r e l a t i o n
holds.
with
and r e p r e s e n t a t i o n s
the r e l a t i o n (13) true
These
relations
can be o b t a i n e d
, and these ~4
= ~n ~ 4
can be i d e n t i f i e d , and the m a s s
are no surprise
by d i m e n s i o n a l
as the N =2,4
reduction
with
6 extra-momenta
relation
M io
= 0
; D = 4 theories
from the N = i, D = 6,10
350 S.S.Y.M.
field
theories.
3. R E P R E S E N T A T I O N S We ( S
shall
= 3;
OF T H E
discuss
~ = i),
SUPERPOINCARE
only
the representations
in t h e m a s s l e s s
s h o w t h a t o n l y o n e of t h e decreases is t h u s
the h e l i c i t y
easily
On Table
Q~
by
case,
The
of t h e
in t e r m s
is r e l e v a n t
1/2.
:
(say
particle
S.P.A.
of f i e l d s . Q~
) and
contents
in D = 4 One
that
can it
of a s u p e r m u l t i p l e t
found. VIII,
supersymmetric exist
ALGEBRA
we
free
see t h e p a r t i c l e
field
theories
u p to N = 2, v e c t o r
multiplets
u p to N = 8 a n d h y p e r g r a v i t y
contents
with
~Ax
of N = i,
~ 5/2.
...,
i0
Scalar multiplets
u p to N = 4, suFergravity m u l t i p l e t s
[7~ax = 5/2)
multiplets
u p to N = i0.
TABLE VIII
XN 1 . . . .
,
6
5
2
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
7
8
9
i0
r -
5/2
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
. . . . . . . . . . . . . .
i
1
1
+__
1
11
1
1
1
i
i0
i0
5
6
8
8
45
45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3/2
1 1
i 2
13
14
i 1 .
.
.
1/2
1 1 .
.
.
.
.
.
.
.
.
.
1 2 .
.
.
.
1,1
.
.
.
.
.
.
.
.
.
i14
6
i0
16
28
28
120
120
4i7
4
ii
26
56
56
210
210
Gi2
618
2
i0
i0
70
70
256
256
S0(3)
S0(4)
1 .
.
ii3 .
.
.
.
21211
.
.
3
.
.
4,3
.
.
1
. . . . . . . . . . . . . . . . . . . . C ? ..............
0
2
4121 i
I
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
G
SO(l)
S0(2)
s0(5) S0(6) S0(7) S0(8) S0(9)
SO (I0)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R e p r e s e n t a t i o n s c o n t e n t s of N = i, ..., i0 s u p e r s y m m e t r y w i t h 0 m a s s . Ogievetsky multiplets ( 0m~a, = 3/2) a r e s h o w n u p to N = 4, b u t e x i s t u p to N = 6 ; h y p e r g r a v i t y m u l t i p l e t s ( ~M~x = 5/2) e x i s t f r o m N = 1 to N = i0 b u t
Explicit theories
based
are
shown only
constructions
for N = 9,
are needed
on these multiplets
but no interacting
field
theory
e x i s t s (15) . T h e v e r y m e a n i n g
i0.
to s h o w t h a t
exist.
This
(hypergravity)
of the w o r d
interacting
is t r u e u p based
on
"hyper"~suggests
to
field
~ ' M ~ = 2,
~;= that
5/2 these
H y p e r b o l e : G r e e k d e m a g o g of S a m o s , t h r e a t e n e d t h e c o m i c p o e t s A l c i b i a d e a n d N i c i a s of o s t r a c i s m , w a s t u r n e d i n t o r i d i c u l e b y t h e m a n d w a s h i m s e l f f i n a l l y o s t r a c i z e d in 417 B.C.
351
theories
may
4. W H A T
IS S U P E R A B O U T
The their a)
super prefix
unification
-
-
-
-
Fields
associated
power
and
of d i f f e r e n t
the p r e v i o u s
with
to t h e i r
Internal
and
and bosons
The
these
? theories
is d u e
renormalizability
first
to
properties.
dichotomy
:
- If t h e B o s e
sector
its B o s e
matter pure
(~"
2 loop
same
representation,
symmetries
are
symmetrical
fields
of a S.S.
and
unified.
roles
sources
field
theory
is a l s o
abolished.
is r e n o r m a l i z a b l e ,
in t h a t c a s e t h e S.S. f i e l d t h e o r y is m o r e s e c t o r (16) . Ex. : t h e N = 4 S . S . Y . M . t h e o r y
supersymmetric = 0,
extended
conformal
in t h e
so
theory
- Further
- Non
play
between
are unified
theorems.
space-time
Renormalizability
than
spins
NO-GO
Fermions
is t h e w h o l e
and
SUPERSYMMETRY/GRAVITY
Unification:
overcoming
b)
not exist.
1/2;
S.G.
theories
i) f i e l d s
theories
f i n i t e (17) . S.C.
g r a v i t y (18)
of g r a v i t y
are one
based
o n the
gravity
loop
in i n t e r a c t i o n
with
non renormalizable,
S.P.A.
is a l s o
convergent
(N = i,
...,
renormalizable,
8) but
while are
1
so is
352
5. E X T R A
SPATIAL
The and
S.P.A.
increase
multiplets
DIMENSIONS exists
/V
~)
for any
we m e e t
with
increase
~x
AND
the
the
S,~
(~
limits
= 1/2,
we m e e t
SS/SG
1,2.
limits
= S ÷ ~ ). If we k e e p
/V = 2,4,8
Similarly,
if we k e e p
~) = 6 , 1 0 , 1 1
~
= 4
for the e x i s t e n c e
of
A/ = 1 and
for the
existence
= 10
= 9,
of the
same m u l t i p l e t s (19) (Z0) An example easy
of a S.S.
theory
to p r o v i d e ( 7 ). L e t us take
theory
given
notations,
in T a b l e
in
~)
and
the M W
which
field
Dimensional 5. One
in w h i c h
case
contents
reduction
only
the e x c i t a t i o n s
M ~ )
~.
reduced
=I, i)
=ii
~
= i,
, and
six
but
the
with
may
in the
~)
are A;"
(~ = I .... 6) = 1 .... 4), theory field to
(4)
theory zero,
are c o n s t a n t
in
(of the type
theory
in
~) = 4 is
be s p o n t a n e o u s l y
broken.
5th d i r e c t i o n . ~/
= i,
= 4 theory (4),
(23)
~"
shrink
which
resulting
that
a S.S.
dependence
S.S.
S.S.Y.M.
= i0 in the
S.S.Y.M.
dimensions
~TMA~ = i,
/%/ = 4,
theory(22)reduced
~
= 1
/Z)
= i) is
scalars 4 ~ ( ~
= 4
of the f i e l d s
6
= 8. T h e v e c t o r
spinors
internal
as the m o m e n t u m are
$-~
= 4, ~)
a simple
~/
provided
since
in s t a r t i n g
of the
invariant,
examples
to the
N
) s u r v i v e (21) . The
appears
Well-known
size
or h a v e
supersymmetric
The m a s s
A~
consists
the
of the
invariant,
of the
(S
= 1 and put
to 4 M a j o r a n a
lets
directions,
~2~x,y)~,'~ still
S.S.
/) = 4 a v e c t o r reduces
I)
example
w h i c h "is p o s s i b l e
spinor ~4
is the
the e x t r a
in
the
in
If we k e e p ~
is still
spinors,
= i0 g i v e s
in ~
III.
the t h e o r y
Majorana-Weyl
N
field
to the
~r=
~)
= i0 t h e o r y
and i, N
the
.~'~= 2,
= 8, ~
= 4
theory. In g e n e r a l i z e d is i n t r o d u c e d ,
dimensional
t h e n be g e n e r a l i z e d which from
has
a rank-4
introduced
S.S. but
starting group
5 to 4 d i m e n s i o n s ,
contain
reduction (7) , a phase
as in the K a l u z a - K l e i n
and
of
from
4 mass
t h e o r y (23)
is s p o n t a n e o u s l y the b r e a k i n g
family
of
~
These
~'MM~= 2 t h e o r y
( ~ = 1,...,4)- can be is o b t a i n e d
(~994 = 0). mass
mass
In t h e s e
degeneracy
relations =
for K = 0,
can
(8) (24) . G o i n g
= 8 theories case
following
= 8,
~p
91~
the F e r m i ' B o s e
as the
~
namely
parameters
~xp"m~dependence
The N = 8 theory
/) = 5,
as a s p e c i a l
broken,
is soft
the
invariance,
a 4 parameter
the C.J.
theory.
which theories
is l i f t e d
still
hold
:
o
i, 2, 3. mass
relations
cosmological
constant(25)is
The
of the ~
spectrum
imply
that
still
= 8 theory
the one
fully
loop
finite
is d e s c r i b e d
correction
to the
(but a p p a r e n t l y in the n e x t
non-zero).
section.
353
6. P H Y S I C A L
CONTENTS
OF THE N = 8 M O D E L
In the s p o n t a n e o u s l y complex,
and the theory has t h e r e f o r e
is a c t u a l l y obtained
a local
field
one.
by r e d u c i n g
In the next
with
b r o k e n ~ = 8 theory,
leads
Its gauge
the m e t r i c t e n s o r
section,
we shall
to antigravity,
the q u a n t i z e d
2
~
there are include
15 + 1 gauge
SU(3)C
we o b t a i n
SU(4) ~q
a model
into
= ~.
bosons.
X SU(2)x~(1), which
(electromagnetism)
X U(1)
The e l e c t r i c
SU(3)C
is a u t o m a t i c
:~=V'~
group
associated "shen"). is U(1)
If we set all
is too small
the weak
(strong
X U(1)
the b r e a k i n g
if we set ~
= ~
to
interactions
interactions)
is taken to be°i/3
~
and at zero mass,
SU(4)
Actually,
s
of this v e c t o r
(read
etc..).
degenerate
if we d i s r e g a r d
(antigravity).
charge
~
of the theory
invariance,
X UCI)
~
are
symmetry
field
the p a r t i c l e
letter
symmetry
As a gauge
but
includes
X U(1)
SU(3)&
the only
coordinate is SU(4)
This
from 5 to 4 d i m e n s i o n s
and we shall d e n o t e
If all Wg~ are unequal,
states
see that the c o u p l i n g
•
the s p e c t r u m
symmetry.
is the v e c t o r
field by the E g y p t i a n
(apart from s u p e r s y m m e t r y , the 9W~ = ~ ,
a U(1)
field
all m a s s i v e
of
( ~ = 1,2,3)
for the triplet
;
of
SJ
= 3/2
"graviquarks
of mass
~
. Once
by taking
the p r o d u c t
The m o d e l and
weak
only one e l e c t r o n surprising
to c o m p o s i t e
of mass
3-rn
and one could
bound
states
gravitino
are d e r i v e d
, but no
~ T
tentatively
as well
attribute
as the SU(2)
of
interactions. A more
a theory.
humble
bosons) (~)
and a
C quark
(~
are
(~ (Q = 0)
a triplet
gravitinos
to leptons,
a massless (~-)
just as a m o d e l
: it is 1 and 2 loop quarks,
graviton,
; a d quark
(~
not as finite
and S U ( 3 ) & X
8 massless = - 1/3)
SU(2)
gluons,
X
a
; a %& quark
= 2/3).
particles
of Fayet
triplets,
coupled
; an e l e c t r o n
The exotic
neutral
theory
is to take the mo d e l
it is not too bad
; it includes
photon
gluinos
attitude
As it stands,
(unlike E i n s t e i n U(1)
=3/2singlet
is fixed,
is not too
structure
~
all m a s s e s and c h a r g e s ++ of r e p r e s e n t a t i o n s .
contains
%2 's. This
the family
of mass 9"~ , 0 for the
this
: the sexy quarks
; the a n t i g r a v i t o n
of ~ -gravi (~(0)),
(~ ; Q
quarks
massless
(~, Q
scalar
; massive
= "i/3), gluons
= 1/3)
; the vector
a set of 2
(2octets),
massless
The symbol ~ s u g g e s t e d itself to the author as an i n v e r t e d ~ , to d i s t i n g u i s h it from the p h o t o n but to suggest its s i m i l a r i t y w i t h it. W. N a h m p o i n t e d out to th~ author that this symbol exists at least in two languages. In the C y p r i o t syllabary, it reads "ro". The E g y p t i a n h i e r o g l y p h s include the signs : ~ , p h o n e t i c v a l u e ~ ~ (shess), , p h o n e t i c v a l u e ~ ~ (shen). T h ~l a r e l i n v e r s i O n s of each other and d e p i c t parts of a rope. (W. Nahm, p r i v a t e communication) ++ These symmetries were discovered b y t h e authors ofref. 24 but were not published.
884
singlet scalar p a r t i c l e s m a s s i v e scalar quarks
7.
A N T I G R A V I T Y AND
(2 of them), m a s s i v e sextets and singlet scalars,
("sarks").
SUPERGRAVITY
:
A CRAZY IDEA ?
Let us consider the scattering of two p a r t i c l e s of mass having also a c o u p l i n g to a m a s s l e s s vector field / ~ the e l e c t r o m a g n e t i c potential) the total potential
w i t h charges
is given by
V(e)
:
~j~P
(distinct from
. In the static limit
:
~Kz,.',m
.~ '
r"
a
eln, ~
E~-~' -
']
~
~
A n t i g r a v i t y is defined by the c a n c e l l a t i o n w h i c h would occur if one had s y s t e m a t i c a l l y the r e l a t i o n b e t w e e n where
~
= +i for a particle,
w h i c h is its own antiparticle,
~
and ~r~
- 1 for an antiparticle,
~
M G~
0 for a p a r t i c l e
2
S.G. there is a
, it had to couple m i n i m a l l y to fields with a gauge c o u p l i n g
constant ~
K'~I . The c o u p l i n g was actually w r i t t e n in lowest order in
in ref(27) I~ultiplet
~ =
like the Z °
In 1977, it was guessed (26) that since i n N = vector
:
. Then in 1978, K. Z a c h o s ( 2 8 ) c o u p l e d N = 2 S.G.
(1/2(2),
antigravity.
0(4)) w i t h a mass ~I
and found
In 1979, the s p o n t a n e o u s l y b r o k e n
~
~
=
to the
M~as
well as
= 8 theory was
found (71) (24) and it was d i s c o v e r e d that a vector A ~
coupled to all the
fields of the model with strength l~I = 2 K~v~, a r e l a t i o n true for the 256 states of the model. ~ If there is such an a n t i g r a v i t a t i o n a l
force in Nature (29), and this
is an inescapable c o n s e q u e n c e of s u p e r s y m m e t r y
if tq>l, why don't we
see it ? If we 10ok at the s p o n t a n e o u s l y b r o k e n find a b e g i n n i n g of an answer. b e t w e e n 2 protons the mass,
( ~
but to ~
Suppose we consider the static force
bound states). As the g r a v i t o n couples not to
it sees the total energy of the quarks and thus
the mass of the proton, and gluons.
which is m o s t l y kinetic energy of the quarks
So its c o n t r i b u t i o n to the force is given by
The a n t i g r a v i t o n
~
is coupled to ~ ~
X
m e c h a n i c a l mass of the quark in question, e.m.current.
Hence,
small
K~n and
~
~n
~zr'~ Mp is the
is the c o n s e r v e d
K~r°%(2"mu+'u~d)~Already,
g r a v i t o n c o n t r i b u t i o n is of the order
( ~ 1 0 -4 ) for the ~ and ~ Finally,
where
t~e ~ sees d i r e c t l y the quark m a s s and not the
proton m a s s and its c o n t r i b u t i o n is ~ that the
~q = 8 model, we m a y
~ _~~ p )% ,
we see
w h i c h is
quarks.
if we compute the relative d i f f e r e n c e between the accele-
ration Of a p r o t o n and a neutron, we find that it is given by
"~n~ ~n~-9,mu)/~~
which is even smaller,
d e p e n d i n g upon the
~,~
mass difference. In the limit of exact SU(2) symmetry for strong ~tIn spite of thefactor2, antigravitystilloccursasthereisaalso aBrans-Dickescalar.
355
interactions,
this difference
vanishes
so that antigravity
is in good
shape. If one looks closer however,
one finds that the
leads to serious problems with the equivalence of atomic numbers A I A ' , having with different accelerations
~'protons
towards
~
exchange
principle.
Two atoms
and of charge z e r o
the earth.
fall
The force between
2
such atoms is given by : The negative
term is due to the
~
exchange
; one has
:
So
Ca, If
~,~'
represent
~) M
Ca',~")
the earth, we can safely replace
M___ ~
(]~')
M
by by
~n__u
SO that the a c c e l e r a t i o n
o f ( ~ l ~ ) towards
the earth is given
M~
:
r' The relative difference
~'('~A~
M (Z,A)
in acceleration
~e
of two atoms
is given by
~'(~',A ~)
-
: (e'A-~ A') a..,,, [',',,..(.,,- ~,,-~',~).'~(",,~-"4¢.,,-",.) Mp --I
, (.,,,,-,.,,~) (,4
In the last bracket so that the
~
the dominant
contribution
H
Mp
H~.W we find a ,
~'~" ,',,,
-7
To save antigravity, mass,
presumably
.,.
At the classical If < ~ > # o i s
~,.,,"
I'4
10 -6 which
~
i
nn,~ ~
.
M"
is unacceptable.
one must assume that the
~
acquires
a
This is rather likely
coupled to scalar fields through the action
F""
~ _
.,.n~ ~b*4, level < ~
due to SU(2)
and with<~--i
is given by
through the Higgs mechanism.
since it is universally
4' -_ _ _.,
"|
is given by
~' For
-
,-,.)] ~ c'~,^),., c2,,~)
-
,,<,,,, A~ ~ ¢, I ~
I('%., -
VC~)
~ 0 and
X U(1)
GeV one finds
_-,
:
~
breaking,
~--I0-19GeV.
~
:
K "t'wI~
< ¢'~
one has typically < ~ > ~ - 1 --100GeV This gives to the
~
a
356
Compton wavelength
of the order
of 1 km w h i c h
case w h e r e ~n~ ~ 0 the p o t e n t i a l is given by
knows
is reasonable.
an atom (~; A)
In the
and the earth
:
Usually,
T.
between
vC~,A): ~__& ~E (l- --C~,^~ a ~Mp e ~ p . ~ ~E ~
E
masses
one
are thought
that they d e p e n d
If one could
to be fixed parameters.
upon e x t e r n a l
"heat the vacuum"
"¢~ = 0 m i g h t
conditions
However,
such as the t e m p e r a t u r e
enough , the phase w h e r e
<~>=
0,
be restored.
An a n t i g r a v i t y
device
field of U F O l o g y ( 3 0 ) o r field of m a t h e m a t i c a l
of this
Science
type h o w e v e r
still b e l o n g s
F i c t i o n (31) , and a p p a r e n t l y
to the
not to the
physics.
SUMMARY
i° SUPERWORD
DICTIONARY
2. S U P E R A L G E B R A S
USED BY S U P E R S Y M M E T R I S T S
3, R E P R E S E N T A T I O N S 4. WHAT 5. E X T R A
OF THE S U P E R P O I N C A R E
IS S U P E R A B O U T SPATIAL
5. P H Y S I C A L
CONTEN~S
7. A N T I G R A V I T Y
SUPERGRAVITY
DIMENSIONS
AND
AND
ALGEBRA
?
SUPERSYMMETRY/GRAVITY
OF THE N = 8 M O D E L
SUPERGRAVITY
This formula w o u l d be correct T a k i n g account of its actual size m u l t i p l y i n g the last term in this ~(~ = ~ X "~ ~ tAX - ~X~ appears rather than the d i s t a n c e upper bound on ~t : rrn~'
: A CRAZY
IDEA ?
if the E a r t h was a point like object. leads (for an h o m o g e n e o u s s p h e r e ) to e x p r e s s i o n by ~ ~ -m~ ~ K ) where . The a l t i t u d e from the surface now from the center. This leads to an ~ ~ . ~ meters.
357
REFERENCES
(I) Le L a r o u s s e en 3 volumes 2) A. Salam and J. Strathdee, Phys. Lett. 51B (1974) 353, S. Ferrara and B. Zumino, Nucl. Phys.B79 (1974) 413 3 D.Z. Freedman, S. Ferrara, P. Van Nieuwenhuizen, Phys. Rev. Dl3., 3214 ( 1 9 7 6 ) a n d DI4 (1976) 912 S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335 4 F. Gliozzi, J. Scherk and D. Olive, Phys. Lett. 65B (1976) 282 ; Nucl. Phys. B122 (1977) 253 5 M.T. Grisaru, W. Siegel and M. Rocek, June 1979 p r e p r i n t 6 E.C. Poggio, H.N. Pendleton, Phys. Lett. 72B (1977) 200 D.R.T. Jones, Phys. Lett. 72B (1977) 199 7 J. Scherk and J.H. Schwarz, Phys. Lett. 82B, 1 (1979) 60; Nucl. Phys. B153 (1979) 61 8 S. Ferrara, M. Kaku, P.K. T o w n s e n d and P. Van Nieuwenhuizen, Nucl. Phys. B129 (1977) 125 9 A. Das and D.Z. Freedman, Nucl. Phys. B120 (1977) 221 (i0 J.A. Wheeler, R e l a t i v i t y group and topology, ed. S.S. and C.M. De W i t t (Gordon and Breach, New York 1964) S.W. Hawking, Nucl. Phys. B 144 (1978) 349 S.W. Hawking, D.N. Page and C.N. Pope, Phys. Lett. 86B (1979) 175 P.K. Townsend, Phys. Rev. D 15 (1977) 2793 (ii) E. Witten, D. Olive, Phys. Lett. 78V (1978) 97 (12) D. Olive, Nucl. Phys. B153 (1979) 1 D. Olive, invited talk at the i n t e r n a t i o n a l c o n f e r e n c e on M a t h e m a t i cal physics, E.P.F.L. Lausanne, (19'79) (13) H. Osborn, Phys. Lett. 83B (1979) 321 (14 V.I. O g i e v e t s k y and E. Sokatchev, JETP Lett. 23 (1976) 58 (15 Free spin 5/2 field theories w i t h o u t ghost exist, see for instance F.A. Behrends, J.W. Van Holten, P. Van N i e u w e n h u i z e n , and B. De Witt Phys. Lett. 83B (1979) 188 ; C. Aragone, S. Deser, Ph. Lett. 868 (1979) 161 (16 J. Iliopoulos and B. Zumino, Nucl. Phys.B76 (1974) 310 (17 For a r e v i e w on the finiteness issue in S.G. see : P. Van N i e u w e n h u i z e n and M.T. Grisaru, Proc. of Orbis Scientiae 1977 in High energy physics, Kursunoglu, P e r l m u t t e r and Scott eds. P l e n u m Press, N.Y., LondOn (1977) (18) K. Stelle, Phys. Rev. Dl6 (1977) 953 (19) L. Brink, J. Scherk, J.H. Schwarz, Nucl. Phys. BI21 (1977) 77 (20) W. Nahn~, Nucl. Phys. B135 (1978) 149 (21) E. Cremmer, J. Scherk, Nucl. Phys.Bl03 (1976) 399 (22) E. Cremmer, B. Julia, J. Scherk, Phys. Lett. 76B (1978) 409 (23) E. Cremmer, B. Julia, Phys. Lett. 80B (1978) 48 and LPTENS 79/6 to be p u b l i s h e d in Nucl. Phys. B (24) E. Cremmer, J. Scherk, and J.H. Sehwarz, Phys. Lett. 84B (1979) 83 (25) B. Zumino, Nucl. Phys. B89 (1975) 535 (26) J. Scherk, La R e c h e r c h e (1977)October issue, 878 (27) S. Ferrara, J. Scherk, B. Zumino, Nucl. Phys. BI21 (1977) 393 (28) K. Zachos, Phys. Lett. 76B (1978) 329 PhD Thesis, Caltech, s u b m i t t e d April 17 (1979) (29) J. Scherk, LPTENS 79/17 preprint, (30) UFOlogy, James McCampbell, LC.C.C.N. 73-93488, J o y m a c - H o l l m a n n (1973)82 (31) A.C. Clarke, Profiles of the future, Paris books (1962) 64
SUPERSPACE
ASPECTS
OF
SUPERSYMMETRY
AND
SUPERCRAVITY
S. Ferrara Laboratori Nazionali di Frascati, INFN, Italy, and International Centre for Theoretical Physics, Trieste, Italy.
Abstract Superspace, as an extension of Minkowski space-time with spinning degrees of freedom is reviewed.
Superspace techniques useful for the construction of globally
and locally supersymmetric field theories are discussed.
I.
INTRODUCTION
Supersymmetric field theories I) involve multiplets of fields with different Lorentz transformation laws.
These fields transform into each other under infinite-
simal transformations involving anticommuting Grassmann parameters
~6
which belong
to the ( I3 ,0) e (0,~) self-conjugate representation of the spinor group SL(2,C).
The
simplest collection of fields having the above properties is the so-called chiral (scalar) multiplet 2) (A,B,x,F,G).
(A,B) are scalars, (B,G) pseudoscalars and
X
a spin-~ MaJorana field.
The infinitesimal transformation law of these fields is A -2 iB , H = F + iG and a (twomost conveniently written using complex fields z component) Weyl spinor
Xa
~z = ~ X
(6 = 1,2) :
6
6
,
~X6 = 2 i ( ~ m ) 8 ~
~
z + 2~
~x m
gH = - i ~ X 8xm
H
c ~
We use Weyl spinors in Van der Waerden notations
3)
1 representations of SL(2,C). refer to (3,0) and (0 ,3) s a8 = _eB6 = -~68 ' ~12 = i (same for dotted indices). X* = X& , X~ = X 6 ehiral projections
(1)
,
6
Undotted and dotted indices 68 The spinor metric s satisfies ( ~ ) 6 { = (l'@i)
and
Weyl spinors are related to four-component spinors through the 3(1 + iT 5) •
The transformation laws in (i) realize the simplest representation of the supersyn~netry algebra. bosonic generators
The latter is the 14-dimensional graded Poincar6 algebra with
Pm ' M mn
and spinorial generators
presentation of the Poincare algebra
Q6 ' ~
' the grading re-
359
{Qm,~&) = 2~m. P
[Q ,Pm ] = [ ~ , P m ] = 0
The multiplet in (i) describes scalar and spinor fields. exist as well. invariance.
,
More complicated multiplets
They are used to describe supersymmetric interactions with a gauge This gauge invariance can be associated with an internal symmetry (Yang-
Mills interactions) or with a space-time symmetry (supergravity) and the corresponding (gauge) field multiplets describe massless particle doublets of helicity content (~,i)
3
and (~,2),respectively, for Yang-Mills and supergravity theories. The most natural mathematical framework for the construction and investigation of supersymmetric Lagrangian field theories has been settled by Salam and Strathdee with the introduction of the concept of superspace and superfields.
Superspace is an
extension of ordinary space-time (Minkowski space) to spinning space-time. manifold of superspace has points parametrized by co-ordinates ZM = (xm,8~,~ ~)
,
m = i ..... 4
,
~, ~
Latin indices denote vectors, Greek indices spinor indices, time co-ordinates and
eu ( ~ )
The base
.
x
1,2 . m
(3)
are commuting space-
anticommuting Grassmann variables:
[xm,x n] = [xm,e ~] = {e~,e v} = {e~,N v} = 0 or more symbolically
(4)
[zM,z M' } = 0
From a group-theoretical point of view, superspace may be regarded as the quotient space
G/H
in which
geneous Lorentz group. (Qj)
C
is the graded Pomncare group and
In extended supersymmetry with
N
H
is the homo-
spinorial charges
Qi
i = I,...,N, the Grassmann variables are supplemented with internal s ~ e t r y
indices as well
ei (~})
and superspace is a h + hN
graded manifold.
However, we
shall not consider extended superspace in this review. Supersymmetry transformations are realized as motions in superspace ~x m = - i £ ~
+ i@o~
,
(5) ae ~ = ~
, a~£ =
~
We shall refer to (5) as a supertranslation of the point ZM = (xm,e~,e ~) On the m o t h e r h a n d , an o r d i n a r y t r a n s l a t i o n with parameter a shifts the superco-ordinate as follows:
6 x m = am The c o m p o s i t i o n r u l e
, ~e ~ = o ~
of supertranslations
of two infinitesimal charges of parameters
) ~g~ = ~
(6)
i s o b t a i. n e d b y p e r f o r m i n g t h e c o m m u t a t o r
~i
[S2,~I]Z M = ([~2,61]x~ , [~2,~i] ¢~ , [~2,~i ] ~ ) = (-2i ~i ~ ~2 + 2i E2o ~l
~2
= ' 0 , O)
(7)
360
The composition law (7) reflects the basic antieommutators given by (2).
It is the
manifestation of the fact that an infinitesimal space-time displacement can be obtained in superspace by performing two infinitesimal supertranslations.
If we regard the
supersymmetry algebra as the "square root" of the Pomncare algebra, we could equally say that superspace is the "square root" of Minkowski space-time.
II.
SUPERFIELDS
We can elucidate the action of supersymmetry transformations in superspace giving more precise concepts•
The superspace previously introduced furnishes a re-
presentation of the algebra given in (2) in terms of differential operators. set
of motions defined by (5) and (6) is obtained by the left-action on an element
of the graded Lie group generated by
Qa ' ~&
Pm
L(x,O,O) = exp i(eaQ + ~ a with
The
another
element
of the
group
- xmPm )
(8)
,
G(am,g,~)
o(am,~,~) L(xm,e,e)
=
n(x + a - i~o~ + ieo~,e + ~ , ~ + ~)
,
(9)
where we have used the Hausdorff formula. The infinitesimal generators of this motion are
Pm = i
~
axTM '
Qe = 8 _ ~ _ . ~e ~
~
io~ ~
--
~xTM ' Q~ =
-
8
m + ie ~ qB&
ae ~
~xTM
(i0)
A scalar superfield is a scalar function in superspace
¢,(x',e')
= ¢(x,e)
,
where in the infinitesimal
~+
~¢=
Because
e~e 8 + eBe a = 0
defined over
.
a
B . , D. ~x a ~
i(em~_~m~)
~
~
(ll)
it is equivalent to a finite collection of ordinary fields
Minkowski space.
and 16 Fermi fields.
~.~+
A scalar field unifies into a single object 16 Bose 3)
Covariant derivative can also be introduced
.
~ .+ i ~ N ~ ~e ~
~ 3m
,
~. ~
=
-
s. 3~=
iS 6
m ~___ G~& ~xTM
They have the property that they commute with all the generators Moreover, they satisfy the algebra
(12)
Pm ' Q~ ' ~&
361
The commutation
relations
(super-)torsion
although its (super-)curvature
vierbein
given by (13) show that "flat" superspace has non-vanishing vanishes.
Note that the "flat" (super-)
(we refer early letters to tangent space and late letters to the basic
manifold)
defined through the relation DA =
~ ~Z M
(14)
where DA = ~a ' Da ' "D& '
a
~_E__= a is also non-trivial
a
~X TM ' ~0 ~ '
~Z M
(15)
(16)
aO~
(13).
due to relation
We can introduce different p~rametrizations
of superspace using group ele-
ments 3)
Ll(X,O,8)
= exp(iOQ -ixP)
L2(x,O,O)
= exp(i~Q
exp(i~Q)
(17) '~(2)
and
L(x,O,O)
~(2)
exp(iOQ)
are connected by the relation
L
The
-[xP)
=
LI(X
+ iOd~,O,~)
motion induced by the left-action
=
L2(x - iOdO,OY)
of a group element
(18)
.
G(a,~,~)
on
is given by x' = x + a + 2i0o ~ + iEo ~ m m m m m
,
8' = e + ~
, ~' = 0 + ~
,
(19)
x'
,
O' = e + 4
,
.
(20)
m
= x
Correspondingly,
m
+ a
m
- 2lEd ~ - i~ m
we can introduce
m
~
superfields
¢1(2)
~, = ~ +~
transforming
according to (19),
(20).
From (19),(20),
it follows that type I (or type II) superspace has complex
bosonic
co-ordinates.
The curved space analogue of (19), (20), has been used recently
by Ogievetsky
and Sokatchev
5) in order to have a geometrical
gravity in terms of an axial vector superfield. II superfields
is that the expression of covariant
in these bases.
derivatives
of super-
of type I or
become extremely
simple
For instance
~&
= -
a~
.
on
¢i
and
~.¢ = 0 ¢l(X,0,e)
D
= a
This shows that a solution of the covariant
is simply given by
description
One of the advantages
on
¢2
constraint
(chirality condition)
= ¢l(x,e),
¢(x,e,e)
a o5
(21)
so we get
= ¢l(X
+ ie~,e)
(22)
362
If we expand
%1
in power series of
8
we get
¢l(X,e) = Z(x) + 8mX (x) + 8~8 H(x) and we obtain the scalar multiplet with transformation A superfield
¢
can be taken to be real
,
(23)
laws as given by (i).
¢ = ¢*; in that case we can no
longer impose a condition like (21) and the superfield is essentially In the literature
it has been called a vector superfield
field) and its general expansion
¢(x,e,e)
= c + iex - i~ -
is
i
i
+ ie~
1 - ~ eeSmx~m~ - i Z ~ e t
-
+ 7 e~(M + i N ) - ~ , e e ( M -
e~m~v
irreducible.
(because it contains a vector
i~)
m
+ 21 ~ e
m~
+ ~1 e e ~ ( D + ½ ~ C )
.
We have so f a r c o n f i n e d our c o n s i d e r a t i o n s Lorentz indices.
Such an i n d e x i s i r r e l e v a n t
can he added w i t h o u t m o d i f i c a t i o n s
(2~) to superfields
for the previous considerations
of the previous results.
analysis to more general superfields
w i t h no ( e x t e r n a l )
as follows.
and
We can e x t e n d o u r
A general
(quantum)
superfield can
he written as
¢(x,e) by definition
¢ = ¢(0,0)
H
(x,8) = (0,0)
H = M
is six-dimensional
For the superconformal H = M If
¢
K m, S
is a representation
whole algebra. ¢(x,8)
algebra
mn '
If
X
L-l(x,8)
according to a (representation)
of the point
inn
L(x,e)¢
,
(25)
is the superfield at the (super) origin.
an object which transforms algebra
=
of
.
The latter is
of the stability
(super)
In the case of the algebra given in (2) (Lorentz algebra)
(26)
is sixteen-dimensional.
(27)
2)
'
D
'
H
H
we can induce on
is a generator of
H
¢(x,e)
a representation
we can compute the action of
of the X
on
as follows: [ X , e -T CeT] = e - T [ Y , ¢ ] e T + e - T [ x , ¢ ] e T Y =
dX eXT[T,X]e -xT
,
,
(28)
T = i(e~Q~ + O ~
- x,F)
,
(29)
0 where the infinite chain of commutators which defines of steps due to the O'Raifeartaigh convenient
theorem.
Y
@tops after a finite number
The superfield
for working out tensor products of supersymmetry
multiplication
of representations
representations.
In fact
is merely reduced to superfield multiplications
which is an almost trivial operation. the concept of Berezin integration
formalism is particularly
6)
Such
a multiplication
over anticommuting
in conjunction with
variables
is one of the es-
363
sential ingredients for the construction of supersymmetric field theories. If we label a chiral
left-handed (right-handed) superfield and vector superfield with subscripts
L(R), V , one can check the following properties of sup~rfield multiplications: @L~L = (@@)L ¢~R
'
@RCR = (¢@)R
+ @R@L = (@~)V(SYMM)
'
'
¢~ = @R
'
i(@L~R - @R@L ) = (@@)V(ANTISYMM)'
(3o) CVCV = (¢¢)V
,
DD¢ = (DD¢) R
,
DD¢L = (DD¢~)R Integration of superfields is defined through the Berezin integration recipe: J deiej = gij
(31)
This implies that
I d~e ¢(x,O,O) = CmsT(x) where ~. by
CLAST
means the coefficient of the
e0o0
,
monomial in the
(32) 0
expansion of
It follows that an invariant action constructed out of a superfield
¢
is given
7)
Eq.(25) is the starting point for the construction of a Lagrangian in the superfield formulation of supersymmetric Lagrangian field theories.
III.
GAUGE THEORIES
The most striking application of superspace is in the context of gauge theories. In fact when a gauge invariance principle is imposed in supersymmetry, it turns out that the resulting geometrical structure of superspace manifests some unconventional features with respect to ordinary gauge theories over Minkowski space. is a constrained geometry.
The geometry
This is the case both for the gauging of an internal sym-
metry (Yang-Mills) and for a space-time symmetry (supergravity). first the gauging of an internal symmetry group
G
Let us consider
Then it is known that the
appropriate superfield 8)'9) for the description of the vector potential is a Lie algebra valued vector superfield chiral superfield
A(~A
= 0).
V
and the gauge parameter is a (Lie algebra valued)
The (finite) Yang~Mills gauge transformation on
V
is given as follows: e
V
÷
e
-iA +
e
V
e
iA
,
(34)
and the superfield strength is given by the following spinorial ehiral multiplet 8).
364
Wa = ~a(e
-V D
eV)
(~&W
It is simple to show that under Yang-Mills transformation W The Yang-Mills
W
(35)
transforms
as follows:
e iA
÷ e- i A W
superspace Lagrangian
= 0)
(36)
is given by
Tr(W I WI )
(37)
and the invariant action is obtained by integration
over
It is important that the vector potential the chiral parameter
A
can be understood
superspace w i t h a constrained potentials
VA
qB the gauge parameter
A
FAB
~C
is an unconstrained
V
that Eqs.(39)
ses the fact that
VA
= e-VD
that the kinematical
It turns out
V) the following con-
Wa
eV
(39)
of the curvature
and t h a t
V .
A to be a chiral superfield
(and
W
=~V
constraints
on
~.A = 0 .
Note
as given by (27)), which expres(horizontal)
description of the supersymmetric the curvatures
to lower the spin content of the vector potential
Fem , F~m ,
the vector potential V A
The remaining gauge t r a n s f o r m =
is a pure gauge along the spinorial
We learn from the geometrical
A .
in such a way that the
= F ~ = 0 .
of a chiral superfield
restricts the superfield V
VA
(in terms of
imply that the remaining components
imply
of
(38) must be imposed i0),ii).
is only a function of a real vector superfield ation on
(38)
"vector" superfield
to the previous formulation
are only functions
theory in
in superspace
F 6 = F~
Fmn
and
= DAY B - DBV A + [VA,VB> + TAB V c
straints on the field strength
These constraints
W
If we define (Lie algebra valued) vector
that in order to reduce the number of field components theory is equivalent
d4x d20 .
, the field strength
in terms of a genuine Yang-Mills
geometry.
and field strengths
V
directions.
Yang-Mills theory
given by (39) are crucial in order
from highest
spin-2 to highest spin-i
as required for a consistent description of the theory.
IV.
SUPERGRAVITY
The theory of supergravity
in the superspace approach can be formulated in
close analogy with the case of the Yaug-Mills
supersymmetric
gauge theory.
One can
start with a general affine superspace whose points are parametrized by co-ordinates Z M = (xm,e~).
(Here
p = i,...,4
and
%~
stands for
Under general co-ordinate transformations ZM + Z M + fM(z) fM(z)
is an arbitrary function on superspace and
8~,~ ~ , p = 1,2.)
we have , M
(40)
refers to a (super) world index.
365
At each point in superspace one erects local tangent frames and defines supertetrads (super-Vierbein) V~M(Z)
,
A = (a,~)
,
M(m,~)
(~i)
,
h
and their inverse
Flat indices are raised and lowered with the flat metric
~
of the group of tangent space.
The appropriate group for the superspace formulation
of supergravity turns out to be the Lorentz group. 12)
In the tangent space there are
two invariant tensors :
'
nAB
'
which have no inverse. cannot be defined.
:
,
qAB
(42)
Y~8
Then it follows that in this approach an invertible metric
This is in contrast with the metric approach considered by Nath
and Arnowitt 13) in which the tangent space group is taken to be the orthosymplectic group Osp(4/4) with invariant (invertible) metric
nAB l (<
"'%~J
being a dimensional constant). One can define on an affine space covariant derivatives
in terms of a superconnection
@M = ¢M ab Xab
, which is Lie algebra valued over the
Lorentz Lie algebra. Covariant derivatives with tangent-space indices [~A , ~ B } = -RAB ah Xab - TABC TABC
and
RABab
mA:VA M ~M
obey
~c
are respectively, the supertorsion and the supercurvature.
(44) Torsion
and curvatures satisfy two sets of Bianchi identities which follow from the Jacobi identities for the ~ A ' S [[~A,~B},~C}
+ cyclic = 0
(45)
Up to now the geometry is not sufficiently specified in order to correctly reproduce the supergravity theory.
As first stressed by Wess and Zumino, additional
kinematical constraints 12) must be introduced in order that the super-Vierbein VA M be suitable for the description of the component fields of supergravity.
The correct
constraints to be imposed are the following restrictions on the supertorsion 14). T ~B e= (~,~,y = 1 .... 4,
c ' 2i ¥~6
T T ~ = T C =ab
~bc = 0
a,b,c = 1 ..... 4).
As a consequence of the Bianchi identities (44) these constraints imply that all components of the supercurvature supertorsion
(46)
T Y T Y ~b ' ab
RAB ah
15)
and the remaining components of the
can be expressed in terms of three superfields
366
G ~ (hermitian) , W ~y which satisfy the following differential
(symmetric)
and
R
,
(47)
identities:
(48)
~ w B Y= 0 , Z ~ R = 0 w~=
~
G~ + ~
G~
,
G~
The analogy ~ith the Yang-Mills case is now clear.
(49)
The constraints (46) are the ana-
logue of the eonstraiflts (39) and the three supermultiplets in (47) are the analogue of the multiplet in (35).
Eqs.(48) and (49) are the analogue of
observe that the chirality condition
~@
= 0
~.W
= 0 .
We
is not in general a covariant statement
in curved superspaee because left (right)-handed spinorial covariant derivatives no longer anticommute (see (44)). torsion given in (46) imply (through Bianchi identities). the sup~rfield
¢
However, the kinematical restrictions on the (super-)
TCag = T ~ = T a
= 0
and also
= 0
(~,g,7,~ = 1,2)
Then the chirality constraint is always invariant if
carries only undotted indices.
This is in contrast with flat
superspace where no restriction on the Lorentz structure of a chiral superfield exists• This result has bee~ also derived in the component formalism in Ref.16. The three multiplets given by (47) are the supersymmetric completion of the multiplets containing the Einstein tensors, the Weyl tensor and the curvature scalar. It is important to point out that the torsion constraints also imply constraints on the s~per-Vierbein
VA M
itself.
In a suitable supersyT~metric gauge a solution of these
constraints can be given in terms of a (real) axial superfield v e ~ o ~ index). to ~ e
V . (endowed with a
This superf{eld, proposed 17) as the natural object which is coupled
Su~er~ur~ent superfield of globally supersymmetric theories 18;, also emerges
im:~ natural way in the unconstrained geometrical approach pursued by Ogievetsky and Sok&tchev 5) and in ~ o t h e r similar (although more general) approach considered by Siegel and Gates 19), ~20) Of %he superfield
V
The axial superfield
V .
in supergravity is the analogue
(see Eq.(34)) of supersymmetric Yang-Mills theories.
Other approaches which u s e a
set of kinematical constraints different from
those of Wess and ZuminO (Eq.(46)) have been considered by and Schwarz 2 1 ) a n d J.G. Taylor 22).
Brink, Gell-Mann, Ramond
All these approaches turn out to be equivalent
as far as the description of pure supergravity is concerned.
However, differences
may emerge when coupling to matter multiplets or non-minimal
interactions are also
considered. Finally it is important to point out that the minimal auxiliary fields which emerge in the component formulation of supergravity 23),24) are nothing but the (Lorentz) irreducible parts of
Ta~(x,8_ = 0), in the formulation of Wess-Zumino
25)
Thi~ has recently been used, by means of a technique developed in Ref.21, to •
s
establish the connection between the gauging of the graded Polncare group in ordinary
26) space and (super) co-ordinate transformations in superspace
367
REFERENCES i)
2) 3) 4) 5) 6) 7) 8) 9) i0) ii) 12) iB) 14) 15) 16) 17) 18) 19) 20) 21) 22)
For complementary reviews, see for instance: P. Fayet and S. Ferrara, Phys. Rep. B2C, 249 (1977)~ A. Salam and J, Strathdee, Fortschr. Phys. 26, 57 (1978). J. Wess and B. Zumino, Nucl. Phys. BTO, 39 ~974). For conventions and notations see: S. Ferrara, J. Wess and B. Zumino, Phys. Letters 51B, 239 (1974). A. Salam and J. Strathdee~ Nucl. Phys. B76, 477 (1974). V. Ogievetsky and E. Sokatchev, Phys. Letters 79B, 222 (1978), and Dubna preprint s (1978, 1979). F.A. Berezin, The Method of Second Quantization (Academic Press, New York 1966). A. Salam and J' Strathdee, Phys. Rev. DII, 521 (1975). S. Ferrara and B. Zumino, Nucl. Phys, B79, 413 (1974). A. Salam and J. Strathdee, Phys. Letters 51B, B53 (1974). J. Wess,Topics in Quantum Field Theory and Gau~e Theories, Salamanca, 1977 (Springer-Verlag, Berlin 1978), Voi.81. M.F. Sohnius, Nucl. Phys. B136, 461 (1978). J. Wess and B. Zumino, Phys. Letters 66B, 361 (1977). R. Arnowitt and P. Nath, Phys. Letters 56B, 117 (1975). J. Wess and B. Zumino, Phys. Letters 74B, 51 (1978). R. Grisaru, J. Wess and B. Ztunino, Nuci. Phys. B152, 255 (1979). M. Fischler, Stony Brook preprint (1979). V.I. Ogievetsky and E. Sokatchev, Nucl. Phys. B124, 309 (1978); S. Ferrara and B. Zumino, Nuel. Phys. B134, 301 (1978). S. Ferrara and B. Zumino, Nucl. Phys. B87, 207 (1975). W. Siegel, Nucl. Phys. B142, 301 (1978); W. Siegel and S.J. Gates, Nucl. Phys. B147, 77 (1978). S.J. Gates and W. Siegel, Harvard preprint HUTP-79/A034. L. Brink, M. Gell-Mann, P. Ramond and S.H. Sehwarz, Phys. Letters 74B, 336 (1978); Phys. Letters 76B, 417 (1978). J.G. Taylor, Phys. Letters 78B, 577 (1978)~ 79B, 399 (1978); 80B, 52 (1978); J. Bedding, C. Pickup, J.G. Taylor and S. Dowries-Martin, Phys. Letters 83B, 59
(1979) 23) 24) 25) 26)
S. K. J. S.
Ferrara and P. van Nieuwenhuizen, Stelle and P. West, Phys. Letters Wess and B. Zumino, Phys. Letters Ferrara and P. van Nieuwenhuizen,
Phys. Letters 74B, 333 (1978). 74B, 330 (1978). TgB, 394 (1978). Stony Brook preprint (September 1979).
SYmmETRIES
OF S U P E R G R A V I T Y
Bernard Laboratoire
de P h y s i q u e
INTRODUCTION
the other
analysis
hand,
point of v i e w more
de l'Ecole
PARIS
mathematical more
CEDEX
supersyrmaetry
there
too.
has been
studied m o s t l y
of q u a n t u m powerful
of these m o d e l s
one must
function
theory
supergravity
theories
and e s p e c i a l l y
the n o n - l i n e a r
I shall discuss
the
" ~
and illustrate
Lagrangian
discovered
series"
in(3)
"dimensional
local
theory
field
simply
so-called
for a d i s c u s s i o n
of the
breaking"
i llustra t e s
this.
exceptional conjectures.
symmetry
groups
w h e re
a
of my talk
symmetries.
of locally
charge
supersymmetric supergravity
coupling
of these
charges
only on the to
dimensional
breakdown. on the
(2) reduclon
One finds fields
a new
and the
from d = Ii s u p e r g r a v i t y
the g e n e r a l i z e d
equations
constant).
(CPT invariant)
elaborate
acting
is
After
g r a v i t y (2),
the s u p e r s y m m e t r y
(straightforward)
obtained
We also r e v i e w
subject
their
one obtains
nor symmetry
a non-central
the classical
; furthermore,
s t r u c t u r e (1) to be
ii d i m e n s i o n a l
We shall
systematic-
of s u p e r s y m m e t r y (4) and refer
"standard"
of N = 8 s u p e r g r a v i t y
The
is the g r a v i t a t i o n a l
states.
not i n t r o d u c e m a s s e s
that p r e s e r v e
the
in four dimensions,
type of algebra with example
construction
r e d u c t i o n "(2)
on zero-mass
"spontaneous
w hich does
(~
tools
form of pure E i n s t e i n
it by giving
By standard
act very
~-model
etc...
and
in s u p e r s y m m e t r i c
a Kaehlerian
with c o m p l e x
become m o r e
investigate
analytical
; on
from the second
will
corrections
possesses
functional
theory and algebra
that analysis
to r e v i e w
actions
Sup~rieure
seem to be using
described
mentioning
Normale
05, FRANCE
than group
For instance,
cancellations
a task that r e q u i r e s
a large class
physicists
frequently
; yet one can p r e d i c t
important
ally p o s s i b l e models,
75231
:
On the one hand, and c o m p l e x
JULIA
Th~orique
24, rue Lhomond,
MODELS
of m o t i o n e~uations.
duality
transformations
and the m y s t e r i o u s We c o n c l u d e
with
three
369
I. N = 0 S U P E R G P ~ V I T Y
: E I N S T E I N ' S G R A V I T A T I O N AS A NON L I N E A R
N d e s i g n a t e s the number of spinor charges, N ~ 1 supergravity th~o~i~
general r e l a t i v i t y can be d e s c r i b e d by two
are the metric
~
it h a p p e n e d that studying
~h~d ~o~enew light on the N -- 0 cage
is the only e s t a b l i s h e d theory of the family).
in the m e t r i c f o r m a l i s m
~-MODEL
(which
It is w e l l - k n o w n that
(locally)
(in first order form)
e q u i v a l e n t formal~i~s :
the independent v a r i a b l e s
and a symmetric c o n n e c t i o n
f ~
and the action
%
reads
:
In the m o v i n g frame f o r m a l i s m £~
(Weyl) the v a r i a b l e s are a m o v i n g frame
and a Lorentz c o n n e c t i o n
~
u)~
and the action is : y
The s e c o n d
formalism
only
the
under
example)
are
Lorentz
different
&0~ ~ ~ and " ~ V fields;
( /~
the
group, from
and its of
global the
of
fermions
%~
6~(16
(i0 numbers)
transforming
properties
first
one.
can be e l i m i n a t e d a l g e b r a i c a l l y
(2) as
Lorentz
introduction
those
but how comes it that
of freedom S0(1,3)
allows
numbers)
In
(topology both
: they are a u x i l i a r y
d e s c r i b e s as m a n y degrees
? The answer lies in the additional
invariance w h i c h identifies all
~ = any Lorentz transformation),
( 6~ ~ ~
~
)'s
i.e. the p r o p a g a t i n g fields are
e x t r a c t e d from a field w h i c h takes values in the coset space SO(3,1)
for
cases,
~(#,~)/
(2); the symmetric p a r a m e t r i z a t i o n of this coset space is just
the m e t r i c field of signature generalized
~-model
(+ - -- -- ). One calls such a theory a
by o p p o s i t i o n to usual
~-(or chiral)
models for
w h i c h the field is a group element.
II. FROM GLOBAL TO LOCAL S U P E R S Y M M E T R Y
i) Global a l g e b r a s
:
:
We shall r e s t r i c t o u r s e l v e s to a d i s c u s s i o n of the s u p e r s y m m e t r y algebras
; they can be e x p o n e n t i a t e d by using a n t i c o m m u t i n g p a r a m e t e r s
to form Lie groups but only i n f i n i t e s i m a l t r a n s f o r m a t i o n s will be n e c e s s a r y for what follows.
The so-called SO(N)
superslnametry theories
370
have invariant actions under the Poincar@ group times the internal group SO(N) are
times a N-vector of spinor charges whose anticommutation
:
6 , ~~
= Majorana spinor indices i, ..., N ; ~( = 0, ..., 3
The "little algebra" be discussed suggests °f
for fixed m o m e n t u m
Lorentz ~
=
the only non-vanishing ~ %~ s
Choosing
matrices
r
with
Z
= 0 here)
can
: ]~>
singlet,
' ]~ >
( w
, 0
are
,
so, assuming
' I~ >
is nothing butl-%> 6 ~ e c•4 e f ~ generally ~, 6 ~ c d e ~
~ ~
states of helicity + $
algebra
of the
field
--9~ --~
(the would-be
for N = 8 we can check :
the CPT conjugate
and - X
aim at a local covariant
the spin
supersymmetry
at most one spin-two
~/* ~ ~ ) - -- ~
,
, and so on, with anti-
forbids N to be larger than 8. Actually
that CPT is built into thensupersymmetry
~ )
• =
sets of indices because of the anticommutativity Thus requiring
~I 0
o
one finds the global
' {~ >
the conventions
~ff = [ "~
~ =
lower helicity by one half unit,
among themselves. graviton)
~
and for
antico~utators
two state to be a SO(N) symmetric
°f the
f3: I
spinor reads
multiplets
( ~
in more detail (5), it fixes the spectrum of states and
(5) f°r a r e p r e s e n t a t i ° n
~
index.
~
the field contents of the Lagrangian.
a Majorana
The
rules
the I-%> ~#¢~e#~{" state
state to
I+ ~; more
~ • ~ k~ )
, exchanges
. This is a necessary
condition
field theory with usual connection
if we
between
spin and statistics. We must now assign fields to these states for spin 0 and 1/2 but for higher invariances p
~)
~
fieldsb
frame field
2) Gauging
N(N-I)
U(1)
=
6~
supersymmetry
is to introduce
groups,
N Rarita-
action for the m o v i n g
- ~
:
in the global
(non-propagating)
supersymmetry fields
case
one must
introduce
in order to realize the supersymmetry
algebra
linearly,in
complex
scalar field one needs a complex auxiliary
also implies
gauge i.e. vectors
2
(~r~'. and the linearized Einstein
K~
Already auxiliary
tensors,
.... The common procedure
at first abelian vector fields gaugin~ Schwinger
this is straighforward
spins we must introduce
if we insist on having conventional
and not bispinors
:
the simplest example of a M a j o r a n a
spinor plus a
scalar field.
that the algebra closes on the classical
This
fields without
using the equations of m o t i o n and that it does not depend On the form
371
of the action.
In the local s u p e r s y m m e t r y
(supergravity)
case, one does
not yet k n o w the a u x i l i a r y fields for N ~ 3, one m u s t c o n s e q u e n t l y be satisfied w i t h an algebra that depends on the action and that closes only on shell. Clearly,
the spin 3/2 part of the N
~6
gauge fields for the N s u p e r s y m m e t r y charges,
fields can serve as the idea is to couple
them to the s u p e r s y m m e t r y currents with a c o u p l i n g c o n s t a n t thenglobal
K
but
s u p e r s y m m e t r y implies a c o u p l i n g of the e n e r g y - m o m e n t u m tensor
to the metric
field
~
. These changes in the action p a r t i a l l y
d e s t r o y the s u p e r s y m m e t r y invariance w h i c h m u s t then be r e s t o r e d order by order in ~
by m o d i f y i n g both the action and the t r a n s f o r m a t i o n laws.
The result is a theory of g r a v i t a t i o n having also a local super -~, symmetry invariance.
Let us detail the algebra for N = 1 s u p e r g r a v i t y
:
the c o m m u t a t o r s of two s u p e r s y m m e t r y t r a n s f o r m a t i o n s w i t h p a r a m e t e r s ~(~)
and
~%(~)
displacement
are sums of a general c o o r d i n a t e t r a n s f o r m a t i o n with
~ ~(x)=
~
~
~[~)
~
~
(x)
, a local Lorentz
t r a n s f o r m a t i o n with field d e p e n d e n t p a r a m e t e r
~W~
= ~ ~
and a f i el d d e p e n d e n t s u p e r s y m m e t r y t r a n s f o r m a t i o n - w i t h p a r a m e t e r £3 = - K ~
. A d d i n g the a u x i l i a r y fields
5,
f , 8~
, one gets an
algebra that closes w i t h o u t using the e q u a t i o n s of m o t i o n and the only modification rotation
is a ~ew value for the p a r a m e t e r of the local Lorentz
:
(we used the c o n v e n t i o n s of
(7) in this last formula).
D i m e n s i o n a l analysis reveals that bosonic fields = ;~ K --~
, K ~
(times
(dimension i)
K%
,
K ~ K.
to ii d i m e n s i o n s
e
I
a p p e a r i n g in E i n s t e i n ' s
) and d e d u c i n g from gauge invariance that only
appears,
functions of the scalar fields (6) . By going
the authors of
which can appear non p o l y n o m i a l l y
(3) avoided the p r o b l e m of scalar fields (in any space-time dimension).
N = 1 s u p e r g r a v i t y in ii d i m e n s i o n s
:
Inspired by the fermionic dual m o d e l d i m e n s i o n s and its c o n j e c t u r e d
s p e c t r u m of states in i0
supersymmetry,
of three gauge fields in ii d i m e n s i o n s M a j o r a n a vector-spinor)
K [~
the only p o t e n t i a l l y t r o u b l e s o m e terms that could
appear are non p o l y n o m i a l
3
~
have d i m e n s i o n 1/2 and can appear at m o s t to the fourth power.
T h u s ~ r e s u m m i n g the spin 2 terms to have only action
:
are d i m e n s i o n l e s s as fermionic c o m b i n a t i o n s
~ MN~
and a m o v i n g frame
£
one is led to a m u l t i p l e t ' ~M ~ A
(32 c o m p o n e n t
. The d i m e n s i o n a l
372 r e d u c t i o n (2) of N = 1 s u p e r g r a v i t y from ii to i0 d i m e n s i o n s to the small slope limit of the dual model(3); d i m e n s i o n s leads to N = 8 supergravity(8!
should lead
its r e d u c t i o n to 4
and to 5 d i m e n s i o n s
it gives
another i n t e r e s t i n g theory (4). The iterative c o n s t r u c t i o n stops after a finite number of steps(in Ii d i m e n s i o n s ) a n d one obtains the f o l l o w i n g Lagrangian
:
L:
_
- [ ~g
z
~.Kz
%
M
19Z
+ ~ 2 - ~- 8
~ c°
F F M# q s~e~
'
^
+
The last term does not v i o l a t e the
(generalized)
the action and is topological in character. structure is abelian 55 U(1)'s
:
; Fm~eq
= 4 t ~m
8 NP~]
~NP~ ~ ~NP~ + ~[~ A@~ 3
.
gauge invariance of
The g e n e r a l i z e d gauge is invariant under
It does not commute with space-
time transformations.
The symmetries of this theory w h i c h have been
d i s c o v e r e d so far are
: 1 s u p e r s y m m e t r y spinor of 32 M a j o r a n a charges, ~Z
subgroup,
We shall now discuss
and local L o r e n t z i n v a r i a n c e SO(I,10).
(~I
~)
general c o o r d i n a t e t r a n s f o r m a t i o n s c o n t a i n i n g
as a
the fate of this action under d i m e n s i o n a l reductions.
III. S Y M M E T R I E S AS SIGNATURES FOR E X T R A - D I M E N S I O N S
1
Simple d i m e n s i o n a l r e d u c t i o n We refer to
:
(2) for details,
but the p r i n c i p l e of standard
d i m e n s i o n a l r e d u c t i o n is quite simple. T M~""
:
One starts w i t h tensor fields
in 4 + p d i m e n s i o n s d e p e n d i n g a priori on the p internal
c o o r d i n a t e s as well as on X ~
~=
0,
.... ,3, the space-time coordinates.
If the o r i g i n a l L a g r a n g i a n is hermitian,
it is a c o n s i s t e n t u n i t a r y
t r u n c a t i o n to r e s t r i c t oneself to fields that are i n d e p e n d e n t of the internal coordinates.
One can assume that the e x t r a - d i m e n s i o n s curl up
to form an internal space of v e r y small radius c o m p a r e d to usual Compton wavelengths
; if it is an hypertorus,
Fourier analysis
i n t e r n a l d i m e n s i o n s leads to a tower of p a r t i c l e s of i n c r e a s i n g d i m e n s i o n a l ) m a s s e s for each %
~
~
4~%
Z
(4+p) d i m e n s i o n a l
field
:
( for a scalar fiel~.
in the (four
373
The
low energy
implies
truncation
a global
SO(p)
So, part of the s y m m e t r y be b l a m e d
on a h i g h e r
extra-dimensions recurrences
can only come
in
share
this
~ ~'s and
(8) resp.
yet strong
from the o b s e r v a t i o n
theory
(4)). could
evidence
for
of the very m a s s i v e
symmetry).
this type of d i m e n s i o n a l
reduction
by the SO(8)
the field c o n t e n t s
in four
:
Number of d e g r e e s of f r e e d o m
ii dimensions
F i eld
£a
off
ancestor,
e x a m p l e (8) . F i r s t of all,
is familiar
all non zero
resp. SO(6)
g r o u p of the four d i m e n s i o n a l
(which do not
supergravity
to ignore
(SO(7)
dimensional
Let us i l l u s t r a t e
dimensions
amounts
symmetry
4 dimensions e(
I0-~
shell
7 . 8~
= 6"
o, l
•
Z~ e~
~ (~-~ :3)
2.
on shell
,~.)
z~
+
Field off
shell
0,, ~l~vi )
!'I6"~_~'5 +~1-1 = t2.0
#Z
on shell
+ 4Z
0~' I
Field
JC~"e
OLJ I
off shell
1_z~r
on shell
(A,M)
; ( W , ~ ) ; (a,i)
i nterna l
(flat resp.
splitting
are
indices
curved)
of a 32 c o m p o n e n t
the e x p e c t e d
in ii dimensions,
and a' spinor
8 spin 3/2 fields,
takes
8 values
and comes
into 8 4 ~ o m p o n e n t
28 vectors,
4 dimensions or
56 spin
ones.
from the
We find
i/2's and 70 scalars
of N = 8 supergravity. Secondiy, mentioned residue
S0(7)
of the o r i g i n a l
the e x p e c t e d work
we a priori
global
global
four dimensions.
(non compact)
2) D u a l i t i e s
fields
The p r o b l e m
by the p r e s e n c e was
of motion,
than the above
times
~L~0~
as a
in (8) was to restore
: it r e q u i r e d
of 8 s u p e r s y m m e t r y
the d i s c o v e r y namely
local
of a m u c h SU(8)
some hard charges
larger
times
in
symmetry
the global
E7(+7).
and first order
It is i m p o r t a n t vector
symmetry
S0(7)
in four d i m e n s i o n s
The o u t c o m e
g roup of the e q u a t i o n s
local
invariances.
SO(8)
strongly motivated
have a larger
: we have
to note
are all a b e l i a n
formalism
that
:
in our v e r s i o n
; furthermore,
of supergravity,
they have
no m i n i m a l
the
374 coupling to matter fields, but only Pauli-type couplings field strengths),
fields look very much like Bianchi identities,
G
=
0
(via their
consequently the equations of motion of the vector
and
linear coupling terms
%~
F ~'=
N
~Z~T
0
where
they are respectively & ~w =
~- ~ V
:
+ non-
Fr~ The manifest SO(N)
invariance of the action of extended supersymmetric
models may sometimes be extended to an SU(N) or U(N) equations of motion.
invariance of the
For real fields a complex invariance is possible
only if one can find a real representation of the imaginary unit i namely
[~ "0~]
for scalars and pseudoscalars, A
spinors, and
~ ~
e~ ~v
~(ff
for vector field strengths
( r e a l ) f o r Majorana
(if one neglects the
couplings) more~ precisely it is the rotation between each ~v
F ~/
• The canonical Lagrangian is manifestlydissymetric
involves only the
~ ~ potentials
for F ~ =
~
~[~ ~ v]
and its
as it . In (8) we
proposed a first order formalism with constraints where the variables are more numerous. potentials
One starts off with potentials
~°~ ~'5 as well as tensors
F
;I/4v and
~'~--/s and dual
~_'~y , related by a
constraint which can be solved to give one as a function of the other. Both the constraint and the first order action are symmetric under SU(8)
local and E 7 global transformations whereas the second order
(usual) action is only invariant under SO(8)
local and
5 ~ ~,~)global.
The dissymmetry at the Lagrangian level stems from the elimination of only one half of the potentials
; the usual Lagrangian comes by elimin-
ating all pseudo-vector potentials
~ !
/ but some other choices of 28
;%
vector fields among the 56
~f~ 's and
5~
's would lead to new actions.
One example of this arbitrariness has been studied in detail the case of N = 4 supergravity
: it is
one form of which has an invariant
Lagrangian under SU(4) (and not only the equations of motion which do not change)(9) study possible
So much for the symmetric theory of "broken" supersymmetries
(8). Let us now
also obtained from higher
dimensions. 3) "Spontaneous breakdown"
of extended supersymmetries
:
The whole idea of spontaneous symmetry breaking is to preserve nice features like renormalizability or predictive power,and yet account for states which do not transform linearly under the symmetry. metry must be broken. in practice,
Supersym-
One does not observe Fermi-Bose mass degeneracies
but a spontaneous breakdown is hard to achieve,
in the case of extended supersymmetry.
especially
A general technique for breaking
375
usual in
supersyn~etry
(i0) u s i n g
has been put forward
a modified
dimensional
by J. Scherk
reduction,
and J. Schwarz
It p r e s e r v e s
a modified
algebra. Let us first discuss from 4 to 3 dimensions. global
invariance,
the former more
chiral
chiral
requirement
general
ansatz
scale)
the m o v i n g
violating
frame
is b r o k e n
not the t r a d i t i o n a l has an i n v a r i a n c e
the algebra
Before
that by this arbitrary
added
an extra p a r a m e t e r
form of b r e a k i n g
: the
which
exhibits
USp(8)
local.
flat
parameters
power
broken"N of
also a large
groups
the m e t h o d
of
,usual
breaking
charge
we m u s t realize
by extra
m u s t be checked
four m a s s
: E6
They
(÷ 6 )
again.
parameters started
in from
to 5 d i m e n s i o n s global
that,more
(2 ~
charge.
in the limit
d' ~
Ell_d, (ii - d')
Finally
the first order
term is finite (4).
only
a little
of E. C r e m m e r
to the rank of the m a x i m a l
and
w a y to see
"electric"
supergravity
they o b t a i n e d
still symmetric
broken
obvious
= 8 supergravity.
to d' d i m e n s i o n s
after
as in the
is a n o n - c e n t r a l
finiteness
of i n v a r i a n c e s (10)
Lagrangian
the
is r e d u c e d
g r o up
it is
and
superalgebra
invariance
in this case USp(8).
symmetry
3
; indeed
of N = 8 supergravity,
ii d i m e n s i o n a l
of the action,
cosmological
Themost
to i n t r o d u c e
It is an o b s e r v a t i o n
equal
"broken"
charges
and that q u a n t u m
reduction
global
generalizing
space
et al. m a n a g e d
d = ii s u p e r g r a v i t y
exceptional
still
to the theory,
breaking
is to note that
to the example
"spontaneously
w hich
of ~
is s p o n t a n e o u s l y
of the vacuum.
trick p r e d i c t i v e
the d i m e n s i o n a l
reducing
to be i n d e p e n d e n t
symmetry
a global
parameters
One
(the lowest mass
J
of s p o n t a n e o u s
has c h a n g e d
going
by a
only up to
fourth dimension.
but it is not the same i n v a r i a n c e
(4) E. C r e m m e r
a modified
fields
only one period
acts on the 2 s u p e r s y m m e t r y
One can r e c o v e r m = 0.
one can replace
in 3 dimensions.
is not an i n v a r i a n c e
operator
In
the c l o s e d
; it is this new i n v a r i a n c e
which
reduced some
on the fourth c o o r d i n a t e
of the Fermi
is s u p p o s e d
we seem to have the spirit
theory has
in our example,
~3
Here,
that
along
the theory by i n t r o d u c i n g
supersymmetry
case
invariance
of N = 1 s u p e r g r a v i t y
dimensional
of i n d e p e n d e n c e
:
Because
example
with periodicity
transformations
truncates
their
If the h i g h e r
times
generally, 8) leads
to
! Then
a number of m a s s
compact
g r o u p of i n v a r i a n c e s
it is e n c o u r a g i n g
quantum
correction
that even
to the
376
IV P H E N O M E N O L O G Y All lead
through
28 U ( 1 ) ' s
t e r m and
into
scalar
dualities be
open
exhibit
dimensional the
action
?
2) A r e
there
non-linear is a fact is the
~-model
full
theory
3) E s p e c i a l l y of a d y n a m i c a l local. degrees
Can
REFERENCES
? Could
spin
for e x a m p l e to b r i n g
then
urgently
topology.
We
fields.
; there
cosmological
rate,
generalizing
coupling
duality more
shall
the
is a
in a l a r g e
(II) • A t any
expects
3/2
be to t r a n s f o r m
zero m i n i m a l
structure
structure
it e x p l a i n
fermionic
70 s c a l a r s
term
of the
to be n o n - l o c a l .
group
theory,
conclude
by p r o p o s i n g
for
corrections
the
action
the
I and
E7/SU(8)
it ;
?
is the p r o b l e m
dimensions
the g e n e r a l i z e d
space
model
group
supergravity
in s e c t i o n
coset
63 a u x i l i a r y
in four
symmetry
ii d i m e n s i o n a l
super-coset
for p h e n o m e n o l o g y term
it to s i m p l i f y
? We m e n t i o n e d
of E i n s t e i n ' s parametrize
and u s e
the e x c e p t i o n a l
in the
symmetries
a supersymmetric
kinetic
quantum
: one
topological
important
of f r e e d o m
would
and
:
of the
the
of n o n
a 12 d i m e n s i o n a l
any more
that
seem
and m o s t
algebraic
reduction
nature
to 8 m a s s i v e
SO(8)
:
by the a b o v e m e t h o d ,
self-interactions
to be d o n e
questions
i) C a n one
group
this w o u l d
in the c a s e
and
be b r o k e n
phenomenology
interesting
remains
differential
three
(i)
because
FOR N = 8 SUPERGRAVITY
mechanism
towards
a non abelian
there
might
Much
could
super-Higgs step
negative
the a b o v e
and
the
crucial
difficulty
more
3 CONJECTURES
8 supersymmetries
Another
vectors
AND
fields
of a p p e a r a n c e gauging
introduce
SU(8)
these
new
?
:
B. Z u m i n o - C E R N p r e p r i n t TH 2733 See also s u b h a r m o n i c f u n c t i o n s a p p e a r i n g in m a t t e r c o u p l i n g s to N = 1 supergravity : E. C r e m m e r et al., Nucl. Phys. B147 (1979) 105 (2) B. J u l i a - E x t r a - d i m e n s i o n s : r e c e n t p r o g r e s s u s i n g old ideas, L P T E N S p r e p r i n t 79/15 and r e f e r e n c e s t h e r e i n . (3) E. C r e m m e r , B. J u l i a and J. Scherk, Phys. Lett. 76B (1978) 409 (4) E. C r e m m e r , J. S c h e r k and J.H. S c h w a r z , Phys. Lett. 84B (1979) 83 and r e f e r e n c e s t h e r e i n . (5) D. F r e e d m a n , L e c t u r e at the 1978 C a r g @ s e I n s t i t u t e on G r a v i t a t i o n and r e f e r e n c e s t h e r e i n . (6) S. F e r r a r a et al., Nucl. Phys. BII7 (1976) 333 (7) P. V a n N i e n w e n h u i z e n , L e c t u r e s at the 1978 C a r g ~ s e I n s t i t u t e on Gravitation . (8) E. C r e m m e r and B. Julia, Phys. Lett. 80B (1978) 48 and Nucl. Phys. B (1979) to a p p e a r . (9) E. C r e m m e r et al., Phys. Lett. 74B (1978) 61 (10) J. S c h e r k and J.H. Schwarz, Phys. Lett. 82B (1979) 60 (ii) J. Scherk, L e c t u r e s at the 1978 C a r g ~ s e I n s t i t u t e on G r a v i t a t i o n
A REVIEW OF DERIVATIONS Derek W. Robinson Department of Pure Mathematics University of New South Wales P.O. Box i, Kensington, Australia
Abstract;
We give a general review of progress in the study of (unbounded)
deriva-
tions with particular emphasis on criteria for the derivations to be generators of continuous groups of *-automorphisms
378
O. Introduction
A symmetric derivation gebra D(~),
~ of a C*-algebra ~ L i s
a linear operator from a *-subal-
the domain of 6, into O~ which satisfies the derivation property 6(AB) = ~(A)B + A6(B)
,
A,B E D(6)
and the symmetry property ~(A*) = 6(A)*
,
A E D(~) •
Since all the derivations we consider are symmetric we consistently
omit the quali-
\
fying adjective.
Note that the range R(6) of a derivation ~ is a subspace of 0"~ but
it is not generally a subalgebra. Derivations of *~automorphisms.
are of interest because they arise as generators of continuous (Recall that such groups are automatically
t E ~ --+T E Aut(0~) denote a one-parameter t assume that T is continuous in some topology.
isometric)
group of *-automorphisms The generator
groups
Let T;
of 0~ and
~ of T is defined as the
linear operator 6; A ~ - + 6 ( A )
= lim (Tt(A) - A)/t t+ 0
where the limit is taken in the topology in which T is continuous and the domain D(6) of 6 is defined to be the set of A E 0 ~ f o r be
constructed
which the limit exists.
from 6 and formally one has T
t
Conversely T can
= exp(t6) but some care has to be exer-
cised in the definition of the exponential. There are various cases of interest.
One can consider C*-algebras
gebras and there are a variety of possible types of continuity
for T.
or W*-al-
For simplicity
we restrict our attention to three cases.
Norm Continuity The group T is norm continuous
if
lITt - III = sup IITt(A) - AII/IIAII --+ 0 A C 0~ t =0 Although norm continuous
automorphism
structure they are rather simplistic cally bounded
(see Section I).
groups do possess much interesting algebraic
in the sense that their generators
are automati-
In this respect the next form of continuity
interesting.
Strong Continuity The group T is stron$1y continuous
if
flit(A) -All --+ 0 t=0
is more
379
for all A 6 ~ L .
Such groups are called C0-groups.
In this context there are two relevant and not at all evident remarks. Remark
i.
Kallman has proved that each strongly continuous one-parameter group of
*-automorphisms of a W*-algebra 0~ is automatically norm continuous (In fact Kallman needed a countability hypothesis on O~ but this was eliminated by Elliott).
Thus
strongly continuous groups are only of interest in the C*-algebra context. Remark 2. Let T be a one-parameter group of mappings on a Banach space X.
There is a
classic theorem of Yosida which states that strong continuity of T is equivalent to weak, or u(X,X*)-, continuity, i.e., is equivalent to the conditions f(TtA) --+ f(A) t=0 for all A E X and all f E X*. The third kind of continuity, U-weak, or U(~I , (YL,)-continuity,
is only of in-
terest for W*-algebras O~ because it depends upon the existence of a pre-dual 01.,. (Recall that a Banach space X is said to have a predual satisfying X = (X,)*.
X, if X, is a Banach space
Moreover Sakai has proved that a C*-algebra has a predual if
and only if it is a W*-algebra).
u-weak Continuity The group T is a u-weakly (O( 6l, O-[,)-) continuous group of automorphism of the W*-algebra ~I if f(Tt(A)) --+ f(A) t=0 for all A E ~[ and f ~ Remark 3.
0~,.
These groups are called C~-groups.
It is possible to study properties intermediate to strong, or O( 61, 0l*)-,
continuity and o ( O q , 01,)-continuity.
For example if F is any subspace of the dual
~L* then one can examine O(O~F)-continuous groups, i.e., the T for which f(Tt(A)) --+ f(A) t=0 for all A E 6 ~
and f E F.
restrict the possible F.
To obtain an interesting theory it is however necessary to This general kind of continuity has been discussed by Bra-
tteli and Robinson. Thus the relevant cases are summarized in the following chart Continuit Norm Algebra
C*- ~
Strong
u-weak
~
X
380
References;
The text of this review is largely based upon Chapter 3 of
Operator Algebras and Quantum Statistical Mechanics by O. Bratteli and D.W. Robinson, Springer-Verlag
Berlin-Heidelberg-New
York (1979).
The specific results we have quoted can be found in the following references Kallman, R.R.;
Amer. J. Math. 91 (1969) 785-806.
Kallman, R.R.;
Proc. Amer. Math. Soc. 24 (1970) 336-340.
Elliott, G.A.,; J. Func. Anal. ii (1972) 204-206. Yosida, K;
Functional Analysis,
Springer-Verlag,
Berlin-Heidelberg-New
York (1968),
Chapter IV. Sakai, S:
C*-algebras and W*-algebras,
Springer-Verlag,
Berlin-Heidelberg-
New
York (1971), Theorem 1.16.7.
i.
Norm Continuity The discussion of norm continuity is greatly simplified by the following classic
Banach space result. Theorem I
Let T = {Tt}tE~be
a one-parameter group of bounded linear operators on the
Banach space X. The following conditions are equivalent i.
T is norm continuous
2.
There is a bounded operator S such that lira II(Tt - l)/t - SII = 0 t~0
3.
There is a bounded operator S such that n = ~ t Sn Tt n>0 ~
This result was proved by Nagumo in 1936.
The proof is relatively straight-
forward and it has one easy consequence which is of interest. Corollary 2
If T = {Tt}t~_ ~
is a C0(or C~-) group of bounded linear operators on the
Banach space X then T is norm continuous if, and only if, there exist g,~ > 0
such
that lITt - III < i - g
for all
0 < t < 6.
If in Theorem i X is replaced by a C*-algebra 0~ and T is replaced by a group T of *-automorphisms
then the bounded generator 6 if I is automatically
follows by differentiating (*)
the relations
Tt(AB ) = Tt(A)~t(B ) ,
Tt(A*) = Tt(A)*.
a derivation.
This
:)81
Conversely if 6 is a bounded derivation of ~ t h e n generates is automatically d ~T_
t (Tt(A)~t(B))
the one-parameter
is group of *-automorphisms,
group T that it
e.g.
= -T_t(~(Tt(A)Tt(B))) = -T_t(6(Tt(A)Tt(B))) =
+ T_t(~(Tt(A))Tt(B)
+ Tt(A)6(~t(B))
0
and d~(T_t (T t (A*) *) ) = -T_t (8 (T t (A*) *) + T_t ( (ST t (A*)) *) =
0
hence the automorphism properties are valid. Thus the simplest algebraic version of Theorem i is the following Corollary 3 i.
Let ~ L b e
a C*-algebra.
The following are equivalent
6 is the generator of a norm-continuous
one-parameter
group of *-automor-
phisms of ~L. 2.
6 is a bounded derivation of ~
There is, however,
.
a much richer algebraic structure which is less evident.
The key point is that each bounded derivation of a W*-algebra is both spatial and inner. Theorem 4
If 6 is a bounded derivation of a W*-algebra M then there exists a self-
adjoint H E M with IIHII < 11611/2 such that ~(A) = i[H,A] ,
A C M,
and hence the group T generated by 6 has the
eovariant form
Tt(A) = eiHAe-itH, This theorem has various implications
for C*-algebras once one establishes that
each bounded derivation of a C*-algebra ~ o f weak closure of 0~ .
A E M.
bounded operators can be extended to the
For example
Theorem 5
If ~ is a bounded derivation of a C*-algebra ~-L and ~ ( 0 % )
tion of ~
then there exists a self-adjoint H
one-parameter
group T generated by ~ is given by ~(~t(A)) = eitH~(A)e-itH~
,
In particular one sees that the norm-continuous
A E~
, t E ~.
one-parameter
morphisms are always unitarily implemented by unitary groups
are no non-trivial norm continuous one-parameter
groups of *-auto-
U~(t) = exp{itH~} which
are weakly inner, i.e. which lie in the weak closure of ~(07.).
C*-algebras.
is a representa-
E ~(0~)- such that the norm-continuous
In particular there
groups of *-automorphisms
of abelian
382
These results are relatively old.
More recently it has been possible to cla-
ssify the C*-algebras for which bounded derivations generate inner automorphism groups if inner is suitably defined. that B(YL C ~ a n d Theorem 6
First recall that a multiplier of ~ i s
a B E 0~** such
~?_B C ~ L .
Let 0L be a separable C*-algehra.
The following are equivalent
i. every bounded derivation ~ generates a norm continuous one-parameter group of *-automorphisms of the form. A E ~-+Tt(A
) = eitHAe-itH
where H = H* is a multiplier of ~L with IIHII < [I~II/2. 2. ~
has the form ~
trace and
~2
=
0~i ~ ~ 2
where
0~ I is a C*-algebra with continuous
i_s the direct sum of simple C*-algebras.
References; Nagumo's result, Theorem i, can be found in the book by Hille and Phillips together with the basic Banach space theory of one-parameter groups. Hille, E. and R.S. Phillips ; Functi0nalAnalysis and Semigroups, Amer. Math. Soc., Providence, R.I. (1957). Theorem 4 and Corollary 5 were proved by Kadison and Sakai. This and much related material is described in Sakai's book. Sakai, S; C*-algebras and W*-algebras, Springer Verlag, Berlin-Heidelberg-New York. Theorem 6 is basically due to Elliott with a subsequent refinement by Akemann and Pedersen. Elliott, G.A.; Some C*-algebras with outer derivations III, Ann, Math. 106(1977) 121-143. Akemann, C.A. and G.K. Pedersen; Central Sequences and Inner Derivations of Separable C*-algebras (To appear in the Amer. Jour. Math.)
2.
Strong Continuity The characterizations of generators of C0-groups of *-automorphism are more
complicated.
We first need two definitions.
Definition 7
An operator S on a Banach space X is dissipative if for each A E D(S)
there exists a non-zero ~ E X* satisfying i.
q(A) = II~IIIIAII
2.
Re q(SA) < 0.
The existence of an N 6 X* satisfying property i follows from the Hahn-Banach theorem. Property 2 can be understood by assuming that S generates a Co-group T of isometries.
Then [~(TtA) I < IIniii}AiI = ~ (A).
Consequently Re n((T t - l)A)/t ~ 0 ,
t > 0
and Re n(SA) < 0. Definition 8
Let S be an operator on the Banach space X.
Then A E X is analytic
:383
(entire analytic)
for S if A E D(sn), n = 1,2,... and n z II SnAil
n>0 is an analytic (entire analytic)
function
It is easy to construct analytic elements for generators of CO- , or C~-, groups by averaging with suitable analytic functions,
i.e. by replacing A E X with
Af = Jdtf (t)TtA There are twelve characterization
of generators which exploit several different
ideas. Theorem 9
Let ~
be a C*-algebra with identity ~ and ~ a norm-densely defined L norm-
closed, operator on 6L .
It follows that 6 is a generator of a C0-grou P of *-auto-
morphisms of 6~ if, and only if, it satisfies one of the twelve combinations
(Ai,Bj,
C k) of the following conditions AI;
6 is a derivation
A2;
~ E D(6) and 8(9) = 0
BI;
(I+~6)(D(6))
B2;
The self-adjoint analytic elements for ~ are dense in the self-adjoint
= 0%,
elements of ~i
~ e ~\{0}
.
CI;
II(I+~6)(A)II
>IIAII, for all ~ E ~ ,
C2;
(I+~6)(A)
C3;
6 and -8 are dissipative
~ 0 implies A ~
A E D(6)
O, for all ~ E ]R and A E D(6)
This theorem synthesizes a variety of different ideas. of the features involved in constructing
We will describe some
the group.
Firstly if A 1 is valid then it suffices to prove that 6 generates a C0-grou p of mappings o f 0 ~ *-automorphisms
viewed as a Banach space since these mappings will automatically be by the argument used in the discussion of Corollary 3.
reduced to a Banach space problem. (AI,BI,CI) ;
This is the Hille-Yosida
vents (I + ~ ) - i ,
~ E ~,
Hence we are
Now let us consider the six possibilities with A I. theorem.
The conditions state that the resol-
are bounded operators with norm one and this suffices to con-
struct the group by strong limits Tt(A) = lim (I - ~)-n(A) n->Oo (AI,B1,C2) ;
These conditions state that the ( I + ~ ) - i ,
~ E ~,
hence bounded, operators which map positive elements of ~ of ~
.
are everywhere defined,
into positive elements
Hence they are contractions and condition C I is satisfied
conclusion one must know that ~ E D(~) and ~(~) = 0. because ~ is a norm-closed derivation
(To draw this last
But the first condition follows
(see Section 4) and the second follows because
384
~(~) = ~ ( ~ " = 2 6 ( ~ ) . ) (AI,BI,C3);
This is the Lumer-Phillips theorem for groups of isometrics.
Condition
C 1 can be deduced from C3, (AI,B2,CI);
This type of theorem was first proved by Lumer and Phillips and indepen-
dently by Nelson. B2;
In fact B 2 can be replaced by the simpler assumption.
6 has a dense set of analytic elements.
Basically one defines T by setting tn Tt(A) =
Z
~n(A)
n>0 ~' for each analytic A and small t.
Next one proves that T is isometric and hence T can
be extended to large t by iteration.
The isometric property follows because
Tt(A) = lim ( l + t 6 ) n (A) n->oo
and hence
IITt(A)III>.lira inf H (I+n~)n(A)II I> IIAII n->OO
by Condition C I.
Consequently
IIAII = ll~_t(lt(A))ll >IIITt(A)II >I IIAII and IITt (A)II
IIAII.
(AI,B2,C3) ; Again C I follows from C 3. (AI,B2,C2) ;
(Hence one could replace B 2 by B2.)
The argument is similar to the above but it relies upon the fact that an
invertible positivity preserving map which leaves the identity invariant is isometric. (It is not known whether B 2 can be replaced by B~). Now consider the six cases for which A 2 is valid. construct a C0-grou p T of isometries of ~
In each such case one can
with generator ~ and condition A 2 implies
that Tt(~) = ~,
t E ~.
It remains to conclude that the T
are *-automorphisms. This is a consequence of the t theory of positive maps and Jordan morphisms. Firstly every invertible isometric map of a C*-algebra which leaves the identity fixed is automatically a Jordan isomorEhism, i.e. both ~ and ~-i satisfy ~(A*) = ~(A)*,
~(AB + BA) = ~(A)~(B) + ~(B)¢(A).
Secondly every C0-grou p T t of Jordan automorphisms is a sroup of *-automorphisms. References; The standard theory of Co-groups on Banach space is described in Hille, E. and R.S. Phillips, Functional Analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31 Provldence (1957). Reed, M. and B. Simon; Methods of Mathematical Physics II, Academic Press, New York
385
San Francisco-London (1975). Dissipative operators were introduced in Lumer, G. and R.S. Phillips; Pae. J. Math. ii 679-689 (1961) This paper also contains a version of an analytic vector theorem. are contained in Nelson, E: Ann. Math. 70 572-615 (1960).
More general results
Parts of Theorem 9 were given by various authors Bratteli, 0. and D.W. Robinson; Con=nun. Math. Phys. 42 253-268 (1975), 46 11-30 (1976). Kishimoto, A.; Commun. Math. Phys. 47 25-32 (1976). Powers, R. and S. Sakai; J. Func. Anal. 19 81-95 (1975), Commun. Math. Phys. 39 273-288(1975).
3.
G-weak Continuity There is an analogue of Theorem 9 for C~-groups
Theorem i0
Let M be a W*-algebra and assume that M is abelian or that M is a factor.
Let ~ be a o(M,M,)-densely defined, D(6) containing ~.
o(M,M,)-o(M~M,)-closed,operator
It follows that ~ is the generator of a
on M with domain
C~-group of *-automor-
phisms of M if, and only if, it satisfies one of the eight combinations (Ai,Bj,C k) of the following conditions. AI;
~
A2;
~(~)
BI;
(I+a~)(D(6))
B2;
is a derivation =
0
= M,
the self-adjoint
~ E ~\{0}
p a r t of t h e u n i t s p h e r e of a n a l y t i c
dense in the self-adjoint CI;
II(I+~)(A)II ~JlAII,
C2;
(I+~)(A)
>0
~ E R,
implies
elements is o-weakly
p a r t of t h e u n i t s p h e r e of M
A>0
A e D(O) for all ~ E I~, A E D(O).
For general W*-algebras the theorem remains true if condition A 2 is removed and B 2 is replaced by the weaker condition i B2;
the set of analytic elements for ~ is o-weakly dense in M.
The construction of the automorphism group in Theorem 9 relied upon Banach space arguments to obtain the group and supplementary algebraic arguments to obtain the automorphism properties.
The same holds true in Theorem i0 but the Banach space arguments
are carried out on the pre-dual ~, of ~.
We will not elaborate further.
Note that M is restricted to be'abelian, or to be a factor, in the first part of the theorem because the previous key result for groups of Jordan automorphisms is no longer valid if strong continuity is replaced by G-weak continuity.
There are C~-
groups T t of Jordan automorphisms which are not groups of *-automorphisms.
These take
the form T = ET (I) + (I - E)T (2) where T (I) is a group of *-morphisms, T (2) is a group of *-anti-morphisms and E is a projection in the center of M.
386 References; Bratteli, 0. and D.W.Robinson, Ann. Inst. H. Poinear~ 25A 139-164 (1976).
4.
General Properties of Derivations There are several interesting and somewhat surprising properties of closed de-
rivations 6 which basically arise from a functional analysis on the domain D(~). There are two approaches, Fourier analysis, or complex analysis, but the following result is useful in both cases. Lena
ii
Let 6 be a norm-close 4 derivation of a C*-algebra ~'L with identity ~
If
= A* E D(6) and ~ is net in the spectrum of A then A(%~ - A) -I E D(~) and ~(A(%~ - A) -I) = % ( ~
- A)-I~(A)(~
- A) -I
If, moreover, ~ 6 D(6) then (%~ - A) -I E D(6) and ~((%~ - A) -I) = ( ~ This is straightforward to deduce.
- A)-I6(A)(X~ - A) -I For example if I%1 is larger than IIAII then
A(%~ - A) -I may be approximated in norm by polynomials in %-i.
The action of 6 on
these polynomials is readily calculated and then the conclusion A(%~ - A) -I E D(6) is obtained by taking limits and using the assumption that ~ is norm-closed. also yields the action of 6.
This
The result for small % is obtained by an analytic con-
tinuation argument. This result has an immediate corollary which we already used in discussing Theorem 9. COrollary 12
Let ~ be a norm-closed derivation of a C*-alsebra 0~ with identity 9.
The following are equivalent
i.
~ e D(6)
2.
there is a positive invertible A E D(6).
In particular if 6 is norm-densely defined then ~ E D(6) i ~ 2
by taking
A =
2 ~ 1
by remarking that
A(g~ + A) -I E D(6)
for all e > 0, IIA(g~ + A) -I - ~II --+ 0 as
~ --+ 0, and
lira sup II~(A(E~ + A)-I)II < lira sup ~÷0 £+0
glIA-IIIII6(A)II = 0
Thus ~ E D(6) because 6 is norm closed and ~(I) = 0. If A = A* E D(~) and f is a function analytic in an open simply connected set containing the spectrum of A then it has a Cauchy representation f(A) = (2~i) -I Sed%f(%)(X~ - A) -I
387
for a suitable contour C. if ~ is norm closed.
It is then simple to deduce from Lemma ii that f 6 D(6),
In particular this type of argument shows that if A E D(6) is
positive and invertible then A~ E D(6). vertible.
This is not generally true if A is not in-
In fact stability of the domain of a derivation under the square root
operation has striking consequences. Proposition 13
If 6 is a derivation of a C*-alsebra ~I with identity ~ 6 D(6) and
if A E D(6), A ~ 0, imply AI~ E D(6) then ~ and -6 are dissipative.
In particular
is norm-closable and satisfies
t](I + ~)(A)II >IIAll,
~ e
An even more striking result is true if ~ is closed. Proposition 14 C*-algebra~
If 6 is a norm-closed norm densely defined derivation of and
A E D(6),
A > 0, imply A ½ E
D(6) then 6 is bounded
There are, however, many derivations of UHF algebras with domains invariant Under the square root operation.
Assume that the UHF algebra ~
of an increasing family {6~ } of full matrix algebra ~ n ~
then 6 satisfies the hypothesis of Proposition 13.
is the norm closure
and 6 is a derivation with
Thus referring to Theorem 9 one
sees that the closure ~ of ~ generates an automorphism group if, and only if R(I + ~ 6 )
=
0L
for all ~ E I~. The second way to analyze the domains D(6) is by Fourier analysis and for this the following result is basic Lemma 15 assume
Let 6 be a norm-closed derivation of a C*-al$ebra~. ~ E D(6).
with identity ~ and
If A = A* E D(~) then exp {tA} C D(6) and ~(e tA) = t lids eStA6(A)e(l-s)tA J0
This allows one to construct functions f such that f(A) E D(6) by Fourier representation.
We will not attempt to prove the most general result but end this
section by mentioning a rather strange discrepancy between the behaviour for abelian and non-abelian ~ L . If ~ is a norm-closed derivation of an abelian C*-algebra, A C D(~), and f is a once continuously differentiable function then it follows by polynomial approximation that f(A) E D(6) (and in fact 6(f(A)) = 6(A)f'(A)). One can find counterexamples if 0~
This is not generally true.
is non-abelian.
References; Functional Analysis of the domain of closed derivation was begun in Bratteli,0., and D.W. Robinson, Commun. Math. Phys. 42 253-268 (1975).
388
Powers, R.; J. Fun. Anal. 18 85-95 (1975). 8akai, S.; Amer. J. Math. 98 427-440 (1976). Proposition 14 is taken from Ota, S.: J. Func. Anal. 30 238-244 (1978). The counterexample mentioned at the end can be found in McIntosh, A.; J. Func. Anal. 30 264-275 (1978).
5.
Spatial derivations Each bounded derivation of a v o n
Neumann algebra M has the form
6(A) = i[H,A] ,
A e M
for some bounded self-adjoint H (and H can even be chosen in M).
Certain unbounded
derivations have an analogous form. Definition 16
A derivation 6 of a £*-algebra of bounded operators on a Hilbert space
ff is spatial if there exists a symmetric operator H, on H~ with domain D(H) such that D(~)D(H) C D ( H )
and 6(A) = i[H,A] ,
A E D(6)
As an example suppose that ~(A) = (~,A~) is a state over 0L .
One can show
that the conditions I~(6(A))I 2 < const.{~(AA*) + ~(A*A)}, are equivalent to ~ being spatial and ~ E D(H).
A E D(6),
In particular this criterion applies
if ~ is invariant under 6, i.e., if ~(6(A)) = 0 ,
A E D(~).
There are various simplifications that occur for spatial derivations especially in association with invariant states, e.g., Proposition 17
Let 6 be derivation and ~ a 6-invariant state.
the cyclic representation
(H,~,~)
is norm-closable and its closure
Moreover ass~me that
associated with m is faithful. ~
It foklows that
generates a C0-grou p of *-automorphisms of Lqh
if, one of the following conditions is satisfies i.
R(I ± ~) =
6L
2.
6 possesses a dense set of analytic elements.
Since ~ is invariant 6 defines a spatial derivation of ~ ( ~ )
and the first,
and easy, part of the proof of the proposition is the deduction that ~ can be implemented by a self-adjoint H.
(*)
Subsequently one has to
e itH ~ ( ~ ) e - i t H
=
~
prove that (0%)
But this follows by applying Theorem i0 to ]~ = /(flu) and then using Theorem 9.
Thus
one is led to study derivations implemented by self-adjoint operators and it is natural to ask to what extent (*) is a natural consequence of algebraic structure. principal result in this direction is the following
A
389
Theorem 18
Let M be a W*-algebra with a cyclic and se#arating vector ~ and let H be
a self-adjoint operator on H such that H~ = 0.
Define
i[H,A] e ~}
D(6) = {A 6 M: The following conditions are equivalent i.
e i t ~ e -itH= M ,
t E
2.
D(6)~ is a core for H and H commutes strongly with the modular operator A associated with the pair ( M ~ )
3.
H is essentially self-adjoint on D(6)~ as an operator on the graph Hilbert space H#associated with A ½.
(The space H# mentioned in condition 3 is the space D(A ½) equipped with the scalar product
(~,¢)# =
~,
A½¢) +
(~,¢).
A reasonably straightforward calculation with the modular operator and the modular conjugation show that H restricted to D(6)~ This result is not easy to prove.
defines a symmetric operator on H#).
Basically one begins with the case that
m(A) = (~,A~) is a trace state and then uses crossed product techniques to obtain t h e general result.
We will not elaborate.
Finally we remark that the automorphism property
(*) can be derived from a
positivity preserving property. Theorem 19
Let 6 be a spatial derivation of a W*-algebra M implemented by a self-
adjoint H.
Assume M has a cyclic vector ~ such that ~ E D(H) and H ~ = 0.
Define
D (~) by D(~) = {A E M; and assume D(~)
is a core for H.
i[H,A] E M}
If
either
H ~ 0
~
~ is separating for M
then the following are equivalent i.
eitHMe-itH = M ,
t 6
2.
eitH M+~ C M+~ ,
t E
where M + denotes the positive elements in M and the bar denotes weak closure. The semibounded case H > 0 is of interest in ground state problems of mathematical physics and a classic result of Borchers which states that under the assumptions of the theorem e i t ~ e -itH = M , is equivalent to
t @
390
e
itH
6~
,
tE
IR
i.e. the automorphisms are inner References; Bratteli, 0. and D.W. Robinson; Ann. Inst. H. Poincar~ 25A 139-164 Math. Phys. 4 6 3 1 - 3 5 (1976). Bratteli, 0. and U. Haagerup; Commun. Math. Phys. 5 9 79-95 (1978).
(1976), Commun.
6. Stability properties To conclude we comment on various aspects of stability of automorphism groups and their generators.
Many results can be obtained by adaptation of known Banach
space results but there are two contexts in which the algebraic structure leads to stronger results. a.
Perturbation Theory The theory of perturbations
the C0-groups of *-automorphisms
of C0-semigroups of C*-algebras.
of contractions
can be applied to
There appears to be no such theory
for C~-groups although some results can be taken over from the C0-theory. if S is the generator of a C0-group, or a C~-group, generator of the same type.
For example
and P is bounded then S + P is a
In the C0-case the result remains valid if P is a dissi-
pative operator with S-bound less than one
i.e. if D(P) D D(S) and
IIP~I < aiIAII + bllSAIl for all A E D(S), some a ~ 0, and a b < I.
No analogue appears to hold in the C$-case.
One basic problem that arises if one applies the theory of relatively bounded perturbations
to derivations
is the problem of existence.
trivial examples of derivations D(~ I) D D ( 6 2 )
There appear to be no non-
61, and 62, satisfying the relative bound relations
and II~I(A)II < aIIAIl + DII62(A)II.
Apart from this difficulty the following result is striking. Theorem 20
Let 6 be the generator of a C0-grou p of *-automorphisms
~]~ and 6' a derivation with the same domain. rive.
Hence 6 + %~' generates a C0-grou P
of the C*-algebr a
Then 6' is 6-boundedaand 6' is dissipa-
of *-automorphisms
for all sufficiently
small %. b. Approximation Theory Approximation
theory is the name we give to the recently developed theory of
comparison of pairs of CO-, or C~-, groups. generators
If T and T' are two such groups with
S and S' one attempts to characterize relationships between S and S' by
properties of T and £', e.g. one can show that for C0-groups S - S' is bounded if, and only if~ D(S) ~ D(S') is a core for S and
391
IIT~ - ~tll = 0(t)
as
t ~ 0
and ~ - S' is S-bounded if, and 0nly if, D(S) A D(S') is a core for S and II(T~ - Tt)All = 0(t) for all A E D (S).
as
t ÷ 0
(In fact these examples do not completely characterize the proper-
ties of S - S' in terms of T - T' because the core condition
is also necessary.)
Instead of attempting to describe the complete development of this field we give two results which depend on the algebraic structure Theorem 21
Let
T,T' be two C~-gr0ups of *-automorphisms
generators 6,6'.
of the W*-algebra M with
The following conditions are equivalent
i.
lITt - ~ I I
= 0(t)
2.
D(6) = D(6')
as
t ÷ 0
and the o-weak closure of 6 - 6' is bounded.
! f M, is separable the following are equivalent i.
there exist 0 < e I < ¼
and 61 > 0 such that
lITt - T~II < e I 2.
there exist 0 < e 2 < ~ , derivation 6
62 > 0 ,
for
It I < 61
an inner automorphism y
of M and a bounded
of M such that P 6 = y(6' + 6p)y -I
and lITt o y o T_t - yll < e 2
for
It[ < 62
References; For Theorem 20 see Longo, R.: Automatic relative boundedness of derivations in C*-algebras (Rome preprinO Batty, C.J.K.; Small perturbations of C*-dynamical systems (Edinburgh preprint). Approximation theory was begun in Buchholz, D. and J.E. Roberts; Commun. Math. Phys. 161-177 Robinson, D.W.; J- Func. Anal. 2 4 2 8 0 - 2 9 0 (1977).
(1976).
The characterizations of bounded S - S' were partly given in this last reference and completely given in Batty, C.J.k.: A characterization of relatively bounded perturbations (Edinburgh preprint). Theorem 21 is taken from Bratteli, 0., Herman, R., and D.W. Robinson;
Commun. Math. Phys. 5 9 1 6 7 - 1 9 6
(1978).
THERMAL EQUILI~{I[~ AND CORRELATION INEQUALITIES
M. Fannes Instituut voor Theoretische Fysica, Universiteit Leuven B-3030 Leuven, Belgium
This paper consists in three sections, the first [I] and the last [2] resulting from a collaboration with A. Verbeure and the second withH.
Spohn and A. Ver-
beure [3]. We show in the first section how one can derive correlation inequalities from the variational principle of thermodynamics for quantum spin lattice systems. These inequalities express therefore a condition of global thermodynamical stability (GTS). As it can be shown that a state which satisfies these inequalities is KMS [4~ one can prove that GTS implies KMS. A similar approach has been used to prove that local thermodynamical stability (LTS) implies KMS [5]. The remaining sections are devoted to applications of the correlation inequalities to more concrete situations. In the second we characterize the structure of the limiting Gibbs states for discrete models of the mean-field type. Our results are sharp enough to yield in specific cases, such as the BCS or Dicke maser models, the existence (uniqueness) of the limiting Gibbs state. Finally we study in the third section the condensed phase of the imperfect Bose gas. The occurence of condensation in and only in the ground state is proven for sufficiently low (high) temperatures (densities). We also show the existence (uniqueness) of the limiting Gibbs state in the case of condensation.
I. Derivation of Correlation Inequalities from Global Thermodynamical Stability [I]. Let A
be the C*-albebra of quasi-local observables for a quantum spin systam
on a lattice Z ~ [6]. Lattice translations are described in a natural way by a group l-~ ~ W e ~
~ of automorphisms of 6~. To fix the dynamics we specify the local ha-
miltonians :
where the potential
i)
~(~
~ ~ ( X~
i ~, C 2~} satisfies the following conditions:
: % ( ~ f ~ o % × , the subaigebra of O% of the elements living on the set X
ii)
%~(~(X)) =
~ ( ~KB
translation invariance
393
Definition 1.I.
A state oa of ~
is GTS at inverse temperature ~ , if
i) co is translation invariant ii) co minimizes the expression
where A @ =
X
~(X) and ~(oJ~ is the entropy density.
In order to derive the correlation inequalities we will 'slightly' perturb the GTS state and express that this does not decrease the value of ~(oa). However the perturbations we need cannot be local as they should preserve the translation invariance of the state. We therefore construct a class of translation invariant semigroups of completely positive unity preserving maps of ~
with generators of the Lindblad ty-
[7].
pe
Theorem 1.2. Let ~ 6 ~ c
be a local element, then
is well defined and generates a translation invariant strongly continuous semigroup ~
~
I ~6 ~
of completely positive unity preserving maps o f ~ .
Let cO now be a GTS state at inverse temperature ~ ; this implies that
By computing ~ o +
~{~(ooo~p~tB - ~(eoB ~
we get our main result of this section:
Theorem 1.3. Let Go be GTS at inverse temperature ~
then for any ~ 6 C ~ o ~
I]. Equilibrium States for Mean-Field Models [9]. Discrete mean-field models are essentially characterized by the invariance of the dynmnics under permutations of the 'spins'. So we take as algebra of observables the, quasi-local albebra O~ bui]t on ~ o group ~
(with W degrees of freedom at each site). The
of permutations of finite subsets of ~}d~ is described on the level of the ob-
servables by a group {~r& I ~ e ~ ] of automorphisms of O~. We specify our local h~i!tonians by
wher~ A~ i~ a co~y ~.t t h e
[th
site
of a selfa~oint
at the ;th and ith site of a selfadjoint symmetric
AeO~C¢ ~ 6 ~ ( ~
Let a ~ . ~ b e , the G-tbs state defined by ~ { % , . o~ ~ a w -!im,it point of the c o ~
)an~ %~i is ~ copy @
(~ CCW~ .
on U ~ , . . , ~ . ~ a n d
~6~
(which exists by w -compactness of the state space o f ~
).
394
Clearly ~o~ is permutation invariant and can therefore be written as [8,9]:
where £ is a density matrix on
~.:l,/
and Loe is the product state of ~
induced by £ .
The state oo~,t can also be c~mracterized by the correlation inequalities [4~:
This leads together with (1) to the following correlation inequalities for ~o~ : Lemma II.S. Let ~ e O~A, then:
" where where
~
m
i He; t and Re~L tea B~ = -V~(~[@ e ~ ~ and F a t
~'~ >.
o%C~
~
~ c'~'~
is a copy at the [th site of ~ e =
(2) /kS B e e ~ [ ~ W )
is the partial trace over the second Hilbert space
In formula (I) the state 6D~ is written as an integral over disjoint states, this allows us to 'concentrate' the correlation inequality (2) on the product states that enter in the integral decomposition (I) leading to our main result in this section: Theorem 11.2.
For~g-almost ill e A
and so Theorem II.2. shows that finding the equilibrium states of mean-field models emo~ts to solve the non-linear one-site equation (3). Some further information about the limiting Gibbs states can be obtained: i) for high temperatures (~ < ~/t~gL~quation (3) admits a unique solution. Therefore all limiting Gibbs states coincide and one obtains the existence of the limit Gibbs state. ii) using the variational principle of thermodynamics one can still further reduce the support of the measure .2%(e) in (I): if 4o = ~
~(eo~ = ~
~(u~- ~/~ ~(O~
where the infimum is taken
over the permutation invariant states of O~ and ~(Oo~
and ~(oo~
are the
energy and entropy density of ~O then
iii) in specific models such as the BCS model and the Dicke maser model (which can be treated in an analogous way by replacing at each site (~(~Lm~ by (B~B
where ~
is a separable infinite dimensional Hilbert space) one finds
for ~ > ~critioal a solution
~norm of (3) corresponding to a normal phase
395
and also solutions ~(i~ o(e [0,21[[ corresponding to a superconducting or superradiant phase. The solutions group ~ (
coe ( ~
~e[0,2~[~ ~
%-~.,
=
oo~
~(~)
transform under the action of a
of automorphisms of C~ (gauge group) as
C o~~- ~, ,~oA ~-~
In this case however the limiting Gibbs state is invariant under the gauge group and one has an example of a broken symmetry. The normal solution can be ruled out by ii) and one finds
which leads again to the uniqueness of the limiting Gibbs state below the critical temperature.
III. The Condensed Phase of the Imperfect Bose Gas [2I. We consider in this section only ~ -dimensional systems with ~ 3
as condensa-
tion can be ruled out for ~ =1,2 [I0]. Let A be the centered cubic box in ~kv with side L , ~ F ( A B
the boson Fock space constructed on A
and ~ ( ~ )
, ~(~)
,
'6 ~t(A~ ~ B the canonical creation and annihilation fields:
The local hamiltonians of the imperfect Bose gas are then given by:
~L('2"~- "} = ~U ---'U'L NL where
"[L
+ __~'
N~
(I)
is the usual kinetic energy with periodic boundary conditions and N
L
is the
particle number operator. The coupling constatnt ~ is strictly positive and the chemical potential .A~L is adjusted in such a way that
co l ~ , L ( N L ) where o ~
e
=
L"
(2)
is the Gibbs state at inverse temperature ~
determined by (I) and
~ > ,o
is the mean particle density. In order to state our first main result we need the following definitions: _ ~ -~ ~ K
- <~hl
I I~ e
~is
~
)7~(the dual of the torus /k)
the orthonormal basis of ~ ( ~
functions ~eA ~-
o
~
~t~=ined ~ the
396
Theorem III. I. (i)
(ii)
~
~u
~-~o
~
C ~.,o~
t.-,, oo
"J"
L'~
~
~ _
~"
o~,~<
~
~
&~
I'~ ( . M L ~ ~ = 0
~,
°O-~L
~-
i,
Theorem III. I. (i) implies that there is a macroscopic occupation of the ground state for sufficiently low (high) temperatures (densities) and (ii) shows that the ground state alone has such an occupation. The essential ingredients for the proof of this theorem come from repeated applications of the correlation inequalities. Take now as algebra of observables the inductive limit of the ( ~ Q ~ F
(h~
and suppose that cOB is a w -limit point as [-~ooof the o0~.L. As one can prove that £oe is locally normal it is sufficient to compute oo~ on some smaller algebra such as the C -algebra generated by ~ % ~ ( ~ = - ~ [ ( ~ ( ~ Q ( ~
~ ~e ~
where S~ is the
space of infinitely differentiable complex valued functions on [ ~ with compact support. It is now possible, as in section II, to derive correlation inequalities for (D~ and to solve them explicitly. This leads then to: Theorem 111.2. Let (i) ~ ~,ll) t.. i f
(o~ =
ao~-
oo~jh~ ~ J t ~ , o ~ - _
eo>O then for ~ e ~
~
co~jk~ ~o
exists
:
where ~( denotes the Fourier transform of ~
and
It is perhaps worthwhile to remark that our proof of lheorem III.2. implies also that the Bogoliubov approximation is valid for the imperfect Bose gas. ~I~ M. Fannes, A. Verbeure; J. Math. Phys., 19, 558 (1978) [2] M. Fannes, A. Verbeure; The Condensed Phase of the Imperfect Bose Gas, Leuven preprint [3] M. Fannes, H. Spohn, A. Verbeure; Equilibrium States for Mean-Field Models, Leuven preprint KUL-TF-79/004 [%] M. Fannes, A. Verbeure; Commtm. Math. Phys. 55, 125 (1977) and Commun. Math. Phys.
57, 165 (1977)
[5] G.L. Sewell; Commun. Math. Phys. 55, 50 (1977) [6] D. Ruelle; 'Statistical Mechanics', Benjamin New York (1969 [7~ G. Lindblad; Commun. Math. Phys. %8, 119 (1976) [8~ E. St~rmer; J. Funct. Analysis__9, 48 (1969) [9] R.L. Hudson, G.R. Moody; Z. Wahrscheinlic~(eitstheorie verw. Gebiete, ~39, 3/+3 (1976) [10] P.C. Hohenberg; Phys. Rev. 158, 383 (1967)
ON THE POSSIBLE TEMPERATURES OF A C*-DYNAMICAL SYSTEM Ola B r a t t e l i , School of Mathematics, University of New South Wales, P.O. Box I , Kensington, N.S.W., 2033, Australia. (On leave from University of Oslo) (This is a report on j o i n t work with G.A. E l l i o t t and R.H. Herman, [ I ] ) There are several good reasons for using the KMS condition as a characterization of equilibrium for an i n f i n i t e quantum system, [ 2 ] .
The main purpose of t h i s note is
to show that the KMS condition alone, without any further specification of the system, may allow almost any structure on the set of equilibrium states at varying temperatures. In t h i s setting the system is represented by a C*-dynamical system (~, ¥), where O(is a C*-algebra with i d e n t i t y , and t E R ~ Yt is a one-parameter group of *-automorphisms which for s i m p l i c i t y is assumed to be strongly continuous. on
I f m is a state
and B is a real number, m is said to be a (y, B)-KMS state i f
~(AYi~(B)) = ~(BA) for all A,B EO~such that B is entire analytic for y.
We use the terminology that
is a (y, + ~)-KMS state i f m is a ground state, i . e .
-i~(A*~(A)) > 0 f o r a l l A is the domain of the generator ~ of ~ (see
[2] for a j u s t i f i c a t i o n of t h i s ) .
The notion of (y, - ~)-KMS state is defined by the converse inequality. The set KB of (y, B)-KMS states is a compact, convex subset of the state-space E ~ o f O ( f o r any B E ~ U {± ~}. = ± ~.
K+=
K~ is a simplex i f
IBI < + ~ and a face in E~for
is in general not a simplex, a certain condition of asymptotic abelia-
ness is necessary and s u f f i c i e n t for t h i s , [2]. The structure of the map
E [0, ~] ~ KB has been analyzed in detail in several models, most notably in quantum l a t t i c e spin systems.
The C*-algebraO~has then a unique trace state T, and there exists a sequence
{Hn}n~ 1 of local Hamiltonians, HE = Hn EO~, such that Yt(A) = lim eitHnAe -itHn f o r a l l A EO~.
I f m6 is a weak*-limit of the sequence m6,n defined by ~6,n(A) = T(e-BHnA)/T(e-BHn)
as n ÷ = , then m~ is a (~, B)-KMS state, and in particular KB # @for all 6.
Under
mild restrictions on the sequence {Hn}nzI one knows that KB consists of only one point for small 8, i . e . for B E [0, T~->where Tc is a c r i t i c a l temperature. For one-dimensional models with short range interactions one has Tc = O. For the two-dimensional
Ising model Tc has a f i n i t e value, and K~ has a f f i n e dimension one for B E< , + ~>, c see Aizenmann's lecture at t h i s conference. For the three-dimensional Ising model, KB is known to be i n f i n i t e dimensional for large 6. K is i n f i n i t e dimensional for the Ising model in a l l dimensions, [ 2 ] .
398 The main purpose of the present note is to show that the general features of the map B ~ KB which are valid for l a t t i c e systems does not hold for general C*-dynamical systems, and presumably almost any " f i e l d " B ~ KB can occur.
Unfortunately we do not
have a precise notion of " f i e l d " for the moment, and w i l l give three partial results from [ l ] .
I f O~is an arbitrary C*-algebra this result would not be too surprisinq,
since one could superpose simpler systems. ple C*-algebras. THEOREM A
Let
However, the result is true even for sim-
F be a closed subset of ~ u {± ~}.
There exists a C*-dynamical
System (~, ~) such that O~is simple, and there is a (y, B)-KMS state i f and only i f E F, and this state is unique. THEOREM B
Let
Fl ~ F2 D . . . D Fn be a f i n i t e sequence of closed subsets of ~ u {± ~}
such that ±~ ~ F2.
Th_ere exists a C*-dynamical system (0(, y) such that O~is simple,
and KB is a k-l-dimensional simplex i f B E Fk\Fk+l, k = O,l . . . . . n.
(use the convent-
ions F0 = ~ U {± ~}, Fn+l = @,the-l-dimensional simplex = 4). THEOREM C ,Let KBI . . . . . KBn be n compact, metrizable simplexes indexed by Bl . . . . . Bn E ~ . There exists a C*-dynamical system (0(, y) such that O~is simple, and there is a (y, B)KMS state i f and only i f B E {Bl . . . . . ~n}, and the set of (y, ~k)-KMS-states is a f f i n e l y isomorphic to KBk for k = l . . . . . n. Furthermore, in all of the theorems A,B,C, the system (0~, y) can be constructed such that O~is separable nuclear with unit, and ¥ is periodic with period 2~. The proof of Theorem A goes in two steps: Step l . One constructs an approximately f i n i t e dimensional C*-algebra~ and an automorphism ~ o f ~ and a projection E i n ' s u c h that I.
I f B E F there exists a unique ( i n f i n i t e , lower semicontinuous) trace TB on~ such
that TB(E) = 1 and TBo~ : e-BTB. 2.
I f B ~ F, no such trace exists.
There are no globally ~-invariant ideals i n ~ (In the case ±~ ~ F , ~ can be taken
to be simple, otherwise not). 3.
~ transforms each nonzero projection i n ~ into a non-equivalent projection.
Step 2 ,
2 and 3 implies that the C*-crossed product C*(~, ~) o f ~ b y ~ is simple.
C*(~, ~) is the closure of the linear span of elements of the form A Un, where A E and U is a unitary operator such that ~(A) = U A U* for all A E~. Define and
Yt(AUn) = eintAu n ,
A E~
OL= E C*(~), ~)E
Then (0(, y) satisfies the conclusion of Theorem A. the projection defined by
2~
In particular i f @ :0~ ~ E ~ E
dtt(A),
is
A OC,
0 then the unique (¥, B)-KMS state mB for B E F is given by ~B(A) = ~(@(A)), To get Theorem B, C one modifies step l , l .
A E OI~, in the obvious way.
To construct~, ~, E one uses the following characterisation of dimension groups
399
[4], of AF-algebras THEOREM, [3].
An ordered abelian group G is the dimension group (= Ko-group) of
an AF-algebra i f and only i f the following two properties are valid. I.
(G is unperforated) I f g E G and ng ~O for some n E ~ , then g ~ 0 .
2.
(G has the Riesz interpolation property) I f g l ' g2' g3' g4 E G and
g l ' g2 < g3' g4 then there exists a g5 E G such that g l ' g2 ~ g5 ~ g3' g4 Bya theorem in [4], the dimension group together with a hereditary subset of i t s positive cone is a complete invariant for AF-algebras.
Therefore, to construct~, ~, E
i t is enough to construct a dimension group G, an order-automorphism ~ of G and an element go E G corresponding to E with the correct properties.
The construction inv-
olves many arbitrary choices, but one can for example l e t G = ~ [x, x - l , (l - x ) ' l ] , i . e . G is the additive group of a l l polynomials in x, x-l and (l - x) -l with integer coefficients.
I f Fl is the set of t E [0, l ] such that t ( l - t) -l = e-8 for some
B E F one defines an order on G by saying that p E G is ( s t r i c t l y ) positive i f and only i f p ( t ) > O for all O < t < l
in a neighbourhood of FI .
that G has the Riesz interpolation property.
I t is then not hard to verify
The order automorphism ~ is defined as
multiplication by x(l - x) -l and the element go as the constant function I . To cope with the cases B and C one consider additive groups of functions from which in the point B E ~ assumes values in the real affine functions over KB. [l]
B r a t t e l i , 0., G.A. E l l i o t t and R.H. Herman, On the possible temperatures of a
dynamical system, Penn. State Univ. preprint (1979). [2]
B r a t t e l i , O. and D.W. Robinson, Operator algebras and quantum s t a t i s t i c a l mechanics,
Vol. I I , Springer Verlag, Berlin-Heidelberg-New York, to appear. [3]
Effros, E., D. Handelman and C.L. Shen, Dimension groups and their affine represen-
tations, Univ. of Penn. preprint (1979). [4]
E l l i o t t , G.A., On the classification of inductive l i m i t s of sequences of semisimple
finite-dimensional algebras, J. Algebra 38 (1976), 29-44.
A REVIEW ON SEMIGROUPS OF COMPLETELY POSITIVE MAPS
David E. Evans School of Mathematics,
The University,
Newcastle-upon-Tyne,
NEt 7RU, England.
We review recent progress on one-parameter semigroups of completely positive maps on operator algebras,
and its application to irreversible Markovian dynamics in
quantum systems. We recall that the positive cone of a C*-algebra A is A + = {a'a: a6A}, and a linear map T between C*-algebra A and B ~ positive if T(A + )C -- B + . If one considers C*-algebras and positive contractions then the isomorphisms for this category are precisely the Jordan isomorphisms, Jordan product a-b = (ab+ba)/2. ator algebra, Instead,
namely the *-linear bijections which preserve the
Thus in order to study the C*-structure of an oper-
it is not enough to look at the positive cone A+, and positive maps.
one can consider the whole sequence of matrix algebras MN(A) ~ A ~ M N, tog-
ether with their positive cones MN(A) +. to b e N - p o s i t i v e
A map T between C*-algebras A and B is said
if T ~ I: A ~ M N ~ B ~ M N is positive,
is ~-positive for all N.
and completely positive if it
Then the isomorphisms in the category of C*-algebras and
completely positive contractions preserve all the C*-structure, linear *-bijections which preserve the usual product ah. 2-positivity here).
i.e. they are the
(In fact,
one only needs
Completely positive contractions arise naturally in quantum theory
as those operations which are probability reducing remadn so in interaction with N-body systems If],
(in the Schrodinger picture) and see also [2].
Moreover if one takes
the viewpoint that reversible dynamics is given by a group {~t: t E ~ } of morphisms (i.e. *-automorphisms)
of a C*-algebra B, and if one restricts to a subsystem repres-
ented by a C*-subalgebra A, via a conditional expectation N:B~A,
then one obtains
T t = N ~tl A (*), which is necessarily completely positive by Tomiyama's theorem [3] on projections of norm on, It].
In general,
ily { T t : ~ O } will not be Markovian.
However,
there will be memory effects and the famrigorous derivation of the s&migroup
law Tt+s=TtTs; t,s ~ O, has been obtained in a variety of models when taking weak or singular coupling limits (see [5] for a review of this). ovian dynamics and define a ~ n a m i c a l
Here we consider only Mark-
semigroup [I] {Tt:t~O } on an operator algebra A
to be a semigroup ofcompletely positive contractions on A (normally with some continuity assumptions). of the semigroup,
We review some dilation problems,
ergodic and spectral properties
but we are mainly concerned here with the infinitesimal generator
L = lim (Tt-1)/t. (For reviews biased in other directions see [~,5,6])First we t~O emphasise that complete positivity is a much stronger property than positivity, However,
if T is a positive map between C*-algebras A and B, with either A or B commute-
* Supported by the Science Research Council
401
tive~
t h e n T is a u t o m a t i c a l l y c o m p l e t e l y p o s i t i v e [7].
the d i s t i n c t i o n does not arise.
Conversely,
A into anothe D ~ is c o m p l e t e l y positive,
F o r r e v e r s i b l e systems,
where e
tL
Thus for classical systems,
if every p o s i t i v e map from a C * - a l g e b r a
then either A or B is c o m m u t a t i v e [8~.
extends to a group of automorphisms,
derivation, as in the talk of D.W. Robinson.
L is a
In this case if A is r e p r e s e n t e d on a
Hilbert space H, one is i n t e r e s t e d in obtaining a h a m i l t o n i a n h (a self adjoint operator on H) such that tL e = Ad(eith),
t E ~,
L = lad(h)
or
(with suitable domain interpretation).
(I)
This can certainly be done if L is b o u n d e d
(with b o u n d e d h); and moreover if the a l g e b r a A is a y o n N e u m a n n a l g e b r a or say a simple unital C*-algebra,
then one
even choose h in A itself [ I0].
can
w h e n L is u n b o u n d e d is more c o m p l i c a t e d
(see[ll]),
w e a k l y continuous group of * - a u t o m o r p h i s m s of B(H), ~ p e r a t o r h on H such that e t L = Ad(eith). dynamical semigroups.
First,
etK(
transform
)e tK*,
t E ~
if {e t L : t E ~}
is a
then there exists a self adjoint
W e seek a n a l g o u s spatial descriptions for
what are the obvious norm continuous semigroups of
completely p o s i t i v e maps o n a C * - a l g e b r a A c B(H)? Lyapunov
however,
The s i t u a t i o n
Let K 6 B(H) be such that the
Lo(x) = K x + xK*, x ~ A leaves A globally invariant. defines a group o f c o m p l e t e l y p o s i t i v e maps on A.
T h e n e tL~ =
Moreover,
if
is a c o m p l e t e l y p o s i t i v e map on A, then e t~ = I + t~ + t2~2/2 + ... is certainly completely p o s i t i v e for t ~ O.
T h u s by the T r o t t e r p r o d u c t formula,
r a t e s a semigroup of c o m p l e t e l y p o s i t i v e maps. In fact, tL tL~ f eeL0 ~ e (t-s)L ds , e = e + o so that e tL~> - e ~L~ = etK( - )e tK*, algebras,
we w r i t e ~ > - I
t > -
0
; w h e r e if ~,
L = L o + ~ gene-
t ~ O,
1 are linear maps b e t w e e n C*-
if ~_ i is" completely positive.
T h u s we see that if
{e t L : t ~ o} is a n o r m c o n t i n u o u s semigroup of c o m p l e t e l y p o s i t i v e maps on a C*a l g e b r a A ~ B(H),
and K E B(H),
then e tL ~ etK(-)e tK*,
L ~ K(.) + (')K*, as maps into B(H). iation).
t ~ 0 if and only if
(The i m p l i c a t i o n = is easily seen by different-
N o w for an a r b i t r a r y ~ n a m i e a l
semigroup T t on a n operator a l g e b r a A ~ B(H)
we seek a strongly continuous c o n t r a c t i o n semigroup G t on H such that Tt
[ N o t e t h a t we do n o t i n s i s t
Gt(-)Gt*
,
t ~ O.
on GtAGt* ~ A ] .
for i r r e v e r s i b l e systems of (I).
(2) We r e g a r d t h i s
as a s u i t a b l e
analogue
N o t e that if T is a n a u t o m o r p h i s m of an i r r e d u c i b l e
C * - a l g e b r a A C B(H) a n d G is a non zero c o n t r a c t i o n on H such that T ~ G(')G*, -~[T(ai)*T(a3 )] ~ ~~F(aiG*)*(a'G*)]3 contraction
V a I . . . . n 6 A, and so there exists a well defined
C on H such that C T(a) = aG*~ a C A.
It is easily checked that G'G,
then
GG* ~ A',
T h u s C = G*, a n d s o G*T(a) = a G.
so that b y irreducibility,
G can be t a k e n
402
unitary and T = Ad(G).
i
Theorem
[ 12].
itive maps
Let {etL: t ~ O} be a norm continuous
on a C*-algebra A c B(H). e
Theorem
~
etK( " )etK*
dynamical
t ~ O.
semigroups
but see [ 15].
for finite dimensional
on a hyperfinite
cases of this have appeared in [ 14].
of positive maps,
of completely poe-
Then there exists K 6 A ~ such that
I was first shown independently
[13] and n o r m a l special
tL
semigroup
W*-algebra
Relatively
matrix algebras
[I],
N o w if ~ is a normal completely positive map f r o m a
yon N e u m a n n algebra A (CB(H)) into B(H),
then there exist V
~(x)
t >-- O} is a norm continuous
=
~vlxv
xEA
[2]
completely positive n o r m a l be given by T h e o r e m
Now if {etL :
~(x)
such that semigroup
6 B(H).
L(x)
= iad
Since e tL are unital maps,
situation
(c.f.
= ½ (~V*xVcxa + VcXV~),
(h)
of
let K 6 A
- (.)K* > O, and so L = K(- ) + (-)K* +
Thus K = ih - v(1)/2 for some s.a.h in A.
In a symmetric
6 B(H)
unital maps on a yon N e u m a n n algebra A C B(H)~
I so that ~ -= L-K(-)
Z V~(.)V , for some V
and some other
less is k n o w n for semigroups
L~I) = 0 or K+K*+7(1)
= O.
Thus for x E A:
(x) + ZV*XVo~oc - ½ [EV'~V , x ] +
(3)
detailed balance [ 16, 17] ) where
we c a n w r i t e :
where 8,50~ are the derivations
8=i ad(h),
L = 6 - -}E(6a5 a ÷ 5c6 8c~ = / ~i d ( c )V
and
5~(x)
~ ,
8 (x*)*.
=
corresponds
to the diffusion equation in classical probability
ups of type
(3) were written down in [ 18, 19] in a study o f quantum stochastic processes.
They also arose naturally mics of an N-level dynamics
(see also [5] for other models)
Such semigro-
as the r e d u c e d
system weakly coupled to a thermal reservoir.
state on B(H),
which is proven under the assumption
and t h e existence
ral results are discussed
of a normal invariant
in [22],
in particular
theory to finite yon N e u m a n n algebras. and generators ible systems,
of the form e.g.
inequalities
As a consequence
T h e o r e m 2 [19, 2/~]. algebra A ~ B(H),
Further of
of T h e o r e m
and
ergodic and spect-
the Perron-Frobenius
We should mention that irreversible
dynamics
in purely revers-
of KMS states using detailed balance
[23]~
to an
lin(V ) s.c.,
(3) have been used to obtain information
characterisation
[17] or correlation
state.
an extension
dyna-
This particular
is amenable to the work of several authors [ 18, 21] o n relaxation
equilibrium {V }' = ~
in[20]
theory.
This
conditions
(c.f. the talk of M. Fannes).
l, it can be shown:
If {etL: t ~ O} is a norm continuous there exists a Hilbert
dynamical
semigroup
on a C*-
space Ha, and a strongly continuous
contrac-
tion semigroup G t on H ~ H e such that etL(x)~l = G t ( ~ I ) G ~ ~ x E A, t ~ O.
403
Proceeding ~ o m this,
dilations of some normal dynamical semigroups on yon Neu-
mann algebras to groups of automorphisms were obtained in [%, 19,2~]. arbitrary families of completely positive of a group were obtained in [25].
maps on C*-algebras,
Dilations of
indexed by elements
This is in some way analogous to constructing a Mar-
kov process f r o m a contraction semigroup in classical probability theory.
There has also been considerable interest in obtaining spatial descriptions of unbounded generators of dynamical semigroups. the Schrodinger picture for B(H). type [6,26-3~,
The following theorem was obtained in
We then proceed to look at generators of quasi-free
concentrating here on the Fermion algebra.
Theorem 3- [31]. (a) Let {eKt: t ~ O} be a strongly continuous contraction semigroup on a Hilbert space H, and V
6 L (Dom (K*), H) such that
+ <~,K*~> + ~ V
%
V ~> = O, V ~ E Dom (K*).
Then there exists a weakly continuous normal dynamical semigroup {Tt: t ~ O} on B(H) such that the infinitesimal generator L. of the predual action satisfies
L,(~)
= K*@ + ~K + ~V 0V ~
on (I-K*) -I T(H)
(i-K)-10
(However in general Tt(1) ~ I). (b) Let {Tt: t 2 O} be a weakly continuous normal dynamical semigroup on B(H), the compacts globally invariant,
leaving
such that <. Q, Q> is a pure invariant state (~ 6 H).
Then if {eKt: t ~ O} denotes the contraction semigroup on H given by xQ ~ Tt(x)~, x E B(H)T there exist
V
6 L(Dom(K*), H) such that the generator L. of
the predual action is given by L.(p) = K*~ + pK + ~ V pV~ o n (I-K*) - 1 T ( H )
(I-K) -I.
Uniqueness of the standard form L = K*(') + (-)K + ~ of a dynamical semigroup on B(H) has been studied in [13,32], and, essentially,
there is uniqueness if re(K) can
be made small in a suitable sense (e.g. when -tr(re(K)) is minimal when dim(H)<~ when H is infinite dimensional,
or
there is at most one K with re(K) compact.
If H is a Hilbert space, let A(H) denote the Fermion algebra built on H, generated by
operators {a(h): h 6 H}, which are the range of a conjugate linear
map a, satisfying the canonical anticommutation relations.
If {eBt: t a O} is a str-
ongly continuous contraction semigroup onlH, commuting with a positive contraction R on H, then there exists a unique strongly continuous unital dynamical semigroup {AR(e tB) : t ~ O} on A(H) such that
404
AR(e tB) [:a~(f i) ... a ~ f n ) : ~ ] = :a~(etBfl) ... a~(etBfn):~ where : :R indicates Wick ordering with respect to the quasi-free state wR, and wR~AR(e
Bt
) = w R.
These examples allow us to study dynamical semigroups,
not n o r m continuous,
and not simply infinite tensor products.
which are
If (~R~FR(H),QR)
is the
GNS decomposition of w R , then, as in Theorem 3, we can define a contraction semigroup FR(eBt ) on FR(H) given by ~R(X)nR ~ ER(AR(e Bt )(x))~ R, x E A(H), such that UR AR (eBt) (') a FR(eBt)UR(') FR(eBt)*' are
-isometrics [28].
as usual.
t ~ O, (~), with equality only when {eBt: t~O}
If R=O, then Fo(H) is Foek space,
If B=ih+k, where h is self adjoint,
and Fo(e Bt) = 1 ~ eBt~ ...
and k bounded and negative,
then it
follows from the proof of Theorem 2 (see[%]) that there exists a Hilbert space H i, a strongly continuous semigroup {Gt: t a O} of co-isometries on Hi, an isometry W:H-4~ I, (so that
A(H) ~C A(H1)) and a positive contraction S on H1 such that WR=SW, [S, Gt]=O ,
and W e Bt = GtW.
Then HsAR(eBt)(x)
= Fs(Gt) ~s(X ) Fs(Gt)* , x 6 A(H), t a 0 is
isometric representation as in Theorem 2. be obtained via Sz. Nagy's theoren~
Dilations to groups of automorphisms
an can
One can next ask: given a strongly continuous
contraction demigroup {eBt: t ~ O} on H, commuting with a positive contraction R, when does there exist a strongly continuous contraction semigroup in the representation ~R' of another quasi-free state w R, s.t.
(a)
First consider the following cases:
By Theorem i, this can be done with K E ~R,(A(H))H if{AR(eBt): t ~ O} is norm
continuous. [27, 283 •
(2) holds?
However {~(eBt):
t ~ O} is n o r m continuous if and only if B is t r a c e c l a s s
In which case B = -i ~ ~n hn ~ ~n - ~ An gn ~ --gn, where {hn}, {gn} are two
complete orthonormal
sets in ~
0 ~ rn ~ I with Rg n = rng n.
~n E R, In > O, and E l ~nl, ~ In < ~' and there exists
Then if 0 is the quasi-free automorphism such that
@[a(h)] = -a(h), the g e n e r a t o r L of {AR(etB): t ~ O} is given by L (x) = i ~ ~n [a*(hn)a(hn)'X]-
+ Z ~n{2[(1-rn)a*(gn)+r -
[((1-rn)a*(gn)+~a(gn))
n a(gn)]@(x)[(1-rna(gn)+rn ((1-rn)a(gn)+ra*(gn)),
a*(gn) ]
x]+}
+ 2 ~ A n ( r n - r n 2) {(a*(g n) - a ( g n ) ) 8 ( X ) ( a ( g n ) - a * ( g n )
) - x}
so that in fact we can take K E ~R,(A(H)) [27,29].
(b)
If w R, is quasi-equivalent
to w R, with R,R ~ projections,
transformed to the representation HRS.
then (~) can be
(i.e. if R-R ~ is trace classY.
For semigroups of Fock type (R=O), we have:
405
Theorem %. [ 3 0 ] .
I f B K O i s b o u n d e d on H, and p a p r o j e c t i o n
trace
cl~s
(i)
The s e m i g r o u p {A ( e t B ) :
pletely (ii)
on H s u c h t h a t
pBp i s
then:
positive
t a O} e x t e n d s t o a w e a k l y c o n t i n u o u s s e m i g r o u p o f com-
n o r m a l maps on ~ ( A ( H ) ) n ( : B ( F ( H ) ) ) . P a strongly continuous self adjoint
There exists
contraction
s e m i g r o u p Gt on
F (H) (Z F(H)) such that ~ (A (etS)(')) ~ Gt~p(')G t, t ~ O. P P
References 1. 2. 3%. 56.
7. 8. 9. 10. 11. 12. 13. 1%.
1516. 17. 18.
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406
27. 28. 29.
30. 31. 32.
G.G. Emch: Commun. Math. Phys. 4 % 191-215 (1976). G.G. Emch: Acta. Phys. Austriacao Suppl XV, 79 (1976). G.G. Emch, S. Albeverio, J.-P. Eckmann: Rep. Math. Phys. 13, 73-79 (1978). D.E. Evans, J.T~ Lewis: J. Funct. Anal. 26, 369377 (1977). G. Lindblad: Rep° Math. Phys. 10, 393-~06 (1976). P. Vanheuverzwijn Ann Inst. H. Poincare (to appear). E.B. Davies: Commun. Math. Phys. 55, 231-258 (1978). D.E. Evans: C o m u n . Math. Phys. (to appear). D.E. Evans: Completely positive quasi-free maps on the Fermion algebra, in Mathematical Problems in the Theory of Quantum Irreversible Processes. ed. L. Accardi, V. Gorini: Proc. Naples Conference 1978 (to appear). D.E. Evans: Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra. Preprint 1978. ~ B . Davies: Rep. Ma~h. Phys. 111 169-188 (1977); J. Funct. Anal. (in press). F~B. Davies: Uniqueness of the standard form of the gemerator of a quantum dynamical semigroup. Preprint 1978.
Pseudospaces, pseudogroups and Pontriagin duality S.L. Woronowicz Department of Mathematical Methods in Physics University of Warsaw, Hoza 7#, 00-682 Warszawa Polska - Poland 0. Introduction The main result of the Gelfand Naima~k theory of commutative C ~algebras can be stated in the following way. The category of commutative C'-algebras with unity and unital C'-homomorphisms is dual to the category of compact topological spaces and continuous maps. It is not difficult to extend this theory in order to include locally compact topological spaces. To this end one has to consider C'-algeb~as without unity. If A is a locally compact topological space, then C.(A ) denotes the C~-algebra of all complex continuous functions on A
tending to 0 at infinity. In this case continuous maps from
AI
into A correspond to C'-homomorphisms from C~(A ) into C(A I ) satisfying certain condition (condition (C) of Section I) and the theory is comparable with that for compact spaces. No theory like those above is known for noncommutative C*-algebra~ However we can formally solve the problem considering the category dual to the suitable category of C*-algebras. This category will be called the category of pseudospaces; its objects are called pseudospaces and its morphisms - continuous mappings of pseudospaces. Clearly the category of pseudospaces includes the category of locally compact topological spaces. One should point out that any theorem concerning pseudospaces is in fact a theorem of the C'-algebra theory and nothing more. It means that using the category of pseudospaces we do not create a new theor~ we only introduce a new language to the C'-algebra theory. But this new language provides us new interesting questions and conjectures some solutions. For example in the last section we show that pseudospaces endowed with an associative algebraic rule form the adequate domain for considering generalized Pontriagin duality problems. The theory, we obtain this way should be considered as a C'-algebraic
408
version of the Kac algebra theory. I. The category of pseudospaces As it was mentioned in the introduction, in the theory of locally compact topological spaces both algebras C~(A ) and C(A ) play an important role. Therefore we have to find a noncommutative analog of the correspondence C~(A ) --~ the well known multiplier functor
C(A ). This problem is solved by A ---~ M(A).
To introduce the multiplier algebra of a C*-algebra A assume for the moment that A is embeded into B(H) : A c B(H). Then we put M(A) = { b ~ B(H): ab, bat A for all a E A} (I) Clearly M(A) is a C*-algebra and A is an ideal of M(A). Let
K
be a
C~-subalgebra of
M(A). We say that
K
contains an
approximate unity for A if the~e exists an increasing net (e~) of elements of K such that 0 ~ e~ ~ I and for any a ~ A, e~a and as~
converge in norm to
Using the definition
a. (I)
we can prove the following
Proposition Let A, B be C*-algebras and
~ : A
~ M(B)
be C*-homomorphism
satisfying the following condition: The image ~(A) contains an approximate unity for B Then there exists unique C'-homomorphism • ding
(C) : M(A)
, M(B) e x t e n -
~ .
In particular we see that any C*-isomorphism AI ~ ' A2 can be extended to the unique C*-isomorphism of the corresponding multiplier algebras. This fact shows that the algebra M(A) introduced by (I) is independent of the way we embed A into B(H). If ~ : A ~ ~(B) and ~ : B - - ~ M ( C ) are C*-homomorphisms satisfying condition (C) then we can construct C*-homomorphism y = ~m : A - - ~ M ( C ) . It turns out that satisfies condition (C). We shall consider a category ~ introduced in the following way. Objects of ~ are C'-algebras; ~ -morphisms (say from A into B) are C'-homomorphisms from A into M(B) satisfying condition (C). The composition of two morphisms ~ ~ Mor(A,B) and ~ ¢ Mor(B,C) is the morphism y ¢ Mor(A,C) introduced by the formula y = ~ , where ~: M(B) ---~M(C) is the unique extention of ~ .
409
The category $ dual to the category @ described above will be called the category of pseudospaces. Objects of $ will be called pseudospaces (or more precisely locally compact pseudospaces), and morplLisms of $ will be called mappings (or more precisely continuous mappings) We shall use the following notation. For any pseudospace A the corresponding C*-algebra will be denoted by C=(A ) and the multiplier algebra M ( C = ( A ) ) by C ( A ) . If r : A ' As , then the corresponding ~-morphism will be denoted by r* : C (At) --~ C(A ). A pseudospace A is said to be finite (finitedimensional, compact) if the corresponding algebra is finitedimensional (finitely generated, contains unity resp.) For every pseudospace A the identity mapping A - - ~ A will be denoted by id (or more precisely by id A ). We have id*(a) = a for any a ¢ C~(A). 2. The cartesian product of pseudospaces Let us remind the definition of the tensor product of C*-algebras. If C*-algeb~as A1 and A2 are embeded in B(HI) and B(H2) resp. then . A I @ A 2 is the C*-algebra generated by operators of the form a 1 @ a 2 ¢ B ( H I $ H 2) , where ak~ Ak, k = 1,2. It is known that this algebra is independent of the choice of the embedings Ak C B(Hk). Moreover if ~ are C*-homomorphisms from A k intq Bk then there exists ur~i'que C*-homomorphism ~1@ m2 from A I @ A 2 into B I @ B 2 such that m l ~ Z ( a I ~ 2 ) = m1(al)@m2(a2) for all ak¢ Ak • Using
(I)
one can easily show that
M(B I)@ M(B 2) for any O*-algebras
BI
c
~(B 1® B 2) and
(2)
B2 •
Now, let AS and A2 be pseudospaces. We define their cartesian product At× A2 as the pseudospace corresponding to the tensor produkt of the C*-algebras associated with A I and A2 :
C.(As× A2) = C=(As)® C=(A2) The canonical projections Pk : AI× A2 ' Ak (k = 1,2) can be introduced as mappings corresponding to the natural embedings p~ : C~(Ak) ~ C(AI× A2) , where p~(al) = a l @ I
al E C ~ A 1 )
410 p2(a2) = l @ a 2
a2E C.(A2)
If rk are mappings from Ak into A~ (k = 1,2), then r~ are C'-homomorphisms from C.(A~) into C(Ak) and r~@r~ is a C'homomorphism from C.(A'sx A~) into C(As)@ C(A2). According to (2) the latter algebra is contained in C(Asx A2) and r ~ @ r~ considered as C'-homomorphism from C.(A~x A 2) into C(Asx A 2) satisfies the condition (C). The mapping from Asx A 2 into A~× A~ corresponding to
r~@r~
will be denoted by
rsx r 2 : Asx A 2 --~ A~× A2
(rsx r2)* = r~@ r~ One c a n p r o v e that r l x r 2 into A~x A'2 such that
is the only mapping from
p ~• ( r h x z' 2) = r ~ where Pk : A l x A2 pro jections
---* Ak
k =
and
Pk : A'~x ~2
-~ A~
Asx A2
1,2,
are canonical
The following statement seems to be interesting: Proposition Let A be a pseudospace. Then a mapping that
pkd = id
space
i.e.
(k = 1,2) C.(A )
d : A
exists if and only if
A
, Ax A
such
is the usual
is commutative.
It means that for pseudospaces (not being spaces) the diagonal map does not exist. To understand better the ideas used in the next section we shall discuss the following result: Proposition Let A and
AI be pseudospaces. We assume that A
is finite
and As is finitedimensional and compact. Then there exist a finite dimensional compact pseudospace A2 and a mapping s : A2x A --~ A S such that for any pseudospace A S and any mapping r : A3x A ~ As there exists unique mapping t : As , A2 such that r = s(tx id). To explain, why this result is interesting assume for the moment that we deal with the usual spaces. Then the mapping s from A2x A into AI can be considered as a family of mappings from A into A S indexed by elements of A2 . In the general case we shall speak about pseudofamily of mappings from A into As . The existence of t means that this pseudofamiiy is universal ioe. contains all mappings and the uniqueness of t means that e v e r y mapping appears in the pseudofamily only once. Therefore A2 can be considered as a pseudo-
411
space of all mappings from Example
: Let
A
complicated object:
= A~ C~(A~)
into A I
.
= two point set. Then
A2 is a quite
is the C*-algebra of all continuous
mapping~
a : [0,1]
Here
denotes the algebra of all complex
M2
: M~
A
such that
a(0)
and
a(1)
2 x 2
are diagonal.
matrices.
3. Pseudog~oups and duality Let S I = { z ¢ C : IzI = I} me that u is a ~ I t a r y element of
and A C(A ).
be a pseudospace. AssuThen, using the functio-
nal calculus of normal operators we have C*-homomorphism C(S I) ~ f
~ : fCu~ ~ C(A )
which in turn corresponds to some mapping from way we obtain from A
into
I - I
A
S I and the set of all unitaries in
correspondin~
to
u
into
S I. This
correspondence between the set of all mappings will be denoted by
C(A ). The mapping
u . W e shall use this cor-
respondence to define pointwise product of mappings into
SI. Let
u, ul, u z : ~ A - - ~ S I. We say that u is the pointwise ~product of mappings Ul and u2 and write u = ulu 2 if ~ = UlU m . This pointwise multiplication is clearly induced by the usual multiplication in
S I. Note however, that pointwise multiplication
in the set of mappings into
S ~ is not abelian in general.
Definition Let
G
be a pseudospace and ~
(G,~) is a pseudo-semigroup if ~ ty law: (idx ~)~ = (@x id)~ .
: GxG
--~ G . We say that
obeys the following associativi-
Assume that (G,~) is a pseudo-semigroup, A is a pseudospace and u : Ax G ~ S I. Denoting by Pl and P2 the canonical projections G x G ---~ G we consider the following three mappings: uo = u(idx ~), Ul = u(idx Pl) and uz = u(idx Pz) from Ax Gx G into S I. We say that (A ,u) is a pseudofamily of characters of (G,~) if
uo
is the pointwise product of
ul
and
u2
:
u(idx ~) = ~(id× Pl) u(idx p2)
(3)
Definition Let (G,~) be a pseudo-semigroup and (~,v) be a pseudofamily of characters of (G,~). We say that (~,v) is the pseudospace of all characters of (G,~) if for any pseudofamily c (A ,u) cters of (G,~) there exists unique mapping r : A ~~ u = v(rx id).
of charasuch that
412
Assume now that (G,~) is a pseudosemigroup and that (~,v) is t~e pseudospace of all characters of (G,~). We denote by ~I and ~2 the canonical projections ~x ~ ~ ~ . Let v I = v(~Ix id) , v 2 = v(~2x id) and v o = VlV 2 (pointwise product) . Clearly v k : ~ x ~x G ---~ S I for k = 1,2,0. It turns out that (~x ~ , ~ ) is a pseudofamily of characters of (G,~) . According to the previous definition there exists unique mapping $ : GX ~ ~ ~ such that v($x id) = V(~lX id) v(~2x id) (~) Remembering that the pointwise multiplication is associative one can easily show that (~,$) is a pseudo-semigroup. This pseudo-semigroup is said to be dual to (G,~).
G
Now, let s : Gx ~ L, ~X G be the natural mapping permuting and ~ (i.e. s ' ( a @ b ) = b @ a). It follows immediately from (@)
that
vs(id×
= vs(id× Pl) vs(id×
It means that (G,vs) is a pseudofamily of characters of (~,$). Therefore, assuming that the double dual ~ , ~ ) exists we have the unique mapping i ~ G - - ~ such that vs = ~(iX id). The main problem of the theory consists in the following: What additional assumptions one has to impose on (G,~) in order to prov e the existence of the dual and double dual pseudosemigroups and t o prove that i is an isomor@]~imm. We belive that these assumptions will distinguish pseudogroups among pseudo-semigroups. For the moment the answer is known for finite pseudogroups and partially for compact pseudogroups. We also have the following THEOREM Assume that G is a locally compact topological group. Then the dual and double dual of G exist and double dual of G is isomorphic to G (the isomorphism is given by the mapping i introduced above). ~ is a usual space if and only if G is abelian and in this case we have the usual Pontriagin duality.
