Lecture Notes in
Physics
Edited by H. Araki, Kyoto, J.Ehlers, MLinchen,K. Hepp,ZSrich R. Kippenhahn,MSnchen,H.A. WeidenmSIler, Heidelberg J. Wess, Karlsruheand J. Zittartz, K61n Managing Editor: W. BeiglbSck
280 Field Theory, Quantum Gravity and Strings II Proceedings of a Seminar Series Held at DAPHE, Observatoire de Meudon, and LPTHE, Universit~ Pierre et Marie Curie, Paris, Between October 1985 and October 1986
Edited by H.J. de Vega and N. S&nchez
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Editors H, J. de Vega Universit~ Pierre et Marie Curie, L.P.T.H.E. Tour 16, ler 6rage, 4, place Jussieu, F-75230 Paris Cedex, France N. S~.nchez Observatoire de Paris, Section d'Astrophysique de Meudon 5, place Jules Janssen, F-92195 Meudon Principal Cedex, France
ISBN 3-540-17925-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-17925-9 Springer-Verlag NewYork Berlin Heidelberg
This work is subject to copyright. All rights are reserved,whether the whole or part of the material is concerned, specif{callythe rights of translation,reprinting, re-useof illustrations,recitation, broadcasting, reproduction on microfilmsor in other ways, and storage in data banks. Duplication of this publicationor parts thereof is only permitted under the provisionsof the German Copyright Law of .September9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violationsfall under the prosecution act of the German Copyright Law. © Springer-VerlagBerlin Heidelberg 1987 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr.; Bookbinding: J. Sch~.fferGmbH & Co. KG., GrSnstadt 215313140-543210
PREFACE
This book contains the lectures delivered in the third year, Paris-Meudon
1985 - 1986, of the
Seminar Series.
A seminar series on current developments
in mathematical
physics was started in
the Paris region in October 1983. The seminars are held alternately at the DAPHEObservatoire
de Meudon and LPTHE-Universit&
encourage theoretical
physicists
ticians to meet regularly. October
1983 - October
Pierre et Marie Curie (Paris VI) to
of different
disciplines
and a number of mathema-
The seminars delivered in this series in the periods
1984 and October
published by Springer-Verlag
1984 - October 1985 have already been
as Lecture Notes in Physics,
volumes 226 and 246,
respectively.
The present volume "Field Theory, the lectures delivered up to October
Quantum Gravity and Strings,
topics of current interest in field and particle theory, mechanics.
Basic problems of string and superstrin~
porary perspective
and quantum field theoretical
cosmology are presented.
cosmology and statistical
theory are treated in a contem-
as well as string approaches
Recent progress on integrable
in two, four and more dimensions
II" accounts for
1986. This set of lectures contains selected
is reviewed.
It is a pleasure to thank all the speakers for their successful delivering comprehensive
and stimulating lectures.
for their interest and for their stimulating Scientific
Direction "Math~matiques
toire de Paris-Meudon
efforts in
We thank all the participants
discussions.
We particularly
- Physique de Base" of C.N.R.S.
for the financial
extend our appreciation
to
theories and related subjects
thank the
and the Observa-
support which made this series possible.
to Springer-Verlag
for their cooperation and efficiency in
publishing these proceedings.
CERN, Geneva, F e b r u a r y 1987
We
H. DE VEGA N. SANCHEZ
CONTENTS
P. Di Vecchia:
Covariant Quantization of the Bosonic String: Free Theory ....
I.A. Batalin and E.S. Fradkin:
Operatorial Quantization of Dynamical Systems
with Irreducible First and Second Class Constraints
.......................
11
Kaluza-Klein Approach to Superstrings ............................
19
Non Linear Effects in Quantum Gravity ..............................
41
Our Universe as an Attractor in a Superstring Model ...............
51
M.J. Duff:
I. Moss:
K. Maeda:
J. Audretsch:
B. Allen:
Mutually Interacting Quantum Fields in Curved Space-Times
.....
Gravitons in De Sitter Space ......................................
D.D. Harari:
R.S. Ward:
Effects of Graviton Production in inflationary Cosmology .......
Multi-Dimensional
N.J. Hitchin:
J. Isenberg:
J. bukierski:
82
97
106
Monopole and Vortex Scattering ................................
117
The Ambitwistor Program ........................................
125
Supersymmetric Extension of Twistor Formalism .................
137
M.A. Semenov-Tian-Shansky:
Supersymmetries of the Dyon ........................
Classical r-Matrices,
Groups and Dressing Transformations
M. Karowski:
68
integrable Systems .............................
E. D'Hoker and L. Vinet:
A.M.
I
Lax Equations,
156
Poisson Lie
.......................................
174
On Monte Carlo Simulations of Random Loops and Surfaces ........ 215
Nemirovsky:
Field Theoretic Methods in Critical Phenomena with
Boundaries ................................................................
229
COVARIANT BOSONIC
QUANTIZATION
STRING:
OF THE
FREE T H E O R Y
P. Di V e c c h i a Nordita,
Blegdamsvej
The b o s o n i c
string
S[x~(~,~),
g~B(~,~)]
t h a t is c l a s s i c a l l y variant
under
on
x~
6x ~ = E ~ ~
8~
a c t i o n I)
g~B~ x . ~ B x
to the N a m b u - G o t o
(I)
a c t i o n 2) and is in-
of the c o o r d i n a t e s induces
of the w o r l d
the f o l l o w i n g
trans-
gee
x~ + ~ y gab
s Y gy ~
+ ~ 8
are two a r b i t r a r y
The a c t i o n
by the f o l l o w i n g
A reparametrization
and
Copenhagen
= _ ~T ~dT ~d~ ~ 0
equivalent
= 87 ~ 6g~B where
is d e s c r i b e d
reparametrizations
s h e e t of the string. formations
17, D K - 2 1 0 0
sY 6
functions
(i) is in a d d i t i o n
(2) gay
of
also
T
and
invariant
~ . u n d e r Weyl
transfor-
mations : 6x ~ = 0 where
6g~8 = 2A(T,O)
A(Y,o) The
is an a r b i t r a r y
invariances
the c o m p o n e n t s However the q u a n t u m
the W e y l
of
(3)
(T,o).
(3) are s u f f i c i e n t
to g a u g e away
all
tensor.
invariance
c a n n o t be in g e n e r a l
maintained
in
theory.
Therefore formal
(2) and
of the m e t r i c
function
g~8
in the q u a n t i z a t i o n
gauge characterized
of
(i) we can o n l y fix the con-
by the f o l l o w i n g
c h o i c e of the m e t r i c
ten-
sor :
gas = P(~)
~B
;
~iI = - ~00 = 1
(4)
where
p(~) Since
critical
is an a r b i t r a r y in w h a t
dimension
choose
p(~)
D = 26
= 1
in
where
the
, where
gauge
second
obtained
ghost
coordinate
of
~
~ (T,o).
we will
the W e y l
the a c t i o n
J0
2--9
minant
dinate
however,
consider anomaly
only
the c a s e
of
v a n i s h e s 3) , we can
(4).
In the c o n f o r m a l
-
function
follows,
e
term
b
(i) b e c o m e s 4 ) :
B
(5)
is the c o n t r i b u t i o n
from having ca
and
fixed
of the F a d d e e v - P o p o v
the c o n f o r m a l
a symmetric
and
gauge.
traceless
deter-
It c o n t a i n s
antighost
a
coor-
b °B
The
conformal
We can
still
gauge.
They
functions
~
are
~(~)
-
=
c
0
+
~
not
Y
fix c o m p l e t e l y
that
transformations
leave
characterized
by two
the c o n d i t i o n :
eY = 0
(6)
+
;
~-
=
~
-+ o
~
;
+
=~
1 /~ ~
~ ~
+ ~
~
the e q u a t i o n s
the gauge.
in the c o n f o r m a l
coordinates 1
c
qeB
(4) does
transformations
satisfying
In the l i g h t - c o n e +
choice
gauge
the c o n f o r m a l
eB + 8Be~
E-
gauge
perform
(6) g e t
the
simple
(7)
/
form:
(8)
2 + ~- = 2 + s- = 0 implying
that
e+[e -]
It is easy formal
is o n l y
to c h e c k
transformations
conformal
fields
with
a function
that Lagrangian
provided conformal
that
x
U dimension
of (5)
~+[~-] is i n v a r i a n t
, b A
and equal
c
under
transform to
0,2
conas
and
-i
respectively. In a c o n f o r m a l (5) w h e r e venient
invariant
the o - v a r i a b l e
to use,
instead
theory
Z = e i(T+O) that
in e u c l i d e a n
as the o n e
described
v a r i e s in a f i n i t e d o m a i n + of ~- , the two v a r i a b l e s
(0,~)
by action it is con-
z = ei(T--O) space
(T ÷ iT)
become
one
(9) the c o m p l e x
conjugate
of the other. A conformal traceless
invariant
energy-momentum
theory tensor
is c h a r a c t e r i z e d with
only
two
by a conserved
independent
and
components
and
T(z)
T(z)
A conformal under
field
a conformal
¢
with
(A,Z)
dimension
transforms
as f o l l o w s
transformation:
[
] [
E'
']
(i0)
~z
For
the
sake
pendence z
can
also
The
¢(~)
=
in the
in m i n d
following
the de-
that w h a t e v e r
we
do w i t h
is o b t a i n e d
(OPE)
~/~ ~(~)- + z-~
tensor
of
by
T(z)
r e q u i r i n g the 5) with ~ :
~(~) + regular (z_~) 2
T(z)
following
terms
is a c o n f o r m a l
(ii)
tensor
with
A = 2 .
implies5) : T(z)
T(~)
additional
-
generators
the From
[L n
where
of
integral (12)
b = b zz
in terms
is d e f i n e d
it f o l l o w s
=
8x.~x
8 -- ~
t e r m can
the c o n f o r m a l of
terms
in g e n e r a l algebra.
be a d d e d
(12)
with-
The V i r a s o r o
T(z) :
(13)
(n-m)
1 + ~
b
they
a way
satisfy
c L n + m + ~-~ n(n2-1) z
and
z
that
dz ~-- = 1 .
the V i r a s o r o
algebra:
(14)
~n+m;0
the L a g r a n g i a n
corresponding
to
to:
' ~ ;
in such
that
of the v a r i a b l e s
N
c/______~2+ reg. (z_~)4
z n+l T(z)
is p r o p o r t i o n a l
L
T(~) + (z_~) 2
singular c - n u m b e r
can be c o n s t r u c t e d
, L m]
In terms
+ 2
the c l o s u r e
L n = % dz
where
8/~ T(~) z-~
more
out destroying
(5)
(i0)
expansion
energy-momentum
This
An
omit
be d o n e w i t h
product
T(z)
we w i l l
z , keeping
transformation
operator
The
of s i m p l i c i t y
on the v a r i a b l e
(b ~ c
~~z =
b zz
+ b
8c)
(15)
and
;
c = c
z
,
~ = c
z
(16)
Since
xU , b
and
-i
and
c
transform
respectively,
6L = ~[e(z)L] implying
that
The lowing
L
fields
with
A = 0,2
is a c o n f o r m a l
density:
(17)
the c o r r e s p o n d i n g
energy-momentum = TX(z)
that
+ ~[~(z)L]
transformations
T(z)
as c o n f o r m a l
it f o l l o w s
on
action
x
tensor
, b
is c o n f o r m a l
and
c
invariant.
are g e n e r a t e d
by the
fol-
:
+ Tg(z)
(18)
where
as
TX(z)
= - yl.{ .k~
xh] 2 :
Tg(z)
=
+ 2c'b
: cb'
it can be
They
:
seen by u s i n g
<xU(z)
x~(~)>
= - g~
c(O>
1 = Z-~
allow
(19)
one
(20)
the
following
contraction
rules:
log(z-~)
(21)
(22)
to c o m p u t e
aiso
the O P E w i t h
two e n e r g y - m o m e n t u m
ten-
sors: D-26 T(z)
T(~)
~/~ =
implying
that
critical
dimension
the c - n u m b e r
As p r e v i o u s l y tities
that
depend
on
taining
for i n s t a n c e
the V i r a s o r o
(23)
algebra
we have In the
limited
case
is v a n i s h i n g
sets
of m u t u a l l y
string
to c o n v i n c e
=
analysis
that
at the
commuting
of the
string
to the q u a n -
string
depend
it is c o n v e n i e n t
of the end p o i n t s c ( z = e iT)
our
of a c l o s e d
for the q u a n t i t i e s
of an o p e n
that
It is e a s y
two
(z_~) 4
.
z .
everything
the p a r a m e t r i z a t i o n implies
of
D = 26
repeat
In the c a s e
+ 2 T(~_____J__} + (z_~) 2
explained
however
This
T(~) z-~
we
on
can z
Virasoro
algebras.
to r e q u i r e is l e f t
ob-
that
unchanged.
~(z=e i~)
oneself
that
In the t r e a t m e n t of the g h o s t we f o l l o w F r i e d a n , M a r t i n e c and S h e n k e r 6) .
for an o p e n
closely
string
we can
the a p p r o a c h
of
use all the p r e v i o u s
formulas with
z = e i~
In the following we
limit for s i m p l i c i t y our c o n s i d e r a t i o n s to this case. Having fixed the c o n f o r m a l gauge we have lost the general invariance
(2) keeping only the i n v a r i a n c e under c o n f o r m a l t r a n s f o r m a -
tions. On the other hand we have gained the invariance under BRST transformations,
that act as follows on the c o o r d i n a t e s of the string:
~X = ICX' ~b = - 21x'
+ l[cb' + 2c'b]
(24)
6c = Icc' where
1
is a c o n s t a n t G r a s s m a n n parameter.
The v a r i a t i o n of L a g r a n g i a n
(15) under the t r a n s f o r m a t i o n s
(24)
is a total d e r i v a t i v e 6L = ~[IcL]
(25)
i m p l y i n g the invariance of the c o r r e s p o n d i n g action. It is easy to see that the product of two t r a n s f o r m a t i o n s
(24) is
i d e n t i c a l l y vanishing. The g e n e r a t o r of the t r a n s f o r m a t i o n s Q = } dz:c(z)
[TX(z)
By using the c o n t r a c t i o n s
(24) is the BRST charge:
+ 1 Tg(z)] :
(21) and
(26)
(22) it can be shown after some
c a l c u l a t i o n that: Q2
1 $ c' ''( = 2-4 (D-26) ] d~ ~) c(~
T h e r e f o r e the q u a n t u m BRST charge is n i l p o t e n t only if
(27)
D = 26
This
implies that our q u a n t i z a t i o n p r o c e d u r e is c o n s i s t e n t only for the critical d i m e n s i o n
D = 26
In this case the BRST charge commutes with the V i r a s o r o generators:
for any
n .
In c o n c l u s i o n if
D = 26
the gauge fixed action
(5) is i n v a r i a n t
under two i n d e p e n d e n t and very important transformations: c o n f o r m a l transformations.
BRST and
It is useful harmonic
to expand
oscillators.
x
(z)
, b(z)
and
c(z)
in terms of the
They are given by: co
X (Z)= q c(z)
=
coz
ip iogz + i n:[I ~nl
~ < l-n CnZ n= 1
+
c
+ n
z
l+n)
(29
co b° + zz
b(z)
The oscillators
~
z-2-n + b + z n-2) n
n
n= 1
satisfy
+ ]=6 am;~
an;~
the following
(anti)-commutation
[q~, pv] =
n,m g ~
relations:
i g~
(30 {c n , b+} = 6n, m
The other
(anti)-commutators
The mode expansion
are vanishing.
for the coordinate
x
(z)
gives
the contrac-
tion:
<0 I x'
Integrating tion
(z)
x'~ (~) I0> =
both sides of
constants
g~ 2 (z-~)
(31) and setting
we get the contraction
In order to derive some more discussion We can introduce
(31
the contraction
to zero the two integra-
(21). (22) from the mode expansion
is needed. the ghost number
current (32
j(z) = :C(Z) b(z): Using the contraction
j (z) j (~) =
Tg(z)
(22) it is easy to show the following
i 2 (z-~)
j(~) = ~ / ~ j(~) z-~
OPE's: (33
+
j(~) (z-~) 2
3 (z-~) 3
Because of the extra term j(~) is not quite a conformal A = 1 . In terms of the mode expansion defined by
(34)
field with
Jn = [ n+l n z
j (z)
(33)
They
and
(34)
L
' Jm
Jm
' Jn]
can
be
(35)
i mp ly :
- m
3n+m
- ~ n(n+l)
(36)
~n+m;0
(37)
= n 6n+m;0
checked
directly
using
the
expansion
in h a r m o n i c
oscilla-
tors: Jm = [ k
: Cm_ k b k
Lg = ~ n m
(n+m)
: b
:
= c
c n b_n
:
n
(38)
and
the
(39)
more
commutator
number
b_]nl
b
(39)
and
~ b +in I
-n
if
n _< 1
if
n > 2
the
one
J0
complicated
c
-n
n
gets
that
J-i
.+ = - 31
lq>
(36)
for
is n o t
(42)
= qlq>
implies ~
is de-
n = ±I
and
antihermitian
as
m = ±i Jl
implies
, but
that
it s a t i s f i e s
the the
relation:
is an e i g e n s t a t e
j01q>
ordering
(41)
,+ J0 + 30 - 3 = 0
If
normal
(40)
+ L1 = L-I
ghost
:
m
:
= - b
and
c
n-m
C_inl ~ C l+ n I a n d as f o l l o w s
where fined
From
(38)
(42)
of
the
ghost
number (43)
that
6 q ; 3 _ q,
(44)
A state with
ghost
number
q
satisfies
bnlq>
= 0
if
n > q-2
Cnlq>
= 0
if
n ~ - q+2
the r e l a t i o n s :
(45)
that
imply 1 = ~ q(q-3) lq>
L01q>
Using
(39)
eigenstate jective
and
(45)
of
J0
subgroup
(46)
it is p o s s i b l e that
to s h o w
is a n n i h i l a t e d
of the V i r a s o r o
that
lq = 0>
is the o n l y
b y the g e n e r a t o r s
of the pro-
algebra:
+ L01 q = 0> = LII q = 0> = LII q = 0> = 0 lq = 0>
is t h e r e f o r e
After cannot ket
these
get
a non v a n i s h i n g
state,
to get
whose
(22)
as it can be Using
ghost
we m u s t
terms
projective
considerations
b(z) simply
the m o d e
in
(22)
differs
by
that,
unless 3 .
because we use
of
a bra
In p a r t i c u l a r
(44),
we
and a in o r d e r
compute: c(~)
lq = 0>
shown
by u s i n g
expansion
of the o s c i l l a t o r s .
Cn
invariant. it is c l e a r
result
number
(47)
-n
(48) (45).
w e can
compute
It is g i v e n
the B R S T
charge
Q
in
by:
n
0J + ~
(49)
n=l where co
co
m c
In+
n,m=l
(n+2m)
c m+ b n + m
- Cn c m
n+m
Cm+ C n + m b + n + C n+ + m
- 2b 0
nc n c n
(5o)
Cmb n
n,m=l From
(49) Q
it f o l l o w s
lq =
In the B R S T ground ever
o>
the
state
lq = 0>
is also
BRST
invariant:
0
quantization
keeping
the
=
that
(51)
one
the m a n i f e s t
space, in w h i c h
treats Lorentz
the s y s t e m
all L o r e n t z invariance
components of the
on e q u a l
theory.
is q u a n t i z e d , c o n t a i n s
states
Howwith
negative must
norm.
require
space.
In o r d e r
that
Its
to c o n s t r u c t
the p h y s i c a l
elements
a consistent
states
span
are c h a r a c t e r i z e d
by
quantum
a positive
theory
definite
the v a n i s h i n g
we
sub-
of the B R S T
charge: Q
IPhys>
Because
= 0
of the n i l p o t e n c y
is a s o l u t i o n II>
is
(51)
type
restrict
known
LnI~>
The
kq = l>
directly
= c~lq
n o t by
used
The m o s t
where
the
that
excitation
(51)
of
reduces
of the
state
to
lq = 0>
oscillators
and by
by
b0 ,
over
(29),
z = 0
the
, is w e l l
on w h i c h
A
lrn
states
of eqs.
integer very
the
annihilated
by
(53)
in t e r m s
for
D = 26
of the
is p r o v i d e s
following
operator: (55)
the of
is n o t
modes,
subalgebra.
defined
Because
the o r b i t a l
24 t r a n s v e r s e
the
log
integral defined acts,
in
z
(55),
only
directions,
appearing
if we
to s a t i s f y
that
in
that
are
e ik'x(z)
is p e r f o r m e d
constraint
,
around
the m o m e n t u m
the r e l a t i o n
• k = - n
an
identi-
states7) :
unless
physical
runs
state,
Two are
no g h o s t
to see
in t e r m s
that,
solution
s t a t e s 8)
i
in
with
with
annihilation
index
see
p
all
• e~ e ik'x(z)
as o n e
the
are
(54)
= ~ dz x' ]
k
of
is g i v e n
to n o t i c e
general
to
can
states
.
(49).
to c o n s t r u c t
orthogonal
the o r i g i n
('52) and
(53)
of the p r o j e c t i v e
transverse
the
I~> + QII>
cO
the g e n e r a t o r s
Ai;n
form
satisfies
the p h y s i c a l Q
it is e a s y
from
by
It is i m p o r t a n t
the
the
= o>
but
by
of
on the p h y s i c a l
lq = i>
it is a n n i h i l a t e d
(54)
words
of
itself
I~> = 0
and
state
state I~>
to s t a t e s
I~> a
conditions
state
any
classes
ourselves
=(n0-1)
it f o l l o w s
Q
state
In o t h e r
lq = l>b, c 8
the w e l l
of
if the
the c o h o m o l o g y
If we
as
of
arbitrary.
fied w i t h
the
(52)
(56) n
.
important
following.
They
properties commute
of the
with
transverse
the o p e r a t o r s
operator Lm
:
(55)
10
L m , An;i] for any integer
= m
0
(57)
and they satisfy the algebra of a non relativistic
harmonic oscillator 9) An, i , Am, j] = n 6ij 6n+m; 0
(58)
as it can be shown by using the contraction In terms of
(31).
(55) we can construct a complete
in the space of physical
states,
and orthogonal
basis
that is given by:
A i n ; -N n 10,p>
n
(59)
n where
N > 0 . n The states (59)
lows from
satisfy
the physical
conditions
(57) and span a positive definite
the s u b s p a c e o f p h y s i c a l
states
is
ghost
space.
(53)
as
it
fol-
This implies that
f r e e 10) .
ACKNOWLEDGEMENTS I wish to thank J. L. Petersen
for many useful discussions
on
BRST quantization.
REFERENCES i) 2) 3) 4)
5) 6) 7) 8) 9) i0)
L. Brink, P° Di Vecchia and P. Howe, Phys. Lett. 65B (1976) 471 S. Deser and B. Zumino, Phys. Lett. 65B (1976) 369 Y. Nambu, Lectures at the Copenhagen Symposium, 1970, unpublished T. Goto, Progr. Theor. Phys. 46 (1971) 1560 A. M. Polyakov, Phys. Lett. 103B (1981) 502 D. Friedan, "Introduction to Polyakov's String Theory" in Recent A d v a n c e d in Field Theory and Statistical Mechanics (Les Houches 1982) M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443 S. Hwang, Phys. Rev. D28 (1983) 2614 A. A. Belavin, A. M. Polyakov and A B. Zamolodchikov, Nucl. Phys. B241 (1984) 333 D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271 (1986) 93 E. Del Giudice and P. Di Vecchia, Nuovo Cimento 7OA (1970) 579 E. Del Giudice, P. Di Vecchia and S. Fubini, Annals of Physics 70 (1972) 378 R. Brower and P. Goddard, Nucl. Phys. B40 (1972) 437 R. C. Brower, Phys. Rev. D6 (1972) 1655 P. Goddard and C. B. Thorn, Phys. Lett. 40B (1972) 235
OPERATORIAL QUANTIZATION OF DYNAMICAL SYSTEMS WITH IRREDUCIBLE FIRST AND SECOND CLASS CONSTRAINTS
I.A. Batalin and E.S. Fradkin Lebedev Physical I n s t i t u t e , Moscow
Abstract Operatorial version is suggested of the generalized canonical quantization method of dynamical systems subjected to irreducible f i r s t
and second class constraints. An
operatorial analog of classical Dirac brackets is realized. Generating equations for generalized algebra of f i r s t
and second class constraints, as well as f o r the unitar-
izing Hamiltonian are formulated. In the f i r s t
class constraint sector new generating
equations are presented d i r e c t l y in terms of operatorial Dirac brackets. Introduction During recent years a method of generalized canonical quantization of constrained dynamical systems has been being developed in the works of the group of authors [ I - 1 0 ] . The cornerstone of the method is the idea [ I ] that constrained systems admit canonical commutation relations in an extended phase space which includes, along with the i n i t i a l
variables, also dynamically active Lagrange m u l t i p l i e r s and ghosts.
The physical u n i t a r i t y and gauge independence are provided within t h i s approach via dynamical compensation of the contributions of Lagrange m u l t i p l i e r s and ghosts, f o r which p o s s i b i l i t y t h e i r opposite s t a t i s t i c s is responsible. Until recently t h i s idea was d i r e c t l y applied as a matter of fact only to the f i r s t
class constraints. The
second class constraints were handled by using canonical measure on the corresponding hypersurface [11] in the path integral and the Dirac brackets in the generating equations of the gauge algebra [4,5]. The lack of a relevant formal scheme that would admit the use of canonical commutation relations in the case when second class constraints are present too, was a serious obstacle in r e a l i z i n g the program of opera t o r i a l quantization in the most general case. In our previous work [12] t h i s obstacle was overcome, and an operatorial version of the method of generalized canonical quantization of dynamical systems subject to second class constraints was formulated. The goal of the present paper is to include a more general case into the framework of the work [12] when f i r s t class constraints are also i n i t i a l l y In the present context both f i r s t
present.
and second class constraints are assumed to be
l i n e a r l y independent ( i r r e d u c i b l e ) .
12
Designations. The same as in our previous works ~(A) designates the Grassmann p a r i t y of the q u a n t i t y A. The supercommutator of operators A and B is defined as [A,B] ~ AB - BA(-I) c ( A ) E ( B ) ,
(0.1)
We w r i t e every canonical p a i r (momentum and co-ordinate) (PA,QA), e(PA ) = c(QA), A = I . . . . .
as
N,
(0.2)
so t h a t the only nonzero equal-time supercommutators f o r them are [QA, PB] = i ~
,
(0.3)
I. Generating Equations of Generalized Algebra of Constraints Let
(pi,qi), be i n i t i a l
~(pi ) = c(qi),
i = 1 .....
n
(1.1)
pairs of c a n o n i c a l l y conjugate operators.
Let a dynamical system be given
in the phase space (1.1) with the Hamiltonian Ho = Ho(p,q), E(H o) = 0 irreducible first Ta' = T ~( p , q ) ,
,
(1.2)
class c o n s t r a i n t s c(T ~) ~ ~a' . a . = I, .
. ..
m'
(1.3)
2m".
(1.4)
and i r r e d u c i b l e second class c o n s t r a i n t s T~ = T~(p,q), " ~ ( "T~ )
~ ~ ",
~ = I , ....
Consider a p a i r of c a n o n i c a l l y conjugate ghost operators f o r each c o n s t r a i n t ( 1 . 3 ) , ( 1 . 4 ) , whose s t a t i s t i c s
is opposite to t h a t of the corresponding c o n s t r a i n t
(,~D~,c,a), ~(~I)~) = E(C ,a) : ~
+ I, a = I . . . . .
m',
(1.5)
(, ~ "~, C''~) , ~(,~ . ) =. ~(C . ''~) . = ~
+ 1, ~ = 1 . . . . .
2m".
(1.6)
Initial
canonical pairs (1.1) form together with the canonical ghost pairs ( 1 . 5 ) ,
( 1 . 6 ) , the s o - c a l l e d minimal sector. Let us a t t r i b u t e some inner c h a r a c t e r i s t i c values to these operators, c a l l e d the ghost numbers. Consider two c]asses of the
13 ghost numbers, (gh I ) and (gh") following the d i v i s i o n of the f u l l set of constraints into those Of f i r s t
and second class:
gh'(q) = -gh'(p) = 0,
(1.7)
gh"(q) = -gh"(p) = 0,
gh'(C') = -gh'(,.~') = 1,
gh"(C') : - g h " ( , ~ ' )
= O,
(1.8)
gh'(C") = -gh'(,5~ ' ' ) : O,
gh"(C II) : - g h l ' ( ~ 'l) : I.
(1.9)
By d e f i n i t i o n , we have f o r every operator having a ghost number gh'(AB) = gh'(A) + gh'(B),
gh"(AB) = gh"(A) + gh"(B).
(1.10)
Consider the following operatorlal equations in the minimal sector I (1.1), (1.5), (1.6) = i41'~,,~,,B,
1.11)
[~"~,o"] = 0, [ n " % ~ ''6] = 0,
1.12)
[n~"]
S(~") = 1, g h ' ( ~ " )
= O, g h " ( n " )
1.13)
= I,
c(fl ''~) = E~" + I , gh'(~ ''~) = O, gh"(~ ''~) = I
1.14)
where meB is a c-numerical inversible matrix, such that II
~(%o) = ~ + ~ ,
~
lJ
(-I)
1.15)
Solution of equations (1.11-1.14) f o r operators ~", ~"~ is looked for in the form o f ~ C - n o r m a l ordered ( i . e . with e v e r y ~ ' , ~ '
placed to the l e f t of every
C',C") 2 series in powers of the ghost operators (1.5), (1.6), the f i r s t term in the
For the sake of u n i v e r s a l i t y and generality we admit here that the generating operators o", ~"~ of the algebra of second class constraints may depend on the ghosts (1.5) of the f i r s t
class constraint sector. Note, however, that there always exists
a solution of the generating equations (1.11-1.14) which does not depend on operator (1.5) and is quite s u f f i c i e n t f o r us. • The same as in our previous works on operatorial quantization we are using the ~C-normal form f o r the ghost operators. Certainly, we might e x p l o i t instead any other normal ordering, e.g. C~-ordering, or the Weyl ordering, since a l l the normal orderings may be related to one another using the canonical commutation relations.
14 expansion of the operator ~" being T~C~. Substituting the~C-expansions of the operators ~", ~"~ into equations (1.11), (1.12), and reducing t h e i r l e f t - and r i g h t hand sides to~)C-normal form we obtain a sequence of relations f o r the c o e f f i c i e n t operators to be solved step by step. In t h i s way structural relations of the generalized algebra of second class constraints are generated within the generating equations (1.11-1.14). Consider now how the gauge algebra of f i r s t t h i s end introduce, f i r s t
[a ''~, a;] = i~'o~, ~11 E(n~) = %,It + 1,
of a l l
.r~"= , ~] gh'(~")
= O,
class constraints is generated. To
operators~"
o, "~"a"~ =~
canonically conjugate to n"~"
-~CII
IIC(
,
gh"(~") : -1.
(1.16) (1.17)
To each operator A we may put into correspondence the solution ~'(~) of the fo]]owing problem with t h e o p e r a t o r A as an i n i t i a l
Or~'= (i~)-I
[~', (j41")-I[~,,,~]],
datum:
~(m:O) : A ,
(1.18)
where m~, ~(m~) = E",
gh'(m) = gh"(m) = O, ~ = I . . . . .
2m"
(1.19)
are c-numerical parameters. The formal i n t e g r a b i l i t y conditions for the problem (I.18) are f u l f i l ] e d
due to the generating equations (1.11-1.14), ( I . 1 6 ) , (1.17).
Operatorial Dirac bracket of any two operators A and B is defined as fo]lows [A,B]~ E (~(~i)~(~21 - ~(mi)~(~2)(-11 ~(AI~(B)
(i .20) x exp
~
ml =m2= O,
where mmB is the matrix inverse to the matrix m~6 from (1.11), (1.15): ~Om~x = o~, B ~ = m~B(_1)(e~+1)(e~+1)
(1.21)
One can show that the Dirac bracket (1.20) possesses every algebraic property of supercommutator defined as (0.1). Using the d e f i n i t i o n (1.20) the generating equations of the gauge algebra of the f i r s t class constraints may be written as
[~',~']~=
O,
[Q"~,~']
= O,
[~',~']~ = O,
(1.22)
15 [H',a']~
= O,
[ a " ~ , H ' ] = O,
[ H ' , a] - "
= O,
(1.23)
~ ( £ ' ) = 1,
ghI(Q')
1,
gh"(Q') : O,
(1.24)
~(H') = O,
g h ' ( H ' ) = O,
gh"(H') = O.
(1.25)
=
Solution of these equations f o r the operators ~' and H' is looked f o r in the form of ,~C-normal-ordered series expansions in powers of the ghosts ( 1 . 5 ) ,
( 1 . 6 ) , the f i r s t
terms of the ~4)C-expansions f o r ~' and H' being T~C 'a and Ho r e s p e c t i v e l y . ing these expansions i n t o equations ( I . 2 2 ) ,
(1.23) and reducing t h e i r
Substitut-
].-h.
sides to
t h e , ~ C - n o r m a l form, one obtains a sequence of recurrency r e l a t i o n s f o r f i n d i n g the coefficient
operators.
algebra of the f i r s t
These r e l a t i o n s are the s t r u c t u r a l
r e l a t i o n s f o r the gauge
class c o n s t r a i n t s .
2. U n i t a r i z i n g Hamiltonian We proceed here by i n t r o d u c i n g new operators. me, E(~ ~) = ~ I,t
gh'(m) = gh"(m) = O,
Consider f i r s t
~ : I .....
the operators
2m",
(2.1
which obey the equal-time commutation r e ] a t i o n s II
[~,~B]
= i~Fm~B ( - I ) ~ B ,
(2.2)
(see also ( 1 . 2 1 ) ) and commute with every operator ( 1 . 1 ) ,
(1.5),
(1.6) as well as
with every operator to be introduced in what f o l l o w s . Second, extend the sectors (I.5),
(1.6) by considering new canonical p a i r s ,
ghosts ( 1 . 5 ) ,
in a d d i t i o n to the f i r s t
class
l e t us introduce the f o l l o w i n g new c a n o n i c a l l y conjugate operator
pairs (i~,x'a),
E(~)
(~,@,a),
~(~)
: E(x 'a) : c~, = ~(.1o.a)
a = I
= E~ + I.
, m'
(2.3)
a = I .....
m',
(2.4)
with the ghost numbers fixed as follows gh'(x'
= -gh'(~')
gh'(,.~' ) = - g h ' ( E ' )
= O, = I,
gh"(x') = -gh"(=')
= O,
gh"(,.~') = - g h " ( ~ ' )
= O.
Analogously, in a d d i t i o n to the second class ghosts (1.6) i c a l pairs
(2.5)
(2.6) l e t us consider new cano-
16 (~",~"~),
~(~)
(~,, ~e . ~. .,., ~
= ~(~"~) = ~ ",
~ = I ,
~(C~, ) = E(# ''~) = ~,, + I ,
....
(2.7)
2m",
~ = I, . . . .
2m",
(2.8)
with the ghost numbers gh'(x") = -gh'(~") = 0,
(2.9)
gh"(x") = -gh"(~") = 0,
(2.10)
gh'(J D'') = -gh'(~") = 0, gh"(;~m'') = -gh"(C") = I.
Let ~(m) be the solution of the problem (1.18), put into correspondence to every operator A, taken as an i n i t i a |
datum. We shall need the following designation
:%(e): ~ ~(~) exp ~ ® I gr ~II
(2.11)
where e~ are operators from (2.1), (2.2). With t h i s designation define the Fermion operator n ~ :~'(e): + ~,a
+ £,, + ~Be6a,,e 4- ~II
I1~
.
(2.12)
Due to (1.11-1.14), (1.16-1.18), (1.22), (1.24) the operator (2.12) is ni]potent: [£,~] = O.
(2.13)
Consider next the i n i t i a l (1.I),
gauge f i r s t
class Fermion depending on the canonical pairs
(1.5), (2.3), (2.4):
~ =~X
'a + ~&X 'a,
(2.14)
where ×,a, ~(x,a) = ~&, g h ' ( x ' ) = gh"(×') = O, a = I . . . . .
m',
(2.15)
are operators that f i x an admissible gauge in the f i r s t class constraint sector. Define a modified (Dirac) gauge Fermion ~' using the equations [~"%~']
= O,
[~',E~]
= O,
gh'(m')
= -I,
gh"(~')
= 0
(2.16)
to be solved by a,~C-normal-ordered series in powers of ghosts with (2.14) as the f i r s t term. With the so]ution of equations (2.16) at our disposal we may define the f u l l gauge Fermion
17 • = :~'(m): + ~",
(2.17)
where ~X
+ C~x
(2.18)
is the second class gauge Fermion, depending on the canonical pairs (1.1), (1.6), (2.1), (2.7), (2.8), while u
x ''~, s(×"~) = s~,
g h ' ( x " ) = gh"(x") = O, ~ = 1. . . . .
2m"
(2.19)
are operators that f i x admissible gauge in the second class constraint sector. The f u l l u n i t a r i z i n g Hamiltonian of the theory is given as [13] H = :#'(m): + (i~f')-11%~] •
(2.20)
Operator (2.12) is conserved owing to (1.11-1.14), (1.16-1.18), (1.23), (1.25), (2.13): [H,~] = O.
(2.21)
Physical states of the theory are selected by the condition ~IPhys> = O,
IPhys> ~ ~ I , , , > ,
(2.22)
where I , , , > stands f o r any state. The physical S-matrix induced by the Hamiltonian (2.20) does not depend on any special choice of admissible gauge operators (2.15), (2.19) and is unitary in the subspace (2.22). References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
E.S. Fradkin, G.A. Vilkovisky: Phys. Lett. 55B (1975) 224 E.S. Fradkin, G.A. Vilkovisky, CERN Report TH-2332 (1977) I.A. Batalin, G.A. Vilkovisky: Phys. Lett. 69B (1977) 309 E.S. Fradkin, T.E. Fradkina: Phys. Lett. 72B (1978) 343 I.A. Batalin, E.S. Fradkin: Phys. Lett. 122B (1983) 157 I.A. Batalin, E.S. Fradkin: Phys. Lett. 128B (1983) 303 I.A. Bata]in, E.S. Fradkin: J. Math. Phys. 25 (1984) 2426 I.A. Batalin, E.S. Fradkin: J. Nucl. Phys. (USSR) 39 (1984) 23 I.A. Batalin: J. Nucl. Phys. (USSR) 41 (1985) 278 i.A. Batalin, E.S. Fradkin: Rivista Nuovo Cimento (1986) [ i n press] E.S. Fradkin: Acta U n i v e r s i t a t i s Wratislaviensis N 207. Proc. Xth Winter School of Theoretical Physics Karpacz (1973) p.93
12.
Lett.(1986)n~inI.A. Batali E.S.press]Fradkin:Preprint P.N. Lebedev Inst. (1986) i'4 132, Phys.
18 13. Generally, we might have used any gauge Fermion in (2.20) depending on the complete set of dynamical variables, under the only requirement that i t should produce admissible gauge conditions in the f i r s t
and second class constraint
sectors. We have preferred here, however, a somewhat more special type of the gauge Fermion, namely the one given as (2.14-2.19), pursuading the f u l f i l m e n t of a natural requirement that l i f t i n g the gauge degeneracy in the sectors of f i r s t and second class constraints should occur in independent ways. Actually t h i s independence is contained in the following two properties. F i r s t l y , we have [:~'(~):,
:~'(~):2 = : [ ~ ' , ~ I ~ :
and, secondly, the operator :~'(~): evidently commutes with the part of operator (2.12), which is marked by two primes indicating the fact that i t concerns the second class constraint sector.
KALUZA-KLEIN APPROACH TO SUPERSTRINGS
M.J. Duff Theory Division~ CERN, 1211 Geneva 23, Switzerland
ABSTRACT
We apply Kaluza-Klein techniques to the bosonic string compactified on the
EsXE 8
group
superstrings, is the
manifold
to
derive
properties
of
ten-dimensional
thus lending support to the idea that the bosonic
fundamental
theory.
We
string
then pose the question of why physical
space-time has just four dimensions.
i.
FERMIONS
FROM BOSONS
The appearance of the rank
16, dimension 496,
gauge groups
E8xE 8
and spin 32/Z 2 as the only available candidates for anomaly free teni) 2) dimensional superstrings prompted Freund to conjecture that the fundamental theory might be the 26-dimensional bosonic string, and that the ten-dimensional
theories emerge after compactification on the torus
T 16 = Rl6/r where F is the even self-dual Euclidean lattice of E8xE 8 or spin 32/Z 2.
In this picture,
the fermions would appear as solitons of
the bosonic
theory.
In addition to the 16 [U(1)] 16 elementary Kaluza-
Klein
bosons,
a further
gauge
480
gauge
bosons
would
also emerge
as
F r e n k e l - K a c 3) solitons since the string can wrap around the torus. subsequent
discovery
The 4) of the E8xE 8 and spin 32/Z 2 heterotic strings
brought the total number consistent superstrings to 5 as in Table i, and only increased the desire for one underlying theory.
20
Table 1 Consistent
TYPE
I [SO(32)]
superstrings
SPINOR
STRING
LOW ENERGY THEORY
Weyl + Majorana
open/closed
N = 1 supergravity + S0(32) Yang-Mills
IIA
Majorana
closed
N = 2 non-chiral supergravity
liB
Weyl
closed
N = 2 chiral supergravity
Heterotic
[SO(32)]
Weyl + Majorana
closed
N = 1 supergravity + S0(32) Yang-Mills
Heterotic
Weyl + Majorana
[EsXE8]
closed
N = 1 supergravity + E8XE 8 Yang-Mills
Noting that the bosonic tification
string can also undergo spontaneous
from D to d dimensions
on the simply-laced
compac-
non-Abelian
group
manifold G of radius R provided 5) R = /~' and
where
c A is the
Nilsson, strings
Pope
second
order
Casimir
and myself 6) proposed
in the
adjoint
obtaining
representation,
the d = i0 heterotic
by choosing G = E8xE 8 or spin 32/Z 2 for which C A = 60 and hence
D = 506, d = i0.
The origin of fermions was (and still is) less obvious
than in the T 16 compaetification
since G is simply connected.
the
spin
appearance
understand
E8XE 8 or
group.
One
is thus to maintain
symmetries
from space-time
which
superstrings
the
32/Z 2 gauge
bosons
is easier
to
i.e., the gauge groups are just subgroups of the D = 506
co-ordinate
approach
the
than the T 16 case, since they are all just elementary Kaluza-
Klein 7) fields, general
of
However,
nice
symmetries
with
feature
the Kaluza-Klein
their
group
manifold
ideal of getting
of
this
internal
in a higher dimension;
primary
Yang-Mills
fields
a feature seemed
to
21
lack.
The main reason why this traditional Kaluza-Klein idea fell out
of favour was its inability to explain chiral fermions. now avoided by the bosonic string;
This problem is
we simply cut the Gordian knot and
dispense with fermions altogether!
Moreover, the old Kaluza-Klein trick
of adding a cosmological constant in the higher dimensional theory with sign and magnitude so designed to cancel the one arising from compactification
is
cosmological string.
now
strings
respectable,
term (D-26)/='
The
coupling
more
Kaluza-Klein
g and
since
the
required
is enforced by conformal relation
the gravitational
R2g 2 =
constant
for which =,g2 = K2 might
D-dimensional
invariance of the
K 2 between K meant
the
that
Yang-Mills
the heterotic
indeed admit of such an interpreta-
tion, but the Type I string for which g2 = KS' would not.
Independently,
at about
the same
time,
Casher et al. 8) took the
idea one step further and showed how all closed superstrings;
Type IIA,
Type liB, heterotic EB×E 8 and heterotic spin 32/Z 2 could emerge from T 16 compactification identification with
of
of the D = 26 bosonic the
the diagonal
transverse
subgroup
string.
space-time
of the
transverse
The key idea was the
SO(8)
of
the
space-time
superstring S0(8)
of the
compactified bosonic string and SO(S) [internal]:
where
S0(8)
[internal]
is a subgroup
GLXG R in the case of Type II.
of GR in the heterotic case and
In this way states transforming as spinor
representations of SO(8) [internal] now transform as fermion representations of SO(8) [space-time,super].
Since the non-linear o-model on the group manifold is equivalent to free bosons and Just one
on the torus 5),
the Frenkel-Kac
it follows that our Kaluza-Klein approach
approach of Casher et al. are in fact equivalent.
like the wave-particle duality of quantum mechanics which picture chooses
is merely
a matter
of
convenience.
So
far,
the
torus
approach has proved more powerful for formal "stringy" results, whereas the elementary nature of the gauge
fields in the Kaluza-Klein approach
lends itself more readily to low-energy field theory considerations. striking
example
A
of this is the derivation of the d = i0 Lorentz and
Yang-Mills Chern-Simons terms summarized in Table 2.
The identification
of spin-connections with gauge potentials 9) is, as discussed by Nilsson,
22
Pope,
Warner
diagonal
~_ in going
already
been
appreciated, Type
II
bosons"
just the field theoretic
choice of space-time
of A with had
10)
and m y s e l f
from the heterotic
employed
however,
(two
is
string
exactly
from bosonic
It is strange,
to the Type
ease with the first still remain
the
(one same
gravitino) "fermions
to from
to heterotic
that physicists
sceptical
II string
What is not generally
(zero gravitinos)
therefore,
of the
The identification
from heterotic
requires
as going
(one gravitino).
going
realization
above.
in the literature II). that
gravitinos)
phenomenon
SO(8) discussed
who
feel at
about the second.
Table 2 Kaluza-Klein o r i g i n of d = I0 C h e r n - S i m o n s terms, connections with torsion ± ½ H and (A,~) are the potentials of (GL,GR).
m+ are the spinYang-Mills gauge
STRING
DIMENSION
CONNECTIONS
(CURVATURE)2 TERMS
CHERN-SIMONS TERMS
bosonic
506
~+,~_
R+ 2 + R_ 2
dH : 0
bosonic on G
I0
~+,~,~_,A
R+2 _ ~2 + R_2 _ F 2
dH = ~'Tr(~ F-F
heterotic
I0
m+ = ~,~_,A
R_ 2 - F 2
dH = u'Tr(R+
R+-F
Type II
I0
m+ = ~, ~- = A
0
dH = ='Tr(R+
R+-R_
We
shall
omitting
now
outline
the details.
the whole
Kaluza-Klein
be
in
found
question: explain
Re fs.
12)
and
the number
2 and
thorough
programme
if the bosonic why
in Sections
A more
13).
string
and
3 how
discussion, its merger
Finally, really
of uncompactified
in
the derivation including
with
Section
space-time
4,
F)
goes
a review of
string
is fundamental
F)
theory
can
pose
the
we
theory does this
dimensions
is just
four?
2.
THE BOSONIC
Our starting
STRING ON THE GROUP MANIFOLD
point is the background
field Lagrangian
=
(3) +.,
where xM(~) d e f i n e s
$ @) . . - .
the embedding of the two-dimensional s t r i n g world-
R_)
23
sheet M 2 in a space-time ordinates metric
on
Yab"
M 2 and
M D (M,N = I,...,D),
R(y)
is the
The g r a v i t o n
curvature
~a = (¢,~) are the co-
scalar
of
the worldsheet
gMN(X), the antisymmetric tensor ~MN(X) and
the dilaton ~(x) correspond to the massless models of the bosonic string spectrum.
The dots refer
to terms describing
modes and the scalar "tachyon". we
are
implicitly
assuming
the higher spin massive
By ignoring these higher modes in (3),
that
in
the
theory, these fields have vanishing vevs.
correct
vacuum
state
of
the
(See, however, the cautionary
remarks 13) about a possible "space-invaders" phenomenon.)
For
consistency,
the
two-dimensional
theory
must
be
conformally
invariant and hence the two-dimensional worldsheet stress tensor must be traceless, the
i.e., there must be no conformal anomaly.
absence
fields
of
gMN(X),
trace
anomaly
~MN(X)
places
and ~(x)
which
restrictions
One can show that on
the
background
are equivalent to the Einstein-
matter field equations obtained from the effective Lagrangian
+o
c4)
One obvious solution to field equations corresponds <~MNP> = 0 and <~HN >_
to <~> = constant,
the flat metric but this is valid only for D = 26.
In this case the possible ground states are given by
K
Y! o
=
M
x T
(5)
when M d is d-dimensional Minkowski space and T k is the k-torus with di= 26-k.
However~Ifor D > 26 the cosmological
term in (4) obliges us to
look for solutions in which some of the dimensions are compactified on a curved
manifold
interpretation.
and
we
can
the
traditional
Kaluza-Klein
U
x ~ (~ = l,...,d) refers to space-time and ym (m = l,...,k) to the
extra dimensions. case
follow
Accordingly, we split in indices
x"= C× where
now
One solution which suggests itself corresponds to the
24
I~ o =
M~x Q
(7)
where G is a non-Abellan group manifold of dimension k given by d = D-k. In this case
(8) c.A where
p~
k (9)
with
fijk the
constants and L i in the left-invariant Killing m c A is the second order Casimir in the adjoint representation.
vectors.
structure
This will indeed be a solution to all orders in ~' provided
where m is a constant size
of
the
conformally titute
invariant
the
Lagrangian
compact
ground
with group
the dimensions manifold.
How
theory to all orders state
values
of ^gMN'
of mass which determines can
we
tell
in ='?
this
the
yields
a
To see this we subs-
~ MN and
~ into
the
string
(3) to obtain
ui= ~+xe~_×~ + 9+~ ~ - ~ - ~ + e ~ )
(12)
where
In (12)
we have used the orthonormal
co-ordinates
~+ = o ± z.
on MdxG with Wess-Zumino
gauge Tab = e ~ a b
and employed
But (12) is nothing but the non-linear
the
o-model
term, a system well studied in the literature 5)
and known
to be conformally
dimension
formula
invariant
provided
we satisfy
and provided that the radius of the group manifold
the critical
is quantized
in units
25
l
of ='-=".
~al where p is an integer. topological
=
I
(15)
The appearance
of the integer p follows
quantization condition on the coefficient
from the
of the Wess-Zumino
term.
The
case
p =
entirely equivalent
i.e.,
1 is rather
remarkable
bosons
to
%-3
I =
l,...,r
on the
torus
L 1 is
i runs over the vector representa-
But L 2 is also entirely equivalent
= where
in this case,
to
a system of free fermions where
tion of G.
because,
where
r
is
the
of dimension
it be simply laced.
rank
r.
See Table 3.
of
(17) G,
i.e.,
a system
The only restriction
of
free
on G is that
In this case
d = D - k = 26 - r.
In p a r t i c u l a r Lagrangian
G = E8xE 8, c A = 60, r = 16, D = 506 and d = 10.
L 3 simply corresponds
establishes manifold
for
the
correspondence
to the MdxT r compactification, between
and this
the D = 506 Kaluza-Klein
group-
approach of Duff, Nilsson and Pope 6) and the D = 26 Frenkel-Kac
torus approach of Casher et al. 7) in going
The
to L 2 works
the equivalence
only
for
the
[In fact, the fermionization S0(16)xS0(16)
of L 1 and L 3 is unimpaired
subgroup
by this.]
required
of E8xE 8 but
26
Table 3 The simply-laced groups
One
G
dim G
r = rank G
cA
S0(2r)
r(2r-l)
r
4r-4
SU(2r)
r(r+2)
r
2r+2
E6
78
6
24
E7
133
7
36
E8
248
8
60
advantage
of
the
group
manifold
immediately write down the Kaluza-Klein the
compactified
theory.
First,
approach
is that we can now
ansatz for the massless modes of
however,
we
should
say
a
few words
about Kaluza-Klein "consistency".
Since the VEVs
the
isometry group of the group manifold
of ~ and
HMN P given
dimensional
theory
For
Kaluza-Klein
generic
ansatz the
will
for the massless
full
isometry
solutions
of
geneous
dimensions,
of
only the
bosons
those
fields
isometry
group.
are only
those
GLXG R.
Moreover,
include
Kaluza-Klein
representation teaches
us
consistent This
can
that
the
of G L. there
however,
For
a
theory
under
solutions
G L ansatz
is in general the
Experience
either
by
original
with
all those
consistent
symmetrized
with
certain
including
the
all
homofields
subgroup
K
the gauge
those of the full isometry group only if we
adjoint
x
d = 11 supergravity, exceptional
ansatz can be achieved without demanding happen
of
for group manifolds
not
in
for which
a transitively-acting
of G L and
may exist
the ~-
Kaluza-Klein
theory
ansatz retains
particular,
scalars
is one
Kaluza-Klein
a consistent
In
a consistent
ansatz
are
generic
invariant
is GLXGR, and since
GL×GR-invariant,
that we keep only a subgroup of
A "consistent"
d-dimensional
also
the Yang-Mills gauge bosons of GL×G R.
theories,
theory.
and
are
sector requires
D-dimensional extra
contain
group.
the
above
fields
which
theories
adjoint however, where
a
this "K-invariance". are
not
K-invariant
27
[e.g.,
the
SO(8)
ansatz
for the S 7 compactification of d : ii super-
gravity] or by omitting fields which are [e.g., the omission of KaluzaKlein scalars in the SO(3) ansatz on $7]. present Ref.
context,
6), where
consistent
consistent, (adjoint
of the
See Re fs. 7) and 14).
latter
phenomenon was
further
provided
GL,adjoint
is summarized
first give
In the
provided
in
we showed that the G L ansatz for the bosonic string was
in spite of omitting Kaluza-Klein
went one step
tion
an example
and showed
that even
we pay the price
scalars. the
In Re f. I0) we
full GLXG R ansatz is
of including scalars SZJin the
GR) representation. This somewhat confusing situa-
in Table 4.
In presenting these results, we shall
the ansatz for GLXG R without including the scalars and then
indicate how their inclusion solves the problem of inconsistency. Table 4
Gauge groups surviving in consistent truncations of theories compactified on the group manifold G, and the corresponding scalar representations.
With KK scalars
Generic KK theory
GL
: (adJL×adJL)sy m
Bosonic string theory
GL×GR: (adJL,adJR)
Without KK scalars
--
GL
Let us introduce the Killing vectors K I on the group manifold G
K
= (. L-~
)
(18)
where L i are the generators of left translations
--
L-
and R i are the generators of left translations
and
(19)
28
[. L ~' R ~]
=
0
(2i)
The corresponding Yang-Mills gauge potentials are denoted by
where
A i are the gauge bosons of GL and Ai the gauge bosons of G R.
The
corresponding field strengths are given by
Fz
--
(.
~
i, ) F '
(23)
where
F"
=
~
+'--+ 5~
-
(24)
;%"
2The
Killing
vector
components
L ai and R aI satisfy
the C a r t a n - M a u r e r
equations
.~~ ~--Jo-
:2-
(26) (27)
2and we a l s o
introduce
the
notation
L4 %
L_~ (28)
We are now in a position to state the ansatz for ~, gMN and BMN and to calculate
the
corresponding
curvatures
and
field
strengths.
For
the
scalar, we write
A (29)
The metric ansatz is
29
•
•
+
v
~r ~ -- ( A t ~ ~ +
~r
~vR "3)
g
(30)
A
The antisymmetric
tensor ansatz is
•
_
gro = •
(31)
V
gr ~ = A
where,
in the absence of scalars, gmn
and
B mn
are just the ground-state
values given by (8) and (ii).
The quantities ~(x), g~)and metric
and antisymmetric
quantities
Ai(x)
the metric
tensors i n
and At(x)
and G R respectively.
B,~)~vwill be interpreted as the scalar, d'dimensional
space-time,
and the
will be the Yang-Mills gauge bosons for G L
Equation
(30) is just the "standard ansatz" for
tensor
familiar to Kaluza-Klein theories, the novel feature
is the ansatz for
N' which also involves the Yang-Mills gauge bosons.
In this Kaluza-Klein interpretation,
the gauge symmetry GLXG R is just a
subgroup of the d-dimensional general co-ordinate group.
To see this in
more detail, consider a general co-ordinate transformation
(32)
and the corresponding
transformations
of ~, gMN and BMN.
Then focus
one's attention on the very special transformation
with i
and~i
arbitrary.
Then from the Kaluza-Klein ansEtze (29)=(31),
30
we
may
compute
We
find
the
not only
transformation
rules
the usual Yang-Mills
for the d-dimensional
transformation rules for ~, g~v'
A i and ~i~, but also that the B v field transforms
I,,
It is now RABCD
and
the
tedious field
lu
but
b
as
b
~'
straightforward
strength
fields•
~ABC"
to compute
In p a r t i c u l a r
b
la
the curvature
the d-dimensional
field strength H is not just dB but rather
where
--
/%
/~
(36)
6
-S'l
:
F
~P,
+~@
..
,,,P'
(37)
,,
6 i.e.,
we
have
there
were
no
covariance cancelled
acquired such
d-dimensional
terms
in
of Q and Q under by the
unusual
that
Yang-Mills
where
transformations
Hence,
although
is e x a c t l y
of (34), ensuring
in D dimensions
dH = 0, in
we have (on using ='m 2 = i) ,
Note
dimensions
terms, even though A H = d~. The non-
transformation rule for B
that H does not transform. d-dimensions
D
Chern-Simons
o
left- and right-handed
gauge
fields enter
and that this equation is exact to all orders in ~'.
with opposite
sign
This is the result
quoted in the second line of Table 2.
In
a
similar
fashion,
into the D-dimensional ponding ensure
d-dimensional
equations equations.
the inclusion for simplicity.
of scalars
substitute
the
Kaluza-Klein
of motion and hence derive Here,
that the ansatz is consistent
requires omitted
we may
however,
the corres-
we must be careful
and, as explained
into
ansatz
to
in Re f. i0) this
the ansatz which we have
Otherwise we obtain unacceptable
so far
constraints
on
31
the other massless fields like F •
i~v3 •
in the scalar field e q u a t i o n , S l ~ N F
3.
= 0 arising from putting S l] = 0
V
~v
-
l~vj.
THE HETEROTIC AND TYPE II STRING
To obtain the corresponding terms for the heterotic string, we
*
Choose G = EsXE 8 for which dim G = 496, r = 16, cA = 60, D = 506 and
hence d = I0
*
Decompose G R ~ S O ( 8 ) ,
*
Following
Re f.
9),
i.e.,
~i
identify
~ab ~ (a,b=l,...,8)"' the right-handed Yang-Mills gauge (i=i,...,496) ÷
potential with the gravitational spin connection
From (38), we obtain
~H
=
But these are just
o~7 "~'v" ( % ~
the heterotic
~+-
~ /% ~')
string Yang-Mills
(40)
and Lorentz
Chern-
Simons terms quoted in third line of Table 2.
Note that in (39) and (40), it is the spin connection with torsion m(+) which appears, where
oa~) To u n d e r s t a n d
this,
orthonormal gauge
=
consider
~3 + ~
~
the h e t e r o t i c
(41) string
o-model II) in the
32
L H
(42)
where
~± = • ± o.
If our previous claims are correct, we must be able
to derive this from the bosonic string o-model
L_ by
(a)
compactifying
on
the
group
manifold
(b)
substituting
in
the
Kaluza-Klein ansatz (c) fermionizing the extra dimensional co-ordinates ym (m = l,...,k) and then making the identification (41).
An interest-
ing question is the origin of the four-fermi term in (42).
This will be
discussed elsewhere 15).
It should be admitted, however, that in common with Casher et al. 8) we
have
as
yet
no
dynamical
understanding
G R ~ SO(8) and the identification (39).
of the d e c o m p o s i t i o n
Nor do we see any justification
for the truncation of the string spectrum which seems to be entailed in reproducing that of the heterotic string.
The idea is that states whose
G R index i (i = I, ....,496) runs over the 8 s spinor index ~ (= = i, .... 8) of S O ( 8 ) i n t e r n a l
transform
as fermion representations of the diagonal
SO(8)
which is identified as the transverse space-time group of the heterotic string. gravitino
Hence, in some sense, the G R Yang-Mills boson A= is really the
33
(45) the scalars S. = are really the gauginos i
6 i
.__>
~ ~
~ and the Yang-Mills parameters ~
---->
(46) .
is really the supersymmetric parameter
~
(47)
but the origin of the Fermi statistics remains obscure.
We would like
to be able to say that there are distinct vacua of the bosonic string relative to one of which all states transform as bosons, but relative to the other some states transform as fermions. yet in a position to make
this more precise.
Unfortunately, we are not Nevertheless, we glimpse
the beginnings of the explanation for supersymmetry by using (39), (45) and (47) to convert the G R Yang-Mil!s transformation rule
into the gravitino transformation rule
(49)
+ on using the property of the E 8 structure constants
~ r~
(5o)
Similarly the Type IIA and Type liB theories are obtained by * *
decomposing both GR ~ SO(8) and G L ~ S O ( 8 ) identifying
both
the
left-
and
right-handed
Yang-Mills
gauge
potentials with gravitational spin connection
(51)
34
to obtain the type II non-linear o-model, with the Chern-Simons term
which is just the final line of Table 2.
The extra 64 bosonic degrees of freedom are then provided by S~ S~
corresponding
(8s,8)
or
to the embeddings (8c,8 s) in the case of Type liA or
in the case of Type
liB.
The second supersymmetry of Type II
has the same origin in GL as did the first in GR, i.e., with A ~ and ~ playing the part of gravitino and supersymmetry parameter.
Thus we arrive at the bizarre picture of a three-in-one world that can
be
described
equivalently
in
I0,
26
or
506
dimensions
as
in
Table IV.
TABLE 4:
A three-in-one world described equivalently by I0, 26, or 506 dimensions.
Bosonic d101mGs06[ i10rankG261 Kaluza-Kle in (496 elementary gauge fields)
ke l-Kac ~
/
(16 elementary gauge
/
~
Fermionic
/
fields + 480 solitons)
35
Of course,
if the bosonic
string really is the fundamental theory
perhaps we should consider compactifications not from d to i0 dimensions but from d to four dimensions,
i.e., on a group manifold for which from
Eq. (I)
rank G = 22
But which G should we choose and why should the string prfer rank 22 to some other rank <26?
4.
FOUR DIMENSIONAL SPACE-TIME FROM THE K3 LATTICE
The major unresolved problem of string theories is that of vacuum degeneracy. almost
no
For
although the higher-dimensional
parameters,
all
predictive
apparent multitude
of different
remains
why
should
a mystery be just
four.
power
string equations have
seems
to be
compactifications.
the dimension
of the
This
based
section,
lost
by
the
In particular,
uncompactified
it
space-time
on a paper by Nilsson and
myself 14), is an attempt to resolve some of these questions.
We begin by r e c a l l i n g
the recent work of Narain 17) who considers
compactifying into tori (10-d) and (26-d) dimensions of the right moving superstring
and left moving bosonic string sectors respectively.
the k-torus
the question devolves the condition even
upon which F to choose.
for modular
Lorentzian
lattices
with
Let us denote
(p,q)
signature.
the
Narain points out that
invariance is equivalent
d i r e c t ions. is
Since
is given by Rk factored by a discrete lattice F(T k = Rk/F),
(10-d)
timelike
to self-duality for
and
(26-d)
spacelike
such even self-dual lattices F(p,q), where
Then
all
such Lorentzian
lattices
are
iso-
morphic to the lattice
where
~-~ for some
=
~
(54)
integer n, where E 8 is the root lattice of E 8 with Euclidean
signature (8,0)
38
2•-i2 - o o0 o0 o0 oO
E8 =
and
-1 0 0
-1 0
0
0
0
-I
0 0
0 0
0 0
-1
-1
where P2 is a two-dimensional
to the group SU(2)xSU(2).
dual
r(q+8n,q)
means
of an SO(8n+q,q)
with
0 0
(55)
-
-! -1 0
2 0
lattice with signature
corresponding lattice
0 -
This means that any even self-
q > 0 can be obtained
transformation.
(i,I)
Distinct
from nE8 ~ q P 2 by
compactifications
are
then characterized by points in the coset
s o There even
,? / s o @ ) ×
is a recent
self-dual
Lorentzian
of a simply-connected Stern 17)
for
theorem due to Freedman 18) that states that all lattices
topological
a readable
are given by the "intersection fourLmanifold
introduction
M 4.
form"
See the article by
to this branch
of mathematics.
Such a manifold will have Euler number
=
where forms.
the
second
Betti
If we denote
intersection
and obviously
~- +
number these
~=
b 2 counts
two-forms
the
by =i
number
of harmonic
two-
(i = l,...,b 2) then the
form is defined by
has rank b 2.
and the Hirzebruch
75
Its signature
(p,q)
is given by (b2+,b2-) ,
signature by
-
b~ + -
b z-
where b2+ count the number of self-dual antiselfdual.
(57)
(Hence
(59)
two-forms
z must be a multiple
of 8.)
and b 2- the number of So the question of
37
which is the right vacuum has been replaced
by which
is the right four-
manifold.
Now Freedman's every
theorem
four-manifold due
to Rochlin 20)
just
for
fun,
manifold
use
is that
the
For example,
~ must
criterion
four-manifolds
but not
a necessary
be a multiple of 16.
of differentiability
condi-
Suppose,
of the
four-
to narrow down the choice of vacuum.
Unfortunately, (i.e.,
topolo$ical
is differentiable.
tion
we
involves
the question
differentiable)
is
an
of which
outstanding
some very interesting results are known.
is not,
even
created
quite
though a
z = 16!
stir
(This
P2 on the other hand corresponds
is smoothable
problem
are smoothable
in mathematics,
but
For example
result,
in mathematical
heart of the proof that R 4 has more
is differentiable.
four-manifolds
due
circles
to Donaldson 21), has
because
it
lies at
than one differentiable
to the intersection
the
structure.)
form on $2×S 2 which
The problem then is to determine whether
for some q > 0.
Note that from the string point of view, our criterion of differentiability perhaps
then means go
to d
differentiable
that
< i0!
we cannot The
amazing
and simply corresponds
go
from D = 26 to d = i0 b u t may
fact
is
that
the
case
to the four-manifold
q =
3 is
K3 for which
b2+ = 19, b 2- = 3 and b 2 = 22.
K3
is
defined
as
a quartic
surface
in complex
projective
three-
space CP 3 by
Applying
Narain's
heterotic
string
low-energy
limit
super-Yang-Mills remaining
rank
techniques theory
with
corresponds
using
this very special
unique
space-time
lattice
dimension
leads to a
d = 7.
to d = 7, N = 2 supergravity
with rank 19 gauge group E8XE8xSU(2)xSU(2)xSU(2). 3 gauge group
simply corresponds
to the
The
coupled
to
[The
three U(1)'s of
38
N = 2 supergravity.] be
obtained
S2xS 2.
Corresponding theories in d = 10-q < 7 could also
by taking the
topological
sum of K3 and
(q-3)
copies
of
to the rank
22
Thus a four-dimensional theory could be obtained from
whose
low-energy
limit
is N = 4 supergravity
gauge group EBXE8x[SU(2)] 6.
coupled
[Once again, the remaining rank six group
simply corresponds to the six U(1)'s of N = 4 supergravity.] nately,
Unfortu-
the "minimal" theory is in d = 7 and there seems no compelling
reason for adding three S2xS 2 manifolds to K3.
So far we have
followed Narain and considered
only the heterotic
string, but the situation becomes much more interesting if we adopt the point of view that the fundamental theory is the bosonic string. again
we
must
compactify
Lorentzian lattice but
on
a
torus
factored
by
an
even
now with signature (26-d,26-d).
Once
self-dual
The "minimal"
theory in the sense described above is now given by
where
K3 corresponds to the four-manifold obtained from K3 by reversing
the orientation and has b2+ = 3, b 2
L =
~-.6
-- % 0
-"
= I0 and ~ = -16.
Hence
+
and we obtain a four-dimensional bosonic
(60) string with gauge group GxG,
where G is the rank 22 group E8xE8xSU(2)6.
Thus our objective is now to repeat the derivation of superstrings from bosonic strings discussed in ~ c t i o n
1 but now compactifying
from
D = 26 to d = 4 on the torus T 22 defined by the intersection form of the four-manifold equivalence, However,
K3+K---3 [or, bearing
in mind
the p r e v i o u s l y
discussed
from D = 518 to four on the group manifold E8xE8xSU(2)6].
the outcome is no longer clear.
whether a chiral N = 1 theory would result.
In particular, it is unclear If a chiral theory does not
emerge directly in this way, it may be necessary to go one stage further and c o m p a c t i f y
not m e r e l y on the torus T 22 defined
T22/~
is a discrete group.
where ~
by K 3 + - ~ but on
Factorings of T6 by ~
have been
39
considered necessary
by Dixon et al. that
"orbifolds". is that
~had
22)
but to obtain
chiral
fermions, it was
fixed points thus leading to singularities,
From our K3 point of view,
i.e.,
a more attractive possibility
advocated by Lam and Li 23) who consider direct compactification
from 26 to 4 via T 2 2 / ~ that T 2 2 / ~
where ~
acts on T 22 without
fixed points,
is a genuine manifold without singularities.
so
These authors
claim to obtain chiral N = 1 theories in this way while still preserving modular
invariance.
(They
consider
E8XE8XSU(3) 3
rather
string
should
than
E8xE8xSU(2)6. ]
The
vital
intersection
question
remaining
is why
forms of differentiable
manifolds,
explain why we cannot remain in d = i0. why space-time
theory
select
but if it does it would
And in answer to the question
has four dimensions we would reply:
because the second
Betti number of K3 equals 22!
ACKNOWLEDGEMENTS
I am grateful
for conversations
with A. Chamseddine,
B. Nilsson,
C. Pope, D. Ross and N. Warner.
REFERENCES i) Green, M.B. and Schwarz, J.H., Phys. Lett. B149, 117 (1984). 2) Freund, P.G.O., Phys. Lett. BISI, 387 (1985). 3) Frenkel, I. and Kac, V.G., Inv. Math. 62, 23 (1980); Goddard, P. and Olive, D., in "Workshop on Vertex Mathematics and Physics", Berkeley (1983).
Operators
in
4) Gross, D., Harvey, J., Martinec, E. and Rohm, R., Phys. Rev. Lett. 54, 502 (1985); Nucl. Phys. B256, 253 (1985). 5) Witten, E., Comm. Math. Phys. 92, 455 (1984); Nemeschensky, D. and Yankielowicz, S., Phys. Rev. Lett. 54, 620 (1984); AltschHler, D. and Nilles, H.P., Phys. Lett. 154B, 135 (1985); Goddard, P. and Olive, D., Nucl. Phys. B257, 226 (1985); Jain, S., Shankar, R. and Wadia, S.R., Phys. Rev. D32, 2713 (1985); Bergshoeff, E. Randjbar-Daemi, S., Salam, A., Sarmadi, H. and Sezgin, E., Nucl. Phys. B269, 77 (1986). 6) Duff, M.J., Nilsson, B.E.W. and Pope, C.N., Phys. Lett. B163, 343 (1985), also published in Proc. Cambridge Workshop on Supersymmetry and its applications (June-July 1985), (Eds. Gibbons, Hawking and Townsend, C.U.P. 1986).
40
7) Duff, M.J., Nilsson, B.E.W. and Pope, C.N., Physics Reports 130, 1 (1986). 8) Casher, A., Englert F., Nicolai, H. and Taormina, A., Phys. lett. B162, 121 (1985); see also Englert, F,, Nicolai, H. and Schellekens, A., CERN preprint TH.4360/86 (1986). 9) Charap, J.M. and Duff, M.J., Phys. Lett. B69, 445 (1977). i0) Duff, M.J., Nilsson, Lett. 171B, 170.
B.E.W.,
Pope,
C.N. and Warner,
N.P.,
Phys.
Ii) Huil, C.M., Nucl. Phys. B267, 266 (1986). 12) Duff, M.J., in Proceedings of the GRII Conference, Stockholm, July 1986, CERN preprint TH.4568/86. 13) Duff, M.J., in Proceedings of the 1985 Les Houches Summer School (Eds. Ramond and Stora). 14) de Wit, B. and Nicolai, H., cERN preprint TH.4359/86 (1986). 15) Chamseddine, A . , Duff, M.J., Pope, C.N., in preparation.
Nilsson,
B.E.W.,
Ross,
D.
and
16) Duff, M.J. and Nilsson, B.E.W., Phys. Lett. 175B, 417 (1986). 17) Narain, K.S., Phys. Lett. B169, 41 (1986). 18) Freedman, M., Diff. J. Geom. 17, 357 (1983). 19) Stern, R.J., The Mathematical Intelligencer ~, 39 (1985). 20) Rochlin, V.A., Dokl. Akad. Nauk SSR 84,221 (1952). 21) Donaldson, S.K., Bull. Amer. Math. Soc. 8, 81 (1985). 22) Dixon, L., Harvey, J.A., Vafa, C. and Witten, E., Nucl. Phys. B261, 678 (1985). 23) Lam, C.S. and Da-Xi Li, McGill University preprints (1985).
NON LINEAR EFFECTS IN QUANTUM GRAVITY
Ian Moss Department of Theoretical Physics University of Newcastle upon Tyne Newcastle upon Tyn e NEI 7RU U.K.
ABSTRACT
Canonical quantum gravity can be reduced in a semi-classical limit to conventional quantum gravity on a curved spacetime background. Changes in the topology of space require a reformulation of the theory which introduces density matrices or nonlinear terms into the semi-classical limit.
I. INTRODUCTION
We are still in the prehistory of a quantum theory of gravity. I shall report here how recent investigations into the origin of the universe, stimulated by the sucess of the inflationary scenario
[ ~ as an explanation of the large scale structure
of theuniverse, has lead to the development of new ideas in quantum cosmology. In particular, we shall see how the Schrodinger equation is recovered from quantum gravity and how changes in the topology of spacetime can fundamentally influence quantum theory and its interpretation. In constructing a quantum model of the universe we need to introduce a fundamental action and initial conditions. The gravitational part of the action presents particular difficuties. We shall use the Einstein-Hilbert action for the time being. It may be that the theory based upon this action can be rescued from some apparent inconsistences. In any case, we expect that we have a good approximation whenever the radius of curvature of spacetime is larger than the Planck length of 10-33 cm. This is analagous to the use of the Coulomb potential in describing a Hydrogen atom where we fix a boundary condition on the wave function at the centre, despite the fact that we know that the Coulomb potential is invalid inside of the nucleus. For initial conditions we shall make use of Hawking's suggestion that "spacetime is finite but unbounded" 12] . This is realised by the Hartle-Hawking prescription [3]
for the quantum state of the universe. This state is a function of the geom-
42
geometry of 3-dimensional hypersurfaces
Z described by a metric tensor gij and
matter fields ~ . The state is defined by
(i) where we sum over all 4-geometries and matter configurations such that the 4-geometry is compact and has no boundary other than Z (fig. i).
E
Figure
Approximate calculations
[4,5,6]
1
of this wave function in various inflationary
models has demonstrated that it is a superposition of states representing universes with a satisfactory large scale structure. This means that they are spatially flat and homogeneous with scale-free density fluctuations. We shall discuss such a decomposition of the wave function in sect.2 . We are confronted, however, with considerable problems of interpretation. The observer is necessarily part of the system as in the "Many Worlds" interpretation of quantum mechanics [7]. In this picture, the collapse of the wave function associated with a measurement becomes a splitting of the wave function into non-interacting branches. With quantum cosmology this leaves us with a problem : which universe from the superposition do we live in and what causes the splitting? Furthermore, the wave function gives us probabalistic information, but the meaning of probability is unclear when we have just one unrepeatable experiment. A remarkable relationship between changes in spatial topology and these questions will be explained in sect. 3.
43
2. CANONICAL QUANTUM GRAVITY
In the canonical approach to quantum gravity we decompose spacetime into 3-dimensional spatial hypersurfaces various matter fields
~t " The phase space consists of 3-metrics gij and
¢ on Z t' together with their conjugate momenta pij and
~.
Under this canonical decomposition the action has the form
where t h e i n d i c e s
on g and p a r e i m p l i c i t .
The s u p e r h a m i l t o n i a m H can be decompos-
ed into gravitational and matter parts,
H(~.,p,}O,=) Einstein's
H~($,p) ÷ H,,,($.,f,~)
= theory
of g r a v i t y
=
where
bar
G ijkl
is the
The for
the
of the
and
metric
shift
N and
HZ=O,
other
(2) with
of c o v a r i a n t
choice
with
Ricci
scalar
R, and
[8],
H=O a n d
action
derivative
functions
The
to the
RI,I
g-covariant
which
respect
Classical
in c o n f i g u r a t i o n
space
N.
1
can
Eintein
derivatives
~ = (NGijkI,NG~p).
ectories fields
DeWitt
equations.
ed in terms metric
the
constraints
Einstein's iation
-
denotes
lapse
corresponds
(3
act be
as
viewed
equations
to g and
Lagrange
p. They
on c o n f i g u r a t i o n solutions
which
are
as
follow
are
multipliers
a subset
of
from
var-
can
the
be e x p r e s s -
space
with
represented
geodesics
when
a
by traj-
the m a t t e r
are m a s s l e s s .
In the
quantum
~(g,¢).
The
H ~
=
H ~P
= O
theory
constraints
states
must
can
be r e p r e s e n t e d
be r e a l i s e d
by
by wave
functions
[8,9]
O
with
p replaced
ion.
There
(7)
(8) by i6/6g.
Eq.
is a n o n - t r i v i a l
7 is k n o w n
factor
as
ordering
the W h e e l e r - D e W i t t
equat~
problem
with
assooiated
44
this equation, covariant
We shall choose a factor
derivative
ordering
which uses the
[i~ . This gives
H~ = -V~ ÷LL
(9)
!
where U =
This
factor
ariant
and
ordering
under If
then
g2R(g)
we c h o o s e
the
implies
coordinate instead
transition
to
the
Wheeler-DeWitt on t h e
quantise
ampl£tude
is
the
given
equation
configuration
theory
by p a t h
is
inv-
space, integrals,
by
41, f]
=
4
where the 4-geometry This amplitude
that
redefinitions
g interpolates
satisfies
use the configuration
between
the Wheeler-DeWitt
the 3-metrics equation
space metric ~ to define
g and g'
provided
that we
the path integral
meas-
ure,
which is invariant Because
of the vanishing
ible to introduce DeWitt
equationl
Wheeler-DeWitt hyperbolic definite
under coordinate
a ~/~t unlike
equation
operator.
The fact that
questions elopment
is a dynamical
equation.
Instead,.the
e q u a t i o n because H forms a
This is only possible
because H i s not a positive
~ does no t depend upon time is simply an expression
covariance,
because
time is a c o o r d i n a t e label.
about time development
of freedom
of the remaining
to measure
have to be addressed
to form clock subsystems
field behaves
the passing
that the W h e e l e r - D e W i t t
inger equation.
against
Physical
by choosing
some
which the time dev-
system can be measured. semi-classically,
of time by the evolving
this limit it has been shown in special [i~
H it is imposs-
term on the right hand side of the Wheeler-
the normal Schrodinger
When the gravitational possible
of the superhamiltonian
Hamiltonian.
of general degrees
redefinitions.
equation
We shall generalise
cases,
reduces
by DeWitt to the
their results
then it is
geometry.
In
[8]~ and Banks
familiar
to include
Schrodthe back
45
reaction
of the matter
Consider
a wave
fields
on the
function
geometry.
of the form
(13) The
parameter %
defined
below,
becomes
the time
coordinate.
We shall
•
construct around and
a solution
then The
Jacobi
6
a gravitational
wave
function
of E i n s t e i n ' s
demohstr~te
that ~
semi-classical
m
~
equa(ions
satisfies
gravitational
which with
is sharply
a back
Schrodinger's field
must
peaked
reaction
term
equation.
s a t i s f y a Hamilton-
equation,
where
represents m
the back
reaction
of matte~,
given
by
#
The t i m e c o o r d i n a t e
~
is
d e f i n e d by
= ~L ~k '
wher e Gijkl function ground
is the DeWitt
S can
field
The
'
(16)
metric.
be identified including
semi'classical
As may
with
the back
be expected,
the g r a v i t a t i o n a l reaction
approximation
term
m
is given
the
principal
action
of the back-
"
by
;
where
(17)
A is a slowly
Wheeler-DeWitt The
leading
vanish
varying
equation
order
terms
function
we neglect vanish
of g. When
the
~2A/~g2
due to eq.
14,
substituted terms,
and
int6
the
but no others.
the next
order
terms
for the choice
(18)
where
go are
inger's
integration
equation,
constants.
The
remaining
terms
give
Schrod-
46
This ground
equation
metric
The
the
of the
to one
upon
components,
importance
the wave
in d e s c r i b i n g WKB
components
has
will
coefficients
semi-classical
how an o b s e r v e r
can
alwready
back-
function ~m.
equation
with
of each
semi-classical
reduce been
develop
which
give
metric
g.
the w a v e
mentioned
funcin
IN T O P O L O G Y
We c o u l d rics,
such
particle which
imagine
theory,
amplitude the
We
the
difference
ions
integral.
path
initial integral
which
imply
between
Figure
2
disconected
represents
would
second
equation
and
final
instead, that
In the breaks
but
there
no L o r e n t z i a n
and
met>
vertex case
of
is dis-
are
theor-
4-geometry
3-geometries.
Lorentzian
In
down be-
geometries
distinct
Euclidean
~],g21g
be an i n t e r a c t i o n
quantisation.
on to t o p o l o g i c a l l y
is a f u n d a m e n t a l of the
the
path
topology
can m a t c h
diagram for
between
2 which
, the W h e e l e r - D e W i t t
between
can use
amplitudes
in fig.
necessity
~l,g21g>
in d i f f e r e n t i a l which
shown
the a n a l a g o n s the
transition
continuous.
exists
transition
as the one
indicates
cause
ems
depends
fixed
introduction.
3. C H A N G E S
the
of W K B
of t h e s e
in the
of the W h e e l e r - D e W i t t
relative
difficulty
tion
in turn
solution
a superposition
an idea
to be s o l v e d
g, w h i c h
A general into
has
This
formulat-
47
There
(i)
are
Sum
two
distinct
over a l l
ways
of the
in w h i c h
unobservable
we
can
proceed:
components
of the
3-geometry (ii)Extend
Case
(i)
leads
the
is based
Hilbert
upon
to t r a n s i t i o n s
the
joint
for
changes
transition
space
to i n c l u d e
an
idea
of H a w k i n g
from
pure
states
amplitude
in t o p o l o g y
with
P(
]g> +
[121 and
to d e n s i t y
]gl>~Igl '>,
a sum
over
the
Iglg2 > +
Page
can
, and Consider
). A l l o w i n g
unobserved
=
We
~
matrices.
..
components,
(2O)
. . . .
represent
on an i n t e r n a l esponds
this
surface
to a E u c l i d e a n
A pure development
state
has
of this
diagramatically shown path the
state
by fig.
by the
dotted
integral
with
form
p =
Each
boundary
[~><~J
is d e s c r i b e d
3, where
line.
where
g is the m e t r i c diagram
metrics ]~> =
corr-
g l , g 2 , g ; , g ~.
Z~(g) Ig > . The
by a s u p e r s c a t t e r i n g
operator
S,
where
From
the
expansion
of P given
in eq.
does
not
factorise
in g e n e r a l
and
pure
state.
I
20 we
can
therefore
I
see
~ p does
that not
"
this
In q u a n t u m be i n t e r p r e t e d state
JRWI> +
universes
1 and
cosmology,
the
as a b r a n c h i n g IRW2>
2 can
evolve
+
of the wave
of pure function.
a superposition
to the
....
3
development
representing
a
I
+
Figure
expression
represent
density
to mixed For
states
example,
can
a pure
of R o b e r t s o n - W a l k e r
matrix
IRWI>
+ iRW2>
48
in w h i c h vanish.
all This
in order Case
(ii)
ised
terms
Some
can
of the
fields
gauge
a second
(8).
2 and
universes
a complicated
can
the wave
a constraint the
observer
Next,
class
classical
at this
function
~(g,O,@i).
their
stage,
and
inter-
is quant-
functions introducing
The
replaces
of F r a d k i n
H 1 and
theory
of wave
Q~ = O, w h i c h
equat-
we i n t r o d u c e the
be fixed
methods
of H,
whose
finally,
a suitable
freedom
Using
the
of the W h e e l e r - D e W i t t
an a c t i o n
equations.
to fig. over
Q in terms
between
to'involve
quantisation
constraint
01into
imposes
(7) and
having
terms
function.
by c o n s t r u c t i n g
the
e and
construct
result
wave
involves
are
interference
us from
the
integration
symmetry
aints
free
corresponding
by path
ghost
quantum
We p r o c e e d
of m o t i o n
action
BRS
may
to split
equation. ions
of the
remaining
the
constr~
Vilkoviski
commutators,
with
[14] we the
that
Q =
d~
H e + H,o~ + ~ v , e o ~
The anticommuting
fields
icall,y
to
conjugate A suitable
Variation
of
is
ei
given
Q~ = 0 w h i c h
is
equation
constraints,
The
in fig.
2,
but
are
antighost
fields
which
are
canon-
0i
in
entum
first
^
and
term
raint
as shown
0 and
action
the
;
~e~(v~e ) o ~ - ~ , ~ e 9 o~ (2~)
second
by
the
action
leads
equivalent term
quattic
to
contains
and
to
the
terms
"free"
Hamiltonian
cubic
higher
the
constand
interaction
could
mom-
terms
also
be incl-
uded. The
action
(23)
the W h e e l e r - D e W i t t limit
is m o d i f i e d
Q? + ?? From
also.
isation
obtain of it.
the
product
the
result
From
of the
the
wave
is no longer
equivalent
gravity.
The
sem~classical
of eq.
23 we
variation
function
equation
particular
of ¢ fields is that
which
of q u a n t u m
to
get
(24>
Schrodinger's The
a theory
~ 0
a decomposition
longer
gives
formulation
form
is defined.
analagous
but we
of this For
the
to eq.
get a non equation simplest
13,
linear
depends choice
we no
generalupon
how
( ~(g,@))3,
49
It is well which
resemble
known
that
such
the collapse
when the~e exist
solitonic
solutions
to the linearised
equation
cosmology,
be possible
without
it may
the i n t e r v e n t i o n
Non-linear a many
terms
electron
a single imation
wave
electron.
moving
are also
important
representing
for
can induce [15] . This
25 which
into.
can happen
typical
In the case
the selection
effects
solutions
of quantum
of a universe
to occur
of an observer.
can also
atom,
arise
of this
in which
in its own charge in solid have
in an analagous
is a p p r o x i m a t e d
An example
photons
terms
function
of eq.
can evolve
function
for a Helium
it where
non-linear
of the wave
a single
electron
physics,
observed
Such
where
where
function
be the H a r t r e e - F o c k
distribution.
state
been
would
situation
by the wave
is viewed non-linear
the soliton
of
approxas if effects
solutions
[16].
4. REFERENCES
l,
A.H.
Guth,
2.
S.W.
Hawking,
3.
J.B.
Hartle
4.
I.G.
Moss
5.
S.W.
Hawking
6.
I.G. Moss, "The New Cosmogony", to appear in the proceedings of the IV Marcel Grossman Meeting, Rome 1985.
7.
B.S. DeWitt and N. Graham, eds. "The Many Worlds I n t e r p r e t a t i o n of Quantum Mechanics", Princeton University Press 1973.
8.
B.S.
DeWitt
9.
J.A. J.A.
Wheeler, Wheeler,
i0.
S.W.
Hawking
ii.
T. Banks,
12.
S.W. Hawking, "The density matrix ( Cambridge preprint 1986).
Phys.
Rev.
Pontif.
and
S.W.
and W.A.
D23
(1981)
Accad.
Sci.
Hawking,
Wright,
Rev.
160
Varia,
Phys.
Phys.
and J. Luttrell,
Phys.
347.
Rev.
Rev.
Nuc.
(1967)
Phys.
Nuc.
Phys.
Page, B249
Nuc.
(1985)
D23
563.
(1983)
(1984) B247
2960.
1067.
(1984)
250.
1113.
in"Battelle Rencontres", Benjamin New York, 1968. and D.N.
D29
48 (1982)
Phys.
eds.
C. DeWitt
B264
(1986)
332. of the universe"
and
185.
50
REFERENCES 13.
(CONT.)
D.N. Page, "Density matrix of the universe" ( Pennsylvania preprint 1986 ).
14.
E.S. Fradkin and G.A. Vilko~iski,
15.
D.Bohm and J. Bub, Rev. Mod. Phys.
16.
A.R. Bishop and T. Schneider, eds. "Solitons and Condensed Matter Physics" (Springer-Verlag, Berlin 1978).
Phys.
Lett.
55B (1975)
224.
38 (1966) 453.
OUR UNIVERSE AS AN ATTRACTOR IN A SUPERSTRING MODEL
Kei-ichi MAEDA
International
Abstract:
Centre for Theoretical
Physics,
Trieste,
Italy
One preferential scenario of the evolution of the universe is discussed
in a superstring model.
The universe can reach the present state as an attractor
in the dynamical system.
The kinetic terms of the 'axions' play an important
role so that our present universe is realized almost uniquely.
I.
INTRODUCTION
A superstring theory is a promising candidate for a fundamental unified theory including gravity (1).
(2)
of view
It may be successful from the phenomenologieal point
Its application to cosmology is certainly important and interesting.
The superstring theory as well as the other unified theories such as the Kaluza-Klein .
iaea
(3)
predict a higher-dimensional space-time, which may play a very important role
in the early universe.
Our world is, however, definitely four dimensional at least
in the macroscopic scale. successful.
The 4-dimensional Hot Big Bang scenario is very
We beleive the Friedmann expanding universe based on the 4-dimensional
Einstein gravity.
Hence, if we take a higher-dimensional space-time seriously, we
must explain how our 4-dimensional universe is naturally realized in the higherdimensional space-time. The present universe must be ~ the 4-dim Friedmann universe (F4) ] x [a very small static internal space (K)] (4).
In the conventional 4-dim theory, the isotropy
and the homogeneity of space-time, which may be deduced from the cosmolo$icalprinciple or from an inflationary scenario
(5)
, guarantee that our universe is a Friedmann space4 In a higher-dimensional theory, however, that is not true because the F x K
time.
space-time is not isotropic at all in higher dimensions. .
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Address after October 1986 : Relativiste,
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Our anisotropic universe .
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Observatoire de Paris-Meudon,
92195 Meudon, France
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Groupe d'Astrophysique
52
(F4x K) must be a special space-time in the dynamical system.
Namely, the F 4 x K
solution should be an attractor in our system. If this attractor is strong enough 4 to guarantee for the universe to reach the F x K solution for a wide range of initial
(6)
conditions, we understand easily why our universe is now in the present state 4 For example, in the 6-dim, N=2 supergravity model, the F x K space-time is a unique 4
attractor and all the space-time
( apart from the time reversal ones ) approach F x K
asymptotically in the later stage of the universe
(7)
The second problem in a higher-dimensional theory is that the reduced fourdimentional effective gravity theory may not be the Einstein theory but the JordanBrans-Dicke (JBD) theory for a Ricci-flat compactification such as a Calabi-Yau manifold.
The JBD parameter is given by the dimension D of the internal space as
•=-(D-I)/D,
(¢.~ >
500
then this theory should be excluded from the astrophysical observations
).
Thirdly, also in
since
inflation
is
theory
for
search our unified
unified
theories,
hand as we l i k e .
very
desirable
inflation.
in
modern c o s m o l o g y
In general,
b e c a u s e we c a n n o t add an i n f l a t o n We have been a l r e a d y
given
a set
of
this
responsible fields
task
(5) is
for
, we s h o u l d not
so e a s y
inflation
by
and we must l o o k f o r
the inflaton among them. Hence, the present main problems in higher-dimensional unified theories,
from
the cosmological point of view, are: (i)
Can the 4-dim Friedmann universe be realized naturally as an attractor in the higher-dimensional space-time ?
(ii)
Can
the 4-dim Einstein gravity be obtained from a higher-dimensional
theory,
rather than the JBD theory ? (iii)
Does inflation really occur in the unified theory ? We investigate the above problems and discuss on~ preferential scenario in the
lO-dim, N=I supergravity model with E x E" Yang-Mills fields and the additional 8 8 curvature squared terms, both of which are derived from the heterotic string model in the field theory limit (8).
The effective 4-dim Lagrangian is given in §.II,
assuming a Rieei-flat compaetification.
In §.III, we consider one simple model 4 without fermion condensations and show that the F x K is always a unique attractor in this system, but the effective gravity theory is the JBD theory with
~2=-i.
In
§.IV, we take into account a gluino condensation of E' gauge fields, which is 8 responsible for the local SUSY breaking.
We show that the minimum of the potential,
53 4
which corresponds to the F x K space-time because of zero cosmological constant, always one of the attractors if the 3-space is expanding.
is
One preferential scenario
in our model and remarks on inflation are discussed in §.V.
II.
FOUR-DIMENSIONAL LAGRANGIAN Assuming a Ricci-flat (e.g. a Calabi-Yau) compactification,
world interval is described by
the lO-dimensional
FI
with
The eonformal factor exp(~/2) string action (8). space.
is from the Weyl rescaling of the iO-dim metric in the
~(x)
is the dilaton and b(x) is the 'radius' of the internal -6 We have factorized out the conformal factor b in the 4-dim metric in order
to obtain the proper Einstein action in four dimensions,
g~N is the metric of a
static Ricci-flat manifold. The bosonic part of the lO-dim, N=I supergravity Lagrangian
(9)
consists of
(2.3-a) with
-ir where
and
2 ~1o/ 8~=
&,.p ..
2.3-b)
GIO and glO are the lO-dim gravitational constant and gauge coupling
constant, respectively,
whilst R(~) and ~
derivative with respect to ~ w
are the scalar curvature and the covariant
i
In the case of a Calabi-Yau compactification,
we need the Riemann curvature
squared term (2) , which is derived in the field-theory limit of a superstring theory (8) Here, we assume the special combination of curvature squared terms; 2
2
in order to have a ghost-free theory of Yang-Miils fields;
(10)
Through the vacuum expectation value (VEV)
54
~.(y)
g~
i
for /~,~ =M,N
N
=
0
otherwise
(2.5)
we obtain a Calabi-Yau compactification in the non-static background (2.1) The curvature squared terms and
~Fm
(11-13)
are rewritten as
(2.6) I (~ma~ ~
+ (totally divergent term),
~
where RMNpQ is the Riemann tensor with respect to gMN" vanishes because of the Calabi-Yau eompaetification, ~l
which depends only on gmn , b and ~
The first term in Eq.(2.6) l(g
,b,~ ) is the term from mn , and it does not contain higher-order
time derivatives~g changes in g
, of b and of ~ . If the time scale or length scale of mn , b and ~ is much smaller than the Planck scale (e.g. in the later
mn stage of the universe), terms such as ~
(iI).
then the I-term can be neglected as compared with the other It is worth noting that this may not be always true for
some combinations of curvature squared terms.
~
Because the structure of the dynamical
system may change completely if higher-order time derivatives appear
(11,13)
The
expression (2.6) is also valid for the simple torus compactification with vanishing MN
and R
MNPQ
The Einstein action is reduced in four dimensions as
where N z = ~ l o /
d y
, and R(g) and
~
are the scalar
curvature
and t h e c o v a r i a n t
derivative with respect to g
mn The VEVs of Hmn p and of its potential BMN provide two 'axions', ~S and ~ T
defined by ~-2~ ~
V~o~
and
BM~
I
Here, we introduce the new scalar fields ~ and ~T ' instead of
and ( ~,
~)
defined by
and ( ~ ,
}
and in b, as
= ~-~'(2 ~/~ g + ~/2 )
(2.9)
~T ) form two complex ehiral superfields S and T in four dimensions;
55
Using ~
and
~T
' the lO-dim world interval
(2.1) is written as
The VEVs of the internal components of H
may also appear through Dirac MNP string singularities for a non-simply connected interal manifold such as a Calabi-Yau manifold.
As for the VEVs of fermions, we consider only the gluino ~
condensation
of E' gauge field, which may give natural SUSY breaking mechanism (14) . This 8 mechanism with the above VEVs of H provides the effective 4-dimensional potential: MNP
6,)
-3.~.,
,,
(2.12) vJhere e o n t a n t s
c and h a r e d e f i n e d by
C Zl'Ip~6"I"JK
Hirff- K respectively,
and
Jr" ~ C J K X
,3
mpL and /6( are the Planck mass and the energy scale of condensation.
b o is fixed by the gauge group. From the above setting, we obtain the four dimensional effective Lagrangian, which is equivalent to that of the lO-dim, N=I supergravity model, as
%
...xJd~:::ll:: i/~
L
_ 1)Ca
(2.14-a) I
-
III.
L
%)
(2.14-b)
THE FRIEDMANN UNIVERSE AS AN ATTRACTOR We, now, consider the cosmological solutions.
assumed to be
The 4-dim metric ds
2 4
is
(15)
56
The basic equations are
(3.2)
(3.3)
~H
-+
~ +;-~
+ a-~
=o
(3.4)
(3.5)
aS~
)V -o
and
4
( °~ c ' ~ * "
)" =
o
(3.6)
,
(3.~)
where H = ~ / a i s the Hubble parameter , Ekin= (~$2+ denotes the derivative with respect to t.
"<+e2~'~2+e28~T~)/2, and a dot
Here, we have also introduced the 4-dim
matter fluid, which energy density and pressure are denoted by
e
and P, respectively.
in this section, we consider the case without fermion condensations, also assuie that H justified if H
MNP
we
= 0 (i.e. e = 0). The potential V vanishes. This might be MNP is induced only when the gluinos condense as discussed by Rohm and
Witten (16) " If V =0, we can obtain the analytic solutions of Eqs.(3.2~7) Introducing the new time coordinate
d~
=
~-34t
~
as follows.
instead of t by
,
(3.8)
Eqs.(3. 4 . 3.7) are easily integrated as
(3.10-a) .p
and
~
(~)~
@
I O T~ ~-
C ~E~%
=
E~
where QS' OT' ES and E T are integration constants. we obtain the analytic solutions as
,
(3.lOb)
Integrating these equations again,
57
¢~×@
2 E# /
"~n~l [2,'k:I ~ ( 7 - ?:) ] = e.,o for Qs#O and QT~O.
' For QS= QT=O,
= (~..O ~ ~'/~E~ ~ (/)T "= ~'f..O ~'~-~ T 8,.%" Here,
~o''
(3.12-a)
= ~S~O
and
~ ~T
.,*
(3.13-a)
,
(3.13-b)
= ~T 0
~f' ~S~0' ~.0'~.O' and % o
(3.14)
are integratiOn cOnstants"
The Einstein equations are, now,
3H~ = and
<
E
•
f
~
,
(3.15)
2 ~'~ = -- ~2 (p_ e ) G6 ,
where ~[=( da/d~ )/a and E=Es+ET.
(3.16)
Using the above solutions (3.11-3.14),
we finally
obtain a complete set of solutions for the following two cases: CASE (I) : Vacuum ( P = ~ =0 ) E (3.17)
CASE (II) : P = ( ~ -
i)~
0 < ~ < 2) i
(3.18) ~ ~o¥0
$8
where a
~a
and
o
are integration constants arid the constant
:x~
a o( t l / 3 , w h i c h
universe,
is defined by
(3.19)
G 3~
For the case (I), 7 -> oo as
~
is
when t -~oo .
The
scale
factor
the same as that of the stiff matter
a expands
(P=~
dominated
because of the massless scalar fields ( ~ , 04~, ~T and ~T ) . , ~
const, but
--~ and --~ 9 - oo
zero, hence this asymptotic
(~ ~
for QS = QT = 0 ). 4 solution is not F x K.
For the case (II), t ~
when
~ ~ ~o-O.
and ~T -~
The internal space shrinks to
a ~ t 2/3~ asymptotically,
is the Friedmann universe with matter fluid of P = ( / - i ) ~
.
which
Since ~.~r ~SI~T-> some
constants, the internal radius approaches some constant. All solutions approach 4 the F x K space-time for t -> ~o. Therefore, the Friedmann universe with a static internal space is a unique attractor in our dynamical system. From the above solutions, kinetic terms of the 'axions' for
~
and
~T
)
we find one important role of QS and QT ( the
(17)
Eq.(3.10)
if QS~ 0 and OTTO,
,
f e,/
shows that there are maxTmum values
i.e.
and
Those kinetic terms provide potential barriers which prevent away to arbitrary large values. stay near the preferential
minimum
~
and Jr from going
This result is important,because (~
the universe must
the Planek scale) when the gluinos condense,
to reach the present state for natural initial conditions. The effective 4-dim action (2.14)
contains the proper Einstein action,
it seems that this model guarantees the'Einstein gravity.
That is, however,
then not true
because the original coupling of a massless scalar to the 4-dim gravity was removed -6 from the action by the Weyl rescaling ( the factor b ), but appeared in the 4-dim world interval
(2.11).
We need a potential for
~
scalar in our model, so that it fixes the value of ~_
, which is the massless JBD and guarantees the 4-dim
Einstein gravity.
IV.
THE EINSTEIN GRAVITY AS AN ATTRACTOR
When the universe expands and the temperature (~/i~),
drops below some critical value
the gluinos of the largest gauge group (e.g. E~) condense.
We find the
59
gluino-condensation potential (14) .
As discussed by Rohm and Witten (16) , the VEVs
of H
may be also induced through an instanton solution at the gluino condensation MNP and its value e is quantized as
C = C)~ - ~ C o
(4.1)
.)
where n is an integer and c
o
is determined by the geometry of the internal space.
If we take into account the Chern-Simon term, however, we find an instanton solution, through which the quantized value c can change to the other value ( c ). Bubbles n n+l may be formed through the quantum tunnelling. Since c changes through the quantum tunnelling,
the potential V also changes with time.
tunnelling ( ~
The time scale of quantum
tQT ) is not yet known because the explicit instanton solution
not been obtained.
has
In this section, we shall discuss the evolution of the universe
mainly for the case that tQT>>
I/H ( CASE (I))
tQT << I/H (CASE (II)) and tQT ~
and briefly for the other two cases
I/H ( CASE ( I I I ) ) J.
( See Eel.
(12) for the
details ).
I CASE ( I ) "
tQT
>> 1/H
I
In this case, we can neglect the quantum tunnelling effect,
c
is actually n constant, then the potential V is fixed during the evolution of the universe. The potential minima are located at
%.o
/% _ ~/q = C7~. -
T
=
i
-
f bo k-~-
2/~ go~
3)t ~
( m: anyinteger
)
(4.2-b)
arbitrary
(4.2-c)
4 Since the potential V vanishes at these minima, it guarantees the F x K as well as the 4-dim Einstein gravity. and 3"17 °r 3"18) with
~=
The analytic solutions are given by Eqs.(3.11-b,3.12-b, ~ p
and
a~ s = ~ m
r"
Since t these o sOlutiOns are a
of the previous solutions,
4 all solutions approach the F x K space-time as discussed
before.
is fixed at
The value of $
$~o'
then the theory at the low energy scale
is effectively the Einstein gravity theory. The potential V, however, has another unpreferential minimum at ~ shown in Fig. 1 for the case of
~=
~
, as
Neither the Friedmann solution nor
the Einstein gravity is not obtained at this minimum. arises.
: ~
Then, the following question
Which minimum is obtained for natural initial conditions of the universe ?
60 In order to answer this question,
we must
V
investigate the dynamics of the universe
,/k
for general initial conditions. Here, we shall show that the preferential attractors
minima (~.o,O~,m) are always
in our dynamical system if the
3-space is expanding,
and those can be
reached with finite probability. Let us introduce the new time coordinate by
Fig. i Define the 'energy'
U( ~ ,
~)
~ and the 'potential'
of the dynamical system for ~ and ~g by
(4.4-a)
e,)-- V ( I , ,oo ¢ ) .
(4.4-b)
From Eqs.(3.4 and 3.5), the equation for ~ is written as
d'C
~'kdU /
with H~=(da/d~)/a.
The constraint equation
If the 3-space is expanding always positive,
~S=~).
(3.2) reads
(i.e. H~> 0 or H~(To)> 0 at some epoch T = T o
then the system is always dissipative
(~s.o,~T~) of the potential U are isolated. for
Therefore,
(4.5-a)
.,,
( d ~/dI
< 0 ).
),
~ is
The minima
( The schematic shape is shown in Fig.2
once the universe is trapped in the shaded region T E
in Fig.2 at any value of ~T ' the universe always approaches along the dotted lines A in Figs. i and 2.
the preferential
On the other hand, if the universe
reaches the region TjB D , then the universe always goes away to ~ =
infinity,
finding the JBD theory, as the dotted line B in Fig.2 . It is worth noting that the 'energy'
minimum
of the total system ' % '
which is
61 defined by
%= Ek, ÷ V
U
(4.7)
A,. ,, B ,.
js
is also decreasing with time if the Sspace is expanding,
jl
i.e.
0
~T = - ~ H E'k; . for H > O.
•
(4.8)
By losing the 'energy'
~T
the universe reaches either the region T
E
or the region T
~T"
The minima
attractors
JBD ( ~,
at some value of ~
Fig. 2
are always
in our system.
In Fig.3, we show the phase diagram of ( ~
,~
) for the case of
9¢.(Qs = 0).
0~=
2.0~
The present universe
( ~ o , O) is a nice attractor. ~ (a few)x condensation,
\
If
~o at the gluino then the universe
l
o.o
~ -:..\ ~-
1.0
may reach the preferential minimum (18) If we take into account the kinetic terms of the 'axions' (Qs and QT )' the
~-field
3.0
0.0
1.0
'~s
2.0
could stay Fig. 3
near
~
=~,o when the gluinos
condense for natural initial conditions,
as discussed in §.IIL
Hence, the universe
can reach the present state ( the Friedmann universe and the 4-dim Einstein gravity for a wide range of initial conditions. We shall give brief comments for the cases I CASE (II): tQT<< I/H
(II) and (III).
[
The phase transition may occur due to the quantum tunnelling immediately after the universe will find the lower potential. universe,
For example
(see Fig.4),
the
which starts with the n=l vacuum, reaches the point A, beyond which the
potential with n:2 becomes lower than that with n:l. occurs just after passing through the point A. vacuum are formed in the old n:l vacuum, changes to n:2.
Then, the phase transition
Many small bubbles with the n=2
and collide with each other.
The effective potential for the
~ - and ~_-fields,
The phase under which
)
62
the universe evolves,
is the minimum of all possible potentials, Uef f, i.e.
(4.9)
os Uef f is shown in Fig.4 for
OS=~m.
Ueff
At each cusp (e.g. the points A and B), '
the universe changes it phase from n to n+l.
n=4
In the case of this figure, the
'
I n=3
...
'
i
,,."
'
i
n=21n=1
~.,,
", ..... "",:
i n=O I
universe eventually settles down to the n=3 vacuum.
If the universe reaches 4 one of the minima of Uef f, the F x K
I~,,y//,d!'d,,,,z;y :' '., ...'.,.. '-..!-:.
space-time is realized asymptotically
",' '. I':',..
and the 4-dim Einstein gravity is also guaranteed as discussed in the case (I).
I,
CASE (III): t Q T ~ 1/H In this case,
$
I
we expect
U
U
4
that
bubbles are formed at each cusp of U
U
3
Fig.
2
U
1
U
0
4
eff
and the bubble structure may not disappear / before the universe will find the next transition point.
For example, the (no+l)-
vacuum bubbles, which are formed in the initial n o vacuum, evolve under the potential Uno+1 and reach the next transition point B.
The (no+2)-
vacuum bubbles are formed in the (no+l)-bubbles, The old n o vacuum evolves under Uno and will also find another transition point. similar bubble formation occurs•
The Finally, the
no-vae~ ~
universe may find itself with a hierarchical bubble structure as shown in Fig.5.
~n ~_~
o-i ~ no- 2
Each Fig. 5
bubble may reach some of the minima of U The Newtonian gravitational constant G
eff and
the gauge coupling constant g4 are
N
given by
hence
G N and g4 take different values in each bubble•
give rise to the so-called
This bubble structure may
'domain wall problem', unless inflation occurs after the
63
universe reaches one of the minima of the gluino-condensation potential V.
V.
suMr~ARY AND REI'{ARKS
Since there is no kinetic term of t h e m e t r i c
of the lO-dim 'target space' in
a tree-level string action, we do not know yet the dynamics of the universe at the string level.
We do not know yet what are the basic equations for the universe at
the Planck time or beyond that stage.
There are some indications of interesting
phenomena (19) , although many things are not clear yet. Therefore, here we have discussed only the dynamics of the universe after the Planck time, in which period the field-theory limit may be justified.
The
dynamics of the universe is described by the Einstein equations with small corrections such as the curvature squared terms. We summerize our scenario.
If the compactification takes place near the
Planek time and the gluinos condense below that scale, then we have two eras; the era before the gluino condensation and the era after that. In the first period, the 4 space-time always approaches the F x K solution, but the effective theory is the JBD gravity theory. scalar fields condense.
If we take into account the kinetic terms of the 'axions', the ~
and
~7
may stay near the preferential minimum when the gluinos
After the gluino condensation,
the universe may find itself in the
present state ( the Friedmann expanding universe and the 4-dim Einstein gravity ) naturally for a wide range of initial conditions.
Our universe may be obtained
uniquely as an attractor. The VEVs of H
MNP
, however, may appear before the gluino condensation too,
i.e. at the compactification. at infinity. scalar fields
The appeared potential V has no minimum except for
Does this change the above scenario completely ? ~
gluinos condense.
and
~
It seems that
the
go away to infinity , where V approaches zero, before the
The universe cannot stay near the preferential minimum
and cannot reach it as discussed in §.IV.
~ =~.o
Our universe may not be realized for
natural initial conditions. However, if we take into account the effect of the 'axions', ~S and ~ T , the equations for the
~-
and
~-
barrier appears and it prevents the universe from rolling down to as follows (17).
on
fields, then we can easily show that the potential ~,~T
infinity
When h =0, Eq.(3.5) can be integrated as before, and then the
64
equations for
~4
and ~T are written as
(5.1)
,3H where V
Q
is defined by
+< This effective potential V The scalar fields ~ barrier.
Therefore,
unit, the ~ - f i e l d
Q
Q
]
(5.~)
provides a potential barrier against
and ~r cannot go away to infinity because of this potential
energy of the 'axions'
V
(5.2)
,
the universe will stay in some finite region.
at the compactification is of order of unity in the Planck
may stay near the Planck scale (~+s.0).
is propotional to a
If the kinetic
-6
This potential barrier
, then it will disappear later when the universe expands
enough. ~Tnen the gluinos condense, ~
=~o'
the universe is still staying around the minima
then will reach one of the preferential minima.
The probability for
the universe to find itself in the present state may become very high. is realized almost uniquely as an attractor.
Therefore,
Our universe
the VEVs of H
before MNP
the gluino condensation do not change our scenario so much. As mentioned in Introduction,
inflation is one of the most desirable mechanism
for the solution of the flatness, horizon and entropy problems in modern cosmology Can we find natural inflation in the present model ? " NO",
so far.
(5)
The answer is, unfortunately,
Here, we shall look at the difficulties.
We know, so far, two types of inflation; i.e. one is the potential type such as the GUT inflation (20) and the other is that due to the curvature squared terms (21) Recently,
inflation of Kaluza-Klein type is also proposed for higher-dimensional
theories (22) .
We search our model for such inflations.
Kaluza-Klein inflation due to a rapid contraction of the internal space does not work in our model.
Because this rapid contraction is caused by the internal
curvature, which vanishes in the Ricei-flat eompactification. the curvature squared terms also does not work.
KK inflation due to
Because the flat potential for the
internal radius is generated again from the internal curvature and the cosmological constant, both of which do not exist in the present model.
65
Inflation of Starobinski type is not possible.
Because if the curvature
2 (2 4) (I0) squared ( R -) terms are the so-called Gauss-Bonnet combination . , those 2 do not provide t~e R -terms in 4 dimensions.
2 Even if the R -terms~turn out to be
the other combination (23) , inflation is impossible due to the coupling to the dilaton field, as discussed the details in Ref.(24).
T h e similar situation happens "for the
lO-dim de Sitter solution (25), i.e. the coupling to the dilaton destroys the de Sitter 2 Forthermore, the R -terms from the string theory have ambiguity"
type solutions.
,
(23)
depending on the renormarization scheme.
Hence, the cosmological application of
2 the R -terms ( except for the Riemann curvature squared term ) might be meaningless. As for inflation of the potential type, we have not known yet what could be the inflaton.
We have not found natural inflation.
It might also be difficult
because of the absense of free ( or small) dimensionless parameter in the string (26) theory, although there are a few proposals
Whether natural inflation can be
obtained in the string theory is one of the most important questions in the superstring cosmology.
ACKNOWLEDGEMENTS The author would like to thank P.Y.T. Pang, M.D. Pollock and C.E. Vayonakis, with whom some part of the present work has been done.
He is also grateful to
Professor N. Dallaporta and Professor D.W. Sciama for their kind hospitality at the International School for Advanced Studies, Trieste
and to Professor Abdus Salam,
the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
FI :
The indeeies ~ , ~ ,... run from 0 to 3 and 5 to i0, while m,n,.., run from 0
to 3.
M,N,... are used for the internal indecies.
Our signature and notations
are the same as those of " G R A V I T A T I O N " , by C. Misner, K.S. Thorne and J.A. Wheeler ( Freeman, San Francisco, 1973 ).
66
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26
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MUTUALLY
INTERACTING
QUANT~
FIELDS
IN CURVED
SPACE-TIMES
JHrgen Audretsch Fakult~t
fHr Physik
Universit~t P ostfa c h D-7750
Konstanz
5560
Konstanz
W.-Germany
I.
INTRODUCTION
The d i s c u s s i o n has d e m o n s t r a t e d with w o r k e d - o u t acting
fields
of the p h y s i c a l
examples
one w o u l d
i.)
What
is an a p p r o p r i a t e
~lat
are the m e a s u r a b l e
ii.)
What
like to answer
i.e.
influence
scheme
is a related
?
calculation
localized
background ? What
to reveal
intuitive
this
advantages
at a certain place
point
to d e s c r i b e way.
point
like to d e s c r i b e of v i e w ? sections
can be o b t a i n e d
and
the cosmic evo-
gravitating
source
such a l o c a l i z a t i o n
Only
of
?
cross
time d u r i n g
near a s t r o n g l y
in an a p p r o p r i a t e information
early
typically
and d e f i c i e n c i e s
one w o u l d
physical
situa-
is the struc--
from a more q u a l i t a t i v e
concepts
How are the m i n k o w s k i a n
Up to now we have no approach process
processes
it reflect what
from a more
rates m o d i f i e d
of the s e m i - c l a s s i c a l
and m a t h e m a t i c a l
? Does
lution or at a certain
creation,
(!) interThe sort of
for such a theory
the g r a v i t a t i o n a l
physics
last but not least:
intera c t i o n
of m u t u a l l y
scheme
of the type:
? What
theoretical
are a p p r o p r i a t e
and to obtain
decay
Universe
a feasible
space-time.
framework
quantities
in w h i c h way does
? What
of a given
And,
are thereby
conceptual
quantum-field
iii.) ~hat are the p h y s i c a l
iv.)
theory
background
characteristics
ture of the u n d e r l y i n g view
to have
?
are the p h y s i c a l
tion,
of the I n f l a t i o n a r y
for q u a n t u m - f i e l d
in a given c o s m o l o g i c a l
questions
scheme
details
that it w o u l d be very u s e f u l
?
of an
for the p r o b l e m of backfrom the e x p e c t a t i o n
69
values of currents
like the stress tensor. For reviews see [1,2]. Fur-
ther l i t e r a t u r e can be found in [3-6], compare also
A p r a c t i c a b l e scheme for the d i s c u s s i o n of mutual S-matrix approach
(in-out approach).
The conceptual
[7].
interaction is the framework,
the mea-
surable q u a n t i t i e s and the related c a l c u l a t i o n scheme have been d i s c u s s e d in detail in [3-6]. This a p p r o a c h is close to the p r o c e d u r e one is used to in flat space-time.
It has the d i s a d v a n t a g e of b e i n g n o n - l o c a l by na-
ture if the b a c k g r o u n d
is non-minkowskian.
On the other hand, when apply-
ing it, one knows rather clearly w h a t one is c o n c e p t u a l l y doing.
There-
fore, the study of this a p p r o a c h is a useful p r e l i m i n a r y stage at which the structure of the g r a v i t a t i o n a l means of exact calculations. thereby,
physically
justified a p p r o x i m a t i o n s may later be found to solve
the l o c a l i z a t i o n problem. with
influence can still be w o r k e d out by
On the basis of the p h y s i c a l insights gained
Naturally,
all q u a l i t a t i v e results o b t a i n e d
the in-out approach should be h a n d l e d w i t h care w h e n using them as
input for a s t r o p h y s i c a l calculations.
In the f o l l o w i n g we give a very brief summary of the results of our papers
[3-6], w h e r e all details and exact calculations can be found. We
discuss two m a i n subjects:
a.) A p p r o p r i a t e l y d e f i n e d t r a n s i t i o n proba-
b i l i t i e s and their a p p l i c a t i o n to d e c a y processes, g r a v i t a t i o n a l l y induced a m p l i f i c a t i o n
b.) The effect of
(and attenuation)
and its demon-
stration by means of the Compton effect.
II.
SEMI-CLASSICAL APPROACH
We study the i n t e r a c t i o n given by
~l(l,g~8, ~, Y) b e t w e e n two types
of neutral scalar p a r t i c l e s d e s c r i b e d by the massive K l e i n - G o r d o n and the m a s s l e s s
d i s c u s s i o n of the scalar q u a n t u m electrodynamics. meter.
field
field ~. This may be regarded as a first step for the ~ is the coupling para-
The fact that the curved space-time b a c k g r o u n d acts q u a n t u m - f i e l d
t h e o r e t i c a l l y already in zeroth order
~I = 0
(for example in creating
p a r t i c l e s out of the vacuum), will t h e r e b y severely influence the outcome of the m u t u a l i n t e r a c t i o n and its registration.
We r e s t r i c t o u r s e l v e s to a t r e a t m e n t in the i n t e r a c t i o n p i c t u r e using an in-out scheme b a s e d on the S - m a t r i x
S = lim T exp{i f~T e-~lql d4x} • The
70
gravitational is the
background
switch-off
a 3-flat
ds 2 = a2(n)
(dq 2 -' dx 2)
is c o n f o r m a l l y
expansion
Robertson-Walker
flat.
Apart
taken into account.
The K l e i n - G o r d o n
particles
(?p v ~ + m
results
Universe
from the e x a m p l e s
law is left unspecified,
the b a c k g r o u n d
exactly
(2.1)
q = + ~) must a l l o w the d e f i n i t i o n
The general
always
parameter.
We c o n s i d e r
which
is thereby
but the
discussed
below,
in- and o u t - r e g i o n s
the
(q=-~ ,
of free particles.
are assumed
+ R/6) ~ =
to be c o n f o r m a l l y
0 (R = scalar
can easily be t r a n s c r i b e d
coupled
curvature
for other
to
, for ~: m=O).
fields
and other
interactions.
The p h y s i c a l energy
called p, q, c reatio n
... for
of massive
number N~°I(_P~I0)' ~ ation :
specification
is not conserved,
there
I~kl~:O
#-fields %-pairs
1 ... for
iv.)
creation
consequences:
3-momentum
~-fields.
iii.)
there
in every mode p with
Because
is
total
of the conformal
of ~-particles
i.)
parameters
situ-
N (0) (_k~I0)
=
.
regarding
is e s s e n t i a l l y
= (ul~• , uOUt, ~},
III.
solutions
Bp = ( u £in,
ADDED-UP
nevertheless
the influence
contained
TRANSISITION
I~£I 2 -IB~
a massless
particle
particles
particles
registered
Y-particles
are p r o d u c e d
equation:
~
=
= i.
out of the vacuum.
and m a t h e m a t i c a l l y
but has solely been
massless
12
backrelating
PROBABILITY
we know that in our case,
background,
coefficients
of the K l e i n - G r o d o n
creates
a physically
of the g r a v i t a t i o n a l
in the B o g o l i u b o v
out*, ) with u_#
The curved b a c k g r o u n d
action:
and k,
following
are c o n s e r v e d
out of the v a c u u m
= I~pl 2 = 14(°) (-_p~Io).
in- and o u t - p a r t i c l e
bility,
there
is no co-rresponding
The i n f o r m a t i o n ground
above has the
ii.)
because
reasonable
created
transition
of the c o n f o r m a l
in the o u t - r e q i o n
by the g r a v i t a t i o n a l
proba-
field equation,
has not come out of the
or i n f l u e n c e d
are ~ood indicators,
To c o n s t r u c t
by the m u t u a l
whereas
background,
massive too.
inter~ -
The new con-
71
cept a d d e d - u p
probability
add w ( s ~ I c ~ r ~) = It answers
of m a s s l e s s
What
then
this q u a n t i t y
(compare
This p e r m i t s
=
?
state
regard-
It is i m p o r t a n t
in w o r k i n g
out in-in
to
amplitudes
in>l 2
technique
only by the r e p l a c e m e n t exact
(3.2)
which
differs
from the one
of the plane ~Taves
solutions
D E C A Y OF A M A S S I V E
The i n t e n t i o n
ui~ --
of the
in
for m a s s i v e
field e q u a t i o n
in
used.
to make
survey of the s t r u c t u r e
Because
particle
Universes we have
increasing
a(q) n ~
The results
statically-bounded In all cases
will
expansion
of the b a c k g r o u n d
We c a l c u l a t e
=
The second
below
(for details
to the process
of m u t u a l
+
quantities
type
[6]).
interaction.
out We
from our results.
for the scale
factor
:
e(~)a o
a i = A/~-B, a0 = A/~, A>B>0
see
is called
gravitationally
can be read off
the p h y s i c a l
n = 0
ao = c o n s t ,
first has been d i s c u s s e d
contribute
and discuss
a(q)n÷~
here.
created
a step at e(-n) ai
The
types
a(~) n + - ~ a i = const,
one with
that p a r t i c l e s
contributions
law
flat in- and o u t - r e g i o n s
a(~) q ÷ - ~> a.] = const,
be d i s c u s s e d
influence
are only two d i f f e r e n t
expect
show how the r e s p e c t i v e
representing
laws:
time-parameter.
is to read off
of the type of e x p a n s i o n
there
not be r e p e a t e d
and will
one w i l l
examples
of the g r a v i t a t i o n a l
approximately
possible,
and the other with
h is an a p p r o p r i a t e
~(n)
independent
to have
definitions
of m o n o t o n i c a l l y
[3].
PARTICLE
of the study of the f o l l o w i n g
complete
in R o b e r t s o n - W a l k e r
with
des~ISlc~r ~
~ I< i n all d
by a p p r o p r i a t e
EXamPLE:
a rather
in
in the out-region,
states
can be o b t a i n e d
that a p a r t i c u l a r
space-time.
IV.
w here
will be found
to the m a s s i v e
a Feynman-diagram
flat s p a c e - t i m e particles
a(h)
(3.1)
[3])
Wadd (s~IcCr~)
curved
situation:
is the p r o b a b i l i t y
particles
less of w h a t has h a p p e n e d note
to this
~ l l 2 all d
the question:
Is ~ out >
is adapted
(4.1)
. We then add the c o r r e s p o n d i n g
rigorous
72
c a l c u l a t i o n of the t a n h - e x p a n s i o n law
a(n) = {A+B tanh bn , b>O
(4.2)
which is as s m o o t h e d - o u t step the p r o t o t y p e of a s t a t i c a l l y bounded expansion law, leading to all the typical p h y s i c a l deviations step situation.
from the
Quantities with hat and tilda will refer to the respec-
tive e x p a n s i o n laws above.
In the following we sketch the exact c a l c u l a t i o n of the decay of a massive scalar ~-particle into two massless order of the
scalar Y-particles in lowest
(-l/a(n)) }y2_interaction with coupling p a r a m e t e r
tor a -I makes the interaction c o n f o r m a l l y invariant, c a l c u l a t i o n below becomes less cumbersome.
I. The fac-
so that the exact
The mass m in the field equa-
tions breaks the conformal invariance.
Figure
;Y
I:
<
-ika4('q)
~-k
Diagrams c o n t r i b u t i n g to the added-up transition probability
To derive the added-up p r o b a b i l i t y for the decay process, we have to work out the p r o b a b i l i t y amplitudes related to the two diagrams of figure I.
We obtain in the step case
~add :
~2~ ~i I Tn 6(re_i)+( i+ i~ajz) Eoo i Tn ~(m-°) ] + ~ (k) 2kip-ki v
(4.3)
and in the tanh case
]]add
_
X2~
2k]p-k]V
I~ I
mmEi (Tn +
finite) d
+(
(w'i)+
½ + I~_pl2)1
(T n +
o
finite)6(~ o)I+ ~(k)
(4.4)
73
Ei/0 = /_p2 + m 2 ai/0 z ticles.
With
energy
in the
the
m-i/0
infinite
quantum
n-time
of the
around
out-region
Figure
=
the e n e r g y
Ei/0 - (kl + k2)
in-out-region T
interaction
behaviour nances
is t h e r e b y
n
A(k)
energy
the
is s h o w n
by
in-
and o u t - r e g i o n s T~+~
in f i g u r e
conservation
2.
in the
q
. The
(!)
introduced
during
which
typical
It shows
in-region
par-
of the m e a s u r e d
m-i/0 = 0 . We h a v e
2 ~ ( o ) = lim T
place:
of the m a s s i v e
the c o n s e r v a t i o n
is g i v e n
between
takes
parameter
the
spectral
smoothed-out
(k = E0/2)
reso-
and
in the
(k = E0/2).
2:
~<~
i
i
E|/2
E0/2 k
The c o r r e c t i o n ~(k) in (4.4) as a f u n c t i o n of the e n e r g y of the m a s s l e s s p a r t i c l e s for d i f f e r e n t v a l u e s of the e x p a n s i o n p a r a m e t e r b for d e c a y i n g p a r t i c l e s at r e s t (p = 0).
Apart
f r o m the
non-vanishing step
leads
interaction nance
fact
that w
outside
formally time
IBpl 2
This
We
turn
two d i f f e r e n t
first.
In the
are
identified
and k, into
respectively.
t u m as w e l l means
out
of the
ones
On the will
as c o n s e r v a t i o n
~-i = 0 for
~ < 0
n > 0 we h a v e
~-0
number
will
origin add
--
of the
A-terms, the
infinite
the o u t - r e s o -
of m a s s i v e
above,
other
massive
in z e r o t h
hand,
the
energy
is f u l f i l l e d . as
law
particles parameters
of a m a s s i v e
if c o n s e r v a t i o n
c a n be w r i t t e n
of
significance
step,expansion
and massless
the d e c a y
order
below.
b u t b y the m o m e n t u m
resonant
of m e a s u r e d
particles
the p h y s i c a l study
of
particle
3-momen-
The
latter
/p2 + m2a 2 = kl + k2
-
For
of the
smoothing-out
that
be g i v e n
and and
quantities,
become
, which
typical
of the b a c k g r o u n d
in w
given
n o t by m e a s u r e d
two m a s s l e s s
the m e a n
resonances
calculations
corrections
it is v e r y
An e x p l a n a t i o n
to a d i s c u s s i o n
of the
because
see t h a t
( l + 2 1 B p l 2) as c o m p a r e d w i t h the in-
contains created
interaction.
we
bounded
by a f a c t o r
factor
cases,
resonances,
to a d d i t i v e
gravitationally
the m u t u a l
is in b o t h
of the
T n. F u r t h e r m o r e
is a m p l i f i e d
resonance.
add
i
= 0 and
therefore
/p2 + m2a 2 --
0
= kl + k2 . A c c o r d i n g l y ,
74
for i n g o i n g
#-particles
gy c o n s e r v a t i o n n Z O,
implies
happening for t h e s e
k I and
k2
of the
~-particles
pearance
tain
either
bounded
total
= ~-#
in the r e g i o n different
~-particles.
But
q ~ O
values
p,
decay
and ener-
or in the r e g i o n of t h e p a r a m e t e r s
in a n y c a s e
the m e a s u r e m e n t
in t h e o u t - r e g i o n . T h i s c a u s e s the ap, add in t h e s p e c t r u m of w , w h i c h are t y p i c a l
expansion
added-up
in the c a s e p
w
a given momentum
is p e r f o r m e d
of t w o r e s o n a n c e s
the
with
two cases
of the o u t g o i n g
for statically
For
prepared
laws.
wt°t = ~ wadd(l_k Ip_~kl1#p )
probability,
, w e ob-
= 0
. Tn ao (~ + I Bp=012t Tq
~tot =TF~X [2_~i Tq + ~01 (1+ ,~pOi2]Tql: + Rfin(ai,ao,l~12) R fin
is t h e r e b y
We How
give
a finite
an i n t e r p r e t a t i o n
is the r e s u l t
transition particles
(4.5)
tot = WMink
probability have
a chance
because
o f the
q -< 0 a f a c t o r up with
the
first
considerations
to d e c a y
of
to
(4.5)
to the
in d i s c u s s i n g
related
q ~ 0
I
the q u e s t i o n :
to the m i n k o w s k i a n
~ In o u r c a s e
of ~ I ' the
instead
t e r m of
tot
one half
in the M i n k o w s k i - r e g i o n
in the r e g i o n
according
related
step-law (~2/4~m)Tt
structure
l a ] -I
~ntroducing finally
correction.
of the t w o w
for t h e
a(q) = ai(and the o t h e r s more,
additive
(4.6)
with
total
of t h e
q ~ 0
with
a(q) = a0 ). F u r t h e r -
interaction
contains
as in the m i n k o w s k i a n
case
for (a = I) .
T t = a i T q the c o n f o r m a l t i m e T q , we end T h e s e c o n d is o b t a i n e d b y c o r r e s p o n d i n g
interval
q > 0. I
^
With number around in t h e third ticles
regard
to t h e t h i r d
of massive p
= 0
particles
which
zeroth
order
term reflects , but
t e r m of
of the mutual the
also
q > 0 . Therefore,
underlying
process, the
and
interaction.
factor
factor
these created as c o m p a r e d I/2 has,
I/E O r e m a i n s
that
with
is the
interval
o u t of t h e b a c k g r o u n d
The
appearance of the
particles
in t h i s
16p=o12
the m o m e n t u m
created
fact that not only one half
the r e g i o n
the
we recall
per unit volume
are gravitationally
in a d d i t i o n
16p=012 w h i i e
(4.5)
the
unchanged.
incoming
are decaying
second
case,
of this
term
parin
and its
to b e r e p l a c e d
by
75
The
three
go back
resonant
m e a n value
proportional
the t e m p o r a l
mean v a l u e
IFL
> =
bounded
familv
il
the r e s o n a n t in ~tot,
terms,
provided
contributions
of the outcome
expansion i.e.
(4.6)
also the
we have to work happening
To do so, we make is true
out
in
use of
for all
we find that also
the terms
we repalce
of
of taking
of the p r o c e s s e s
relation
laws,
in Q t o t
But instead
of t a n g e n t - s p a c e s .
~ .L~i.~z" +~--|.Because0 ) . this
monoton!c
to T
processes.
of the two m i n k o w s k i - t y p e
the continuous <
terms
to the m i n k o w s k i - t y p e
statically
in the ceneral
case
proportional
to T , agree w i t h those n c o e f f i c i e n t s in the usual
the B o g o l i u b o v
way.
The a p p e a r a n c e important
of the a d d i t i o n a l
generic
consequence
ence of the i n t e r a c t i o n
process
such t e r m in Q t o t b e c a u s e the one p o i n t of time
The t r a n s i t i o n nite d u r a t i o n (a(n)
T
~-1
that
~tot
is the
the g r a v i t a t i o n a l
is only of finite duration. influence
influ-
There
is no
happens
only at
contain
the infi-
n = O.
probabilities of the m u t u a l
by the time T n = Tt,
quantity
term R fin in
fact,
the g r a v i t a t i o n a l
n = I) the usual p r o c e d u r e
lities
finite
of the
the r e c i p r o c a l
(4.5)
and
(4.6)
interaction. would
be
to divide
thus o b t a i n i n g
lifetime
still
In M i n k o w s k i
space-time
the r e l a t e d p r o b a b i -
as a p h y s i c a l l y
of the m a s s i v e
particles
relevant at rest
= 12/4~m
Mink Referring
applied
in the f o l l o w i n g
in the step-case
%2
(4.5)
c iproca l
= (~)
n-lifetime
minkowskian
1 2E 0 ) "
g o i n g back
cause we had to base
which
are created
It is p o s s i b l e term in
can be
to
(4.7)
(2 i
contributions
sition p r o b a b i l i t y
leading
the same p r o c e d u r e
,,l p_-o1 -
m 1
~<Mink>
n-time,
2
i _ I__ + ~<Mink> Z ~ o
with
to the
~ The
to the t e m po r a l
as d e s c r i b e d
our d i s c u s s i o n
(3.1-2),
this
represents
mean value
in the p r e c e d i n g
on the c o n c e p t
the d e c a y p r o d u c t
out of the b a c k g r o u n d
to e l i m i n a t e
latter
en t e r
influence
the re-
of the
local
paragraph.
of the a d d e d - u p
of those
tran-
O-particles
the c a l c u l a t i o n
in o m i t t i n g
Be-
the
of $.
IBD=0 I2-
(4.7). ~
On the other hand,
to w o r k
in other n o n - s t e p - c a s e s ,
out the H - l i f e t i m e
we have
T in the tanh-case
to d r a w a t t e n t i o n
to the fact,
or
that in
78
these
cases
the d u r a t i o n
of the m u t u a l
gravitational
influence
characterized
by two d i f f e r e n t
cause
of the adiabatic
c ontra s t
to this,
time scales
time
switch-off
influence
scales.
scale w h i c h
on the m u t u a l
to be a c h a r a c t e r i s t i c a l splce-time:
in the a s y m p t o t i c
in- and out-regions,
We have
thus
to stress,
ness there seems
In order
implying
however,
the
that
influence
to their
respective
time T
finite
It
of these
to introduce
and g e n e r a l l y
Nevertheless,
- according
is finite.
The a p p e a r a n c e
the curved part
ideas divide
time T . In n the d u r a t i o n
trait of an S - m a t r i x
second
are
is,be-
the infinite represents
to be able
fine such a Tg ray rigorously.
~tot by the i n f i n i t e
scale
two
approach particles
of the s p a c e - t i m e
time
scale.
from the p o i n t of view of the exact-
to be no u n a m b i g u o u s
about the g r a v i t a t i o n a l
and the
on the other,
first time
interaction
time T grav.
in a given curved
must be localized,
The
in S , again
time
the g r a v i t a t i o n a l
seems
on one hand,
caused by the curved b a c k g r o u n d
the second
of the g r a v i t a t i o n a l will be called
interaction
applicable
to obtain
some q u a n t i t a t i v e
of the d e c a y process, origin
and the finite
- the d i v e r g e n t rest
way to de-
R fin by
we m a y part of
T gray to
obtain: : =
_ _
Rfin ~2 1 +Z~TI~p=o 12 + -Tgrav ~<Mink> 0
1 T
V.
GRAVITATIONALLY
INDUCED
We turn now in the second effect w h i c h The mean
governs
number
N(Eml a) =
quantum
n(~la)
The m e a n i n g
AND A T T E N U A T I O N
part of this paper field
theory
of o u t g o i n g m a s s i v e
to another
in given
B-particles
curved
important space-times.
in the mode ~ is
Z ll 2n(£~I b) was
Ia in > .
zeroth order e x p r e s s i o n
where
AMPLIFICATION
(5.1)
all b
if the in-state
rI(°)(~la)
(4.8)
Parker
[8,9]
has shown
: N(°)(E~IO) + n(zmla) + N(°)(E~IO)~(E~Ia) is the number of the three
that the r e s p e c t i v e
has the structure
of B-particles
terms
in
(5.1)
+ n(-£~la)]
occupying
is: p a r t i c l e
(5.2)
the z - m o d e creation
of
la>.
out of the
77
vacuum, p a r t i c l e s w h i c h have passed through and,
finally,
induced a m p l i f i c a t i o n of the ingoing p a r t i c l e content. results in additional o u t g o i n g pairs. a p p e a r a n c e of n ( - ~ l a ) ,
gravitationally
This a m p l i f i c a t i o n
The latter fact is i n d i c a t e d by the
a c c o r d i n g to w h i c h ingoing p a r t i c l e s in the mode
-p induce c r e a t i o n in the mode ~.
Fermions,
on the other had,
show a t t e n u a t i o n
(negative third term).
For complex fields the -~-mode is an a n t i p a r t i c l e mode.
For higher orders of the mutual i n t e r a c t i o n we obtain c o r r e s p o n d i n g l y (for details see
N(z)(_p@la) =
[4])
~ l
blSla
+ N(°)(p<~lO ) -
in>l~z)n(p~ib)+
~ I
in>[~z)[n(p~Ib)+n(-~Ib)]
all b
+
(5.3)
+ Re(6p C~p)
with
C~p = -2c~* ~ -P a l l b P -p The second term is again the amolification, tual interaction.
(5.4)
now b e i n g a part of the mu-
The third term has no c o r r e s p o n d e n c e
in the zeroth
order formula.
The structure of matic rule:
(5.4) may be v i s u a l i z e d using the following d i a g r a m -
"Let the mutual
region only and d e s c r i b e
i n t e r a c t i o n happen c o m p l e t e l y w i t h i n the in-
it a c c o r d i n g l y by in-in t r a n s i t i o n ~mplitudes.
Process now the c o r r e s p o n d i n g
p a r t i c l e outcome as in zeroth order in a
twofold way: At first the p a r t i c l e s in the p-mode pass through into the o u t - r e g i o n as in
(5.2) to o b t a i n the first term of
p a r t i c l e s are a m p l i f i e d in the same way as in
(5.3). Secondly,
(5.2) w h e r e b y a p o s s i b l e
outcome in the -~-mode c o n t r i b u t e s in a symmetric manner. the second term in
these
(5.3). F i n a l l y the a-term of
(5.3)
This leads to
is to be added".
W h e n using this rule it m u s t not be f o r g o t t e n that the c a l c u l a t i o n is in fact
b a s e d on one single coherent in-out process.
W r i t i n g the p a r t i c l e number a c c o r d i n g to
78
N(Z)(-P}la) = a]]~ b I< in bISla in >l(2z)n(p~Ib){l + N(°)(_p}[O)[l +
n(-m+Ib)Tin(p~ib ) j
(5.5)
+ Re(~p Op ) we can r e a d o f f t h a t g r a v i t a t i o n a l amplification a c t s as a m o d e - d e p e n d e n t an~lification factor (!) and n o t as an a d d i t i v e t e r m . T h i s means t h a t t h e minko~skian contributions contained in the in-in transition a m p l i t u d e are altered in a m u l t i p l i c a t i v e way, w h i c h may lead to c o n s i d e r a b l e modifica-, tions. Fermions show for n o n - z e r o order of the mutual i n t e r a c t i o n again a t t e n u a t i o n instead of amplification.
See the appendix
of reference
[4] for details.
An immediate consequence of
(5.2)
and
(5.3) is
N(p~l a) - N ( - S I a ) = ~ l < i n blS[a in>12[n(£{Ib)-n(-P¢Ib )] all b
(5.6)
A s y m m e t r y in the p a r t i c l e content of the o u t g o i n g p- and -p-mode can therefore be solely caused by the structure of the mutual i n t e r a c t i o n as r e p r e s e n t e d by S. term in
(5.3)
A m p l i f i c a t i o n and the process leading to the o-
all happen as C r e a t i o n of
3 - m o m e n t u m conservation.
In the charged case we would find particle-
and antiparticle pairs.
In reference
.........
[4] we have studied the example of p a r t i c l e creation out
of the v a c u u m for the i n t e r a c t i o n law
(p, -p)-pairs. Thus r e f l e c t i n g
~I
=
- -/~'(i/a2(~))#~ and the expansion
a2(~) = 1 + e 2b~ • This has been done exactly up to the order
~2
The
m u l t i p l i c a t i v e a m p l i f i c a t i o n and the role of the o-term are d i s c u s s e d in detail.
VI.
IMPROVED I N D I C A T O R C O N F I G U R A T I O N S
We can improve the p r e d i c t i v e power of the concepts i n t r o d u c e d above in specifying the massive part of the end state too and in using less extensive summations as compared with
(3,1)
and
(5.1). Good m a s s i v e indi-
79
cators they
are all c o n f i g u r a t i o n s
consist
only.
of m a s s i v e
The sum t h e r e b y
pairs
states
guarantees
finite
number
Feynman
that
all p o s s i b l e
are
indicated
for any finite
ending with
of m a s s i v e
rules w h i c h
seems
(6.1)
states
Q which
by capital
order
of m a s s i v e
letters.
(compare
[5])
(6.2)
contain
A perturbation
as simple
as close
consist
of the m u t u a l
Q which
In the f r a m e w o r k
to be a c o n c e p t
or
Probability.
in>i 2
states
particles.
are again
fore be established. w inc
has o c c u r r e d
out of the b a c k g r o u n d
can again be b u i l t up out of in-in a m p l i t u d e s
sum over Q stops,
because
from the i n t e r a c t i o n
such a t r a n s i t i o n
w~nC(d~s~Ic~r ~) = ~ 1 < i ~ O ~ I S ] ~ # all Q This
pairs,
~ [I 2 all Q
goes over
Such pair
inc
originate
that
of m a s s i v e
particle
is the pair-includin_.q~Ltransition
winC(d~s~Ic~r~ ) :
pairs.
massive
which
probability
of the c r e a t i o n
interaction
w
particles
The c o r r e s p o n d i n g
regardless
without
interaction
scheme
based
as in flat s p a c e - t i m e
sketched
the
only a p a r t i c u l a r
above,
on
m a y there-
this p r o b a b i l i t y
as it can be to w h a t we are used to
in flat space-time.
Specifying out-state lating
again not only
and a l l o w i n g
the p a r t i c u l a r
to the f o l l o w i n g
the in- but also
for the p r o d u c t i o n
transition
concept
process
of a s p e c i f i e d
the u n p a i r e d
of pairs
part of the
as above
as far as possible) m e a n n u m b e r N(÷)
(thus isowe are
N(o%Id~s~+--c~r ~) = 7_. lL~n(p~iQ~s ~) all Subtracting creation
leads
and
be
can
to
(6.3)
Q
the c o n t r i b u t i o n Nint(÷)
transcribed
of p a r t i c l e s
which
refers
into ( c o m p a r e
which originate
to the m u t u a l
led
:
from v a c u u m -
interaction
only,
[5])
~int(p~ldms~ +__ c~r~)= ~ lI2n(_p~IQ~d~s ~1 + all Q + N(°)(~(O) + Re(B~
~o)
~ l[21n(p~IQ~d~s~)+n(_p¢IQ~d~s~)l +
all Q
-
-
(6.4)
80 with -
~p
:
- 2~
-
-
~#
all
# ^~ ~
~
Z
(6.5)
L ^
Summation over d reproduces (5.2) and ( 5 . 3 ) . The f i r s t term, t h e r e f o r e , i s a weighted p a r t i c l e c r e a t i o n out o f the vacuum. The second term i s again the amplification which shows its specific
structure
already on
this level.
VII.
EXAMPLE:
COMPTON-EFFECT
IN THE ¢2~2-MODEL INDUCED AL~p LIFICATION
GRAVITATIONALLY
We study the Compton
scattering
outside of forward scattering.
REFLECTS
in the interaction
We disregard
=
_/7~1%2~z
the contribution
resulting
from pair creation out of the vacuum and concentrate mean numbers which refer to the mutual agrees
because of the conformal
~-case.
Discussion
interaction
coupling
of the amplitudes
in
on the specified
only Nint(+).
with N(+)
(6.4)
~i
then leads directly to
Nint(p~ll%l~ ) + 0(~3) p k <-- 14 q l i~) = Nint(k~ - i~i p ~k + _ I ¢qi ~'~+N(°)(p~lO l) _ where
II~k ~
scattering,
is the end-state
and
I~ I~!
as has been shown in [5].
Compton effect the gravitationally ticles and massless
~-Particles
going out in e q u a l number, flat space-time
is the initial
(7.1)
leaving the mutual
m a s s i v e par%icles
(7.1)
state of Compton
clearly demonstrates
induced amplification:
Massive
for the Z-par-
interacti0n- are not
as one would expect from .the. situation
or from the Feynman diagram.
This
in the massless
Rather the
is a m p l i f i e d by a m o m e n t u m - d e p e n d e n t
in
number of
factor
"~1'+ N(O)(p¢'O)~"--"-~il + '621~
REFERENCES
I)
N.D. Birrell, in: Quantum Gravity 2, eds. C.J. Isham, R. Penrose and D.W. Sciama (Oxford University Press, Oxford, 1981).
81
2)
L. Ford, Bristol,
in: Quantum theory of gravity, 1984).
3)
J. Audretsch and P. Spangehl,
Class.
4)
J. Audretsch and P. Spangehl,
Phys.Rev.
5)
J. Audretsch and P. Spangehl, Improved concepts for the discussion of mutually interacting quantum fields in Robertson-Walker universes, preprint University of Konstanz (1985).
6)
J. Audretsch, A. R~ger and P. Spangehl, Decay of massive particles in Robertson-Walker universes with statically bounded expansion laws, preprint University of Konstanz (1986).
7)
L. Ford, Phys.Rev.
8)
L. Parker,
Phys.Rev.
9)
L. Parker,
Phys. Rev. D3
D31
Quantum Grav.
(1985), 704.
183
(1969),
1057.
(1971), 346.
ed. Christensen
(Adam Hilger,
2 (1985), 733.
D 33 (1986),997
GRAVlTONS IN DE SITTER SPACE
Bruce ALLEN Department of Physics and Astronomy Tufts University Medford, MA 02155 U.S.A.
Everyone knows that "Einstein's greatest mistake", the cosmological constant ~ , is very close to zero 111. There have been many attempts to explain why A
must be
exactly zero 121, but none of these e f f o r t s has succeeded. In fact i t is now fashionable to believe that during the very early h i s t o ry of the universe the value of A was quite large 131. This so-called " i n f l a t i o n a r y " epoch would have been a long period of e x p o n e n t i a l l y rapid expansion, and would elegantly explain two otherwise mysterious observational truths : the universe is uncannily f l a t , wave background r a d i a t i o n
and the cosmic micro-
has no r i g h t to be as i s o t r o p i c as i t is.
There are three things that we would l i k e to know. F i r s t , why is A zero today ? Second, could A have been very big in the past ? And f i n a l l y ,
if~
was very big in
the past, what consequences would that have today ? Unfortunately this paper w i l l not answer any of these questions, but I hope that i t w i l l nevertheless accomplish something useful. I am going to show that one of the answers that has recently been given to the f i r s t question above - Why is A zero ? get into the technical n i t t y - g r i t t y ,
is not correct. However, before I
l e t me give you a synopsis of the kinds of ans-
wers that have been suggested to these questions. One answer which has been given to the question - why is A
equal to zero today?-
has been that zero is the only consistent answer. Let me reveal my predJudices at once and say that I don't believe t h i s . F i r s t of a l l ,
i t i s n ' t borne out by careful calcu-
l a t i o n . For example, someone once decided that A must be zero, because i f i t was not zero then a certain scatteringamplitude would not be unitary. However a more careful i n v e s t i g a t i o n showed this to be false ked on by several people, is that i f
141. A d i f f e r e n t argument, which is being worA
is not zero then p a r t i c l e s get created out
of the vacuum, and damp the value of A to zero
151.
The problem is this : What quantities to you calculate to see i f
A r e a l l y decays (or
has to be zero from the outset) ? One may show, f or example, that a scalar p a r t i c l e propagating in a background with
A
nonzero w i l l emit additional scalar p a r t i c l e s ,
which w i l l continue to do the same thing, and so on, ad i n f i n i t u m . Now this sounds
83
unstable. However i f you calculate the energy-momentum tensor of this process, you find that i t only s h i f t s the value of
A
a little
bit
I6L.
Another example : you can calculate the corrections to the stress-tensor due to the Plank-scale quantum f l u c t u a t i o n s of the vacuum. Indeed, these corrections s h i f t the value of
~
, and one can study the semi-classical back-reaction to find out what
e f f e c t this has on the metric tensor. Do t h i s , and you find that the metric is great l y a l t e r e d . I t sounds l i k e
a tremendous physical i n s t a b i l i t y u n t i l you r e a l i z e
that the i d e n t i c a l argument implies that f l a t space with A = 0 can't be stable e i ther ! 17i. So here i t is clear that something is wrong with the argument i t s e l f , since l o c a l l y our spacetime is very f l a t ,
and shows no signs of decaying away be-
neath our f e e t ! One of the basic problems with these arguments is that the natural ground state (or vacuum state) f o r de S i t t e r space is time-reversal symmetric i81. In this socalled Gibbons-Hawking vacuum state, i t is impossible f o r p a r t i c l e creation to occur, because i f the number of p a r t i c l e s were increasing, that would break time-reversal symmetry. I f i t could r e a l l y be established that
A = 0 was the only consistent value,
I
don't think that i t would be a good thing . The cosmological constant is a measure of the local energy density of the empty vacuum state. I f i t was t r u l y zero then there would be no way to generate cosmological i n f l a t i o n , which would be very unfortunate. (This is another reason why I am inclined to believe that there is nothing which is inconsistent about
A
nonzero). Because A
is simply a measure of the vacuum or
l a t e n t energy, i t can change during phase t r a n s i t i o n s , and i t seems certain that i f symmetry in gauge theories is restored at high temperatures then such phase t r a n s i tions must have taken place, as our universe cooled and expanded 191. So i t seems quite possible that A
was nonzero in the past-and this leads to our f i n a l question
above. What kinds of effects would be associated with a large p o s i t i v e A ? Well, f i r s t there would be classical " g r a v i t a t i o n a l " effects. Separated p a r t i c l e s , f r e e l y f a l l i n g , would accelerate away from one another. At a certain distance the recessional v e l o c i ty would become unity, and there would be a cosmological particle horizon Ii01. A given observer could not see farther than this distance. In addition to these classical effects there would be quantum effects. The best known of these is the GibbonsHawking effect -a freely f a l l i n g observer would see a thermal spectrum at temperature (I~'~'I~A
l j & r a d i a t i n g from the imaginary surface which we have Just described-
the observer's p a r t i c l e horizon i81. There are probably other i n t e r e s t i n g effects too, but we don't know what they are yet. The subject of quantum f i e l d theory in de S i t t e r space is s t i l l
in i t s infancy. In fact the only real results are that we know how to
construct the Fock space of states, and how to f i n d the c o r r e l a t i o n functions f o r spin O, ½ and i I i i i . space I121.
We also know a l i t t l e
b i t about i n t e r a c t i n g f i e l d s in de S i t t e r
84 Now l e t me t e l l you the idea which I intend to spend the rest of t h i s t a l k t r y i n g to destroy. The idea was to show that ~ tons when A
must be zero because of properties of gravi-
is nonzero. I f we were t r y i n g to prove that ~ =
O, then, in the ab-
sence of any exact symmetry or invariance which would force ~ be the next best thing. The reason is t h i s :
to vanish, t h i s would
A is only observable through r e l a t i v i -
t y , because i t represents an otherwise completely a r b i t r a r y zero-point for measuring the energy-density of space-time. Were i t not for general r e l a t i v i t y ,
or g r a v i t y , then
any background energy density would be completely unobservable, and i t could be set to zero with the stroke of a pen. However in the presence of g r a v i t y , the vacuum energy density A
does become observable, f o r example through the classical effects des-
cribed above. I t would therefore be nice i f the only thing ( g r a v i t y ) that enables us to observe
A
in the f i r s t
place would also carry with i t some quantization consis-
tency condition that would force
A
to be zero. This would be an elegant solution to
our problem : i f g r a v i t y , the only force that allows us to observe A , demands that /~
vanish for reasons of consistency. An argument of t h i s type has recently been made by Antoniadis, l l i o p o u l o s and
Tomaras J13J. They claim that i f one quantizes g r a v i t a t i o n a l f l u c t u a t i o n s in the presence of a background energy density A , then the r e s u l t i n g theory shows a p a r t i c u l a r kind of inconsistency called an " i n f r a - r e d divergence". Now these words can refer to any one of several d i f f e r e n t problems. For example saying that QED has i n f r a - r e d divergences generally refers to the fact that external lines in Feynman diagrams emit an i n f i n i t e number of low-frequency photons 1141. However t h i s is not a real problem because the energy carried away by t h i s process is not i n f i n i t e .
S i m i l a r l y , in the
theory of massless scalar electro-dynamics, the e f f e c t i v e action has an i n f r a - r e d divergence, and a mass-scale must therefore be introduced into the theory. The gauge f i e l d s thus aquire a mass, and the gauge symmetry is broken jl5J. In the recent work by Antoniadis, l l i o p o u l o s and Tomaras I i 3 j ,
i t was claimed t h a t ,
because of an i n f r a - r e d divergence, the two-point function of g r a v i t a t i o n a l f l u c t u a tions was i n f i n i t e this infinity
(regardless of the separation of the two points). They argued that
caused certain t r e e - l e v e l scattering amplitudes to be i n f i n i t e ,
and
thus rendered the theory of quantum g r a v i t y inconsistent unless A equaled zero. What I am going to do in t h i s t a l k is quite straightforward. F i r s t ,
I am going to
t a l k about the graviton propagator, and explain why i t is not, in and of i t s e l f ,
a
physical object. In fact i t depends upon the choice of gauge (by which, as I w i l l s h o r t l y explain, I mean the choice of a gauge-fixing term). What t h i s means in p r a c t i ce is that physical q u a n t i t i e s ( f o r example scattering amplitudes, or the expectation value of the curvature tensor) depend only upon certain components of the propagator. This dependence is Just subtle enough so that the d i f f e r e n t graviton propagators, a r i sing from d i f f e r e n t choices of gauge, give exactly the same physical r e s u l t . The next thing that I w i l l do is to show that i t is indeed true that for certain choices of gauge, the graviton propagator is indeed i n f i n i t e ,
exactly as claimed by
Antoniadis et al. However I am then going to show that there are other choices ofgau-
85 ge for which the propagator is completely f i n i t e
! Then I w i l l explain why t h i s is so.
The point w i l l be that the gauge-fixing terms of Antoniadis et al. do not completely f i x the gauge because they s t i l l
allow a f i n i t e
number of gauge transformations. I t
is for t h i s reason that the propagator that they find is i n f i n i t e .
But t h i s i n f i n i t y
is not a real physical divergence ; i t is an a r t i f a c t of how the gauge-fixing was done. For a better choice of gauge, the propagator is completely f i n i t e
! To make t h i s
point absolutely c l e a r , I w i l l then show that i f one calculates scattering amplitudes in the Antoniadis et al. gauge, one s t i l l
obtains a p e r f e c t l y f i n i t e
of the fact that the graviton propagator is i n f i n i t e .
graviton propagator in t h e i r gauge is the sum of an i n f i n i t e f a c t ) term and a f i n i t e
r e s u l t , in spite
The reason f o r t h i s is that the (unphysical gauge-arti-
part. The i n f i n i t e term does not contribute to the scattering
amplitude of any i n t e r a c t i o n whose stress-tensor is conserved, and thus physical scatt e r i n g amplitudes remain completely f i n i t e , al.
contradicting the claims of Antoniadis et
Although I w i l l not show i t here, t h i s cancellation takes place at higher orders
as w e l l . The point is that for good choices of gauge, the Fadeev-Popov ghosts are w e l l behaved ; for a bad choice of gauge they introduce additional i n f r a - r e d divergences in Just the r i g h t way to cancel those a r i s i n g from the gravitons. So the real point of a l l t h i s technical i n v e s t i g a t i o n is t h a t , at least so f a r , there doesn't seem to be any i n t r i n s i c problem with
A
~ O. Of course i t ' s
entirely
possible that something else w i l l turn up in the future that w i l l render de S i t t e r space inconsistent ; as things stand at the moment, i t seems that we have to keep on thinking about i t . I.
Gauge-Fixing Terms and the Choice of Gauge. This is a straightforward subject, but one that a great many people seem to be un-
clear about. The source of most of the trouble is confusion about the r e l a t i o n s h i p between the classical process called "choice of gauge" and i t s analogue in quantum f i e l d theory, which is called "choice of a gauge-fixing term" in the action. Let us begin by considering these two ideas, and the connections between them. Suppose that hab is a small perturbation of some background metric gab. Then there is a whole class of metric perturbations hab that represent exactly the same physical perturbation. This is because under the i n f i n i t e s i m a l coordinate transformation xi--)X i + Vi the metric perturbation transforms into hab +~aVb). Since coordinate transformation does not cause any changes to p h y s i c a l l y observable q u a n t i t i e s , we can conclude that the perturbations hab and hab +~(aVb) are gauge-equivalent. Any physical q u a n t i t y , f o r example the perturbation of the curvature induced by hab +~(aVb), w i l l not depend upon Vb 1161.
86
For this reason, in classical perturbation and s t a b i l i t y theory, i t is very common to "impose gauge conditions". The metric perturbations obviously l i e in equivalence classes ; two perturbations w i l l be deemed equivalent i f f they d i f f e r by ~(aVbl for l
l
some vector Vb. "Imposing gauge conditions" is a way to pick out one p a r t i c u l a r member from each equivalence class. For example one can impose the following conditions on h ab' Vahab = 0 a ha = 0
(transverse), (traceless),
tahab = 0
(i.i)
(synchronous),
to r e s t r i c t the gauge freedom. Here t a is some a r b i t r a r y vector f i e l d (usually choosen to be t i m e l i k e ) . These are not the only conditions that one could impose ; there are c l e a r l y an i n f i n i t e number of other p o s s i b i l i t i e s . Now what about the quantum f i e l d theory of gravitational perturbations ? Well the action is a scalar and i t is thus invariant under coordinate transformations, so the perturbations hab and hab +V(aVb). have exactly the same action ; they are gauge equivalent. In t h i s situation the standard thing to do is to add to the action an a r b i t r a r i l y choosen term which is not invariant under the above transformation. This a r b i t r a r i l y choosen gauge-fixing term breaks the gauge invariance. For example i t could be, Sgauge = ~ ( ~ a h a b ) 2
+ ~(h~ )2 + ~(tahab)2 ] d(Vol),
(1.2)
where at least one of the positive constants ( d l ~, l~ ) was nonzero. Now of course we have made a very a r b i t r a r y choice here ; the Fadeev-Popov procedure allows us to compensate for this choice in Just such a way that the scattering amplitudes are u l t i mately independent of i t . Now suppose that we have determined the euclidean propagator, which we could write for example as the path integral I171 Gabc,d,(X,X' ) = yd[hab]hab(X)hc,d,(X')exp(-S[hab]-Sgauge[hab]).
(1.3)
This propagator obviously depends upon the gauge-fixing parameters ~ l ~ and ~
. Now
suppose that we considered the divergence ~a Gabc,d' as a function o f ~ l ~ and ~" . In general i t would not vanish. However i f ~ vanish. The reason is that i f
~
was taken to i n f i n i t y , then i t would
is very large then the f i e l d configurations in the
action which don't have ~/a hab = 0 are exponentially suppressed by the gauge-fixing term. In the l i m i t = ( - ~
such configurations would make no contribution to the pro-
pagator. Similarly, i f ~ l
~
and ~'
were simultaneously taken to i n f i n i t y , then the
propagator would satisfy the "classical gauge conditions" (1.1) in that
a
~ Gabc,d,,
87
Gaac,d ,, and taGabc,d, would all vanish. So we can see that the classical gauge conditions are obtained in quantum theory by singular choices of gauge. This is analogous to the situation in QED. There, i f we wanted to have a transverse propagator sat i s f y i n g ~a = 0 we would need to choose Landau gauge ; ie use a gauge-fixing term
~(~A~)
Now
2 in the action and take the l i m i t ~ - ) ~
.
what gauge was used by Antoniadis et al. 1131 ? In fact there were two pos-
s i b i l i t i e s that they considered. The f i r s t one was equivalent to the three conditions given before (1.1) where t a is a vector f i e l d orthogonal to a family of f l a t spatial surfacer which are the standard k = 0 f o l i a t i o n of de S i t t e r space. In this choice of gauge the propagator for gravitons can be related to that of a pair of minimally coupled scalar f i e l d s in a simple manner. Indeed in this gauge (corresponding to taking ~l~
and ~'
to i n f i n i t y in (1.2)) the propagator is i n f i n i t e . However i t was not
clear i f the reason for this was because the three gauge-fixing parameters were becoming i n f i n i t e , or i f i t was because the introduction of the vector f i e l d t a into the action was breaking de S i t t e r invariance, or i f i t was because the gauge-conditions did not e n t i r e l y determine the gauge. At that point in their work, Antoniadis et al, were not themselves certain i f the infra-red divergence that they had discovered was a gauge a r t i f a c t or not. To resolve this uncertainty they then considered a second choice of gauge for which the gauge-fixing term was Sgauge =
~a(hab - 1/4 gab hCc)]2 d(Vol).
(1.4)
In this case they also found an infra-red divergence in the propagator. They then carried out a tree-level scattering-amplitude calculation, and found an i n f i n i t e result. This, they claimed, was proof that the infra-red divergence that they had found for two different gauge choices was t r u l y a physical effect and not merely a gauge a r t i fact. We are going to concentrate on the second choice of gauge-fixing term (1.4) and w i l l reach very different conclusions
than those of Antoniadis et al. We are going
to consider gauge-fixing terms of the following form, with ~ = i / 2 , Sgauge = ~ [ ~ a
(hab _ ~gab hCc)]2 d(Vol)
(1.5)
for a l l values of the constant ~ . We are going to prove the following three statements : 1. The graviton propagator is f i n i t e i f
does not equal one of the following
"exceptional values" (1/4, 7/10, 5/6 . . . . ) exceptional : (n2 + 3n-3) / (n 2 + 3n)
(1.6)
88 fo r n = 1, 2, 3 . . . .
If
~
has one of the exceptional values, then the propagator
is i n f i n i t e . 2. I f
~ takes one of the exceptional values, then the propagator diverges because
f o r that value of ~ the gauge-fixing term is not " s e n s i t i v e " to a gauge transformation corresponding to a p a r t i c u l a r ( f i n i t e set of) vector f i e l d s Va. 3. The scattering amplitude is f i n i t e and independent of the value of ~ . While the gauge f i x i n g term that we consider has ~ = 1/2 and not ~ - ~ ( ~ ults apply equally well to the
~=~
because f o r the exceptional values of ~
the gauge-fixing-term f a i l s to f i x the gau-
ge f o r any value of ~ . In other words, when ~ (regardless of ~
, our res-
gauge of Antoniadis et a l. The reason why is takes one of the exceptional values
) then there e x i s t certain perturbations hab which are pure gauge
hab = ~(aVb) ~ 0 and such that Sgauge~(aVbl] vanishes. Thus our conclusion w i l l be that Antoniadis et a l . found an i n f r a - r e d divergence only because they had the bad %
i
luck to choose an i n e f f e c t i v e gauge-fixing term, and not because the graviton propagator in de S i t t e r space has any i n t r i n s i c physical i n f r a - r e d divergence. For most choices of gauge the propagator would have been completely f i n i t e . 2.
How to find the Graviton Propagator. The basic idea of this section is to find a form f o r the graviton propagator which
w i l l make i t easy f o r us to see how i t depends upon the choice of a gauge-fixing term. For this purpose i t turns out to be very convenient to represent the propagator as a mode sum. Such mode sums are very f a m i l i a r in the context of Lorentzian space-time calculations of ( f o r example) the commutator and symmetric functions f or a scalar f i e l d . Here we are doing something s l i g h t l y less f a m i l i a r - a Euclidean mode sum. The point is this : in the Hawking-Gibbons vacuum state, which is de S i t t e r i n v a r i a n t , the two-point function only depends upon the geodesic distance between the two-points. I t is also an a n a l y t i c function f o r spacelike-separated points. Therefore i f we can find this function f o r spacelike separations, i t s analytic continuation to t i m e l i k e separation w i l l y i e l d a l l information about the physically i n t e r e s t i n g function, which is the Lorentzian two-point
function.
Thus we w i l l look fo r the two-point function only f o r spacelike separated points. One way to do this is to carry out the c a l c u l a t i o n on a Euclidean (++++) metric 4sphere of radius a, whose scalar curvature R has the same constant value as the curvature of the physical Lorentzian de S i t t e r space R = 4 A . On this four-sphere the distance between two points is always p o s i t i v e , so that spacelike separation is the only p o s s i b i l i t y . I t can be e a s i l y shown 1181 that f or such spacelike separation the two-point function on the sphere, considered Just as a function of geodesic distance,
89
is exactly the same as the Lorentzian two-point function for spatial separations. For that reason, from this moment on, we w i l l perform all calculations on a foursphere of radius a and volume 81T2 a4/3. The cosmological constant is then A = 3a-{ To find the two-point function, we need to know the quadratic part of the gravitational action for a small metric fluctuation hab. This has been calculated in many places
J191. When we add to i t the gauge-fixing term previously given (1.5), we find
that the total gauged-fixed action is S +
Sgauge = ( 6 4 ~ G ) - 1 S
habWabCdhcd d(Vol) •
Here the second-order d i f f e r e n t i a l
(2.1)
operator Wabcd is given by
A
Wabcd = ([l-2~2]gabgcd -gc(agb)d ) ~
+ (2gc(agb) d + gabgcd) ~
(2.1) + (2~-Z)(gab~(c~d)
+ gcd ~ ( a ~ b )
)
where ~ is the gauge-fixing parameter. Now the propagator is defined by the different i a l equation Wabcd
GCda'b '
(X,X')
= ~(a
a'
~ b)
b'
'
(2.3)
together with the boundary conditions that Gaba'b' depend only upon the distance from X to X' (in the sen6e of 1181) and that i t only be singular i f X = X'. This equation can be solved in many ways. One method that we have already exploited is to actually perform the path integral (1.3). This is done by choosing lO"coordinates" in the space of all metric perturbations hab, and then integrating all of them from - ~ t o
~
120J. A simpler method w i l l be followed here ; i t involves using
an ansatz which is J u s t i f i e d by the previous method. To use the simplest method, i t is only necessary to have an orthogonal expansion of the delta function appearing on the right hand side of (2.3). This orthogonal expansion is ~(aa'~
b)
b' = S h ~ , ( X ) h ~ ' b ' ( x ' ) + ~ V~,(X)v~'b'(x')+n~ 2 W~b(X)w~'bix') n=O au n=l au , + =
a
b (X)X n b(x') "
Here the tensor f i e l d s hab Vab n , wab and~ ab are all eigenfunctions of I ~ and they n ' n form a complete set for the representation of any symmetric rank-two tensor. What this means is that any such tensor, Qab = O ~ n
Qab' can be represented as a sum of the form
hnab +~iPnV~b +~2~nW~b +o~n](~b
(2.5)
90 f o r a unique set of constant c o e f f i c i e n t s
{~.j#,,~m,~.~._
I f the eigenfunc-
tions are appropriately normalized, so that = Sh
b(X)hab(x)dV = . . .
,
(2.6) O:
then i t is easy to show from equations (2.5) and (2.6) that the delta function defined by (2.4) satisfies b'
Qa'b'(x') =
Qab(x) ~ a l a ( X , X ' ) ~ b ) (X,X') dVx
(2.7)
for any symmetric tensor Qab" The nice thing about t h i s method is that we don't have to e x p l i c i t l y construct ab ab any of the eigenfunctions hn . . . . . ~ n " We w i l l only need basic information about their eigenvalues and m u l t i p l i c i t y which can be found in many articles 1211 and which can be obtained entirely from the group representation theory of S0(5). The basic facts are as follows. The details, including the normalizations and eigenvalues of these modes, can be found i:n reference 1201. The ten degrees of freedom in the symmet r i c tensor Qab (2.5) are divided among the different modes. There are five degrees of freedom in the tensor modes hab which are transverse and traceless (TT) n '
0 = hab ab n gab - ~ahn r~hab = n ~(~)
_
~(2) ham -,n n
(2.8)
A (n 2 + 7n + 8)
3 There are three degrees of freedom in the modes Vab which are the symmetrized derivan t i v e s of transverse vectors.
[½ a
+^>] v(a~nb)
0 :~afn
(2.9)
[]vab = ~X(~)+ ~^ ) vab n
n
)~(1)= _--~(n2 + n
3
5n
+ 3)
There is one degree of freedom in the modes W ab which are the traceless derivatives n of longitudinal vectors
91
Wab = IX(On) (3/4 ~(0n) +A)] -I/2 (~avb-¼gab I-l)~n (2.10)
[] wab : (X(On)+8/3A) Wab n n " Finally there is one degree of freedom in the pure-trace scalar modes ~ab n" ~b = 1/2 gab ~n F]'xab
: X(On)Xabn
(2.11)
~(On) = -~--n (n+3) 3 From a detailed examination of these modes and t h e i r normalizations (for which see 1201) i t can be shown that ~a~Zb
wab n
:
~ ( 0 ) 3 4 ~ (0) +A) ~n zl n ( / ,, n (2.12)
(gab
d + gcd~a~b ) XCdn
n + 2~(On)~n b . = 2[~ (0) n (3/4% (0) n +/~ )]½ Wab
This is all that we w i l l need to know about the mode functions. As we said e a r l i e r , the path integral calculation of the propagator 1201 J u s t i f i e s our use of the following ansatz for the propagator : Gaba'b'(x'x')
=~
I ~ n h~b(X) ha'b'(X')n
0
+
~n V~b(x) va'b'(X')n + ~n W~ b(X) W~ 'b'(x') + ~n
~ab(x ) ~ a ' b ' ( x , ) n " " n
rn LnEab(x)wa'biX') +n
(2.13)
+
b(x)
(X,)] 1
Applying the wave operator Wabcd to the propagator (2.13) and demanding that i t yield the delta function (2.4) gives linear equations that uniquely determine the coefficients ~n . . . . . ~'n to be ~n
= K(-~(n2) + 2 / 3 • ) - i
n = K ( - ~ ( ~ ) - ~ )-1 ~n
for n > 0 for n >~ I, else zero.
= K(-~(On)(1-~)+A)-2[(2~2-E-1/4)~(On)-A/21
for n>2, el se zero. (2.14)
92
~n
:
K(1/4)k(On) + A / 2 ) ( ) ~ (0) n ( -1~ )
=
K()k(On)(1-~) +A)-2(~-1/2)(,.n)~(O)(3/4)k(~ ) + A ) ) ½ f o r n>~2, else zero.
+/~) - 2
f o r n >~0
(2.14)
Here the constant K = 64~G where G is Newton's constant. With the propagator now determined by (2.13) and (2.14), we can discuss i t s i n f r a - r e d behavior. 3.
I n f r a - r e d Behavior of the Graviton Propagator. We begin the discussion of the i n f r a - r e d behavior of the graviton propagator by
asserting that the propagator is f i n i t e the c o e f f i c i e n t s
~n .....
(I n are f i n i t e .
f o r separated points i f and only i f a l l of This is indeed the case, provided t h a t the
g a u g e - f i x i n g parameter ~ does not have one of the values exceptional =
f o r n = 1,2 . . . . . The f i r s t
i +
~/~(~)
-
n2+3n-3 n(n+3)
(3.1)
{m . Now l e t us prove our assertion.
terms in the mode sum f o r the propagator, corresponding to ~nhnhn' and
~ n V n ~ , have been evaluated by A l l e n and Turyn 1221 and shown to be completely f i n i te. This leaves the f i n a l for different
three terms, which can be related to the scalar propagator,
values of the scalar mass. Thus to understand the i n f r a - r e d behavior of
the graviton propagator, a l l we have to do is understand the scalar case. Here the s i t u a t i o n is very simple. For two points X and X', separated by a geodesic distance
j(~(X,X'),
G(m2,jL~) = ~
~n(X)~n ( -rl
n
the massive scalar propagator is I18,20,221, X'
)
P(3/2 + V)~(3/2-V) 16~r 2 a
F(3/2+V,3/2_V;2;cos2~W,/2a)).
(3.2)
The r i g h t hand s i d e of t h i s e q u a t i o n , and hence the mode sum, is c o m p l e t e l y f i n i t e provided t h a t 3/2-V is a not a n o n p o s i t i v e i n t e g e r . Since V= (9/4 - a2m2) ½, t h i s means the p r o p a g a t o r is f i n i t e
m2 = _
provided t h a t m2 does not take one of the ( n e g a t i v e ) v a l u e s
i n(n+3) a2
f o r n = O, 1 2, ' ....
(3.3)
But these are e x a c t l y the values of m2 f o r which the d e n o m i n a t o r - ~ ( ~ ) + m2 in the mode sum vanishes ! Exactly the same analysis applies to the " s c a l a r " parts of the g r a v i t o n propagator. We have thus proved t h a t p r o v i d e d t h a t i f ~ i s
not given one of the
"exceptional" values given above ( 3 . 1 ) , the propagator is completely f i n i t e . will
now
do is to show why t h i s i s .
What we
93 4.
How Can The Gauge-Fixing Term Fail ? The infra-red divergence that occurs in the propagator for the exceptional values
of ~ can be e a s i l y understood. Imagine expressing the propagator as a path i n t e g r a l , or average, over a l l f i e l d configurations. I f the gauge-fixing term was not present, then this integral would y i e l d i n f i n i t y ,
because i t would include an i n f i n i t e number
of gauge-equivalent f i e l d configurations which had the same value of the action. The purpose of the gauge f i x i n g term is to make the integral converge by giving gaugeequivalent f i e l d configurations d i f f e r e n t values of the action. Thus the gauge-fixing term " f a i l s to do i t s duty" i f there e x i s t a d i s t i n c t pair of configurations which are physically gauge equivalent and which have the same value of the gauge-fixed action. Let us now show that this is exactly what happens i f
~ is given one of the
"exceptional" values. We can write the gauge f i x i n g term (1.5) in the following form, a f t e r integrating by parts.
Sgauge :
_=(~(hab _ ~gab he)~a Vce
(hbc -
gbc h~) d(Vol).
(4.1)
Now consider the following gauge transformation : hab-~h ab +~(avb) where vb=v b~n for the scalar mode ~ n ' and n > i . I t is easy to v e r i f y that for n > O, ~/(avb) is nonzero, and Sgauge[~(avb)] : 2 W [ ~ ( ~ ) ( I _ ~ ) + A ] 2 : 2 ~ A
Thus, i f
~
2 n2(n+3)2[~_ 9
n2
+3n - 3]2 n(n+3) "
(4.2)
takes on one of the exceptional values - say the n'th exceptional value-
then the gauge-fixing term f a i l s to be sensitive to the gauge transformation hab")hab + ~ a V b
~ n induced by the n'th scalar mode~ because the r . h . s , vanishes!
This is the source of the infra-red divergence that occurs for the exceptional values of ~ .
We w i l l now show that this infra-red divergence, should i t happen to arise be-
cause of a bad choice of ~ ,
is a harmless gauge a r t i f a c t and makes no contribution
to scattering amplitudes. 5.
The Infra-red Divergence is a Gauge-Artifact. Consider the tree level scattering process where two matter f i e l d s , which we deno-
te
~ ,
i n t e r a c t by exchanging a graviton. Here ~
Just a scalar f i e l d . Schematically this looks l i k e :
could be any kind of matter, not
94
The amplitude for this process is determined by the stress tensor Tab of the matter. I t is A = I ' ] T a b ( x ) G a b c , d , ( X , X ' ) TC'd'(x')dVdV'
,
(5.1)
where dV denotes the invariant four-volume element g J ~ ) d 4 X
at the point X, and
dV' denotes the same thing at X'. Let us assume only that ~7a Tab = 0 ; ie that the operator Tab , which is quadratic in the f i e l d ~ , is conserved. This is true even in the presence of trace anomalies, for the renormalized operator, provided that i t is a matrix element between physical (on-shell) states
1231. We w i l l show that this
amplitude is f i n i t e regardless of the value of the gauge-fixing parameter ~ , and in p a r t i c u l a r for the "exceptional" values of ~ , for which Gabc,d, contains infra-red divergences. The amplitudes A is a sum of f i v e terms arising from the propagator (2.13). The first
two terms are independent of ~ .
The f i n a l three terms, upon integration bypart~
can be expressed as A~ + As + A~
: ~IT(X)p~(X,X')
T(X') dVdV'
(5.2)
where T(X) = Ta a(x) is the trace of the stress tensor. The function
p~(X,X')
is
of the form
~(X,X') = CI + O2(E-1/4)-2 ~I(X)~I(X')
+ C3 n~_-2 ~n(X)~n(X') :
X(O)+ 4 n
Here C1, C2 and C3 are nonzero constants. What matter is that there appears to be a single term in the amplitude that depends upon ~ .Howeverfrom gauge-invariance we known that the amplitude can not depend upon ~ at a l l ! We w i l l now show that the second term above contributes nothing, even in the l i m i t
~1/4
!
1241. The reason why is simple : the mode(s) ~)I(X) obey ~ / a V b 9 1 = - 31~ g a b ~ 1 Thus replacing Ta~la by T a b V a V b ~ l , and integrating by parts 1251 to get (~a Tab)~b~l,_ we see that the ~,-dependent term vanishes as long as the stress tensor is conserved. What this means is that even in those cases where the two-point function has an infra-red divergence, the scattering amplitude is f i n i t e . This shows
95 that in those cases where i t occurs, i n f r a - r e d divergence is a h a r m l e s s g a u g e a r t i f a c t . 6.
Conclusion. What has been shown in t h i s t a l k is that the graviton propagator in de S i t t e r spa-
ce is OK. I f one makes a bad choice of gauge ( - f i x i n g term) then the propagator is i n f r a - r e d divergent. However t h i s is not a problem. You can e i t h e r make a better choice of gauge (of which there are an i n f i n i t e pletely finite,
number), f o r which the propagator is com-
or else you can go r i g h t ahead and use the i n f r a - r e d divergent one.
We demonstrated that i t doesn't matter. Gauge-invariance is the o v e r - r i d i n g p r i n c i p l e , and i t ensures that even i f the propagator has an i n f r a - r e d divergence, the physical scattering amplitudes are f i n i t e . A more detailed discussion of these points can also be found in an e a r l i e r published paper 1201. The complete closed form f o r the graviton propagator with ~ = 1/2 has also been found 1221. F i n a l l y a closed form in the de S i t t e r - n o n - i n v a r i a n t gauge (I.I)
has
been
recently obtained 126i. This form applies to any s p a t i a l l y - f l a t
Robertson-Walker model. Acknowledgements. I would l i k e to thank S. Coleman, J. l l i o p o u l o s a n d M . T u r y n f o r h e l p f u l discussions.
REFERENCES 1 C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freedman, San Francisco, 1973) p. 410. 2 I . Antoniadis, J. l l i o p o u l o s , T.N. Tomaras, Nucl.Phys. B261 (1985) 157. I. Antoniadis and N.C. Tsamis, Phys. Lett. 144B (1984) 55. E. Baum, Phys. Lett. 133B (1983) 185. 3 G. Gibbons, S.W. Hawking and S.T.C. Siklos, The Very Early Universe, Proceedings of the N u f f i e l d Workshop (Cambridge UP, 1983). R. Brandenburger, Rev.Mod.Phys. 57 (1985) 1. 4 B. deWit and R. Gastmans, Nucl.Phys. B128 (1985) 1. 5 N.P. Myhrvold, Phys. Lett. 132B (1983) 308. N.P. Myhrvold, Phys. Rev. D28 (1983) 2439. E. Mottola, Phys. Rev. D31 (1985) 754. E. Mottola, Phys. Rev. D33 (1986) 1616. E. Mottola, NSF-ITP 85-33 p r e p r i n t UCSB. E. Mottola and P. Mazur, NSF-ITP 85-153 p r e p r i n t UCSB. S. Wada and T. Azuma, Phys. Lett. 132B (1983) 313. P. Anderson, U n i v e r s i t y of Florida at Gainesville p r e p r i n t , 1985. Gary T. Horowitz, Phys.Rev. D21 (1980) 1445. G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2738. B. A l l e n , Ann.Phys. 161 (1985) 152. B. A l l e n , Nucl.Phys. B226 (1983) 228. 10 S.W. Hawking and G.F.R. E l l i s , The Large Scale Structure of Spacetime (Cambridge UP, 1980). 1 1 0 . Nachtmann, Commun.Math.Phys. 6 (1967) 1. N.A. Chernikov and E.A. Tagirov, Ann.lnst. Henri Poincar~ IX (1968) 109.
96
12
13 14 15 16 17 18 19
J. G~h#niau and Ch. Schomblond, Bull.Cl.Sci., V.Ser.Acad.R.Belg. 54 (1968) 1147. E.A. Tagirov, Ann.Phys. 76 (1973) 561. P. Candelas and D.J. Raine, Phys.Rev. D12 (1975) 965. Ch. Schomblond and P. Spindel, Ann.lnst. Henri Poincar~ XXV (1976) 67. T.S. Bunch and P.C.W. Davies, ProcoRoy.Soc.Lond. A360 (1978) 117. B. Allen, Phys.Rev. D32 (1985) 3136. B. Allen and T. Jacobson, Commun.Math.Phys. 103 (1986) 669. B. Allen and C.A. LUtken, Commun.Math.Phys. 106 (1986) 201. O. Nachtman in reference 11. O. Nachtman, Z. Phys. 208 (1968) 113. O. Nachtman, Sitzungsber. Oesterr.Akad.Wiss.Math.Naturwiss.Kl. 167 (1968) 363. G.W. Gibbons and M.J. Perry, Proc.R.Soc.Lond. A358 (1978) 467. I. Antoniadis, J. lliopoulos and T.N. Tomaras, Phys.Rev.Lett. 56 (1986) 1319. C. Itzykson and J.B. Zuber, Quantum Field Theory (McGraw-Hill, NY, 1980). S. Coleman and E.J. Weinberg, Phys.Rev. D7 (1973) 1888. This of course is the infinitesimal form of the gauge transformation. To generate f i n i t e transformations we have to go to higher order in V. The Fadeev-Popov determinant lhabl does not depend upon hab at one-loop, and thus does not contribute to the tree-level propagator. We have therefore l e f t this Jacobian out of the formula for Gaba,b,. B. Allen and T. Jacobson in reference 11. S.M. Christensen and M.J. Duff, Nucl.Phys. B170 (1980) 480. N.H. Barth and S.M. Christensen, Phys.Rev. D28 (1983) 1876. B. Allen, Phys. Rev. D34 (1986) 3670.
20 B. Allen in reference 19. 21S.L. Adler, Phys. Rev. D6 (1972) 3445, D8 (1973) 2400. R. Raczka, N. Limic and J. Nierderle, J.Math. Phys. 7 (1966) 1861, 7 (1966) 2026, 8 (1967) 1079. G.W. Gibbons and M.J. Perry, Nucl.Phys. B146 (1978) 90. S.M. Christensen, M. Duff, G.W. Gibbons, and M.J. Perry, Phys. Rev. Lett. 45 (1980)161. A. Higuchi, Yale Preprint YTP 85-22 (1985). A. Chodos, E. Meyers, Ann.Phys. (NY) 156 (1984) 412. M.A. Rubin and C.R. Ordonez, J.Math.Phys. 25 (1984) 2888, 26 (1985) 65. 22 B. Allen and M. Turyn, The graviton propagator in maximally symmetric spacer, Tufts University preprint (1986). 23 R. Wald, Phys.Rev. D17 (1978) 1477. R. Wald, Commun.Math.Phys. 54 (1977) 1. S.A. Fulling, M. Sweeny and R. Wald, Commun.Math.Phys. 63 (1978) 259. 24 In fact the mode that we have labeled 41 is degenerate. There are five such modes with the same eigenvalue. I f the four-sphere is X2 + . . . + X2 = 1 then the modes ~i are proportional to the i ' t h coordinate Xi. 1 5 25 T~e boundary terms can be shown to vanish in the Lorentzian spacetime case -see reference 20. 26 B. Allen, The graviton propagator in homogeneous and isotropic spacetimes, Tufts University Preprint TUTP 86-14 (1986). (submitted to Nucl. Phys.)
EFFECTS OF GRAVITON PRODUCTION IN INFLATIONARY COSMOLOGY Diego D. Harari 1 Physics Department Brandeis University Waltham, MA02254
ABSTRACT
A quantum derivation of the spectrum o£ gravitons created in an inflationary cosmology is discussed, and the way in which they can affect the isotropy o£ the cosmic microwave background is briefly reviewed.
INTRODUCTION An inflationary cosmological model [I], in which the early universe underwent a period of exponential expansion, solves in a very attractive way many longstanding cosmological puzzles, such as the large scale isotropy and spatial flatness of our presently observed portion of the universe. picture of the early universe correct,
Were this
then two regions located in opposite
directions in the sky that just recently entered into our Hubhle sphere would have been in close causal contact in the past.
So close,
in fact, that quantum
effects acting on such scales at early periods during the inflationary epoch, when they were small enough, may have had important consequences much later on the history o£ the universe.
It was indeed suggested that quantum fluctuations
in the energy-density during inflation might be the origin o£ the primordial seeds essential to explain galaxy formation [2].
At least, a Zeldovich spectrum
of gaussian fluctuations naturally arises in most inflationary models, although it is not clear at present what natural inflation- driving mechanism will provide them with the adequate amplitude (~p/p ~ 10-4 at horizon crossing).
1Supported by a F e l l o w s h i p from the Consejo N a t i o n a l de I n v e s t i g a c i o n e s Cient~ficas y T~cnicas, Rep6blica Argentina. P r e s e n t Address: 32611.
P h y s i c s Department, U n i v e r s i t y of F l o r i d a ,
Gainesville,
FL
9B
Inflation
predicts
fluctuations
not only in the energy-density,
the background m e t r i c of the s p a c e - t i m e i t s e l f provide an important constraint stochastic
on a n y model o f i n f l a t i o n .
Indeed, any waves c a n i n d u c e
i n t h e c o s m i c microwave b a c k g r o u n d , s i n c e t h e r a d i a t i o n
through these "ripples"
in the gravitational
potential
ways a c c o r d i n g t o t h e p a t h f o l l o w e d .
period around 1 year can also affect millisecond pulsars
[7,S,9].
used to p l a c e c o n s t r a i n t s inflation,
but also in
These metric fluctuations
background of v e r y long wavelength g r a v i t a t i o n a l
anisotropies
different
[3-6].
travelling
c a n be r e d s h i f t e d
Gravitational
in
waves w i t h a
i n a s i m i l a r way t h e o b s e r v e d " t i m i n g " o f
The o b s e r v e d b o u n d s on t h e s e q u a n t i t i e s
on t h e a l l o w e d m e t r i c
fluctuations
c a n be
p r o d u c e d by
a n d h e n c e on t h e p a r a m e t e r s of t h e model ( e s s e n t i a l l y
on t h e v a l u e o f
t h e Hubble c o n s t a n t d u r i n g i n f l a t i o n ) . In the present article
a quantum d e r i v a t i o n
of t h e m e t r i c f l u c t u a t i o n s
p r e d i c t e d by i n f l a t i o n
w i l l be d i s c u s s e d [ 1 0 ] , a n d t h e n t h e l i t e r a t u r e
their potential
on t h e c o s m i c microwave b a c k g r o u n d w i l l be r e v i e w e d .
effect
about
GRAVITON PRODUCTION
I n t h e a p p r o a c h to be p r e s e n t e d h e r e , g r a v i t a t i o n a l b e c a u s e t h e de S i t t e r
invariant
vacuum s t a t e
period appears as a multiparticle relevant
state
the creation
is used.
of p a r t i c l e s In o t h e r words, an
of a g r a v i t o n d u r i n g the r a d i a t i o n -
m a t t e r - d o m i n a t e d e r a s i s a combination of both g r a v i t o n c r e a t i o n annihilation
today
during the inflationary
when t h e d e f i n i t i o n
to the p r e s e n t m a t t e r - d o m i n a t e d u n i v e r s e
operator representing
waves e x i s t
established
operators as defined during inflation.
or
and
The c o e f f i c i e n t s
of t h i s
m i x i n g a r e known a s B o g o l y u b o v c o e f f i c i e n t s . Using for convenience a conformal time v a r i a b l e , the inflationary
the background m e t r i c of
model r e a d s
ds 2 = S2(T)[-dT 2 + d~-d~]
with
s(v)=
f-1/~T
during inflation,
while
(1)
T < - T
4T Tm/T02
during radiation-domination,
w h i l e T~ < T < Tm
T2/T0 2
during matter-domination, while T > 2T
Here ~ i s t h e v a l u e o f t h e Hubble c o n s t a n t d u r i n g i n f l a t i o n ,
(2)
m
TO i s t h e p r e s e n t
c o n f o r m u l t i m e (T O = 3 t 0 ) , Tm i s t h e c o n f o r m a l t i m e a t t h e end o£ t h e r a d i a t i o n
99
- d o m i n a t e d e r a (Tm = 2 - 1 / 2 32/3 t lm/ 3 inflationary
Tw = TO/2(XTm)I/2.
period finishes,
t h e m e t r i c i s t h a t S(T) and i t s transitions
between different
derivatives,
first
The o n l y r e q u i s i t e
regimes.
The u n p h y s i c a l d i s c o n t i n u i t i e s
in higher
m o d e l s , have no
f o r long w a v e l e n g t h s , which a r e t h e r e l e v a n t s f o r t h e
to be d i s c u s s e d l a t e r .
Notice also that the definition
time u s e d i n (2) i s s u c h t h a t i t continuous).
imposed on
d e r i v a t i v e be c o n t i n u o u s a t t h e
t h a t s h o u l d be a v o i d e d i n more r e a l i s t i c
c o n s e q u e n c e on t h e r e s u l t s effects
t2 o / 3 .) and --Tw i s t h e c o n f o r m a l time when t h e
jumps a t t h e t r a n s i t i o n s
These d i s c o n t i n u i t i e s
of c o n f o r m a l
( t h o u g h S and S a r e
have t o be t a k e n i n t o a c c o u n t when computing
t h e Bogolyubov c o e f f i c i e n t s . The s m a l l m e t r i c p e r t u r b a t i o n s
a r o u n d t h e R o b e r t s o n - W a l k e r background can be
w r i t t e n in terms of g r a v i t o n c r e a t i o n a n d a n n i h i l l a t i o n transverse traceless
gauge) a s
hij = ~
k +
f *.~.
~ d3k [ax(k)eij(k.k) ei ~ - ~ (2~)3/2S(T)V~-k "
~ .~..
ak(K)eij[K,AJe
-i~-~
~(kT) +
(3)
~*(kT)]
where k r u n s o v e r t h e two p o s s i b l e p o l a r i z a t i o n s tensors.
operators (in the
and e i j ( ~ , k ) a r e p o l a r i z a t i o n
Each i n d e p e n d e n t d e g r e e o f freedom can be q u a n t i z e d a s i f i t were a
minimally coupled scalar field
[11].
The f u n c t i o n ~ ( k r ) i s g i v e n by
~(kT) = ~ikT(1 - i / k T )
during inflation
(43
and m a t t e r - d o m i n a t i o n , and by -ikT ~(kT) = e
(5)
during the r a d i a t i o n - d o m i n a t e d era. The g r a v i t o n c r e a t i o n a n d a n n i h i l l a t i o n related
operators at differents
eras are
through
4nat(~) = a2Ck)a~nf(~ }
These Bogolyubov c o e f f i c i e n t s
"
t
can be e v a l u a t e d m a t c h i n g b o t h h . . and i t s 12
first
100
derivative
at both transitions.
h o r i z o n a t T~ o r v m r e s p e c t i v e l y
The r e s u l t
is,
f o r modes w e l l o u t s i d e
the
( k T . << 1 o r k r m << 1)
--2~Tm a l ~ - ~1 ~ (lrr0)2
2k3T02
The quantum s t a t e
of t h e s y s t e m w i l l b e c h a r a c t e r i z e d
gravitons as defined during inflation. operator
( t4 n -f )
quantum s t a t e
(7)
- - 4n f w i l l b e c o n s i d e r e d . [4) may seem a r b i t r a r y ,
i n t e r m s o£ t h e number of
I n o t h e r words, e i g e n s t a t e s T h i s way of c h a r a c t e r i z i n g
but recall
of t h e the
t h a t t h e w a v e l e n g t h s of
r e l e v a n c e f o r t h e cosmic microwave b a c k g r o u n d a n i s o t r o p y a r e of t h e o r d e r of t h e s i z e of t h e u n i v e r s e
today,
t h a t i s lcr 0 ~ 1.
For t h e s e waves, e a r l y d u r i n g t h e
inflationary
period
the horizon,
i n w h i c h c a s e t h e modes g i v e n by (4) a r e j u s t
[tcr[ )> 1, which means t h a t t h e y were o r i g i n a l l y
Minkowski d e f i n i t i o n
of p o s i t i v e
I n t h e quantum s t a t e can be characterized
f r e q u e n c y modes.
[~>, t h e f l u c t u a t i o n s
k3 1 fd3~ i~*~ (2~) 3 • ~ e
w i t h a sum on i and j u n d e r s t o o d . C h o o s i n g t h e quantum s t a t e
<~lhij(x,v)hij(~,v)l~>
W a v e l e n g t h s much l a r g e r
observable universe have to be cut-off
fluctuations
i n h .1j . a t time T and wavenumber
by
Ah2(~) ~
invariant)
well inside
the ordinary
when a c t u a l l y
[4) t o be t h e vacuum s t a t e
(s)
than the present
computing this expression. (which i s de S i t t e r
and using the relevant Bogolyubov coefficients
given in (7),
the
f o r l o n g w a v e l e n g t h s ( k v <(1) t u r n o u t t o b e
(9) 211-2
N o t i c e t h a t i n b o t h c a s e s t h e a m p l i t u d e o£ t h e f l u c t u a t i o n s v a l u e , 6b(2/27r2, f o r w a v e l e n g t h s o u t s i d e alternative
derivations
[3-6].
the horizon,
tend to a constant
in agreement with
101
I f i n s t e a d o£ c h o o s i n g t h e s t a t e particles
is considered, the result
]4> a s t h e vacuum, a s t a t e w i t h Nk(k ) (9) would a p p e a r m u l t i p l i e d by a f a c t o r
[I+NA(~)+NA(-~)]. S i n c e a t t h e s t a r t
of i n f l a t i o n
c o n s i d e r a t i o n were w e l l i n s i d e t h e h o r i z o n , i t assume Nk(k)<
the wavelengths under
seems more t h a n p l a u s i b l e
to
O t h e r w i s e t h e e n e r g y d e n s i t y i n what a t t h a t time were h i g h
f r e q u e n c y modes would b e enormous.
EFFECTS UPON THE COSMIC MICROWAVE BACKGROUND
Small p e r t u r b a t i o n s
o£ t h e m e t r i c around a R o b e r t s o n - W a l k e r background can
i n d u c e an a n i s o t r o p y i n t h e o b s e r v e d t e m p e r a t u r e o f t h e cosmic microwave background.
The e f f e c t
i s d e s c r i b e d by t h e S a c h s - W o l f e f o r m u l a [12,
5To TO -
13]
~ O-TE • ~ - i j J0 dz h i j ( z e , T O - z ) e e
(10)
where v 0 and TE a r e t h e c o n f o r m a l time f o r t h e e m i s s i o n and r e c e p t i o n o f t h e radiation
respectively,
TO i s t h e t e m p e r a t u r e o b s e r v e d a t p r e s e n t t i m e s and ~ i s
a unit vector pointing in the direction
of t h e o b s e r v a t i o n .
Knowing from t h e p r e v i o u s s e c t i o n how t o e v a l u a t e e x p e c t a t i o n v a l u e s of t h e field hi j,
the fluctuations
i n t h e t e m p e r a t u r e of t h e cosmic microwave
b a c k g r o u n d i n d u c e d by t h e quantum f l u c t u a t i o n s inflation
can be computed.
o f t h e m e t r i c p r e d i c t e d by
The t e m p e r a t u r e a n i s o t r o p y can b e expanded i n a
multipole decomposition as
6TO(e,~) To
- ~ a~J~m(e,,)
(11)
$,m
I t i s more c o n v e n i e n t t o work w i t h t h e r o t a t i o n a l l y
invariant quantities
ae
d e f i n e d by
ae
2
=
e 7. la~m 12 m---~
Each a~m can be t r e a t e d a s a g a u s s i a n random v a r i a b l e
(12)
[3,14,15]
(since hij
is a
f r e e f i e l d ) w i t h r . m . s , d e v i a t i o n from t h e z e r o mean g i v e n i n t e r m s o£ t h e 2 vacuum e x p e c t a t i o n v a l u e o f aem. The a s w i l l t h e n a l s o be g a u s s i a n v a r i a b l e s ,
102
with standard deviation determined by ,'" which c a n be computed.
There
will be no dipole contribution to the anisotropy coming from gravitational waves.
The quadrupole term gives [3-6]
<0{a22[0> ~ G~2
(the integration the horizon after
(13)
i n t h e S a c h s - W o l f e f o r m u l a i s done f o r w a v e l e n g t h s t h a t e n t e r e d recombination).
B e i n g a 2 a g a u s s i a n random v a r i a b l e
e x p e c t t h a t a22>O.3 w i t h a 90% c o n f i d e n c e l e v e l . prediction.
The o b s e r v a t i o n a l
one c a n
That is the
bound i s [16] a22 < 5 . 2 x 10 - 8 .
Consequently,
w i t h a 90% c o n f i d e n c e l e v e l
(Gx2) 1/2 =
Equivalently, reheating
X
< lO-d
(14)
t h i s bound c a n b e t r a n s f o r m e d i n t o a bound on t h e maximum
t e m p e r a t u r e o£ t h e u n i v e r s e when i n f l a t i o n
e n d s . Assuming a p e r f e c t
reheating
TRtt(Max) % 1017 GeV
Predictions relevant
(15)
f o r h i g h e r m u l t i p o l e moments c a n a l s o b e made [ 3 , 6 ] .
They a r e
b e c a u s e a n e v e n t u a l measurement o r bound on h i g h e r m u l t i p o l e
anisotropies
can also test
the spectrum predicted
by i n f l a t i o n
[13,14].
GENERALIZED INFLATION It has been pointed out [3] that the homogeneity and spatial flatness of the present universe can be explained by a long enough early period of "accelerated expansion", during which the physical distance between two points o£ fixed coordinate distance grew faster than the horizon size.
Such property of the
conventional exponential inflation is shared with many other possible scale factor evolutions [3,17.18].
For instance,
if R(t) ~ tp with p>l, the
requirement R'> 0 (that defines these generalized inflationary models) is met. The effects o£ metric [3] and energy-density fluctuations [18] in these and other generalized models havebeen studied.
For completeness, here it is shown
briefly how to apply the method o£ the previous sections to evaluate the metric fluctuations in these generalized models and the corresponding bounds on the reheating temperature.
The scale factor during the generalized inflationary period reads ~(t)=h'-' tP. Assuming a transition to a radiation dominated universe, the scale factor and its first derivative matched continuously, then in the conformal time variable:
(
1 - p ) ~ ] while ~ ~
T
< - P-1
I*
where
The normal modes of the graviton field decomposition (3) now are, during the generalized inflationary period,
They behave like ordinary positive frequency modes in the limit lk~(>>l.The Bogolyubov coefficients that relate the graviton creation and annihilation operators during a radiation-dominated period to those of the inflationary regime behave, when k ~ ,<< 1, as
The expression (8) can now be used to evaluate the vacuum expectation value of the metric fluctuations for long wavelengths during the radiation-dominated period with the result
-
,~h~?'(v) 22u-2
~h~(k)
T3(p-~)
1-2" 2u-3 k
sin k~ 2
(7'
or, equivalently, if we define k(k) as the value of the Hubble constant when of coordinate wavenumber k left the horizon during inflation 1)
104
The s p e c t r u m i s s t r i c t l y (p-~o).
scale-invariant
only for the exponential
I n d e e d , when v = 3 / 2 t h e e x p r e s s i o n (19) h a s no e x p l i c i t
the a m p l i t u d e of the f l u c t u a t i o n s that scale
left
of the spectrum i n the e x p o n e n t i a l
inflationary
t o t h e c o n s t a n c y o f t h e Hubble c o e f f i c i e n t .
be done f o r t h e t r a n s i t i o n regime.
k-dependence: t h e t i m e when
t h e h o r i z o n . The e x p r e s s i o n (20) shows more t r a n s p a r e n t l y
the scale-invariance is related
was t h e same f o r a n y s c a l e a t
inflation
A similar analysis
from t h e r a d i a t i o n - d o m i n a t e d
The a m p l i t u d e o f t h e f l u c t u a t i o n s
that
cosmology can
to a m a t t e r - d o m i n a t e d
for wavelengths that enter
the
h o r i z o n d u r i n g m a t t e r - d o m i n a t i o n i s t h e same a s i n ( 2 0 ) . The S a c h s - W o l f e e f f e c t
c a n now be computed i n t h e s e g e n e r a l i z e d
models, and the r e q u i r e m e n t f o r the f l u c t u a t i o n s
a n i s o t r o p y on t h e c o s m i c microwave b a c k g r o u n d beyond o b s e r v a t i o n a l imposed.
inflationary
n o t to i n d u c e a q u a d r u p o l e bounds
Then a bound on HHC, w h i c h c a n be t r a n s f o r m e d i n t o a bound on t h e
maximum r e h e a t i n g c o u r s e , on p.
temperature after
inflation,
is obtained.
The u p p e r bound on TRH i s s m a l l e r a s p ~ 1.
10 S GeV i f p = 2.
See t h e r e f e r e n c e s
I t depends, of For e x a m p l e , TRH <
[3,18] for details.
CONCLUSIONS The bound on t h e v a l u e of t h e Hubble c o n s t a n t d u r i n g i n f l a t i o n , GeV, i s a n i m p o r t a n t c o n s t r a i n t instance,
one s h o u l d n o t e x p e c t i n f l a t i o n
taking place at scales generalized
to be a q u a n t u m g r a v i t y
comparable to the Planck s c a l e .
inflationary
evolutions
be u s e d t o r e s t r i c t p o s s i b l e
the reheating
temperature can
t h r o u g h b a r y o n number non
[3].
larger
the signal,
The g r a v i t a t i o n a l
than the observed r.m.s,
after
proper motion. inflationary
fitting
timing
waves s h o u l d n o t i n d u c e a n o i s e i n t h e of the r e s i d u a l
the data for period,
of the a r r i v a l
period derivative,
t i m e o£
position
The bound on t h e s q u a r e d a m p l i t u d e of m e t r i c f l u c t u a t i o n s
and in
c o s m o l o g i e s d e r i v e d i n t h i s way i s weaker t h a n t h e one coming from
t h e i s o t r o p y o f t h e c o s m i c microwave b a c k g r o u n d by a t magnitude.
b a c k g r o u n d o£
waves o f p e r i o d a r o u n d 1 y e a r u s i n g m i l l i s e c o n d p u l s a r
measurements [7-9]. signal
process,
t e m p e r a t u r e i s t o be l a r g e
A bound c a n a l s o be p l a c e d on t h e a m p l i t u d e o f a n s t o c h a s t i c gravitational
For
In the c a s e of the
t h e bound on t h e r e h e a t i n g
models, if
e n o u g h t o e x p l a i n t h e b a r y o n number o f t h e u n i v e r s e conservation processes
~ < 1015
on c a n d i d a t e m o d e l s f o r i n f l a t i o n .
least
However, w i t h a n improvement i n t h e e a r t h - b a s e d
t e c h n o l o g y t h e m i l l i s e c o n d p u l s a r bound may i n p r i n c i p l e power of t h e d u r a t i o n o f t h e o b s e r v a t i o n s
[8,9].
four o r d e r s of timekeeping
improve a s t h e f o u r t h
105
ACKNOWLEDGEMENTS
My work on t h i s a r e a was done i n c o l l a b o r a t i o n w i t h L a r r y A b b o t t , t o whom I am g r a t e f u l
f o r i n t r o d u c i n g me t o t h e s u b j e c t .
the hospitality
I t i s a p l e a s u r e t o acknowledge
e x t e n d e d t o me w h i l e I was v i s i t i n g
t h e P h y s i c s Department o£
Brandeis University.
REFERENCES
I.
A. Guth, Phys. Rev. D23 (1981) 347.
2.
A. Guth and S. Y. Pi, Phys. Rev. Lett. II5B (1982) 189. S. Hawking, Phys. Lett. 11713 (1982) 295. A. Starobinskii, Phys. Lett. 117]] (1982) 175. J. Bardeen, P. Steinhardt and M. Turner, Phys. Rev. ])28 (1983) 679.
3.
L. Abbott and M. Wise, Nucl. Phys. B244 (1984) 541.
4.
A. Starobinskii, JETP Lett. 30 (1979) 682 and Sov. Astron. Lett. 9 (1983) 302..
5.
V. Rubakov, M. Sazhin and A. Veryaskin, Phys. Lett. 11513 (1982) 189.
6.
R. Fabbri and M. Pollock, Phys. Lett. 125B (1983) 445.
7.
S. Detweiler, Astrophys. J. 234 (1979) ii00. B. Mashhoon, Mon. Not. R. Astr. Soc. 199 (1982) 659. B. B e r t o t t i , B. J. Cart and M. Rees, Mon. Not. R. Astr. Soc. 203 (1983) 945.
8.
L. Krauss, Nature 313 (1985) 32.
9.
M. Davis, J. Taylor, J. Weisberg and D. Backer, Nature 315 (1985) 547.
I0. II.
L. Abbott and D. Harari, Nucl. Phys. B2@4 (1986) 487. L. Ford and L. Parker, Phys. Rev. D16 (1977) 1601. B. L. Hu and L. Parker, Phys. Lett 63A (1977) 217.
12. 13.
R. Sachs and A. Wolfe, Astrophys. J. 147 (19@7) 73. L. A b b o t t and R. S c h a e f f e r , p r e p r i n t LHEA 86-003 (1986). R. F a b b r i , F. L u c c h i n a n d S. M a t a r r e s e , P r e p r i n t Univ. o£ Padua, DFPD-11/86 (1986).
14.
L. A b b o t t and M. Wise, Phys. L e t t .
135B (1984) 279.
15.
L. A b b o t t and M. Wise, A s t r o p h y s . J . 282 (1984) 647.
16.
P. Lubin, G. E p s t e i n and G. Smoot, Phys. Rev. L e t t .
50 (1983) 616.
D. F i x e n , E. Cheng and D. W i l k i n s o n , PHys. Rev. L e t t . 17. 18.
50 (1983) 620.
F. L u c c h i n and S. M a t a r r e s e , Phys. Rev. D32 (1985) 1316. F. L u c c h i n and S. M a t a r r e s e , Phys. L e t t .
164B (1985) 282.
R. F a b b r i , F. L u c c h i n and S. M a t a r r e s e , P h y s . L e t t .
I~B
(1986) 49.
M U L T I - D I M E N S I O N A L INTEGRABLE SYSTEMS R.S.Ward D e p a r t m e n t of M a t h e m a t i c s Durham U n i v e r s i t y Durham, DHI 3LE, U.K.
I.
Introduction
This lecture is concerned with integrable systems of d i f f e r e n t i a l equations. Frobenius'
I am not using the word theorem:
the equations.
Rather,
equations have.
"integrable"
in the sense of
it does not refer to the existence of solutions to it refers to a special p r o p e r t y which certain
In the case of classical mechanics,
for example,
implies that one can t r a n s f o r m to action-angle variables. to classical m e c h a n i c s
(ordinary d i f f e r e n t i a l equations),
be c o n c e r n e d with partial d i f f e r e n t i a l equations;
it
In addition I shall also
examples of these
which I include in the class of integrable equations are soliton equations such as KdV and sine-Gordon,
and integrable
such as self-dual Yang-Mills and 2 - d i m e n s i o n a l say anything about techniques
field-theory equations
a-models.
for solving these systems;
is in trying to u n d e r s t a n d what i n t e g r a b i l i t y is,
I shall not my interest
in some fairly general
sense. Integrable equations are very special: every equations is not integrable.
roughly speaking,
Nevertheless,
integrable
almost systems
turn up over and over again in the study of non-linear phenomena;
and
in addition they are a s s o c i a t e d with some very beautiful mathematics. In section 2 b e l o w I shall discuss how one might try to define what i n t e g r a b i l i t y is.
Section 3 describes
some classes of integrable
systems which arise as reductions of the self-dual Yang-Mills equations. I shall then c o n s i d e r g e n e r a l i z a t i o n s of this scheme,
in p a r t i c u l a r to
higher dimensions. It is w o r t h r e m a r k i n g that the notion of i n t e g r a b i l i t y also applies to q u a n t u m field theories,
and an interesting q u e s t i o n is how this is
related to classical integrability. restrict to classical systems.
But in what follows,
I shall
107
2.
W h a t is I n t e g r a b i l i t y ?
Let us first recall the s i t u a t i o n with regard to o r d i n a r y differential equations. the classic
For H a m i l t o n i a n systems w i t h n degrees of freedom,
"Liouville"
d e f i n i t i o n of i n t e g r a b i l i t y
e x i s t e n c e of s u f f i c i e n t l y
is in terms of the
m a n y c o n s t a n t s of motion.
Namely,
there
should exist on phase space n i n d e p e n d e n t functions K i w h i c h Poissoncommute with the H a m i l t o n i a n and w i t h one another.
It is important to
emphasize that such functions always exist locally on phase space (except p o s s i b l y around critical points of the Hamiltonian).
So inte-
g r a b i l i t y is the s t a t e m e n t that they should exist globally, appropriate
in some
sense.
One can tie things down with a stronger d e f i n i t i o n such as "algebraic complete integrability" system,
(ACI).
This involves c o m p l e x i f y i n g the
so that the p h a s e - s p a c e v a r i a b l e s become c o m p l e x - a n a l y t i c
functions of complex time.
The c o n d i t i o n of ACI is that the flow is
linear on abelian v a r i e t i e s constant.
(complex algebraic tori)
(See van M o e r b e k e
1985 for a review.)
d e f i n e d by K i = This implies that
the p h a s e - s p a c e v a r i a b l e s have the "Painlev~ property", are m e r o m o r p h i c
functions of complex time.
system has the P a i n l e v ~ property,
i.e. that they
And conversely,
if a
then it seems to stand a good chance
of being ACI. One p r o b l e m with ACI,
and with the P a i n l e v ~ test,
somewhat c o o r d i n a t e - d e p e n d e n t , example, H(p,q)
is that they are
and therefore rather restrictive.
For
a System with only one degree of freedom, with H a m i l t o n i a n
= ½p2 + V(q), may fail to satisfy ACI or the P a i n l e v ~ p r o p e r t y
(if the p o t e n t i a l
function V is not s u f f i c i e n t l y w e l l - b e h a v e d ) .
such a system i_ss integrable
in the c l a s s i c a l
But
sense.
It has been suggested that one should c o n s i d e r a w i d e r class of dynamical
systems w h i c h are in general n o n - i n t e g r a b l e
but n e v e r t h e l e s s and C h u d n o v s k y
"solvable".
See,
(1984), who discuss
b i l l i a r d s in the plane",
for exmaple,
(in fact ergodic),
E c k h a r d t et al
systems a s s o c i a t e d with
(1984)
"rational
or geodesics on surfaces of n e g a t i v e curvature,
as examples of such solvable systems.
I shall not consider
such
systems here. The subject of o r d i n a r y d i f f e r e n t i a l equations will come up again later, but for the r e m a i n d e r of this section I shall study the q u e s t i o n of w h a t is m e a n t by i n t e g r a b i l i t y
for a system of partial d i f f e r e n t i a l
equations.
(a)
By analogy w i t h the L i o u v i l l e definition,
one may specify that the
108
system admit an infinite number of conserved currents. idea is relevant to soliton equations
C e r t a i n l y this
such as KdV, where one gets an
infinite number of commuting flows on an i n f i n i t e - d i m e n s i o n a l
space,
so
there is a close analogy with the f i n i t e - d i m e n s i o n a l L i o u v i l l e picture. But there are a number of objections to using this as a d e f i n i t i o n of integrability.
First,
it does not seem to apply to systems such as
the self-dual Y a n g - M i l l s or self-dual E i n s t e i n equations, in any natural way.
Secondly,
sum" of a n o n - i n t e g r a b l e many conserved densities. served densities
c o n s i d e r the "direct
and an integrable system with infinitely
The combined system has i n f i n i t e l y m a n y con-
(which happen to depend only on the first summand),
is not integrable
(because of the second
rules out d e f i n i t i o n s transformations,
and more seriously,
system,
at least not
summand).
but
The same argument
involving i n f i n i t e l y many symmetries,
B~cklund
and the like.
It is i n t e r e s t i n g to note that any c o n f o r m a l l y - i n v a r i a n t sional system has an infinite number of c o n s e r v e d currents
2-dimen-
(Goldschmidt
& Witten 1980), which are important in the c o r r e s p o n d i n g q u a n t u m field theory.
But such systems are not in general integrable.
For example,
consider the "harmonic map" or "~-model" p r o b l e m for functions ~2
L
=
6~v ( ~
where gij is the metric on M. not be integrable.
%i) ( ~ # j )
And geodesic
gij (4) ,
This is c o n f o r m a l l y invariant,
(For example,
but may
the solutions which depend on only
one of the C a r t e s i a n coordinates of ~ ,
(b)
# from
to M, defined by the L a g r a n g i a n
c o r r e s p o n d to geodesics on M.
flow on a R i e m a n n i a n m a n i f o l d is seldom integrable.)
One may require that the system has the P a i n l e v 6 property, w h i c h
(roughly speaking)
w o u l d say that its solutions should be m e r o m o r p h i c
functions of the c o m p l e x i f i e d i n d e p e n d e n t variables W a r d 1984).
(see,
for example,
This can be useful as a test for integrability,
revealing p r o p e r t i e s of integrable systems. a d e f i n i t i o n of integrability,
and for
But it is not m u c h good as
since for one thing it is not p r e s e r v e d
under a t r a n s f o r m a t i o n of the d e p e n d e n t variables. (c)
I n t e g r a b i l i t y seems to be a p r o p e r t y which can be said to hold
locally in space-time.
For example,
the sine-Gordon equation is just
as integrable on a flat 2 - d i m e n s i o n a l torus as it is on flat space-time ~2.
In particular,
boundary conditions are irrelevant as far as inte-
g r a b i l i t y is c o n c e r n e d
(although o b v i o u s l y they m a t t e r for the e x i s t e n c e
of p a r t i c u l a r kinds of solutions such as solitons). not define i n t e g r a b i l i t y
Thus one should
in terms of ideas such as the existence of
109
multi-soliton
solutions,
or the inverse scattering transform,
depend on a p p r o p r i a t e global conditions.
which
(Also, these ideas do not
seem to apply to systems such as the self-dual Y a n g - M i l l s equations.)
(d)
M a n y integrable
systems are closely a s s o c i a t e d w i t h Lie algebras,
and in p a r t i c u l a r with i n f i n i t e - d i m e n s i o n a l Algebraic
(Kac-Moody)
Lie algebras.
structure is also i m p o r t a n t in integrable q u a n t u m field
theories and s t a t i s t i c a l - m e c h a n i c s how fundamental
systems.
But it is not clear to me
such algebraic b a c k g r o u n d is, and w h e t h e r
applies to a specific class of integrable
it only
systems with specific b o u n d a r y
conditions.
(e)
One can study the reductions of a given system of partial differen-
tial equations
to o r d i n a r y d i f f e r e n t i a l equations.
For example,
Y a n g - M i l l s e q u a t i o n s and E i n s t e i n ' s equations can be reduced by tially)
saying that the fields depend only on time
variables). behaviour
And the r e s u l t i n g
("chaotic cosmology"
1984 for the Y a n g - M i l l s case). that E i n s t e i n ' s
(or definition)
reliable,
since the original
Finally,
systems can exhibit chaotic
in the E i n s t e i n case;
see e.g. Savvidy
From this one should c e r t a i n l y conclude
of integrability,
"enough"
reductions.
one could require that the system of equations be the
c o n s i s t e n c y c o n d i t i o n for an o v e r d e t e r m i n e d
system of linear equations.
this linear system has to have some special property.
example,
Dubois-Violette
(1982)
linear system, but as noted above,
linear systems are; of a l l o w a b l e systems.
for a
these equations are not integrable.
So one has to r e s t r i c t to "allowable" to integrable equations.
For
showed that both the E i n s t e i n and the
Y a n g - M i l l s equations can be w r i t t e n as c o n s i s t e n c y conditions
3.
But
this m e t h o d may not be very
system may not possess
However,
grability"
(essen-
(and not on the space
and the Y a n g - M i l l s e q u a t i o n s are net integrable.
as a test
(f)
"mechanical"
the
linear systems w h i c h do give rise
I do not know what the most general allowable
in w h a t follows,
I shall m e r e l y give some examples
But it seems that the best way to define
"inte-
in general, w o u l d be to specify what is m e a n t by "allowable".
R e d u c t i o n s of the S e l f - D u a l i t y E q u a t i o n s
A basic example of an integrable system is that of the self-dual Y a n g - M i l l s equations
in four dimensions.
Its solutions can be des-
cribed in terms of h o l o m o r p h i c vector bundles,
or, equivalently,
in
110
terms with
of the
"Riemann-Hilbert
the v a r i o u s
the s y s t e m s
solution
that
problem".
techniques
c a n be o b t a i n e d
But
here,
f r o m the
I shall
merely
n o t be c o n c e r n e d
with
listing
self-duality
some o f
equations
by
reduction. Let A t i m e C4,
denote
taking
a gauge
values
in some
F a n d the
self-duality
potential
=
complex
Lie
- ~
A
+
=
F
~ A
equations
are a s y s t e m
equations linear
for A
F ~B
of s e m i - l i n e a r
.
equations
They
which
are
Minkowski
a l g e b r a g.
space-
Its c u r v a t u r e
is
[A ,A ]
are
½~ which
on c o m p l e x i f i e d
first-order
the c o n s i s t e n c y
partial
differential
conditions
for a p a i r
of
m a y be w r i t t e n (Dy - {D v) ~
=
0,
(I) (D u + ~D z) ~ Here D
=
8
+ A
is a c o m p l e x and u,v,y,z dydz
parameter
are n u l l
+ dudv).
tions
denotes
follow
First,
"factoring
of t h e s e
yu
=
F
~.
=
zv
one
out"
To r e d u c e
can reduce
is in f a c t P o i n c a r ~ - i n v a r i a n t , form.)
discussion
this
of w h a t
Secondly,
imposing
straints equations
one
the
corresponding
"spectral
the
to A
,
paramater"),
(the m e t r i c
coordinates,
0,
to t h e
A complete
F
that
this
is
ds a =
self-duality
equa-
0.
(2)
involves
this means
(1980)
(I)
it h e r e
in
for a g e n e r a l
entails.
of d e p e n d e n t
variables
in a c o n s i s t e n t to i m p o s e
in e f f e c t
by
(The s y s t e m
not written
& Manton
the n u m b e r on t h e A
variables
group.
I have
for all v a l u e s
two processes.
of independent
reduction"
also be allowed
, since
(i) b e c o n s i s t e n t
system
although
=
uv
of the P o i n c a r ~
"dimensional
can reduce
+ F
yz
See Forgacs
conditions
should
on t h e A
a large problem, (B).
one
algebraic
know whether
and
called
the n u m b e r
by a subgroup
Poincar4-invariant
(B).
derivative
on space-time
f r o m the r e q u i r e m e n t
of the p a r a m e t e r
(A).
(sometimes
0.
are F
These
the c o v a r i a n t
coordinates
In t e r m s
=
way.
differential
adding
extra
by (I d o n ' t
con-
differential
system.)
analysis because
I shall merely
of all r e d u c t i o n s
of t h e
there
possibilities
are
exhibit
so m a n y some
examples.
system
(i)
looks
for b o t h
like (A)
111
EXAMPLE
I.
lations
~iu and
function
reduce
~v-
as in
for m o r e
(1983).
different
details
satisfying
the u s u a l [Ha,H b]
that Ap be a
(B), we shall use some background,
is a n a l o g o u s
the one p r e s e n t e d
that the g a u g e
out the two trans-
the c o n d i t i o n
as in
of the a l g e b r a i c
to,
see O l i v e
although
here.)
Ea,
E _ a } a = l , 2 ..... n
commutator =
&
somewhat
a l g e b r a ~ is a simple Lie a l g e b r a
(think of s%(n - i))~. L e t {Ha, of ~,
impose
To r e d u c e
(Their t r e a t m e n t
from,
Suppose
(A) by f a c t o r i n g
In o t h e r words,
of y and z only.
algebra; Turok
First,
0,
of rank n
be a C h e v a l l e y
basis
relations
[Ea,E_ b]
=
6abHb, (3)
[Ha,Eb]
=
KbaEb ,
A p by r e q u i r i n g Ay
=
Au
=
that t h e y h a v e X a
=
-KbaE b.
We can r e d u c e
H e r e K b a is the C a r t a n m a t r i x . variables
[Ha,E b]
the n u m b e r
of d e p e n d e n t
the form
faHa ,
Az
=
X gaHa , a
~ eaE a, a
Av
=
[ eaE_ a, a
(4)
where
fa' ga,
shows that the
ea are
functions
self-duality
of y and z.
equations
Then a simple
(2) r e d u c e
to the
calculation
"Toda m o l e c u l e "
equations ~ y ~ z #a where KII
#a = 2 log e a.
The s i m p l e s t
= 2 and one gets the L i o u v i l l e For a s e c o n d e x a m p l e ,
where
the i n d i c e s
s y s t e m of s i m p l e commutation
roots
relations
is n = i, a l g e b r a
equation
~y ~z ~
let us use the e x t e n d e d
(0 c o r r e s p o n d s
to m i n u s
(3) r e m a i n valid,
in m o r e detai l .
(4), w r i t i n g
s£(2), w h e r e
= -2e~"
C a r t a n m a t r i x Kab , an e x t e n d e d
the h i g h e s t
w i t h Kab r e p l a c e d e a = exp(½#a ) .
root).
The
by Kab;
and the E a are l i n e a r l y
=
~ ½ea(~y~ a + 2fbKab)E a a,b
independent,
so F yu=
0 gives
~y~a
=
-2 ~' K a b f b . b
(6)
y~a
=
2 [ K a b g b. b
(7)
Fzv = 0 gives
And f r o m Fy z + F u v = 0 we get
and
I shall do this
First,
Fy u
Similarly,
case
(5)
a and b n o w run f r o m 0 to n, l a b e l l i n g
we use the same r e d u c t i o n case
= -[ K a b exp %b b
112
(exp
~a - ~zfa + ~ y g a ) H a
=
0.
(8)
a Because
the Ha are not l i n e a r l y
expression
in p a r e n t h e s e s
in
independent,
(8) v a n i s h e s ,
[ Kba (exp #a - ~zfa + a N o w from
(6),
(7),
are the s O - c a l l e d
is w h e n n = I, a l g e b r a
but we can d e d u c e
~yga )
=
_ ~ Kab(~zfb b
=
- [ Kab exp #b b
"Toda l a t t i c e "
=
0.
(9)
(i0)
equations.
The s i m p l e s t
case
= -2
(I0) b e c o m e s
- ~ygb )
s~(2), w i t h
Kab
so that
that the
(9) we get ~y~z~ a
which
we c a n n o t d e d u c e
(essentially)
2
the s i n h - G o r d o n
equation
~y~z ~
=
-4 sinh ~. A further equations",
generalization
which
include,
~y~z ~ = -2e~ + e-2~"
EXAMPLE now,
2.
Cf.
As before,
Olive
suppose
i mpose the c o n s t r a i n t
equations
(2) r e d u c e
gives
the
"generalized
for example, & Turok
are the 2 - d i m e n s i o n a l
in a m o r e u s u a l
chiral
f o r m as follows.
A p w h e r e g takes v a l u e s
Eqn.
(llb)
=
=
Eqn.
(lla)
they can be w r i t t e n
says that 2Ap has
zero
in the g a u g e g r o u p G.
(The index
p is n o w 2-
space-time with metric
dydz.)
then gi v e s
algebraic
as c p n - m o d e l s symmetric
(llb)
0.
½g-l~g,
we are in 2 - d i m e n s i o n a l
is the u s u a l
further
(lla)
0,
form
~p(g-l~g) which
=
field equations;
and t h e r e f o r e m u s t h a v e the
dimensional:
But
T h e n the s e l f - d u a l i t y
to
~yA z + ~zAy
curvature,
only on y and z.
A v = -Ay.
~yA z - ~zAy + 2 [ A y , A z]
These
equation
(1983).
the A p d e p e n d
A u = Az,
Toda lattice
the B u l l o u g h - D o d d
f o r m of the c h i r a l constraints
and n o n - l i n e a r
spaces,
=
field equations.
on the Ap u-models.
0,
(or on g) gives Such r e d u c t i o n s
and h a v e b e e n c o m p l e t e l y
classified.
Imposing examples
such
correspond
to
113
EXAMPLE
3.
example,
of f a c t o r i n g
factor out r o t a £ i o n s .
autonomous appear
Instead
equations,
explicitly.
relativity,
i.e.
In g e n e r a l
equations
An example
which
out t r a n s l a t i o n s ,
this w i l l
in w h i c h
is the
can be w r i t t e n
"Ernst equation"
in the
stationary
being
a 2 × 2 matrix.
axisymmetric
(12) by a r e d u c t i o n
except
that one
factors
two t r a n s l a t i o n s . Ward
of g e n e r a l
0, of
(12) (12)
correspond
equations.
One can
to that of the p r e v i o u s
out a r o t a t i o n
Some m o r e d e t a i l s
and a t r a n s l a t i o n
to
example,
rather
m a y be found in W i t t e n
than
(1979)
and
(1983).
EXAMPLE
4.
differential achieved
Finally,
therein).
some o n e - d i m e n s i o n a l
equations.
by s a y i n g
Nahm equations
for example,
of s t r u c t u r e w h i c h
ordinary
=
of w h a t h a p p e n s
condition
[Az ~ + A u
seems
if we r e d u c e to
(13)
there
1980).
the
cited
(I) r e d u c e s
' - A v ~ + Ay].
about w h i c h
(Adler & v a n M o e r b e k e
really
for
is
this gives
1985 and the r e f e r e n c e s
with parameter",
theory
grable mechanical
only on y + z:
g i v e an e x a m p l e
(Az~ + Au)
is a "Lax e q u a t i o n
Clearly
Ward
T h e n the c o m p a t i b i l i t y
sive m a t h e m a t i c a l
i.e.
One r e d u c t i o n w h i c h has b e e n of i n t e r e s t
L e t me r a t h e r
d dy
reductions,
that Ap should depend
(see,
by A ~ = Ap(y).
This
=
of E i n s t e i n ' s
analogous
variables
f orm
The s o l u t i o n s
solutions
obtain
for
lead to non-
the i n d e p e n d e n t
~z(j-i ~zj) + p-i ~p(pj-I ~pj) w i t h J(p,z)
one could,
is an exten-
It is this k i n d
to lie at the h e a r t of c o m p l e t e l y
inte-
systems.
(13),
or a l t e r n a t i v e l y
(2) w i t h A~ = A~(y),
is e q u i v a l e n t
to [Az,A v] A'z
=
[Az,Ay] A u,
where
the p r i m e
denotes
the A p by s a y i n g
the a i and b i b e i n g m i n e s Ay in terms
d/dy.
+
0,
(14a)
[Av,A u] ,
(14b)
[Au,Ay] ,
(14c)
Let us i m pose
algebraic
skew-symmetric
constraints
n x n matrices,
on while
with Az
=
d i a g ( a l , .... an),
Av
=
diag(bl,...,bn) ,
constants.
of Au, M~j
where
=
t h a t A u and A y are
A v and A z are d i a g o n a l
=
and =
Then (14c)
(14a)
is s a t i s f i e d ,
(14b)
deter-
then becomes
[ (~ik - ~kj)MikMkj , k
(15)
114
lij
=
(b i - bj)/(a i - aj)
for i # j
and where Mij denotes the ij-th element of the matrix A u. is the E u l e r - A r n o l d - M a n a k o v equation for an n - d i m e n s i o n a l body, or a l t e r n a t i v e l y the e q u a t i o n of geodesic
Equation
(15)
spinning rigid
flow on SO(n), with lij
d e f i n i n g a l e f t - i n v a r i a n t diagonal metric on SO(n).
4.
G e n e r a l i z a t i o n s of the S e l f - D u a l i t y Equations
One can g e n e r a l i z e the linear system ways.
Observe that
single spectral p a r a m e t e r (i) (ii)
~.
So one could:
increase the number of equations;
or
allow each equation to be a polynomial meter(s),
(iii)
(I) in a number of d i f f e r e n t
(i) consists of two equations, each linear in the
of degree greater than i;
in the spectral para-
or
increase the number of spectral parameters;
or any c o m b i n a t i o n of these. izations in Ward
(1984b),
There is a d i s c u s s i o n of these general-
and I shall m e r e l y m e n t i o n here some of the
features that arise. First of all, dimensions.
there is an increase in the number of "space-time"
For example,
if one replaces
(i) by a pair of equations
quadratic in ~, namely
(Dy
~D v + ~2Dx)~
=
0
(Du + ~D z + ~ZD w)~
=
0,
-
(16)
then one is dealing w i t h a gauge field in 6 - d i m e n s i o n a l space, with coordinates
(u, v, w, x, y, z).
The c o n s i s t e n c y conditions
for
(16)
amount to a set of linear relations on the curvature F~v, g e n e r a l i z i n g the s e l f - d u a l i t y equations
(2).
Not much is known about the r e d u c t i o n s of tency conditions. is well-known,
However,
namely the n o n - l i n e a r S c h r o d i n g e r
linear system for NLS, Fordy & Kulish
(16) and of its consis-
there is a 2-dimensional r e d u c t i o n w h i c h (NLS)
equation.
(1983), have the form
(16).
Similarly,
system for the KdV equation and its g e n e r a l i z e d versions Fordy 1986)
is a pair of equations w h i c h are cubic in C.
point of view,
The
and also for the g e n e r a l i z e d NLS equations of the linear (Athorne & So from this
the KdV equation is a reduction of a set of g a u g e - f i e l d
equations in eight dimensions
(although for practical purposes this may
not be a very useful way of dealing with the KdV equation!). Possibility (i),
namely increasing the number of linear equations,
also increases the number of space-time dimensions.
Now, however,
the
115
c o n s i s t e n c y c o n d i t i o n s become
(not surprisingly)
is also a 2 - d i m e n s i o n a l r e d u c t i o n of this with
(ii)) w h i c h is well-known,
an o v e r d e t e r m i n e d "many times"
(i) c o m b i n e d This is
Date et al 1983).
(iii), namely increasing the number of spectral para-
In this case, however,
system in higher dimensions.
it does not seem that there are any interesting
to low dimensions.
Clearly,
all the 2 - d i m e n s i o n a l
the AKNS or Z a k h a r o v - S h a b a t d e s c r i b e d here.
soliton equations w h i c h fit into
framework,
also fit into the general scheme
One i n t e r e s t i n g q u e s t i o n is w h e t h e r one could deal
with the "complete" KdV hierarchy, commuting
of
There
involves c o m m u t i n g flows with respect to
also leads to an o v e r d e t e r m i n e d
reductions
(or rather,
namely the "KdV hierarchy".
(see, for example,
Possibility meters,
system which
overdetermined.
w h i c h involves infinitely many
flows on an i n f i n i t e - d i m e n s i o n a l
W i l s o n 1985).
It would have
space
(Date et al 1983,
to involve an i n f i n i t e - d i m e n s i o n a l
analogue
of the structure d i s c u s s e d here.
5.
Other Types of Linear System
One integrable
system w h i c h does not fit into the above scheme is
that of the self-dual E i n s t e i n equations. similar in nature.
It is, however,
rather
The linear system can be w r i t t e n in the particularly
elegant form VAA' where
VAA,
=
0,
(17)
is the c o v a r i a n t space-time derivative,
ponent spinor.
The c o n s i s t e n c y c o n d i t i o n for
dual E i n s t e i n equation, "twistor"
~B'
and ~B' is a 2-com-
(17) is indeed the self-
and this d e s c r i p t i o n leads to Penrose's
c o n s t r u c t i o n of its solutions.
(1976)
The spectral p a r a m e t e r
in
(17) is the ratio of the two c o m p o n e n t s of ~B'' All the c o n s i s t e n c y conditions d i s c u s s e d up to now have been "the v a n i s h i n g c u r v a t u r e of a connection".
But there are some equations
that should be r e g a r d e d as integrable,
but that have not
ledge)
These are e x e m p l i f i e d by the KP
been r e p r e s e n t e d
equation
in this way.
(Date et al 1983),
(to my know-
the linear system of which can be w r i t t e n
in terms of h i g h e r - o r d e r d i f f e r e n t i a l operators or p s e u d o - d i f f e r e n t i a l operators,
but not
(apparently)
operators of "connection" at the end of the previous dimensional
space.
type.
in terms of first-order d i f f e r e n t i a l This may be related to the comment
section,
about the need to go to an infinite-
But in any event,
the scheme d e s c r i b e d by Date et
al and W i l s o n is so elegant that one may not want to replace it with something else.
116
6.
Conclusions
I began
this
is to define one based able".
allowable,
linear
integrable
settled.
to e s t a b l i s h
But one could
described
linear
With
this
to be
is "allow-
3 and 4 are c e r t a i n l y
can be o b t a i n e d
in this
so the
sort of definition,
or not a given e q u a t i o n
all the integrable
type of linear
seems
system w h i c h
in sections equations
the best way
definition
such as the KP equation,
whether
try to c l a s s i f y
from a certain
of w h a t
promising
overdetermined
are exceptions,
is not yet
be d i f f i c u l t
the q u e s t i o n
The most
systems
and m o s t known
But there
questio n
by posing
on an a s s o c i a t e d The
way.
lecture
integrability.
it w o u l d
was
equations
integrable.
which
arose
system.
References. Adler,
M.
Athorne,
& P. van M o e r b e k e C.
& A.P.Fordy
symmetric Chudnovsky,
spaces.
D.
& G.
equations. lems, Date,
eds.
Non-Linear
Fordy,
B.,
A.P.
Forgacs,
P.
D.
(Marcel Dekker),
& T. Miwa
In:
eds.
and A r i t h m e t i c pp.
of RIMS
M. Jimbo
& T. M i w a
& F. Vivaldi
B 119,
1984 P h y s i c a
D i_33, 339-356.
Phys.
899, 427-443.
& N.S.Manton
1980 Comm.
Math.
Phys.
72,
E~.Witten 1980 Phys.
& N. Turok
Savvidy,
G.K.
1983 Nucl.
1976 Gen.
van Moerbeke,
Rel.
1984 Nucl. P.
Phys.
1985 Phil.
1983 Gen.
Rel.
R.S.
1984 Phys.
Ward,
R.S.
1984b Nucl.
Ward,
R.S.
1985 Phys.
Lett.
7,
Trans.
Lett.
Wilson,
G.
1985 Phil.
Trans.
Witten,
L.
1979
Rev.
470-494.
302-334. Lend.
A 315,
379-390.
279-282. 381-396. 3-5.
R. Soc.
D 19,
15-35.
392-396.
105-109.
B 236, A 112,
B 215,
R. Sec.
15,
A 102,
Phys.
B 91,
31-52.
B 246,
Grav.
Ward,
Phys.
Phys.
Grav.
Lett.
Lend.
718-720.
A 315,
393-404.
on
(World
157-161.
Math.
R.
groups
Symposium
39-119. Lett.
Prob-
99-115.
1983 T r a n s f o r m a t i o n
Proceedings
Systems,
1982 Phys.
J. Ford
with
system of d i f f e r e n t i a l
1983 Comm.
Penrose,
R.S.
associated
& P.P.Kulish
Goldschmidt,Y.Y.& Olive,
pp. M.
3_~8, 267-379.
and Q u a n t u m Models
M. Jimbo
Integrable
Scientific),
Ward,
Classical
equations.
Dubois-violette, Eckhardt,
on E i s e n s t e i n ' s
& G. C h u d n o v s k y
E., M. Kashiwara, for seliton
in Math.
KdV equations
(Leeds preprint).
1984 Note
In: D.
1980 A d v a n c e s
1986 G e n e r a l i z e d
M O N O P O L E AND V O R T E X S C A T T E R I N G
N.J. H i t c h i n M a t h e m a t i c a l Institute 24-29 St. Giles Oxford England.
§i.
This lecture is c o n c e r n e d with the R i e m a n n i a n g e o m e t r y w h i c h
arises by considering
the m o t i o n of soliton-like
Y a n g - M i l l s - H i g g s equations.
solutions to the
These are the equations derived from
the action d e n s i t y a = (FA,FA)
+
(DA~,DA~)+I(i
-1412) 2
in M i n k o w s k i
space for a c o n n e c t i o n
Higgs field
~
FA
A
w i t h gauge group
in some r e p r e s e n t a t i o n of
is the curvature of
A
and
DAB
G.
G
and a
In this e x p r e s s i o n
the covariant d e r i v a t i v e of
~.
The full Y a n g - M i l l s - H i g g s equations m a y be considered as the time e v o l u t i o n of a c o n n e c t i o n
A
and Higgs field
such they have the following m e c h a n i c a l the space of e q u i v a l e n c e
classes
C
of the group of gauge t r a n s f o r m a t i o n s
~
on
interpretation.
of pairs G.
(A,~)
~3
and as
We consider
under the action
The tangent space at a
point on this i n f i n i t e - d i m e n s i o n a l m a n i f o l d can be i d e n t i f i e d w i t h the i n f i n i t e s i m a l d e f o r m a t i o n s orbits.
Then
C
(i,$)
o r t h o g o n a l in
L2
to the gauge
acquires a R i e m a n n i a n metric~
3 There is also a p o t e n t i a l f u n c t i o n defined on
C
by the gauge-
invariant q u a n t i t y
V = [
%
(FA,E A) + ( D A # , D A ~ ) + ~ ( i
-I~12) 2
3
a s s o c i a t e d to the c o n n e c t i o n and Higgs field on
~3.
The Y a n g - M i l l s - H i g g s e q u a t i o n s can then be viewed as the m o t i o n of a p a r t i c l e on this i n f i n i t e - d i m e n s i o n a l
configuration
kinetic energy term given by the m e t r i c and a p o t e n t i a l Manton
space w i t h V,
[103 took this i n t e r p r e t a t i o n further by suggesting that,
just as a ball bearing rolling inside a bowl will with small v e l o c i t i e s roll around the base of the bowl,
so small v e l o c i t y soliton-like
118
solutions motion there ment
to the Y a n g - M i l l s - H i g g s
on the absolute m i n i m u m are a n a l y t i c a l
rigorous
of cases
problems
special
shall
see.
§2.
There
are two situations
minimized
in a known m a n n e r
The first
is the 3 - d i m e n s i o n a l G
is
representation, is the
cases
value
for the
ditions. satisfy
I#I ÷ 1
potential
this m a p
sphere
giving
as Jaffe
corresponding points,
k
k.
of this
and Taubes whose
at least
(A,#). where
field
[93.
which provides boundary
circle
of the q u a r t i c
sphere
of radius
~2
are solutions
energy density
for large
we also
invariant.
to the
is c o n c e n t r a t e d
separations,
R
of
to the unit
is again a t o p o l o g i c a l there
to
and the degree
in
a
con-
is assumed
last vestige
SU(2)
These
In the case of a v o r t e x
show,
the
second
I = ~.
suitable
from a large
from a large
is
The
and Taubes
given
of
V
G = U(1), and
invariant
is the
a map
charg 9
a map
equations
providing
V,
above.
where
if the Higgs
which
defines
term
of Jaffe
in the Lie algebra
and the degree
Moreover,
of
case,
R ÷ ~,
is the m a g n e t i c
I~I ÷ 1
circle,
as
term, then ~
to the unit
have
absolute m i n i m u m
In the m o n o p o l e
geometry
function
monopoles,
representation
is a t o p o l o g i c a l
the argu-
is in the adjoint
of vortices,
in the book
there
field
Clearly
configurations
in the p o t e n t i a l
situation
are both d e s c r i b e d In both cases
Riemannian
case of m a g n e t i c
one-dimensional
V.
leads us in a number
the p o t e n t i a l
the Higgs
I = 0
2-dimensional
lies in a complex
where
geodesic
function
in order to m a k e
it is one w h i c h
by p a r t i c l e - l i k e
SU(2),
and
approximate
finite-dimensional
as we
gauge g r o u p
will
to be solved
but n e v e r t h e l e s s
to a rather
system
of the p o t e n t i a l
around
k
a particle-like
description. The e q u a t i o n s equations.
w h i c h realize
For magnetic
this
monopoles
absolute m i n i m u m
they
are first order
are the B o g o m o i n y
equations:
F A = *DA# and for v o r t i c e s
~A # = 0 ; There time.
the v o r t e x equations:
i
F A = ~(I~
is a d i f f e r e n c e Whereas,
12
- i)
of status b e t w e e n
through
a variety
these
of m e t ho d s ,
equations
at the p r e s e n t
the B o g o m o l n y
equations
119
may
be r e d u c e d
to the
explicit
solutions
although
some
the
is known.
description
[3]
geometry
of the v o r t e x
analytical
Nevertheless, classes
algebraic
equations
information
structure
space
space
(k - i)
unknown,
available
of g a u g e
it is p r o v i d e d
of r a t i o n a l
of g e n u s
are e s s e n t i a l l y
is c e r t a i n l y
of the
For m o n o p o l e s
as the
of a c u r v e
[i13.
equivalence
by D o n a l d s o n ' s
maps
k-i a 0 + a Iz+. . .+ak_ 1 z f(z)
= b 0 + b I z + . . . + b k _ 1 z k-l+z k
of d e g r e e factor),
k
(i.e.
where
existence
where
ai,
theorem
of
denominator
bj
are c o m p l e x
[9]
shows
and n u m e r a t o r numbers.
that
the
have
no c o m m o n
For vortices,
space
is the
complex
the space
of
polynomials k-i p(z)
-- a 0 + alz +
a k-dimensional The
the n a t u r a l equations
§3.
kind
K
the
of t h e s e
which
satisfy
automatically (positive
~I(X,Y)
which
lead
= g(IX,Y) ;
us
In o r d e r recall acts form
into
the
map
symplectically e,
the d u a l
then
on a s y m p l e c t i c ~
g
which I,
metric
is J
should
geometry. be h y p e r k l h l e r ,
if a Lie g r o u p M
soluThe
= g(KX,Y)
with
G
symplectic
is an e q u i v a r i a n t
such t h a t
are
forms
~3(X,Y)
geometry: manifold
so t h e y
equations.
symplectic
: M ÷ g* G
is of a v e r y
structures
tensor, vacuum
symplectic
of
the m o n o p o l e
of
f r o m the
etc.
= g(JX,Y) ;
in s y m p l e c t i c
algebra
arises
identities:
Ricci
to t h r e e
map
to is to find
is a m e t r i c
complex
Einstein
consideration
the m o m e n t
of the Lie
zero
e2(X,Y)
to see w h y
the m o m e n t
have
rise
which
This
constant
definite)
give
reduces
this m e t r i c
IJ = - JI = K
tions
structures
case,
the q u a t e r n i o n i c
Such m e t r i c s
complex
spaces
metric.
covariant
12 = j2 = K 2 = -i;
to the
problem
parametrize.
in the m o n o p o l e
three
,
space.
they
- a hyperkahler
with
+ z
scattering
on e a c h
solutions
Fortunately,
special
vector
which
metric
whose
compatible and
complex
question
k
... + ak_iZ
map
to
2
,
120
= i(X~)m
where
X~
is the v e c t o r field generated by
W i t h i n symplectic qeometry,
if
G
~ ~ g .
acts freely on
M,
then the
manifo~Id -i
(0)/S
is again symplectic: result
the M a r s d e n - W e i n s t e i n quotient.
holds in h y p e r k ~ h l e r g e o m e t r y E83.
freely on a h y p e r k ~ h l e r manifold, Z2
and
~3
An analogous
If a Lie group
we obtain three m o m e n t m a p s
corresponding to the symplectic forms
~i'
~2
G
acts Z!'
and
~3
and then 3 -i n Pi (0)/G i=l is again hyperk~hler. The relevance of this result to m a g n e t i c m o n o p o l e s is the follow -~ing i n f i n i t e - d i m e n s i o n a l point of view. A
and Higgs field
9
on
~3
We consider the c o n n e c t i o n
as a v e c t o r in an infinite d i m e n s i o n a l
q u a t e r n i o n i c vector space:
9 + iA 1 + JA 2 + k A 3 which with the
L2
inner product is a flat h y p e r k l h l e r m a n i f o l d .
The group of gauge t r a n s f o r m a t i o n s
G
acts on this and the three
m o m e n t m a p s turn out to be the three components of the B o g o m o l n y equations.
Thus
3 -i N Zi (0)/G i=l is the space of e q u i v a l e n c e classes of solutions to the B o g o m o l n y equations,
§4.
and its natural m e t r i c is hyperkahler.
In the vortex case,
suppose we consider 91, 92
SU(2)
life is not quite so simple. connections
in the adjoint representation.
A 1 + iA 2 + j91 + k92
A
on
~2
However,
and Higgs fields
Then w r i t i n g
121
we have an i n f i n i t e - d i m e n s i o n a l h y p e r k ~ h l e r m a n i f o l d gauge t r a n s f o r m a t i o n s
in the same way as for monopoles.
Now for a symplectic m a n i f o l d -i (0)/G also
M
we need not take simply
to create a new symplectic manifold,
p-l(x)/Gx
is symplectic,
where
Gx
x i<
but we can c o n s i d e r
is the stabilizer of
and likewise the c o r r e s p o n d i n g
klhler situation.
acted on by
x e g*.
This again
result in the hyper-
Taking
o1
-½i
-1 ~Pi ( x ) / % w h i c h is the space of i solutions of the e q u a t i o n s
we obtain a h y p e r k ~ h l e r m e t r i c on equivalence
classes
~A # = 0 F A + ½E%,~*] = x where
~i
and
%2
Here e q u i v a l e n c e x
J
are i n c o r p o r a t e d
is with regard to the gauge t r a n s f o r m a t i o n s
invariant i.e.
U(1)
A
reduces to a
}.
leaving
gauge transformations.
C o n s i d e r the special case where nection
into one complex Higgs field
U(1)
~ = [i
connection.
~]
and the con-
Then the above equations
reduce to the v o r t e x e q u a t i o n s
SA# : 0
FA
=
½(l~I
2
-
i)
An i n v a r i a n t way of c h a r a c t e r i z i n g
them is the solutions
(A,~I,~ 2)
w h i c h are invariant in the space of e q u i v a l e n c e classes under the circle action
# ÷ ei8%.
In other words those for w h i c h this circle
action is g e n e r a t e d by a gauge transformation. This way the natural m e t r i c on the space of e q u i v a l e n c e of v o r t i c e s is the r e s t r i c t i o n of a h y p e r k a h l e r m e t r i c point set of a circle action
[7].
classes
to the fixed
122
~5.
Reverting to the m a g n e t i c m o n o p o l e
situation, general information
about this m e t r i c m a y be derived from the m e t h o d s of solving the B o g o m o l n y equations t h e m s e l v e s [2].
In the p a r t i c u l a r case of
k = 2
the h y p e r k ~ h l e r structure itself is sufficient to lead to an e x p l i c i t solution.
This is because,
if we reduce to the situation of 2 m o n o -
poles which are centred about the origin, classes is 4-dimensional,
the space of equivalence
h y p e r k ~ h l e r and has an
isometries arising from the r o t a t i o n group of
S0(3)
action by
~3.
Such m e t r i c s were studied by Gibbons and Pope [5] and can be put in the form:
ds2 = WlW2W3d~2 where
~i'
~2
and
w2w3 2 W3Wl 2 WlW2 2 W 1 ~i + w 2 ~2 + w 3 ~3
+ o3
invariant 1-forms on
are a standard o r t h o n o m i a l basis of left-
SO(3)
and the functions
Wl,
w2
and
w3
satisfy the equations:
+w
=-2wlw 2 ;
2w2w 3
w 3' + w I' = -2w3w I These equations were,
in fact,
solved by Halphen in 1881 [6] with
elliptic integrals.
In this m o n o p o l e
situation the elliptic curve
w h i c h g e n e r a t e s the solution of the B o g o m o l n y e q u a t i o n s yields a reason for this. C o m b i n e d with the r e q u i r e m e n t s of n o n - s i n g u l a r i t y and completeness one obtains a unique solution to the above p r o b l e m and the m e t r i c is c o m p l e t e l y d e t e r m i n e d
[2].
As to the g e o d e s i c motion, constants of the motion,
the
S O ( 3 ) - i n v a r i a n c e g e n e r a t e s two
the m e t r i c a third but there has not yet
a p p e a r e d a fourth commuting integral to m a k e the flow c o m p l e t e l y integrable.
It is worth p o i n t i n g out that although there are v a r i o u s
Killing tensors
of higher rank on this manifold,
none of t h e m appears
capable of c o n t r a c t i o n to give such a fourth integral,
analogous to
that w h i c h exists for the Kerr solution in general relativity. Nevertheless,
the m o t i o n on some totally g e o d e s i c surfaces of
r e v o l u t i o n m a y be analyzed
[i],
[2] to give interesting scattering
behaviour of the m o n o p o l e s thought of as particles.
123
§6.
In the case
equivalence
classes
the fixed p o i n t manifold
is
for c e n t r e d v o r t i c e s ,
~2,
but this
set of a circle
case there
arising
÷ eie~ these
k = 2
is no
from p h y s i c a l
which
space
as d e s c r i b e d
of
in §4,
action on a 4 - d i m e n s i o n a l
two actions m u s t
represents
the action)
SO(3)
M
over
leave
in
or gauge
invariant DI'
V2'
as a circle
~3
action,
rotations
is an i n t e r n a l
and ~3 to y i e l d m o m e n t m a p s This
lies,
the
as
hyperkahler
M.
In this action
of
~2
the ~3'
is a circle
and the circle
action.
bundle
and leads
but there
action
Some c o m b i n a t i o n
symplectic
forms
and a m a p
~
(outside
the
to the following
~i'
of ~2
: M ÷ ~3.
fixed p o i n t s
ansatz
of
E43 for such
a metric:
ds 2 = V d x - d x
w here
+ W) 2
grad V = curl The m e t r i c
V,
+ V-I(dT
itself
which because
symmetric either
on
explicitly about
~3.
because
Beyond
or because
boundary
We cannot is one
consequence
a geodesic picture
SU(2)
conditions
on
just on the h a r m o n i c
rotation we have
invariance little data
qualitative
equations
information
introduced
to d e t e r m i n e
in §4
scattering share,
through
ones.
as in [i]
of the m e t r i c
an axially
which move
scattering
solution
collision,
which
studied
is a of the whereby
leads to the
and
off in d i r e c t i o n s
is something
and has been
90 °
symmetric
underqoing a direct
This
but there
which
and the s m o o t h n e s s of
subsequently orthogonal
can be seen d i r e c t l y
by P. Ruback
in the case
of v o r t i c e s .
References
[ i]
M.F.
Atiyah
& N.J.
Hitchin,
abelian m o n o p o l e s ,
Phys.
Low e n e r g y Lett
107A
to
it.
and v o r t i c e s
is the p h e n o m e n o n
V
equations
both m o n o p o l e s
into two p a r t i c l e s
to the o r i g i n a l
V
to d e t e r m i n e
the vortex
about v o r t e x
This
passing
vortex
function
is a x i a l l y
then,
of the e x i s t e n c e
space.
for m o n o p o l e s
depends
we do not have e n o u g h
of two p a r t i c l e s
splitting
this,
say v e r y much,
feature w h i c h
parameter
spatial
we do not k n o w h o w to solve
the g e n e r a l i z e d
impose
therefore
of the
scattering (1985),
of non-
21-25.
124
[2]
M.F. Atiyah & N.J. Hitchin, magnetic monopoles",
[33
S.K. Donaldson, monopoles,
"The geometry and dyDamics of
Princeton University Press
(to appear).
Nahm's equations and the classification of
Commun. Math. Phys. 96
(1984), 387-407.
[4]
G. Gibbons & S.W. Hawking, Gravitational multiinstantons,
[53
G. Gibbons & C. Pope, The positive action conjecture and
Phys. L e t ~ B78
(1978), 430-432.
asymptotically Euclidean metrics Math. Phys. 66 [63
G.H.Halphen, Acad.
[73
in quantum gravity,
Commun.
(1979), 267-290.
Sur un systeme d'equations differentielles,
C.R.
Sci. Paris 92 (1881), 1101-1103.
N.J. Hitchin, Metrics on moduli spaces, in "Contemporary Mathematics",
Volume 58, Part I (1986), American Mathematical
Society, Providence. [8]
N.J. Hitchin, A Karlhede,
U. Lindstrom & M. Ro~ek, Hyperk~hler
metrics and supersymmetry, [9]
A. Jaffe & C.H. Taubes, Boston
[i03
"Vortices and monopoles",
(to appear). Birkhauser,
(1980).
N.S. Manton, A remark on the scattering of BPS monopoles, Lett. ii0 B
[ii]
Commun. Math. Phys.
Phys.
(1982), 54-56.
H.J. de Vega & F.A. Schaposnik,
Phys. Rev. DI4
(1976), ii00.
The A m b i t w i s t o r
Program
J ames
Isenberg
Dept.
of M a t h e m a t i c s
University Eugene,
of O r e g o n
OR
97403
USA
The goal of the
of the a m b i t w i s t o r
fundamental
constructions correspond
on a m b i t w i s t o r
of analysis,
some
which
The a m b i t w i s t o r programme arising
program
twistorial
on two major
geometric
structures
geodesics), analytic)
cases
closely
parallels
However,
analyses
as well.
fields
with
ambitwistor
the twistorial with
of s e l f - d u a l
spaces
allows
research
parallel
Ambitwistor
approach
it s h o u l d
be noted,
out.
the hope
and
both
some
of
indicate
is heading. of the twistor
sense,
in both p r o g r a m s
on c o n f o r m a l l y planes
representation
invariant
and null
on h o l o m o r p h i c
More p a r t i c u l a r l y ,
(i.e.
complex
as we shall
of certain
see,
fields
representation. twistors
usually
spaces
has a p o t e n t i a l l y
spaces,
to more
to the
the use of
readily
Hence
Currently, to carry
to examine
these a n a l y s e s
handle
the
application.
begin
one
(non self-dual)
are more d i f f i c u l t
if m a t h e m a t i c i a n s
ambitwistorial
fields,
spaces.
w i de r
analyses
restricts
general
also seem
than do twistor
is that
the basic
we discuss
those used
null
to represent
ambitwistor
insights,
ideas and t e c h n i q u e s
In a b r o a d
(e.g.,
in this
to new t e c h n i q u e s
an o u t g r o w t h
of the
points
is that
to date,
and a n t i s e l f - d u a ]
one
supersymmetry
sketch,
(i) Focus
analysis.
while w o r k i n g
fields
will
more become
and more practical.
Before represent spaces
many
analyses.
the a m b i t w i s t o r i a l
representation ambitwistor
But
current
(2) Rely as much as p o s s i b l e
in many
carefully
accessible
is of course
themes:
The hope
has a c h i e v e d
in s p a c e t i m e
mathematical
are spaces w h o s e
thus be led to new
Indeed,
in a m b i t w i s t o r i a l
are b a s e d
simpler
in w h i c h
of Penrose. [I]
corresponding
become
In this brief
the p r o g r a m
of the d i r e c t i o n s
which
a representation
of g e o m e t r i c a l
in spacetime.
one will
and m a t h e m a t i c a l .
is to o b t a i n
in terms
spaces,
the fields will
and perhaps
the s u c c e s s e s
program
fields
to null g e o d e s i c s
n e w formulation,
physical
physical
discussing
fields,
h o w to use a m b i t w i s t o r i a l
we shall
themselves.
of the p r o g r a m - - t h e
Then
briefly
in Sec.
techniques
describe
(in Sec.
2, we shall
discuss
representation
of Y a n g - M i l l s
to
I) the basic the major
fields.
success
We follow
126
this
(in Sec 3) by c o n s i d e r i n g
research.
Finally
ambitwistorial
in Sec.
techniques
gravity,
space
of the groups
itself
SL(4,C)
of the conformal
of
s p a c e t i m e [2] CM, w h i l e of
~M).
B a s e d on T, one defines [3]
Twistor
Twistor T
real Minkowski
modulo
complex
PT
of each
is the 4-fold
cover
complex M i n k o w s k i cover
spacetime
of the conformal
M
(sitting
inside
conformal
scale
factors}
Space:
= {linear maps W:T ~
Projective
SL(4,C)
action
Space:
PT = {twistors, p3(¢)
Dual
on T.
is the 4-fold
group
Projective
for
The Spaces
(compactified)
SU(2,2)
(compactified)
a focus at
role
and superstrings.
is T = ~4, w i t h a s p e c i f i e d
and SU(2,2)
group
is p r e s e n t l y
on a p o s s i b l e
in the s t u d y of strings
i. Twistor
which
4, w e - c o m m e n t
Dual
Twistor
= {dual
t:Z~
<.WIZ>)
Space:
twistors,
modulo
complex
conformal scale
factors}
=. p3(¢) Ambitwistor
Space:
A = ((Z,W)
~ T x T
7 dimensional
Projective
quadric
Ambitwistor
PA = {([Z],[W])
describing objects
fields
it appears points
Z =
quadric
x AA'
(-A.I .. J
= 0}
in 8 4 x ~4
such
in s p a c e t i m e
that
in p3(~)
to the u t i l i t y
in ~M and g e o m e t r i c
are e a s i l y d e r i v e d
<WIZ>
Space
~ FT x PT
5 dimensional
One of the keys
such that
<[W]I[Z]> x p3(z)
of these various
is the c o r r e s p o n d e n c e
objects
= O}
in these spaces.
spaces between These
in geometric results
a l g e b r a i c a l l y [4] from a certain spinorial e q u a t i o n - AA' A in the form ix "A' = ~ -- r e l a t i n g
in the literature in ~M and
twistors
We s k e t c h
the important
ones
in the following
table|
127 CM
PT
point
PT
projective line
projective
[PI(C)]
null g e o d e s i c
PA line
projective
[PllC)]
-
-
p r o j e c t i v e plane
point
[PI(c)] x
line [PI(C)
point
line
self-dual
p r o j e c t i v e plane
[P2(G)]
null plane
[p2(c)]
("~-plane")
anti-self-dual
point
p r o j e c t i v e plane
p r o j e c t i v e plane
[p2(c)]
plane
[p2(c)]
("~-plane")
In this table, we find the key result that PA ~ (null g e o d e s i c s (Note that for convenience, the s u b t l e t i e s p o i n t e d out,
here and throughout
however,
rather w i t h PT ~ p3(~)
_ pl(~),
are g e n e r a l l y not compact.
(such as for the
or some s u b s p a c e thereof. we shall
(noncompact),
space.}
we ignore
It should be
one works not w i t h PT ~ p3(~) but
Similarly,
to denote all of, or p o r t i o n s complexified Minkokski
(I)
that for most a p p l i c a t i o n s
c o r r e s p o n d e n c e s d e s c r i b e d below)
Twistor and A m b i t w i s t o r
Such spaces
(ambiguously)
use " ~ M "
of c o m p a c t i f i e d
In all of these spaces,
s t r u c t u r e s are c o m p l e x analytic,
2.
this paper,
r e g a r d i n g points and lines "at m"
in ~M}
the important
rather than Hermitian.
Representations
of Yang-Mill_ss
Fields Perhaps the most programme
important success thus far of the twistor
(as well of its a m b i t w l s t o r offshoot)
of Y a n g - M i l l s
fields.
Yang-Mills
is the r e p r e s e n t a t i o n
fields, we recall,
fields A on s p a c e t i m e w h i c h satisy the Y a n g - M i l l s
are G - c o n n e c t i o n
field e q u a t i o n s
D ~ F = 0, w h e r e F is the curvature of A, D is the exterior covariant d e r i v a t i v e based on the c o n n e c t i o n A, * some Lie group. the c o n d i t i o n
m F = F) or a n t i s e l f - d u a l
follows from the Bianchi
identity
For these,
(satisfying
(satisfying
* F = -F) then it
(DF = O) that the Y a n g - M i l l s
e q u a t i o n is a u t o m a t i c a l l y satisfied. dual c o n n e c t i o n s
is the Hodge dual, and G is
W h e n a c o n n e c t i o n field A is self-dual
Hence self-dual and antiself-
form a special subclass of the Y a n g - M i l l s solutions.
one has the basic Ward T h e o r e m [5] w h i c h establishes
the
128
following one-one Self-dual
correspondences:
Connections
A}
f
on ~M
<
@n Vector Bundles
'> I ~[Tr~vial
over PT on all PI(C)
E
t subspaces]]
(I)
and
e
0onne }on on
-- {
Vcor un
[Trivial
on all P1(@)
Note that the dimension
n of these vector bundles
that of the Lie algebra
in which the Yang-Mills
These correspondences, holomorphic
functions,
SU(n))
fields
involve connections
GL(n,~).
be real
on an S 4 subspace
E corresponds
Remarkably,
(e.g.,
of @M can be readily
of
which take values
however,
taking values
to
take values.
which rely strongly on properties
generally
in complex groups--e.g., that the connection
subspaces]/
the condition
in SL(n,~),
incorporated
or in
into the
twistor side of these correspondences. admits a symplectic
In particular, if the bundle E 2 structure ~:E * E (with ~ = -i) which corresponds
to a lift of the reality structure p on PT determines (self-dual)
(that map p which
real and imaginary parts of PT), Yang-Mills
connection
then the corresponding
is real on S 4.
Schematically,
the
result [6] takes the form
'Self-dual
SU(n)
Connections
on S 4
A1 ;
( <
C n Vector Bundles
> ~
E
over PT* !
with instanton number k
![Trivial I
As remarkable vector bundles
Consequently,
they correspond
are also well understood,
constructed.
and other self-dual
out to be very important quantum
I ]
is the fact that the ~n
the special Yang-Mills
These are the Yang-Mills
Instantons
with symplectic ~
on the right hand side of
understood. [7]
subspaces]
and second Chern number k
as this correspondence,
E described
on all pl(~)
(3) are wellfields to which
and can be explicitly
instantons
Yang-Mills
(on S4).
connections
have turned
in mathematics, [8] and may play a role in
field theory calculations
as well. [9]
However,
one wishes
to
129
also u n d e r s t a n d
Yang-Mills
antiself-dual)
fields w h i c h
restriction.
the s t u d y of such u n r e s t r i c t e d if one c o n s i d e r s over
PT and PT
of GL(n)
in general solve
holomorphic as in
connections neither
(2),
complex
self-dual
the Y a n g - M i l l s
s p a c e was
Yang-Mills
vector
(I) and over
have no such
Ambitwistor
connections.
bundles
One
then a g a i n Minkowski
or
to aid
finds
over PA a n a l o g o u s
in
that
to those
one has a r e p r e s e n t a t i o n
space.
nor a n t i s e l f - d u a l ;
equations.
(self-dual
introduced
But now they are
nor do they g e n e r a l l y
S c h e m a t i ca l l y ,
one has [I0]
{A11Connectons}{nvectorBundle } over
~M
>
over PA [Trivial
To u n d e r s t a n d connections
while
w i t h no such exactly Now
why
those w h i c h
are
s pace
PT
(therefore
, or PA)
the c o n n e c t i o n s
are
planes
(4),
in that
in S e c t i o n
on ~M are
one
flat over
in ~M.
finds
that
twistor-type
the g e o m e t r i c
twistor-type
I, we see that b u n d l e s
flat over a n t i s e l f - d u a l
are self-dual),
which
null
and
flat over
to points
the table
to c o n n e c t i o n s
over a g i v e n
are n e c e s s a r i l y
correspond
to s e l f - d u a l
connections
(2),
to bundles
to c o n n e c t i o n s
connections
(I),
subspaces]
correspond
flat on all a n t i s e l f - d u a l
recalling
over PT c o r r e s p o n d
determine
bundles
(4)
x pl(¢)
correspond
that s e l f - d u a l
correspond
in ~M w h i c h
Hence,
note
the c o r r e s p o n d e n c e s which
(e.g., PT,
bundles
the a m b i t w i s t o r i a l
the c o n n e c t i o n s
space.
the twistorial
restriction,
in v e r i f y i n g
structures
on all PI(c)
while null
bundles
lines
null
planes
over PA
(therefore
no
restriction). As n o t e d
above,
the c o n n e c t i o n s Yang-Mills
as a q u a d r i c
But
(codimension-one
su(2)
Among
extend one
satisfies
antiself-dual,
satisfy
the Y a n g - M i l l s
bundles
over PA w h i c h
over
subset
determined
the c o l l e c t i o n
finds,
a number
the into
that PA sits
by an a l g e b r a i c
of all b u n d l e s
of other
over PA,
The c o n n e c t i o n s
conditions
[If,
equations. w h i c h are
for example,
it is n e c e s s a r i l y
To o b t a i n
and n o t h i n g
are e x t e n d i b l e
satisfy
equations
the Y a n g - M i l l s
these conditions, or Abelian].
restriction,
Recall
over PT x PT
satisfy
to be of interest.
equations
PA needn't
the Y a n g - M i l l s
of c o n n e c t i o n s ?
to bundles
they s a t i s f y
too r e s t r i c t i v e
connection
self-dual,
.
to these,
in addition,
generally
to bundles
representation
there are some w h i c h
free of any s e l f - d u a l
How do we b u i l d
in PT x PT
corresponding
being
corresponding
equations.
the a m b i t w i s t o r i a l
condition)
besides
conections
stronger,
to PT x PT" only
we
an either
which
look at
to third order
130
in Taylor series.
That is, the transition
functions which define the
bundle over PA may be expanded parameter
in a Taylor series involving a * 3 to PAc--)PT x PT ; but only to order s are
"s" transverse
the expansion
terms consistent
satisfied in a neighborhood have[10, 11 ]
Yang-Mills Connections on
with the cocycle conditions
of PA in PT x PT .
I A
l[Trivial
on all PI(~)
x Pl(¢)
[3rd order extendible This correspondence Yang-Mills self-dual
could be as useful
fields.
representation
as is PT[~
two other representations
The first of these,
step of our proof of c o r r e s p o n d e n c e
metric).
of
however,
of Yang-
which are closely related to the a m b i t w i s t o r i a l one
and focuses on complex Minkowski dimensional
subspaces]
This has not been the case,
We wish to briefly describe just discussed.
(5)
to PT x PT*]
because PA is not as familiar a space to m a t h e m a t i c i a n s p3(¢1 _ pl(¢)].
Mills connections
I
in studying nonself-dual
solutions as has been the Atiyah-Ward Yang-Mills
form, we
~n Vect°r Bundles ~ over PA
<----->
CM
being
In schematic
diagonal
which appears as an intermediate (5) [10], eschews
twistor-like
spaces
space ~M embedded as a 4 complex
in @M 8 (which is just ~8 with an orthogonal
Let us assume
that an orthogonal
split of ~M 8 has been
chosen so that we have ~M 8 : ~M~- x CM~- ["Physical[' complex Minkowski space,
~M,
is everywhere
We define a connection to be bidual satisfies 8F~
We then discover connection
as in the diagram].
~M:I//
~M~
8F
the conditions (where "3" indicates
TCM~)) 8F~ ~ is antiself-dual ab TCM~) 8F~ = 0 ab
order
to this split,
8A on CM 8
if its curvature
is self-dual
satisfies
transverse
(where "a" indicates
that a connection
the Yang-Mills is extendible
in a parameter
restriction
equations
restriction
4A on the diagonal on the diagonal
to a connection
transverse
to ~M,
to vectors
in
to vectors
spacetime
in
~M
if and only if this
8A on ~M 8, and that to first
the connection
8A is bidual.
131
This ~M r
> ~M s r e p r e s e n t a t i o n of Y a n g - M i l l s
our schemes
fields m o t i v a t e s some of
for trying to find an a m b i t w i s t o r r e p r e s e n t a t i o n of
gravitational
field equations,
as d i s c u s s e d b e l o w
Another r e p r e s e n t a t i o n of Y a n g - M i l l s the a m b i t w i s t o r i a l
correspondence
(in Section 3).
fields,
closely related to
(5), is that w h i c h W i t t e n has
d e v e l o p e d using a s u p e r s y m m e t r i c v e r s i o n of PA.
Recall
that one may
regard s t a n d a r d p r o j e c t i v e a m b i t w i s t o r space as the c o l l e c t i o n of null geodesics
in complex M i n k o w s k i
space ~M.
s t a t i o n a r y points of the action massless
These g e o d e s i c s are the dx ~ dx~ (gMv dT d-~) for the m o t i o n of a
dT
free p a r t i c l e w i t h t r a j e c t o r y XM(~)
metric gMv'
in CM w i t h orthogonal
Based on this i n t e r p r e t a t i o n of PA, one may define
(following W i t t e n [Ill) a g e n e r a l i z e d a m b i t w i s t o r
space PA[n,s ]
c o n s i s t i n g of the t r a j e c t o r i e s of free m a s s l e s s s u p e r s y m m e t r i c p a r t i c l e s m o v i n g in a super
(complex) M i n k o w s k i
space ~M[n,s ] w i t h n
c o m m u t i n g d i m e n s i o n s and s super ones. W i t t e n has s t u d i e d two such s p a c e s - - P A [ 4 , 1 2 ]
and P A [ 1 0 , 1 6 ] - - a n d
found that they both lead to interesting r e p r e s e n t a t i o n s of Y a n g - M i l l s fields. [11'12]
A p p r o p r i a t e vector bundles over PA[4,12]
N = 3 super Y a n g - M i l l s s o l u t i o n s on @M[4,12],
c o r r e s p o n d to
while appropriate
bundles over P A l 1 0 , 1 6 ] c o r r e s p o n d to N = I super Y a n g - M i l l s s o l u t i o n s on ~M[Io,16 ].
The first is n o t e w o r t h y because the N = 3 super Yang-
Mills c o n n e c t i o n s nonself-dual)
induce s t a n d a r d Y a n g - M i l l s
on ~M 4.
connections
(generally
So we have a r e p r e s e n t a t i o n of Y a n g - M i l l e
fields w h i c h avoids any explicit s t i p u l a t i o n s of bundle extendibility. The second is n o t e w o r t h y because the recent p o p u l a r i t y of s u p e r s t r i n g theory has led to interest
in s u p e r s p a c e t i m e s with I0 real dimensions.
How do these superambitwistorial
r e p r e s e n t a t i o n s of Y a n g - M i l l s
fields avoid bundle e x t e n d i b i l i t y r e q u i r e m e n t s ?
Recall,
in these t w i s t o r - t y p e r e p r e s e n t a t i o n s of connections, guarantees
again,
that
the c o n s t r u c t i o n
that the c o n n e c t i o n s will be flat over the s t r u c t u r e s
in
s p a c e t i m e w h i c h c o r r e s p o n d to points in the t w i s t o r - t y p e space. the points null
in PA[n,s ] (for s # 0), though often r e f e r r e d to as "super
lines",
are m u l t i d i m e n s i o n a l
number of s u p e r d i m e n s i o n s ) .
objects
PAr4,121.. and P A [ 1 0 , 1 6 ] - - t h i s
requirement
e x t e n d i b i l i t y c o n d i t i o n is s m u g g l e d
supermanifolds
that a c o n n e c t i o n be flat
In the cases m e n t i o n e d above--
It should be noted that,
representation.
(one n o n s u p e r d i m e n s i o n plus a
The requirement
on these is t h e r e f o r e nonvacuous.
equations.
Now
imposes the Y a n g - M i l l s
in a certain sense,
the
into the s u p e r a m b i t w i s t o r
One sees this w h e n one relates bundles over to bundles over o r d i n a r y manifolds.
field
132
3.
Twistor and A m b i t w J s t o r R e p r e s e n t a t i o n of G r a v i t a t i o n a l Fields The g e o m e t r y of complex Minkowski space is built into the
s t r u c t u r e of p r o j e c t i v e twistor space PT and p r o j e c t i v e a m b i t w i s t o r space PA. general
Hence PT and PA are not themselves useful
(curved)
spacetimes.
As shown by Penrose,
d e f o r m a t i o n s [13] of these spaces are in fact useful Specifically,
for r e p r e s e n t i n g
however, for this task.
he shows [14] that d e f o r m a t i o n s of PT w h i c h p r e s e r v e
f i b r a t i o n over pl(~) and also p r e s e r v e a certain
(deformed)
its
vertical
two form M can be used to represent self-dual s p a c e t i m e s w h i c h are d e f o r m a t i o n s of complex Minkowski space.
[A self-dual s p a c e t i m e is
one for w h i c h the Ricci curvature vanishes and the Weyl curvature self-dual].
Schematically,
s°,dooSpaoet,me I f
i
<""+'g)
t <->
[Deformation of CM])
1 I
Bond e°
I
l(c) w i t h vertical
2-form F
PT w i t h lX [Deformation of pl(¢) These self-dual spacetimes,
Penrose,
admit
as "=-planes"].
Indeed,
@]
called "nonlinear gravitons"
flat self-dual n u l l p l a n e s
corresponding space~
is
this c o r r e s p o n d e n c e may be w r i t t e n as
by
[referred to in the literature
for a given n o n l i n e a r g r a v i t o n spacetime,
the
is e s s e n t i a l l y the c o l l e c t i o n of these =-planes,
and some of the g e o m e t r i c c o r r e s p o n d e n c e s w h i c h hold between PT and ~M survive: together
For example,
the p r e s e r v e d fibratJon of ~ over pl(~)
(in fibres) all parallel = - p l a n e s
this fibration c o r r e s p o n d to points i n ~ 4. do not g e n e r a l l y exist flat a n t i s e l f - d u a l
jn~4;
ties
and the sections of
On the other hand, null planes
there
("~-planes")
in
~]~4, and this is r e f l e c t e d in the structure of any n o n t r J v i a ] l y deformed~.
Note that,
roughly speaking, ~ carries the i n f o r m a t i o n
r e g a r d i n g the conformal scale of the s p a c e t i m e In a s p a c e t i m e w i t h nonself-dua] = or ~ null planes.
(~4,g).
curvature,
there are g e n e r a l l y no
However there are always g e o d e s i c curves,
and so
one is led to consider a m b i t w i s t o r l i k e spaces for r e p r e s e n t i n g general curved spacetimes.
Y a s s k i n and I [15] have chosen to consider
d e f o r m a t i o n s of a m b i t w i s t o r space which, characteristic
pl(~).
foliation
(for PA,
like ~ , p r e s e r v e the 1 this foliation is over P (~) x
Our d e f o r m a t i o n s also preserve a certain c o l l e c t i o n of
"contact forms", which we label as
{~p).
We get
133
rTeleparallel
I
Spacetimes ~
Bundles
(~4,g,v)
pi(¢)
[Deformations
o f CM]
with contact [Deformations
PI(C)
x
forms
{~p}
of PA with ~p]
pl(~) ~ p1(@)
Note that a teleparallel compatible generally
spacetime
is one which has a metric
connection whose curvature
vanishes,
torsion
does not vanish.
We seem to be guilty of false advertising have a c o r r e s p o n d e n c e
for the c o n f i g u r a t i o n
geometries
(presumed torsion-free);
spacetimes
with teleparallel
(parallelizable) arbitrary geometry
but whose
torsion
('~L4,g,v):
{e } for {~%4,g}
free spacetime
geometry
can be represented
is for
(~4,g)
with
by a teleparallel
One simply chooses an orthonormal
frame field for which
The relation between the torsion of v and the
of g [more properly,
connection
to
But in fact any
and then defines v to be that connection
{e } is parallel. curvature
We c/aimed
instead our c o r r e s p o n d e n c e
geometries.
Riemann curvature
here:
space of all spacetime
the curvature
R of the Levi-Civita
for g] is then given by
R t~m nab = Va km nb _ V b k m n a
+ kmpbkPna
_ k m pa kp nb
where kabm = ~ (Qmb a + Qbma - Qamb), and Qa bm is the torsion. Going the other way, a teleparallel geometry determines a torsion-free geometry:
One simply
forgets v and works with the Levi-Civita
connection
(metric-compatible,
torsion-free).
many-to-one,
with the class of teleparallel
torsion-free
geometry parametrized
fields
for the given metric.
correspondence
{spacetimes
(~4,g)}
deformations
of ambitwistor
space,
solutions
of Einstein's
general
deformations
form but g e n e r a l l y guarantees
and we believe However,
Eastwood,
that they will not
correspondence
one may consider
Baston,
other
geometries.
a global
Their global
Our fibration guarantees
for
and Mason. [16]
of PA which preserve
lose the fibration.
no torsion.
From our perspective,
and well behaved class of
space w h i c h avoid teleparallel
This is the program of LeBrun, consider
connections?
a Penrose-type
equations.
of a m b i t w i s t o r
frame
of the
> {~,{~p}}.
of a w e l l - m o t i v a t e d
get in the way of our obtaining deformations
by the set of orthonormal <
is
for a given
This is the "gauge freedom"
Why bother with teleparallel they are an artifact
The relationship
geometries
contact
They
contact form
teleparallelism.
134
(Each fibre is a collection of parallel null geodesics). As of yet,
there is no firm result concerning the incorporation
of the Einstein equations field equations)
(or any alternative set of gravitational
into the structure of deformed ambitwistor spaces
(either of the sort considered by Yasskin and me, or the sort studied by LeBrun,
Eastwood,
Baston, and Mason).
led us to a plausible conjecture, deformations of PT x PT
*
The Yang-Mills results have
however.
p1(~) along with certain other structures preserved in the nonlinear g r a v i t o n ~ .) generically~P.
The idea is to consider pl (¢) x
which preserve the fibration over
(analogous to the two-form M Let us call these spaces
Then, one conjectures that a given spacetime will
satisfy the gravitational
field equations if and only if the
corresponding deformed ambitwistor s p a c e ~
is embeddible--perhaps
only
to certain order--in some ~ ~p Testing this conjecture directly is formidable. chosen to study it indirectly,
via an intermediary.
Hence we have Recall that a G-
connection on complex Minkowski space satisfies the Yang-MJlls equations
if and only if when this connection
is extended into ~M 8, it
is bidual to second order in the extension parameter Now let us consider a spacetime
[see section 2].
(~Ti4,g,v) and assume it embeds into
some eight complex dimensional space 9~18 with metric 8g and connection 8v.
We call such a spacetime
(~8,8g,8v)
"bidual"
if and only if
(I)
it admits a pair of four complex dimensional distributions TL°8%Band TRr/~ (not necessarily integrable) which split the tangent space at each 8 8 and (2) the metric g and connection v split orthogonally
point;
relative T,~@and~ TR~918~ with the curvature and torsionebeing selfdual relative to TL@Yi~and antiself-dual relative to T~)%~at~ each point.
(These conditions are described more precisely elsewhere[I7]).
Requiring that ( ~ 4 , g , v ) very strong condition. requirement
embed in a bidual spacetime is presumably a Our conjecture is that the appropriate
is that (#~4,g,v)
bidual only (to some order)
embeds in some (@118,8g,8v)
which is
in a neighborhood of (9~14,g,v).
We also
conjecture that such embeddibility occurs if and only if the corresponding ~q can be embedded
~.
(to some corresponding order)
in some
Note that Yasskin and I have proven that there is a one-one
correspondence between bitwistor spaces. ~
4. To date,
~
and bidual spacetimes
Ambitwistors and Strinqs
the suspicion that ambitwistorial
(or twistorial)
techniques might be of some use in studying superstring theory is
135
largely based upon two sets of interesting results: Witten [12]
(and as noted above),
space PA[Io,16 ] . .
corresponding
and sixteen superdimensions; space.
there is a well-defined ambitwistor
to super Minkowski
space with ten real
and one can use bundles over PAl10,16 ] . .
represent connections satisfying super Minkowski
i) As shown by
the super Yang-Mills
to
equations over
2) Developing and extending work by
Weirstrauss [19], Shaw [20] has shown that solutions of the classical string field equations
[i.e., minimal surfaces]
in three,
four, six, and
ten dimensions may be represented by pairs of curves in generalized twistor spaces. Either of these results may well lead to fruitful studies of superstring theories. [21] different space.
The approach I prefer,
however involves a
Recalling the definition of PArn,s ]..
geodesic trajectories of massless particles
in terms of null
in super Minkowski space,
we define
B[n,s ]
:= {world sheets of strings in CM[n,s]}
[A similar space may be defined for worldsheets of strings in nonflat spacetimes.]
The properties of these infinite dimensional
far from understood. structure,
spaces are
The hope is that in studying their holomorphic
and perhaps also in examning certain holomorphic bundles
over B[n,s ], one might obtain some insight regarding either the full (all-energy)
superstring theory,
gravitational
and Yang-Mills
or the low energy theory
fields).
(manifest as
The former study could involve
studying appropriately defined "string instantons".
The latter may
involve establishing relationships between B[n,s ] andPA[n,s ]. This is of course just speculaton. ambitwistorial
I believe,
however,
that
ideas could play an interesting role in the development
of superstring theory. Acknowledgements: I thank Professors N. Sanchez and H. deVega for inviting me to speak at their seminar in Meudon. References I.
Comprehensive reviews of twistor theory appear in:Penrose, R. and Ward, R.S., "Twistors for Flat and Curved S p a c e t i m ~ i n General Relativity and Gravitation (ed. A. Held), Plenum, 1980~ Hughston, L.P. and Ward, R.S., Advances in Twistor Theory, Pitman, 19791 Wells, R.O., Complex Geometry in Mathematical Physics, SMS #78, Les Presses de l'Univ, de Montreal, 1982; Penrose, R., and Rindler, W., Spinors and Space-tim e (2 volumes)
136
Cambridge, 2. 3.
4. 5. 6. 7.
8.
9. I0.
11. 12. 13.
14.
15. 16.
17. 18. 19. 20.
21.
1984-1985~
Hugger,
S.A.,
and Tod, K . P . ,
A__nn
I n t r o d u c t i o n to Twistor Theory, Cambridge, 1985. For a d i s c u s s i o n of ~M, see Penrose and Rindler, ref. [1]. These spaces, as well as others w h i c h p l a y an important ~ole in the twistor programme, are all "flag manifolds" of T = C-. See Wells, R., Complex G e o m e t r y in M a t h e m a t i c a l p h y s i c s , Les Presses de L ' U n i v e r s i t e de Montreal, 1982. See references above, or Newman, E.T. and Hansen, R., Gen. Rel, Gray. 6, 361, 1975. Ward, R., Phys. Lett. A6!, 81, 1977. Atiyah, M., and Ward, R., Comm. Math. Phys. 5__99, 117, 1977. Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., and Manin, Yu. I., Phys. Lett. A65, 185, 1978. Atiyah, M.F., G e o m e t r y of Y a n ~ - M i l l s Fields, Scuola Normale Superiore, 1979. We refer e s p e c i a l l y to the work on "Fake R4s ". See Donaldson, S., ~ Diff. Geom. 18, 269, 1 9 8 3 ; ~ F r e e d , D., and Uhlenbeck, K., Instantons and Four-Manifolds, Springer, 1984. Coleman, S., "The Uses of Instantons", lecture notes from 1977 International School of Subnuclear Physics, 1977. Isenberg, J., Yasskin, P.B., and GreeD, P., Phys. Left. 78B, 462, 1978. Isenberg, J. and Y a s s k i n P.B., "Twistor D e s c r i p t i o n of N o n - S e l f - D u a l Y a n g - M i l l s Fields, in Complex M a n ~ f o l d T e c h n i q u e s in Theoretical Physic s (eds. D. Lerner and P. Sommers);P,*m~,z~7~. Witten, E., Phys. Lett. 77B, 394, 1978. Witten, E., Nucl. Phys., B266, 245, 1986. A "deformation" of a complex m a n i f o l d may be u n d e r s t o o d as a s m o o t h l y p a r a m e t r i z e d set of t r a n s f o r m a t i o n s of the complex t r a n s i t i o n functions (on patch overlaps) w h i c h define the h o l o m o r p h i c s t r u c t u r e of the manifold. The p a r a m t r i z e d set must include the identity t r a n s f o r m a t i o n . Penrose, R., Gen. Rel. Grav. 7, 31, 1976. Curtis, W.D., Lerner, P.F., and Miller, F.R., Sen. Rel. Grav. 10, 557, 1976, Hitchin, N.J., Math. Proc. Camb. Phil. Soc. 85, 465, 1979. Yasskin, P.B., and Isenberg, J., Gen. Re/. Grav. 14, 621, 1982. LeBrun, C., Trans AMS 278, 2 0 8 1 9 8 3 , Eastwood, M., Twistor N e w s l e t t e r 17, 1983. BQsten, R., and Mason, L., Twistor N e w s l e t t e r 21, 1986. Isenberg, J., and Yasskin, P., Twistor N e w s l e t t e r 22, 1986. Unpublished. Weirstrauss, K., Monats, B e r l i n e r Akad., 612, 1866. Shaw, W., Class and Qtm Gray 2, 113, 1985. Shaw, W., to appear in M a t h e m a t i c s in General Relativity, (ed., J. Isenberg) AMS Contemp Math. It w o u l d be interesting if someone were to a p p l y some of S e g a l ' s ideas on q u a n t u m field to the space of classical s t r i n g field s o l u t i o n s w h i c h arises in Shaw's work. &ee Segal, I., J. Math. Phys. I, 468, 1959.
S U P E R S Y M M E T R I C E X T E N S I O N OF T W I S T O R F O R M A L I S M
J. L u k i e r s k i I n t e r n a t i o n a l Centre for T h e o r e t i c a l Physics, 34100 Trieste,
Italy
Contents I. I n t r o d u c t i o n 2. D=4 s u p e r t w i s t o r s 3. D=4 SUSY a m b i t w i s t o r s 4. D>4 5. Final remarks
I. I n t r o d u c t i o n
In the twistor p r o g r a m m e of replacing D=4 Minkowski metry by more e l e m e n t a r y complex g e o m e t r y of twistors
space-time geo-
(see [I]) one can
d i s t i n g u i s h two approaches: a) one introduces totally null r-planes as p r i m a r y geometric objects in complex M i n k o w s k i
space
(CM). These n u l l - p l a n e s in CM, called ~-pla-
nes, are d e s c r i b e d by single p r o j e c t i v e
point in PT
~
twistors
(PT), i.e.
s-plane in CM
(A)
If we hold the point in CM, and vary the
(nonprojective) twistors
(T),
we get the linear 2-space in T, i.e.
2-plane in T
point in CM
(B)
(line in PT)
The c o r r e s p o n d e n c e
(A) describes
the twistor transform.
i) The d e s c r i p t i o n of YM e q u a t i o n s
c o n d i t i o n s on s-planes, w h i c h select only self-dual tions
(see e.g.
It appears that
in PT implies the i n t e g r a b i l i t y (instanton
solu-
[2-3]).
*)On leave of absence from the Institute for T h e o r e t i c a l Phys±cs, v e r s i t y of Wroclaw, ul. C y b u l s k i e g o 36, 50-205 Wroclaw, Poland
Uni-
138
ii)
the g e n e r a l i z a t i o n
to general
4-dimensional
on its conformal -dual.
However,
analytic ral
so l u t i o n
there were
b) one
in order
introduces
we interprete the SU(2,2) space
ZA6T
the Weyl efforts
M imposes
tensor
to g e n e r a l i z e
null
lines
(A=I...4)
scalar
Ambitwistor
where by w A c T
product
space:
we d e n o t e d dual
and r e s p e c t i v e l y
dulo complex
point
scale
in PA
gene-
existing
geometric
spinors,
,
field
theories objects. If
one can introduce
and define
the a m b i t w i s -
PA=(z,w) 6 PT×PT
Because
i.e.
complex
null
(C) describes
metrics
(without and with
line
T~C
:
(PT) are d e f i n e d mo-
(c)
in CM
complex
transform.
lines
is less
a l l o w to d e s c r i b e
stringent
nonself-dual
[6,7]
and with
sources
relativity
[10,11].
It appears
characterizing
the solutions
are complicated,
of flat a m b i t w i s t o r
linear maps
twistors
the a m b i t w i s t o r along
of general
sources)
defining
dual
YM fields w i t h o u t
that the conditions
(1 .1)
= 0
show that
the a m b i t w i s t o r s
general
non-self-dual
formation
One can
the i n t e g r a b i l i t y
than on s-planes,
twistors
projective
factor.
~
The r e l a t i o n
ver,
of complex-
able to describe
in CM as p r i m a r y
as conformal
tions,
the class
self-
as follows:
Projective
~
space
restrictions
equations.
to be able to describe
- invariant
from flat severe
should be e s s e n t i a l l y
of ~-planes I) , none were
of E i n s t e i n
Therefore
tor
geometry:
deformations
of the notion of s-planes
complex m a n i f o l d
space w h i c h
solu-
[8,9], and howe-
of Y M e q u a t i o n s
and the r e s t r i c t i o n
describes Einstein
on de-
equations
is even not known. Both a p p r o a c h e s Minkowski c onside r
described
space-time.
the i n t e g r a b i l i t y
c) the
above
require
In SUSY theories
lines
(light-like
to
of in real M i n k o w s k i
if we put in PA z=w, we obtain
the space of null
In p a r t i c u l a r
denoted by N, or in p r o j e c t i v e
points
of
useful
rays)
twistors
submanifold
extension
however
on the SUSY g e n e r a l i z a t i o n
space of real null
space M.
such
the complex
it appears
of PA describes
in PN
version
real null
lines
real null
PN.
One can show that
in M,
lines
i.e.
in M
(D)
(null twistors)
Because
the scalar p a r t i c l e s
in m a s s l e s s
limit p r o p a g a t e
along
the
139
light rays,
one can also
orbits
show that the phase
free m a s s l e s s
scalar
tot
[12]).
(see e.g. can
plexified)
sless
[6];
were
geometry
also W i t t e n
with
tion of real
null
by the
fermionic
contains
(geodesic) fermionic
SUSY m a s s l e s s
[14].
orbits
which
theories
null
lines
line.
does
null
dimensions. there have
lines,
this
which
between
null
twistor
gauge
invariance,
the SUSY e x t e n s i o n
by
SUSY
for masobserved
of the no-
(E) in SUSY case
is sup-
twistor
fermionic
sector
real null
line
formalism
to s u p e r s y m m e t r i c
explanation
along
of SUSY YM and super-
real
along
on c o n v e n t i o n a l
is changed
are s u p p l e m e n t e d
integrability
constraints,
The i n t e g r a b i l i t y
any r e s t r i c t i o n s conclusion
(E')
of superspace
of the f o r m u l a t i o n
of SUSY
with
SUSY
if we
real
YM and
introduce
additional
- extended
SUSY
fermionic
null
lines
been d e r i v e d
SUSY YM c o n s t r a i n t s 3)
ii)
N=2
D=6
[18]
and N=I
D=I0
[14]
SUSY YM c o n s t r a i n t s
iii)
N=I
D=4
[16]
and N=I
D=I0
[14]
SUGRA c o n s t r a i n t s
supersymmetric
A) was c o n s i d e r e d
sively
(com-
of m a s s l e s s
D=4
and its
formalism
and
that the action
N=I,2,3
tions
twis-
counterparts.
i)
The
of
and s u p e r - a m b i t w i s t o r s
The r e l a t i o n
in superspace.
not p r o v i d e
Using
by a single
the d e s c r i p t i o n
It appears
for
feature
theories
theories 2) , but
-extended
(E)
sector.
is a g e o m e t r i c
(SUGRA)
gravity
in M
supersymmetric
[13],
particles
are a p e c u l i a r
gravity
(A-E)
their
describes
One of the aims of e x t e n d i n g (SUSY)
as
(i.e. all observables)
the fermionic
[15], w h i c h
gauge
have
by Ferber
first r e l a t e d
supertwistors
by Siegel
plemented
twistors
lines
can be d e s c r i b e d
introduced
SUSY p a r t i c l e
firstly
particles
space-time
particles
null
space
s h o w that all the r e l a t i o n s
Supertwistors Witten
real
null
particles
one can
One
physically
of scalar
massless
Indeed,
interprete
solutions
for D=4
studied
extension
of the
for the d e s c r i p t i o n
integrability
of the
in [19-21] 4) . The
SUSY YM t h e o r i e s
in m a t h e m a t i c a l l y
[6,16,17]
supersymametric
has been p r o p o s e d
rigorous
on Q - p l a n e s
super-selfduality
in
way by Man±n
(see
equa-
ambitwistor [6] and exten[9]
(see also
140
[23]). R e c e n t l y also the c o n s t r a i n t s for D=3,4,6 and 10 SUSY YM and SUGRA theories have been derived from the c o n s i s t e n c y of the superstring action with the D=2 g e n e r a l i z a t i o n of Siegel fermionic invariance (see E')
[14,24,25].
Such result led to an idea
it should be useful an i n t r o d u c t i o n
of the
tor space T, g e n e r a l i z i n g the relations
(see e.g.
[14,26]) that
infinite-dimensional
(D,E)
as follows:
points of T
orbits of m a s s l e s s
twis-
(F)
( ~ -dimensional)
strings in M
Finally it should be m e n t i o n e d that the i n t e g r a b i l i t y along real null lines has been also used for the constraining of the D=2 local string s u p e r a l g e b r a s 5) We see that in order to relate the d i s c u s s i o n of constraints described above with twistor formalism it is desired the following threefold e x t e n s i o n of conventional twistor methods:
i)
SUSY e x t e n s i o n
ii)
Multidimensional
(Kaluza-Klein)
e x t e n s i o n to at least D=6 and
D=10 iii) E x t e n s i o n to i n f i n i t e - d i m e n s i o n a l m a n i f o l d s describing
the
string c o n f i g u r a t i o n s
In this paper I shall discuss m a i n l y the first, D=4.
SUSY extension,
for
In Sect.2 we shall discuss the supersymmetric e x t e n s i o n of purely
twistor approach;
in Sect.3 the s u p e r s y m m e t r i c a m b i t w i s t o r
be i n t r o d u c e d and the d e s c r i p t i o n of graded null lines. shall only briefly discuss the m u l t i d i m e n s i o n a l f o r m a l i s m for D>4.
space will
In Sect.4 we
extensions of twistor
It should be m e n t i o n e d that the constraints
for ma-
ximally e x t e n d e d SUSY YM theory have neat i n t e r p r e t a t i o n as the integr a b i l i t y c o n d i t i o n s along SUSY - e x t e n d e d null lines in D=I0 so3)).
(see al-
In Sect.5 we provide remarks on related problems.
2. D=4 Supertwistors. Let us recall
[I] that the basic e q u a t i o n e x p r e s s i n g the incidence
b e t w e e n points z 6CM and points in twistor space tA=(~e,n~6T A=I,2,3,4; ~=
~=I,2)
iz&Bz B
(B=I,2,3,4
is z~B = I ~O~B z
(2.1)
141 where the
~ =(~i,12)
are
Pauli
correspondences - if we h o l d
tisfying
(2.1)
the
z o~
complex
fixed
parameters.
the M i n k o w s k i
For
metric
describes
the
following
one can p a r a m e t r i z e
self-dual
z
sa-
2-plane
(2.1) i and X~ d e s c r i b e two z , z u +Az on the same ~ - p l a n e
two p o i n t s
and v a r y
the
(2.2a)
twistor
2-space
in T, p a r a m e t r i z e d
2-plane
z &~ by a p a i r
Introducing
~I;
any
satisfying
linear
(r=I,2).
-
in CM
(2.2)
complex
the
wl;
I '
2;
can e x p r e s s
coordinates e.g.
by
in
~6" One
of n o n p a r a l l e l
(2.1), can
twistors
tA; r
2x2 m a t r i c e s
2 ~
f
/
~ = \
~2; I '
both
vanishes
z &~ f i x e d
describe
one
(2.1)
ds 2 = dz dz ~ = 0
- if we h o l d
= (
(w e ' ~ ) f i x e d
the
point
= 0
we o b t a i n
equation
z ~6 = z~ 8 + X~z B
is any
Az~6~
The
(B), b e c a u s e
twistor
describing
e-plane:
where
matrices.
(A) and
2
~1
~1
;I
Z={z ~6}
(2.3)
/
72;2
the m a t r i x
;2
. ~2;2
in t e r m s
of two
twistor
coordinates
as f o l l o w s
Z = £:~-1
The
formula
coordinates
(2.4)
(2.4)
describes
on b i t w i s t o r
In t w i s t o r
composite
space
space
T one
can
ver
of the c o n f o r m a l
group
of CM),
ver
of the
group
of M),
scalar
of C M in t e r m s
of
the
define
the
action
of
or in p a r t i c u l a r and
introduce
SL(4;C)
(4-fold
co-
SU(2,2)
(4-fold
co-
the U ( 2 , 2 ) - i n v a r i a n t
product
where
conformal
structure
T × T 6) .
GAB
GAB
= UAGABt B
is a H e r m i t e a n
(0 - 12
and putting
(2.5)
metric
with
the
signature
(÷+--).
Choosing
) (2.6) 0
u=(~e,p~)
one
obtains
142
= -p ~ ~ + [ ~" z
The form
(2.7)
(2.7)
of the scalar product
of D=4 twistor as SO(4,2) Lorentz group 0(3,1).
(2.5) exhibits the d e c o m p o s i t i o n
spinor into Weyl spinor and Weyl cospinor of
The i n t e r p r e t a t i o n of twistor as a D=4 conformal
spinor permits to express the m a n i f o l d of complex 2-planes as the following H e r m i t e a n coset space
SU(2,2) CM --~ G2(C 4) = S(U(2)×U(2))
The formula
(2.8)
(2.8)
permits to derive the conformal
Z=ZnO n as 2×2 m a t r i x M o b i u s t r a n s f o r m a t i o n
Z
A + BZ C + DE
/ABh \CD]
(see e.g.
[31])
E SU(2,2 )
confirming the i n t e r p r e t a t i o n of z fied)
t r a n s f o r m a t i o n s of
(2.9)
as the coordinates of
U
(compacti -
CM.
The twistors can be defined in several ways, e q u i v a l e n t for D=4, e.g.
as
i)
fundamental D=4 conformal
ii)
the solution space of the "twistor equation", mal ~illing spinors formula
iii)
spinors
(such a d e f i n i t i o n is related closely with
(2.1).
four complex c o o r d i n a t e s describing the phase space of free m a s s l e s s conformal particles
v)
(see e.g.
[12,32,33]).
twistor bundle over space-time M with the fibre d e s c r i b e d by all complex structures on M
F o l l o w i n g Ferber first definition.
The N - e x t e n d e d conformal
internal U(N)
generators
r e s e n t a t i o n of SU(2,2;N) space of s u p e r t w i s t o r s (commuting)
(see e.g.
andNodd
[34]).
[13] we shall extend here s u p e r s y m m e t r i c a l l y the
o b t a i n e d by adding to SU(2,2) N 2 7)
defining confor-
superalgebra SU(2,2;N)
is
g e n e r a t o r s 4N complex supercharges and (see e.g.
[34,35]). The fundamental rep-
is d e s c r i b e d by (4+N)-dimensional complex super(tA,~i) 6 T(N ) =C 4;N
(anticommuting)
(i=I ,.. . N) , with 4 even
coordinates.
The U(2,2)
norm
(2.5)
is e x t e n d e d as follows:
= U A G A B t B + ~i~i
where U(N)=(UA,qi).
The superalgebra of SU(2,2;N)
(2.10) is r e a l i z e d on T(N )
143
by (4+N)×(4+N) matrices
[13,35], and SU(2,2;N) matrix supergroup is o ~
tained by the exponentiation map with commuting parameters in the b o s ~ nic sector, and anticommuting in the fermionic one In supertwistor space the correspondences
(see e.g.
[36,37]).
(A) and (B) becomes non-
unique because one can introduce N+I superspaces by the following SUSY extension of the formula
(2.8)
S C ~ N) _~ G2;k(C 4;N) =
[38-40]
SU(2,2;N) S(U(2,k) x U(2,N-k))
(2.11)
where SCM~0)~ CM, and k=0,1...N. The SUSY version of the relation
(B) can be written separately for eve-
ry N-extended superspace SCM~ N) as follows
linear (2;k) subspaces in T(N ) ~ points in SCMi N) (linear (1;k) subspaces in PT(N)) where linear subspaces
(2.12)
(n,m) are parametrized by n even and m odd coor-
dinates. The basic formula
(2.1) is extended for the superspace CM~N)-
in the following way [40] •
~
k
°
,
= ize~w~ + j=1@~3~j (2.13)
k
~i = @elW~ + [ 113 ~j j=1
where l=k+1,...N, and the coordinates of CMI N) are ScM~N):
(z
, 11J ; 0~I , @~J)
i.e. 4+k(N-k) even and 2N odd coordinates.
(2.14) We see that only for k=0 and
k=N the even sector is given by CM. In such a case the formulae
(2.13)
are simplified, and the equations for super-s-plane are 8) chiral superspace (k=0) antichiral superspace
~
= iz~6~6
~i = @ e i ~
~ = iz~6z 6 + @~i_~i
(2.15a)
Q.15b)
(k=N) where we introduced different chiral and anti-chiral complex Minkowski
144
coordinates,
because
C ~ N) for different The formulae tes
(e~,~
in the formulae values
(2.13)
,~i ) describe
k=N one obtains chiral
(2.14)
a priori
the coordinates of
of k are not related.
for fixed values
of the supertwistor
the super-a-plane
the following
(k=O)
(2,k)
parametric
in C N~ ) -
equations
coordina-
For k=0 and
for super-m-planes:
z$~ 6 = Z(o)+&6 + (2.16a)
super-m-plane (2 ,N)
@?z
antichiral
(k:N)
a = 01(0)
z ~_
z(0)_ + Q.16b)
super-m-plane (2,2N)
@&i =
where
the coordinates
with
1 ~ are complex,
si' s& i that @$ occuring
ressed
+ Si Z a
0
&i (0)
subscript"0"
&i +
(2.16b)
'
denote particular
complex-Grassmann in
~
parameters.
is a chiral
solutions and
It should be st-
coordinate,
defined
by
1
(2.15a) 9) . it makes described
therefore
sense
to consider
nonchiral
(z+, z _ ,0 ai' @e" i J" It appears
by the coordinates
superspace,
from
(2.15a,b)
that
L0
iz~+6~ 6
=
:
i(z a6 - i0azG6i)~ 6 --
(2 17)
i.e. one can identify 10) z &6
=
+
z &B
-
io&iO6
(2.18)
--
or define
z~ 6
=
1
the "symmetric"
za6 ¥
z ~6 by 11)
i oaio~
The complexified
(2.19)
nonchiral
space coordinates the eq. (2.15a).
CM coordinate
superspace
(z~,0~i,0~).
Using
symmetric
:
+ ½(Mi
by the super-
in SCM (N) is determined
CM coordinates
+
nonchiral
SCM (N) is described
The a-plane
(see
(2.19))
by
we obtain
_
z (0)
super-m-plane (2,3N)
6@~ = 0~i -0~i(O)
= El. ~ (2.20)
o&i
6@ai =
where
_ @&i (0):
l&'ei and s ~i are the parameters.
&i
145
The i n t e g r a b i l i t y ral
super-a-planes
self-duality
3. D=4
of SUSY Y M s u p e r s p a c e
(2.20) w a s u s e d
equation
(see
connection
forms on nonchi-
for the SUSY g e n e r a l i z a t i o n
of the
[21]).
SUSY a m b i t w i s t o r s
In o r d e r a dual
to d e s c r i b e
twistor UA =
geometrically
(pa,r~)
(where
an a m b i t w i s t o r
pa=(pa),,
r~=r~)
we i n t r o d u c e
the dual
for
incidence
equation
p~ = - ir.z ~B a which
determines
(3.1)
in C M the a n t i - s e l f - d u a l
B-plane,
parametrized
as fol-
lows B-plane
where
:
z ~B = z~ B + r~l B
18 are two c o m p l e x
L e t us a s s u m e a-plane dual
~B one o b t a i n s
r.~ a The
parameters.
n o w t h a t the c o m p l e x
described by twistor
twistor UA
&
solution
ric e q u a t i o n
tA
the c o n s i s t e n c y
Multiplying
z aB lies
s i m u l t a n e o u s l y on
and a - p l a n e
d e s c r i b e d by
(2.1) by p~, and
(3.1)
by
condition
= 0
(3.3)
of the eq. (2,1)
and
for
null
the
point
(see eq. (2.1))
(see eq. (3.1)).
a + p ~ a =
(3.2)
complex
(3.1) w i t h
(3.3) p r o v i d e s
the p a r a m e t -
line
~B z~
= i
.
a
+ cra~ B
(c c o m p l e x )
(3.4)
ro~ a
i.e.
we o b t a i n
the
eq.(2.1)
and
the
the and
eq.(3.4)
correspondence (3.1)
have
takes
the
(C).
If
common r e a l
the
points
coordinates in
M only
z if
are
real,
tA = uA ,
f o r m 12)
a~B X aB = i
g_
+ l~az a
(Ireal)
(3.5)
a
a n d we a r r i v e The
at the c o r r e s p o n d e n c e
SUSY e x t e n s i o n
(D).
of a m b i t w i s t o r
(1.1)
is g i v e n
by
[6,9,41] m
D=4 c o m p l e x s u p e r a m b i t w i s t o r space :
P A ( N % = ( U ( N ) , t I N•) ) C P T f N,) X P T f N %,, ,
146
C : 0 where the scalar product is given by
(3.6)
(2.10), and projective
twistors
are defined modulo complex scale factor. Explicitly we have r .w~ + P ~ ze + i ~i : 0
(3.7)
where I i = ~.. The SUSY extension of the eq.(3.1) l
6 : _ir.z~
+
looks as follows
N l=k+1
(3.8) • ~J = @~3r~ +
We see from roduce
N
1
~ l=k+1
l]I q
(2.13) and
(3.8) that in every superspace CM~~ N) one can int-
(2;N-k)-dimensional
B-planes.
In particular
super-B-planes
and
(2,k) dimensional
super-
for k=0 and k=N one obtains the following equa-
tions for super-~-planes: chiral superspace
p6 : _ir&z~B + 6~ i
(3.9a)
(k:0) antichiral
superspace:
pB= _ir.z~B
qi = @~ir.
(3.9b)
(k=N) which can be parametrized as follows chiral
z~6_ = z+(0 )&6
(k:0)
" 6
+
r~l
-
" B i@~ZSl (3.10a)
super-B-plane (2,2N)
antichiral
@~ =@~ + s~ i i(0) i (k=N)
z~B _
=
Z _~$ (o
)
cr&16
+
(3. I0b)
super-6-plane (2,N)
@i
where IB,E i and ~
=
@(0)
+
~
r
are the parameters. l
The 6-plane in complexified nonchiral with additional requirement nonchiral
super-~-plane
(2,3N)
space is defined by the eq. (3.9b)
(2.19). We obtain z ~6=z(0 ~ ) + r~l 6 + ~(i~0~i@~ _ 0~i6@~)
147
6691 = @i - @1(0)
= ei (3.11)
6@&i
@&i
=
•
It can be checked that the point plane
(2.20)
(3.7)
is valid.
extension
and super-B-plane
•
ze6 = z(0 &B )
+
"
cra~ B
6@~ = S . ~ ~ 1
+ ilo~i~6
~,0u
e
&
( z ~ , @ i , G i) can lie on super-s-
(3.11)
The superambitwistor
of complex null line
i & = s z
&i - 9(0)
simultaneously (3.6)
describes
ui
(3.12)
@@~i = sir~
i
are complex parameters.
super-B-plane
i.e.
w e• B w ~
it follows
super-s-plane real SUSY-null
from the eq. (2.15a)
superspace
SM=(x
and super-6-plane
tor "collapses"
and
SUSY-extended
(3.9b
,9e,@ i ~i = (@~)*)
Min-
(3.13)
that the points
lie simultaneously
on
i.e. when SUSY ambitwis-
The null supertwistors
by 2N real Grassmann
dimensions
describe the for-
as follows:
+ ~(60 ~i
It was observed
*
,
firstly by Witten
lutions of SUSY YM equations
SUSY YM system
(3.14)
null lines
is equivalent
(0~N~3)
can be obtained as permitting
(3-N)-th order of the infinitesimal
of SUSY-extended
(3.14)
[6] and further investigated
detail by Manin;" [9] that for N - e x t e n d e d
lity on
super-s-plane
(3.12) are SUSY-
@~id@~)
if U(N)=t(N),
to a null supertwistor. line, extending
= x(0 )
sions to
along
along
" _ ~(d@ i ~i @i8 ~d~ = dz e~
of real chiral
(3.5)
and consequently
of the following (compare with 2.2a) 13)
= 0
Finally
(3.11)
they lead to vanishing
kowski metric
mula
SUSY
(I,2N)
_ @~i6@~)
It is easy to see that all the translations null,
the following
(3.4), with the complex dimension
1
where c, si and s
(2.20),
if the condition
(3.14).
neighbourhoo~
In particular
to the equations
in
the so-
the extenin the space
if N=3 the integrabi-
of motion.
In the case
148
N=4
the
self-duality
~13
~ ~-i3kl#kl
obstructed
way
D=4
if we consider
interesting
reason
tor f o r m a l i s m
4.
interpretation N=I
SUSY YM e q u a t i o n s
it became
Another
in internal
0(4)
space
(3.15)
the twistorial
culty d i s a p p e a r s vides N=4
condition
D>4
theories
However,
SUSY YM theory
via d i m e n s i o n a l
to consider
for c o n s i d e r i n g and string
D=I0
in D=4.
this diffi-
[14] w h i c h
reduction.
the SUSY twistor
is a p os s i b l e
pro-
In such a
formalism
in D>4.
relation between
twis-
[41-44].
D>4
The D=4
twistors
can be d e f i n e d
in several
equivalent
ways,
for
example i)
as the f u n d a m e n t a l
SU(2,2)=S0(4,2) ii)
representation
of D=4 conformal
as the p a r a m e t r i z a t i o n
xified Minkowski iii)
of the
four-fold
covering
group. of the totally
null
2-planes
in comple-
space C 4
as a bundle
over
S 4 describing
all possible
complex
structu-
res on S 4 We
shall discuss
i) T w i s t o r s We define
briefly
as conformal
twistors
the e x t e n s i o n
as fundamental
In such a way one can introduce on the choice described
linear
spinors
twistors
of D) they can be real,
by the following
of these
definitions
to D>4.
s~inors.
complex
vector
of SO(D,2). for any D, and
(depending
or quaternionic.
spaces
for 4~D~I0
They are
(see e.g.
[45])
D
4
5
6
7
8
9
T
C4
H4
H4
H8
C 16
R 32
Table
I. D - d i m e n s i o n a l spinors
It should bed by a pair l exifie d
be added
R 32
as the f u n d a m e n t a l
that for any D the c o n f o r m a l
of Lorentz
rotation
twistors
10
groups
spinors. as well
This
conformal
spinors
decomposition
are descri-
is valid
as for the real one,
with
for comp-
arbitrary
signature. In such a f r a m e w o r k
the d i m e n s i o n s
D=6 and D=10
are s e l e c t e d
becau-
149
se they correspond
to quaternionic
complex descriptions following
table
extensions
of D=4 spinors and D=4 twistors.
(see e.g.
spin covering
Let us write
D=6
the
D=I0
SL(2,C)
SL(2;H)
DL(2;0)
C2
H2
R16~02
(4;H)
U (4;0)
H4
R32=04
spinors
spin covering
SU(2,2)=
of Conf.group
=U
U
(4;C) C4
Conf.fundamental
of the
[46])
D=4
of Lorentz group Weyl
and octonionic
spinors Table 2. The relation of complex numbers with D=4,quaternions with D=6'and octonions with D=I0
where
U
(F=C,H or O)
(4;F)
:
qAHABqB
i.e. U a describes ii) Twistors Following D=2k twistors planes dence
= inv
antiunitary
H + = -H group.
In particular
one can chose H=
0
as pure spinors.
[1,42-44]
one can adopt the view that in even dimensions
are pure conformal
spinors,
in C 2k. We obtain the following
describing
totally null k-
generalization
of the correspon-
(A) for even D>4:
point
in T(2k)
where T(2k) Pure
denotes
~
totally null k-planes
the space of twistors
spinors are obtained by imposing
ints on 'brdinary"projective dimension n=2k-1
fundamental
in C 2k
(A')
for D=2k. r=2 k
k(k+1) linear constra2 spinors, with complex
SO(D+2;C)
i.e. they are described by quadric Qq, with complex di-
mension q= k(k+1______~) We obtain the following complex manifolds 2 bing "ordinary" projective spinors and twistors: "ordinary"
projective
conf.
spinors
descri-
twistors
D=4
CP(3)
CP(3)
D=6
CP(7)
Q6
150
"ordinary"
projective
conf.
spinors
twistors
D=8
CP(15)
QI0
D=I0
CP(31)
Q15
Table
3. From o r d i n a r y
We s e e
that for D=4 o r d i n a r y
for D=6 one needs r=16.
to pure
to impose
In p a r t i c u l a r
for D=6
can be identified,
for D=8 r=5,
[42] that the p u r i t y condi-
as the c o n s i s t e n c y extended
Twistor
twistors
it can be shown
tion follows
iii)
and pure
spinors.
one constraint,
uation
(2.1)
conformal
condition
and for D=10
for the Penrose
incident
eq-
from D = 4 to D=6.
space as bundle
over
S 2k d e s c r i b i n g
all p o s s i b l e
comp-
lex structures. In the d e s c r i p t i o n space one can e x t e n d the e x t e n s i o n
of self-dual
the gauge
fields
connections
space as fiber bundle
cally by CP(1)=
SU(2) Because U(J] "
extended
(compactified)
from S 4 to CP(3)
is pure gauge. Such a c o n s t r u c t i o n
tion of twistor
that CP(1)=
on
SO(4) U(2)
' i.e.
over
leads
S 4 with
SO(4)=SU(2)xSU(2)
locally
Euclidean provided
to the introduc-
fibers
described
one can w r i t e
P T ~ S 4 X u( SO(4) 2 ) " This
relation
to any even k and we obtain locally for D=2k (see e.g. [47-49])
Counting
real
(4.1)
dimensions
with the d i m e n s i o n s
iii)
also
can be
PT ~-- S 2k x SO(2k) U(k)
as pure
lo-
spinors).
2k+k(2k-1)-k2=k(k+1),
obtained
From
from our second
the i d e n t i f i c a t i o n
we get the a g r e e m e n t definition
(i.e.
of the d e f i n i t i o n s
twistors ii)
and
one gets
D=4
D=6
D=8
D:I0
D P (3)
Q6
QI 0
QI 5
+CP(1)
+CP(3)
+ Q6
+ QI0
S4
S6
s8
s I0
Table
4. Twistor
The twistor of selfdual Finally
bundles gauge we
bundles
written
fields
shall
in D=2k
above
in D=2k,
consider
(k=2,3,4,5).
should be u s e f u l k=2,3,4,5.
for the d e s c r i p t i o n
151
iv)
Supersymmetrization
Only
for D=6
classical group
for D>4
the
spin c o v e r i n g s
SO(D,2)
Lie group,
and S0(6,2) = U
(4;H)
can be
Ua(4;H)
where
of t w i s t o r s
supersymmetrized
÷
as follows
(D>4) [50].
is d e s c r i b e d
by a
The D=6 c o n f o r m a l spin
[51]
U U(4;H)
the b o s o n i c
(4.2
sector
of the
SUSY e x t e n s i o n
of D=6
conformal
group
is
U
(4;H) × U(N;H)
The D=6 H4
conformal
H4 =
where We
see t h e r e f o r e [53]
c o n formal
purity
can be
mal a l g e b r a the
÷ H 4;N =
that
spinors
described
i) There
exists
variables
to make
in N=2,3
point
jectories".
Lie
the
a close
matrices
superalgebras
following
relation
SUSY
harmonic
as
"ordi-
of D=I0
[46]
(see
of
confor-
leads b e y o n d
[54]).
strings
superspace
by actions
selecting
In p a r t i c u l a r
twistorial and D=4
string
one can
[58],
exploiting in
minimal
scalar
Q2N_3 =
some notions
lines,
world
of twis-
and their
sheets
as
dynamics
"string
in any d i m e n s i o n
by a pair of null
of null
and the p a r a m e t r i c
space 15)
[49,57].
show that
Eisenhart
and harmo-
bosonic
lie on the quadric
is d e s c r i b e d
[60],
methods
recalls the a m b i t w i s t o r
along null
parametrization
for D=3 and M o n t c h e u i l
twistor
The additional
superspace
propagate
comments:
between
[55,56].
uivi=0) , w h i c h
of h a r m o n i c
of a b o s o n i c
[59]
supersymmetrization
of view has been p r e s e n t e d
Massless
is d e s c r i b e d
D=3
Z2-graded
the t w i s t o r s
It is not clear how the c o n d i t i o n The
by 4×4 a n t i h e r m i t e a n
to e x t e n d e d
The d i s c u s s i o n
The
c o o r d i n a t e s 14
of c o n v e n t i o n a l
supersymmetrize
for D=6.
((ui,v i) 6 CP N-I x CP N-I,
43].
(4.4
quaternionic
framework
of
Remarks.
nic a p p r o a c h
gation
in the
of a s s o c i a t i v e
like
ii)
by the SUSY e x t e n s i o n
(q1...q4;@1 .... @N )
one can only
We w o u l d
torial
are d e s c r i b e d
supersymmetrized.
framework
5. Final
(4.3
0° r er are G r a s s m a n n - v a l u e d i + Oi
=
superalgebras nary"
(8) × Sp(2n)
superspinors
(ql...q4)
@i
= SO
tra-
the propa-
curves
[58,
curves
has been g i v e n
for
formulae
of W e i e r s t r a s s
[59]
[61]
for D=4 has been;derived.
152
The e x p l i c i t e
parametrization
of null
red by H u g h s t o n
and Shaw
the d e s c r i p t i o n
of s u p e r s t r i n g s
has been given iii)
The
placement space
in
of QFT
zed only
nature
nature
complex
of s u p e r t w i s t o r s
nonchiral
to
superspace
ap p r o a c h
space by QFT
Such a p r o g r a m m
spinor
space
in twistor was also
approach between
of space-time
coordinates,
(see
is the re-
or a m b i t w i s t o r
investigated
[64,65].
is an analogy
of e l e m e n t a r y
to physics
in the
It should be al-
which
can be locali-
(2.4))
objects
with u n o b s e r v a b l e
consti-
(quark or p r e o n models).
It is t e m p t i n g tuents
in D=4
in terms of two twistors
- composite tuents
63]).
that there
- composite
and the a p p l i c a t i o n
of twistor
in M i n k o w s k i [62,
of R z e w u s k i ' s
so o b s e r v e d
for D=6 has been c o n s i d e -
[41].
"strong version"
(see e.g.
framework
[42],
curves
to describe
as due to the
stor space. [66-68],
Some
where
they were
the c o n f i n e m e n t
fact that they are
investigations
the notion
r e l a t e d with
in some
r e l a t e d with
of q u a r k - t w i s t o r
strings
of u n o b s e r v a b l e sense
locali~d
in twi-
such an idea were made
variables
on conformal
consti-
was proposed,
supergroup
in and
manifold.
Acknowledgments The author w o u l d le discussions, tional
Centre
completed.
and Prof.
talk was p r e s e n t e d
This paper ki on his nates
to thank
dr. L. H u g h s t o n
A.
for the h o s p i t a l i t y
Salam
for T h e o r e t i c a l
We w o u l d
perunification
like
like also
Physics,
where
to m e n t i o n
these
I would
like
70 -th birthday,
should be more
valuab-
at the Interna-
lecture
notes were
that the first version
at the F i r s t Torino Meeting
(23.IX - 27.IX
for several
on U n i f i c a t i o n
of t h ~
and Su-
1985).
to dedicate
to my teacher,
who teught me first
fundamental
that the
than the ones
prof.
J. R z e w u ~
spinor
described
coordi-
by space-time
fourvectors.
FOOTNOTES I. These g e n e r a l i z a t i o n s vitons);
see e.g.
were
defining
und
fields
states
(photons,
gra-
[4,5].
2. One can show that the coupling and.gravity
"googly"
does
not impose
of m a s s l e s s
particles
any r e s t r i c t i o n s
to external
YM
on these b a c k g r o -
fields.
3. For N=4 D=4 tegr~bility
SUSY YM theory
along
it is not k no w n how to derive from the in-
SUSY-extended
null
lines
the
internal
sector
selfdu-
153
ality c o n s t r a i n t for the field strenght superfield. 4.It should be m e n t i o n e d however, of SUSY in real E u c l i d e a n
that in these papers
the structure
space has not been taken into account.
For
the d i s c u s s i o n of s u p e r - s e l f - d u a l i t y with more e x p l i c i t e d i s c u s s i o n of E u c l i d e a n SUSY see
[22].
5. For local string s u p e r a l g e b r a
see
[27]
, for i n t e g r a b i l i t y see
[28].
6. This formula is due to Penrose, but some authors did put forward earlier ideas that space-time coordinate can be e x p r e s s e d as composite in terms of spinor c o m p o n e n t s - see e.g. 7. For N=4 one gets as internal
[29]
(see also
symmetry groups SU(4)
[30]).
[34].
8. We call superspace
SCM~N)" chiral because the v a r i a b l e s @~ and 0~" can u 1 1 be o b t a i n e d from 4 - c o m p o n e n t complex Dirac spinor by chiral projecticns ~(I±Y5)
(for Y5 diagonal).
9. From
(2.15b)
one gets for the. last. term of 6z &~ the e q u a t i o n i 6 z ~+
+@8 ~i:0 w h i c h is solved by 6z~8=i@@~8~ if we put ~i=@iz8. 10. The formula flag m a n i f o l d 11. In formula
(2.18)
(see e.g.
can be e x p l a i n e d g e o m e t r i c a l l y as s u p e r s y m m e t r i c [9]).
(2.19) one can recognize the known r e l a t i o n between the
real, chiral and antichiral 12. From
(3.3)
superspace coordinates
follows that ~ ~
is p u r e l y imaginary.
13. The S U S Y - e x t e n d e d null lines defined in [6] in fact do not lie on the s u p e r - l i g h t - c o n e
(3.13), because the part d e s c r i b i n g the transla-
tions along G r a s s m a n n directions
is missing.
not invalidate however the c o n c l u s i o n s 14. The q u a t e r n i o n i c morphisms
supergroups
This s i m p l i f i c a t i o n does
in [6].
as q u a t e r n i o n i c n o r m - p r e s e r v i n g endo-
in superspace were c o n s i d e r e d r e c e n t l y in [52].
15. In ref.
[57] even the name "isotwistor
superspace was p r o p o s e d as
more a p p r o p r i a t e than "harmonic superspace".
REFERENCES
I. R. Penrose and W. Rindler, "Spinors and Space-Time", V o l . 2 , C a m b r i d g e Univ. Press, 1986, and the literature quoted therein 2. R. Ward, P h y s . L e t t . A 6 1 , 8 1 , 1 9 7 7 3. M.F. Atiyah, V.G. Drinfeld, N . I . H i t c h i n and Yu.I.Manin, Phys.Lett. A65,185(1978) 4. R. Penrose in "Advances in Twistor Theory", e d . L . P . H u g s t o n and R.S. Ward, P i t m a n , L o n d o n , 1979 5. L.J. Mason, Twistor N e w s l e t t e r s , N o . 1 9 and 20 6. E. Witten, P h y s . L e t t . 7 7 B , 3 9 4 ( 1 9 7 8 ) 7. J.Isenberg, P . B . Y a s s k i n and P.Green, P h y s . L e t t . 7 8 B , 4 6 2 ( 1 9 7 8 ) 8. G.M. Henkin and Yu.I.Manin, P h y s . L e t t . 9 5 B , 4 0 5 ( 1 9 8 0 ) 9. Y u . I . M a n i n "Gauge fields and complex geometry", ed. Nauka, M o s c o w 1984 (in Russian) 10. J. Isenberg and P. Yasskin, Gen. Rel.Grav.14,621 (1982)
154
11. C.R. Le Brun, Class.Quantum Gray.2,555(1985) 12. L.P. Hughston, "Twistors and Particles", Lect.Notes in Phys.No 79 (Springer),19~9 13. A. Ferber, Nucl.Phys.B132,55(1978) 14. E. Witten, Nucl.Phys.B266,245(1986) 15. W. Siegel, Phys.Lett. 128B,397(1983) 16. J. Crispin-Romao, A. Ferber and P. Freund, Nucl. Phys.B182,45(1981) 17. I.V. Volovich, Teor.Math.Fiz.54,89(1983) (in Russian) 18. C. Devchand, "Integrability on light-like lines in six-dimensional superspace", Freiburg Univ.preprint,1986 19. Yu.I. Manin in "Problems of High Energy Physics and QFT, Proc. Protvino Seminar 1982,p.46 20. A.M. Semikhatov, Phys.Lett.120B,171(1983) 21. I.V. Volovich, Teor.Math.Fiz.55,39(1983) (in Russian) 22. J. Lukierski and W. Zakrzewski, to appear as ICTP preprint 23. A.A. Rosly, Class.Quantum Gray.2,693(1985) 24. M.T. Grisaru, P.S. Hove, L. MezTncescu, B.E.W. Nillson and P.K. Townsend, phys.Lett.162B,116(1985) 25. E. Bergshoff, E. Sezgin and P.K. Townsend, Phys.Lett.169B,191(1986) 26. J. Isenberg and P. Yasskin, Ambitwistors (and strings?), preprint 1986 27. W. Siegel, Nucl.Phys.B263,93(1985) 28. P.G.O. Freund and L. Mezincescu, preprint EFI 86-11(1986) 29. J. Rzewuski, Nuovo Cim.5,942(1958) 30. J. Kocik and J. Rzewusk[, "On prQjections of spinor spaces onto Minkowski space", to be published in "Symmetries in Science II", ed. B. Gruber, Plenum Press, New York, 1986 31. R.O. Wells Jr. Bull.Am. Math. Soc. (New Serie) 1,296(1979) 32. W. Lisiecki and A. Odzijewicz, Lett.Math.Phys.~,325(1979) 33. I.T. Todorov, "Conformal description of Spinning particles", SISSA preprint 1/81 34. R. Haag, J. ~opusza~ski, and M. Sohnius, B88,257(1975) 35. S. Ferrara, M. Kaku, P. van Nieuvenhuizen and P.K.Townsend, Nucl. Phys.B129,125(1977) 36. F.A. Berezin, ITEP preprint ITEP-76,1977 37. F. Gursey and L. Marchildon, J.Math. Phys.19,942(1979) 38. J. Lukierski, "From supertwistors to composite superspace", Wroc~aw Univ.preprint 534, 1981 39. L.B. Litov and V.N. Pervushin, Phys.Lett.B147,76(1984) 40. M. Kotrla and J. Niederle, Czech.J.Phys.B35,602(1985) 41. W.T. Shaw, Class.Quantum Grav.3,753(1986) 42. L.P. Hughston and W.T. Shaw, "Minimal Curves in Six Dimensions",MiT preprint, 1986 43. P. Budinich, "Null vectors, spinors and strings", SISSA preprint 10/86 44. W.T. Shaw, "Classical Strings and Twistor Theory: How to solve string equations without using the light-cone gauge", Talk at VC Santa Cruz AMS meeting, June 1986 45. T. Kugo and P. Townsend, Nucl.Phys.B221,357(1983) 46. A. Sudbury, J.Phys.A17,939(1984) 47. R.L. Bryant, Duke Math.J. 52,223(1985) 48. A.M. Semikhatov, JETP Letters 41,201(1985) 49. A.M. Semikhatov, "Harmonic superspaces and the division algebras", Lebedev Inst.preprint N ° 339(1985) 50. R. Gilmore, "Lie groups, Lie algebras and some of their applications", Willey, New York, 1984 51. Z. Hasiewicz, J. Lukierski and P. Morawiec, Phys.Lett. 130B,55(1983) 52. J. Lukierski and A. Nowicki, Ann.of Phys.166,164(1986) 53. V. Kac, Comm.Math.Phys.53,31(1977) 54. Z. Hasiewicz and J. Lukierski, Phys.Lett.145B,65(1984) 55. V.I. Ogievetski and E.S. Sokhaczew, Yadernaja Fiz.31,205(1980) 56. A. Galperin, E.Ivanov, S.Kalitzin, V.O.Ogievetski and E.Sokhaczew
155
Class. Q u a n t u m Grav.1,469(1984) 57. A . A . R o s l y and A.S.S~hwarz, " S u p e r s y m m e t r y in a space with auxiliary dimensions", ITEP p r e p r i n t 39/1985 58. W.T. Shaw, C l a s s . Q u a n t u m Grav.2,L113(1985) 59. K. W e i e r s t r a s s , M o n a t s . B e r l . A c a d . 6 1 2 ( 1 8 6 6 ) 60. M. Montcheuil, Bull. Soc.Math. France 33,170(1905) 61. L.P. Eisenhart, Ann. Math. (Ser. II),13,17(1911) 62. M.A.H. Mac Callum and R. Penrose, Phys.Rep.6,241(1972) 63. A.P. Hedges, Proc. R . S o c . L o n d . A 3 9 7 , 3 7 5 ( 1 9 8 5 ) 64. J. Rzewuski, Acta P h y s . P o l o n . 1 8 , 5 4 9 ( 1 9 5 9 ) 65. J. Rzewuski, Rep. Math. Phys.22,235(1985) 66. J. Lukierski, L e t t . N u o v o Cim.24,309(1979) 67. J. L u k i e r s k i in "Hadronic Matter at Extreme E n e r g y Densbty", ed.by N. C a b b i b o and L. Sartorio, Plenum Press, 1980,p.187 68. J. Lukierski, J.Math. Phys.2_~1,561(1980)
S U P E R S Y M M E T R I E S OF THE D Y O N +
Eric D'Hoker Department of Physics Princeton University Princeton, New Jersey 08544 U.S.A. Luc Vinet Laboratoire de Physique Nucleaire Universite de Montreal C.P. 6128 Succ. "A" Montr(~al,Quebec H3C 3J7 Canada
Contents
Introduction A. Spectrum supersymmetries of particles in a Coulomb potential I. 2. 3. 4. 5.
The The The The The
4-dimensional system 3-dimensional system quantum numbers OSp(2,1 ) representations spectrum of HI)
B. Hidden symmetries of a spinning particle in a dyon field I. 2. 3. 4.
Symmetries of H (generalization of the Runge-Lenz vector) Supersymmetries of H Structure relations Spectrum analysis a la Bargmann
Acknowledgements References
* Seminar delivered by Luc Vinet in January 1986 at the Laboratoire de Physique Theorique et des Hautes Energies, Universite Pierre et Marie Curie (Paris VI).
157
INTRODUCTION Over the last two years or so, we have investigated the rble of superalgebras as dynamical algebras in Quantum Mechanics[l]. The first problem we analyzed[2,3 ] was that of a Non-Relativistic spin-I/2 particle in the field of a Dirac magnetic monopole which was shown to possess an OSp(1,1) dynamical superalgebra. We also observed [4] that this system can be generalized to accomodate a I/r2-potential and further noted the presence of an N = 2 superconformal symmetry in such instances. These interesting observations allowed us to obtain the spectrum and wave functions of the above systems from group theory alone. A famous problem with dynamical symmetries is certainly that of a spinless charged particle in a Coulomb potential. It possesses an 0(4) invariance algebra which explains the "accidental" degeneracy of the spectrum and all its states fall into a single irreducible representation of 0(4,2). A natural question that one can ask then, is the following: Can we find supersymmetries in the presence of a I/r-potential? We came up with the following answer. A. Consider the Hamiltonian D 2
-
-
(i)
(~-q)2- q~ + ~F.2
~ri~ i
42r 2
i= 1,2,$
r3
where ADi is the vector potential for a magnetic monopole of unit strength,
' - ( ° ' ° I 0o
.
0 ) :,0
(2)
2 and ~. is a free parameter. ItD describes the quantum dynamics of two spin 0 particles and one spin I/2 particle with electric charge - 11e in the field of dyons with electric charge e and magnetic charges respectively (q¥ I / 2 ) / e and q/e. We have found that HI) admits an OSp(2,1) spectrum supersymmetry which we used to obtain its spectrum and eigenfunctions[5]. B. In the special case i~ = 2q, the two lower components of HI) read
HI = Ho
112- q Bi°i
Bi - ri
rs
(3)
with
=
_
~+2r
2
(4)
158
It happens that the spectrum of H1 possesses high degeneracies. These are understood by viewing HI as the supersymmetric partner of H01]2 which is known to have the same spectrum structure as the Coulomb problem (with q=0). The constants of motion responsible for the accidental degeneracy of HI were obtained and embedded in an 0(4)~U(212) invariance superalgebra of the combined H0~2~)H 1 system [6], Their knowledge allowed for an analysis ~z la Bargmann of the spectrum of H1 [7] It is these results that I would like to expand upon in the course of this talk.
A. SPECTRUMSUPERSYMMETRIESOF PARTICLESIN A COULOMB POTENTIAL
In order to derive the spectrum supersymmetries of HD we shall use dimensional reduction to establish its connection with a 4-dimensional oscillator-like Hamiltonian. The supersymmetries of our 3-dimensional problem will then be inferred from those of this 4-dimensional system. It will be convenient to coordinatize ~4 with 2 complex variables za, a = 1,2 and their complex conjugate z a. We shall denote the corresponding vector fields by Oa = 0/O~ , ~a = O/Oza • Let r i, i = 1,2,3 be the standard Cartesian coordinates on ~3. Dimensional reduction will be effected via the Hopf map "
{2\ ri=,
{o} ---,
,IIi(Z)
=
3 \{o}
~'ao'ibzb
i- t,~,z
(s)
where a i stand for the usual Pauli matrices. This projection defines ~4 k{0} as a U( 1 )bundle over ~3\{0}. (Summation over repeated indices will be understood throughout.)
I. The 4-dimensional svstem
Consider the supercharges
(5) Izl"
159
with ~. a free parameter and the rls verifying
{~,.~b}=
0
(7)
=
We shall use the following realization of this Clifford algebra •
,(o
t(
,oo,)
rlz=-'~
~t="~ )_os
o o)+to 2)
-o 1-to 2
(8)
0
The anticommutators involving Q and Q~ are given by
[ Q , 0 } = {Q+,0+} = o
(9)
and H
= ½{~ .0 )} -
0a~'a +
(t0) T-~-z(~.-c)
-
z~,
izl4
where C = X +5:_
--
X = Zaa a - Za8 a
X-
' and
(
03 0 0 0
)
o)
(I)8)
(11b)
0 o_i 2
Note that X is the generator of the U( I )-action on the fibers of ~4 \{0} ~ ~3 \{0}. Now it is not too difficult to see that we can adjoin to Q, Q* and H, two more odd generators (S, St) and 3 more even generators (D, K, Y) to form an OSp(2,1 ) realization. Indeed one can check that
S =
Zaqa
S*=
Zaq)a
(superconformal)
(12a)
o =½(z°o,+:.:~+21
(dilations)
(!2b)
K = ~'aZ a
(conformal)
(I2e)
160
together with Q, Q) and H satisfy the structure relations that characterize the superalgebra OSp(2,1). These are
{Q,Q)} = 2H
{ S , S t } =, 2K
{Q,S )} = - 2 D - 2 i Y
{ Q+, s ] - - 2D + 2iY
[H,S] = -|Q
[H,S t ] ---iQt
(13c)
[K,0] =
[K ,0 t] = is +
(!~)
is
(1~)
(13b)
[D.Q]=-~'~ [D.S1= ~S [D.Q'1=-~O' [D.S+1=~S+ ['~.Q1=½~ [.~.s]= ½s [,,,,Q+].. -~ Q+[y.s+]--~s + [H.D] = iH
[H,K] = 2LD
[D.K] = iK
( 1u)
(1~j)
with all the other { ] equal to zero, We remark that all the above charges are invariant under the generated by
ji - - ' (~-z
i (~b - -ZaOabi ~ b ) + a oab
~i
i = 1,2,3
SU(2)-action
(14)
We also note that C = 2× + ~ commutes with ji and with all the 0So(2.1) eenerators. This observation will play a crucial r61e. In summary, the full symmetry algebra of the 4-dimensional problem that we have just defined is OSp(2,1) (~ SU(2) (~ U(I)
(is)
2. The 3-dimensional system Let us take the following superalgebra element :
(16) mB
",.
~,,a'a. (-
+ Iz) 2 +
b
- 2~
Izl 4
161
and introduce the eigenvalue equation :
R~,
=
(-2E) ~ •
(~7)
In order to project this equation from [~4 \{0} to ~3 \{0}, we shall require that the 4-dimensional wave function • be equivariant under the U(1)-action generated by X. More precisely, we shall take • in the U( 1) representation with weights ¢-~ ai,g ( q -
= di,g
q,
(
,
I
q
Equivariance under this representation is expressed by the condition X~= (q - : / 2 ) ~ or equivalently C ~ = 2q~ (tg) Let us point out here that the symmetries of the projected system shall be those of the 4-dimensional system which preserve this constraint. Since C is central, it means that the basis elements of our OSp(2, I ) ¢ SU(2) realization all generate symmetries of the 3-dimensional problem. To carry out the projection it is convenient to introduce the Euler coordinates O
O~
O<'n'
,
O~
4)< 21f,O~(o<41T
(20)
for IR4 \{0} ~ R*×S3 ~ R)×SU(2) in terms of which (21o)
and rt=
rcosOsin$
r2-
rsinOsinO
r3-
rcose
(21b)
In these coordinates the generator X takes the form X ---t ~
(22)
and the solution to equation (19) is given by"
~,(r,e,~,=) = ei(q'~z)=)(r,e,,)
We shall henceforth always designate the 4-dimensional wave function by an upper case psi (~) and the 3-dimensional wave function by a lower case psi (~).
(23)
162
Note that q must be an integer or a half-integer for • to be single-valued. Using these wave functions in equation (17), the :0-dependence can be separated out and one is left with the following eigenvalue problem •
(%l,R) ~# == (-2E) "~ )
(24)
where I) 2
(k-q)2-
q x +¼X 2 r
(25) 2Xri~i ] r2
Multiplying (24) by 11r and rescaling according to r i --, (-2E)~r i we further find that satisfies the Schrodinger equation
HI)*
(26)
= E*
with HI) given by equation (I). As already mentioned, this equation describes the quantum dynamics of two spin 0 and one spin 1/2 particles in dyon fields. We have shown that it admits an OSp(2,1)@ SU(2) spectrum supersymmetry.
3. The quantum numbers
The eigenstates of HI) can now be obtained by constructing bases for representations of OSp(2,1)@SU(2). One choise of quantum numbers for the basis states of the irreducible representations of OSp(2,1)@SU(2) is provided by the eigenvalues of the Casimir operators associated to the canonical chain of maximal subalgebras:
OSp(2,1)@ SU(2) D 0(2)~0(2,1)@ U(1) D 0(2)(~0(2)(~U(1) C2,C3
j2
y
Co
j~
R
The Casimir operator CO of O(2,1 ) is well known and given by
Co = ½ ( H K + K H ) - D 2
(27)
163 The quadratic and cubic Casimir operators of OSp(2, I) have respectively the following expression"
c,- co- Y"+ ¼[o,s'] • ¼[o', s] C~ -
<2=>
-~(c~ ,, ~[o. s+] ,. -~ [Q+,s]-½) (2sb) -
11oo .s"]D ,, ~[o".s]D- -~[~.o+]): --~[s,s+].
In our realization, C2 and C3 are completely determined in terms of j, q and ~. so that at fixed angular momentum the system is described by those representations of OSp(2,1) for which
Cz -
} (j+ 1) - in(in + 1)
C3= ( q - x ) [ j ( j + I)
Jo = Iql - ½
j o ( J o + I)]
(2~) (29b)
As usual, the eigenvalues of j2 are written in the form j(j +I), J-J0, Jo+I .... and those of J3 denoted by m=-j,-j+I ..... j. Now, in order to characterize the states belonging to these irreducible representations, it is convenient to replace Co and Y by 0 ,5 = (10 =l)
(3.)
and
x = ½(i+~5)~
+ ½(1-~s)=~(i[o.:]
• i[0+.s])
<30b>
which represent an equivalent pair of labelling operators. It is not difficult to check that Co and Y can unambiguously be reconstructed in term of 5/5 and A which both have their eigenvalues (X and ~) equal to +- i. As a matter of fact
Co=([J2-Jo(J+1)+(q'~)2]½-¼()-'Is)x}2-±4
(31,>
'r = ¼ ( ) + ~ s ) Z
(31b>
+(q-x)
Finally, from the representation theory of 0(2,1), we know the eigenvalues of R to be given by
= (~.~. + .)
°.0.,.2
~32>
164
if those of CO are written in the form (33)
Here. A I , & , X . [ I ( j + I ) - I o ( J o + I ) + ( q . ~ ' ) 2 ]½__ 1 ( I - X ) ~ ' + 2
1
(34)
]n summary, we have the following eigenvalue equations to characterize the states of our system ' j21j,m,~,X,n>
= j(j+l)lj,m,~,~,n>
(3se)
}3 I j , m , ~,X, n> = In I j , m , ~,X, n>
(3Sb>
,~ l j , m , & , X , n >
(3Sc>
ySl
= & Ij,m,~,,X,n>
j,m, ~ , ~ , n> -
R lj,m,&,X,n>
X I j,m, &,)~, n>
= (A
(35d)
+n)lj,m,&,X,n> ,X-
(3se>
4. The 0SD(2.1) reoresentations The action of the remaining 0Sp(2.1) generators on the I j. m. &. ~. n > state vectors has been obtained recently [6]. This is most easily achieved by going to a Cartan-type basis for 0Sp(2.1 ). Introduce the following ladder operators B+"' ½ [ K - H + 2iD l L
F, = -~rLs + 0]
(36e)
F? = ¢F~)+
(36~>
The 0Sp(2. I ) structure relations then become
[R,6,_] = ± B,
{F"'~F"'"}- o
[B+.B ] = -2R
OF:_,F~}- B.
¢3~.>
C~+'_.F:}" ~ +--"
_L,R L,R L L | R [)~,~,. 1- ±½F_~ [~',F.]=-½F_. [Y,F~]- ~F+ L,R
lB.,. F,. ]-o
L,R
[B+ F"'"] =-T-F,.
~'"'
<3,°> <37d>
165
After a little work, one find that B± and F±(L,R)act as follows on our basis states: 8+lj,m;~.,X,n>
L
- [
(~ ;°)(~ ;o+,) ,&,
,&,
-
(3~)
^
F+ [ j , m ; e , - 1 , n> = (38b)
).m;-,,...± -~>
a aa[(,±a> ~.~,~ ½()-a)+.] R
F± I j,m;
^
n>
o~,-1,
(300)
-
,,~ ½^ a_£[(i±&)Aj,R,x + I g( I - ~,)+ n] i,m; 11, , n± -~> L
F±Ii,m;
t,l,n>
=
-
1 :t 1 + n
+a [<'-+'>~.~,.~ ~-~ I + +
R
F ± i j , m ; 1, 1 , n ) L F±lj,m;-!,1,n>
s + n
]~l j,m;
!,-1
I i,m,-I :I .
'½ )
n+~+
(3~1)
n+~-~> _
-
0
(3~)
-
0
(3o0
R
F_+ i j,m;-1,1, n> ffi
a+[ ()~))A),~,,x+ ~I + ~1 + 0 ,]~' Ii,m' -a_ [~'")5.~.~
I ± 1+n
~
,
' ½~>
1,=1 , 1').+-~+
(300)
i j,rn;-1,-1,n±~-~>
where
(39)
= o~
2A i,&,X
By going to a coordinate realization, solving for the ground state and applying repeatedly yhe ladder operators, the wave functions can then be obtained simply (see reference [6]).
166 5, The spectrum of HD
The spectrum of HD can now be straight forwardly gotten. We have arranged our equations for E to be the eigenvalue of HD. Now from eqs, (17) and (35e) we have R == ( - 2 E i ~ =
(Aj,&,~ll)
(40)
It trivially follows that
En4'~J~
•,
=I
2 (Aj,&,~ n) 2
(41)
In the special case ~, - 2q the two lower components (~C= - I ) of HD become H, -
½~2_
_I
r
+ q2
~'2
_ eB I'Oi
(42)
with V " -- (p =eR) =
~×~,
.~, = g R D =
(43)
g[-'. ro
We readily see that H' is the Pauli Hamiltonian that describes the dynamics of a spin- I/2 particle with gyromagnetic ratio equal to 4 and electric charge equal to e in the presence of a 1/r2-potential and in the field of a dyon with electric charge - I / e and magnetic charge g. It is immediate to check that when ~, = 2q •
A},&,X=I"
} - ½ 8, + I
(44)
and thus, n
E
,,
-1
2(n+i_ ½a+)) 2
,,,
~-1
(45)
2: p=q,q+l ....
Interestingly enough, H' manifestly exhibits accidental degeneracies similar to those of the Coulomb problem. Indeed when p /: q we find a 2(p2-q2)-fold degeneracy at the level p, (In the ground state p = I q I and the 21 q l-fold degeneracy is accounted for by the rotational invariance of the problem,) We shall discuss the hidden symmetries that explain these accidental degeneracies in the second part of this talk.
167 B. HIDDEN SYMMETRIES OF A SPINNING PARTICLE IN A DYON FIELD
In order to study the invariance of
H'- Ho - qri3oi
(468)
where Ho =
{~'2-
q2
l
(46b)
~+~72
it is convenient to form the following 4 x 4 block diagonal Hamiltonian which simultaneously describe two spin 0 and one spin 1/2 particles •
[
(Ho+~)ll~
H
.
o
o
]
( 47e )
(H'+ ~,)
In equations (47),
U(r) =
zi. (°i°) 0 o,
.
¢
r
I
(4oa)
q
~,L =
½('-+v,)
(48b)
I. Symmetries of H (~eneralization of the Run~ze-Lenz vector)
Since the symmetries of H include the symmetries of H' as a subset, we may as well look for these operators that commute with H. The rotational invariance of H leads evidently to the conservation of the angular momentum
j i I) + IPLY.i L i " Eijk
[j!j)] . iE,~jk
r Jv k _ ¢ r i
r
(49,) (491))
168
Now the spin motion " i
Ao
=.
0 Hamiltonian
H0 is known to admit the following constant of
½Eijk(vJLk-LJvk) - ?
ri
(SO) -
q2ri ri r i v 2 - v i r J v J + y L i + "~2 - "~
We extended [7] this result by showing that H commutes with
Ai
-
_ A oi II4 Pc{ Etikzi k-
q
i
IE|
+
Z i}
(Sl)
thus providing a generalization of the Runge-Lenz vector to a system with spin. A i clearly transforms as a vector •
[j!A )] = ¢EiJkAtc
(s2)
and quite surprisingly, upon commuting A i with itself, we see that there is yet another vector quantity f~i which is conserved. Indeed,
[A!A i] = -I~ ill [2(H - ~ 2 ) d k + f2k]
(s3)
with ~ l = Pt[ ½ ~ i ¥ 2 - z j v j v i
+ EUkU(r) ~ j v k - } Uz(r)2i]
(54)
Note that every term in (~i is affected by a spin matrix. Note also that pRzi trivially commutes with H since H0 is spin independent. Now a problem remains unsolved : Why have we added a term I/2q 2 to the Hamiltonian ?I The answer to this question will become apparent in the next section. Let us just say as a hint that with this addition the spectrum of H' precisely starts at zero and is positive. 2. Suoersvmmetries of H At this point, we would like to obtain the commutation relations satisfied by the symmetries of H' : PLJi, PLAi , PL~ i. The evaluation of these commutators can be greatly simplified by making use of the fact that H actually possesses an invariance suoeral~ebra. First, one can check that the supercharges
o, -V~2(l'/Iv( + "1°u(r))
(sse)
169
satisfy {Oo~,O~} ffi 2 8 ~ H
o~,p =1,2
(56)
(We use a standard chiral representation for the Dirac matrices.) Since Qcx and A i both commute with H, so must their commutator and indeed we find that [ Ai, QQ] gives two new conserved vector supercharges. These are i = Oo~ 2iq [A!Oo~ ]
(57)
O~ v = ~i I, ,:,,o i i + l EiJk'YJvk -i. "fiU(r))
(s~)
with
i
Q2" i
y~
i
(sob)
Q|
Note also that
[ PR Y. i ,Qg] = - i Q g i
(59)
We can now form the quaternion ~=
(6O)
Qoc~12 + ( Q i o i
to see that the supercharges transform according to a spin I/2 representation •
[ji
=
This is in contradistinction with the situation encountered in part supercharges were rotation scalars.
(61> A where the
3. Structure relations We are now in a position to describe the structure relations satisfied by all the above invariant operators. In order to arrive at these relations, it suffices to evaluate the anticommutators of odd generators and the commutators between odd and even operators. Once these quantities are known, the Lie product on the "bosonic" subalgebra can easily be deduced. Since the conserved supercharges are of first order in the velocities, the procedure for determining the commutator of PLJi , PLAi and PLr~i is clearly simpler than a brute force calculation.
170 For the sake of illustration, we shall take H ~ 0. (Although the ground state needs to be considered separately, its treatment is completely analogous to what will be presented below,) Let us define
MI" ½(o'+ ~ ~' + p(-,,A' + ~ ~'1)
<,~,>
M~- ~( i ji + ~ e
<,2b>
Mi- -Z~Q
- ~(-qAi+hei))
i
i
(,20
M4i .. ½ p)¢y.i
<62d)
with
p = (,_2q'H)-½
<,3>
In terms of these operators, one obtains the following set of structure relations for our constants of motion • {Oo~,Op} ffi 280c13H
{Qi0~' Q~3] J
I
2 80~p 8
{Oo~.oiO}
i =" "4H8o~ p (N3i + ['14)
ii H - 4E0~DEi)k H(M 3k_
M~)
[ji. ¢)o~] = ½¢)o~<~'
[M~,©o~ ] - [!'1~.¢)oJ-, o
[M,,* ~o~] ,,- ½ ¢l~.c'i
[M,( , ~o~] =
[ji, oJ ] . Lsijk.] k
• [j',M i ] I. |Eijk[,l, k
i
_½<~i
(6~) (641))
0o~
i Eijk k [Ma,M~] = tS.b M,
(640
i] =o
[,/5, 00~] = -ziO= a -
(6,~)
1,2,3,4
Let us observe that' i) At fixed energy, the operators ji, Mai, ~(~ and ~5 form a closed set under the graded product. The superalgebra thus obtained is identified as being SU(2/2)~)SU(2)~)SU(2).
171
ii) The symmetries of the Hamiltonian H', namely { PLJi, PLAi , PLni } or equivalently { plMIi , pLM2i , pLM3i } form by themselves the algebra SU(2)eSU(2)eSU(2). 4. Spectrum analysis a la Bargmann We will now apply the Pauli-Bargmann method to show that the spectrum of H' is given by -I E -
~"~2
t>=q'q+t . . . .
(65)
In the process, the 2(pZ-qZ)-fold degeneracy of the p-level will be completely accounted for. Here also we shall restreint to H # 0.
From the definition of the operators Mai, we have
(6~)
+ p2q2A2 - ~ p()* P)A~ -2qpJ-A} M22 =
M~(P --)-P)
2
~2
M3 = ~2
(66~)
(6e)
With a little bit of effort, it is possible to show that £=22. 3 H2
J.£'2= A'~
(67o)
AH * - ~ J . A = ½ H
+
= = 2(J'A)H + -~J'~
2 - 2 + 2qH +
(67b)
H
A 2 = 2H(J2-q2+~) - ¼2J 2+2HA+2HA 2 • 2H * ~ - J . A
(67c)
(67d)
with A
=
XiL"÷zq XiLi+) r
(6e)
Substituting in equation (66), one then finds
Ma =
%-0(%+1)
(69)
172 with !
% =
''½÷ I] -
$3 = 2
(70¢)
From the representation theory of SU(2), we know that these Sa are the dimensions of the three SU(2) representations generated by ['lai, a = I, 2, 3. From eq. (70a), it is immediate to obtain I
H =
~-~2
!
2(Sl + q)z
('/1)
Setting
s t = (P = q)
(/2)
and substracting the I/2q z that had been added, we therefore find H
S m
~|
2p 2
p= ¢ + l , q +2. . . .
(73)
As for the degeneracies, these are given by the product of the dimensions of the corresponding three SU(2) representations, For the level p, we have
sl. sZ.s 3 = s 1. (Sl+ z¢).2 = (P - qXp + q),2 = 2(p 2- q2)
(74)
We should point out that the degeneracies of the ground state can similarly be analyzed (see ref. [8]).
AcknowledRements One of us (L.V.) would like to thank H.J. de Vega, N. Sanchez and R. Kerner for their kind invitation to give a talk in their seminar series. We also want to ackowledge the help of M. Mayrand in the preparation of the manuscript. This work has been supported in part through funds provided by U.S. Department of Energy (D.O.E.), the Natural Science and Engineering Research Council (NSERC) of Canada and the Quebec Ministry of Education.
173
References
[ 1]
For a review see E. D'Hoker, V.A. Kosteleck~ and L. Vinet in "Dynamical Groups and Spectrum Generating Algebras" edited by A. Barut, A. Bohm and Y. Ne'eman, World Scientific (to appear).
[2]
E. D'Hoker and L. Vinet, Phys. Lett, 137B. 72 (1984).
[3]
E. D'Hoker and L. Vinet, Lett. Math. Phys., 8,439 (1984).
[4]
E. D'Hoker and L. Vinet, Commun. Math. Phys., 97. 391 (1985).
[5]
E. D'Hoker and L. Vinet, Nucl. Phys., B260, 79 (1985).
[6]
L. Benoit, M.Sc. Thesis (Universite de Montreal) unpublished (1986). L. Benoit and L. Vinet, A supermultiplet in a Coulomb potential : states and wave functions from the representation theory of OSp(2,I ), LPNUM preprint (to appear).
[7]
E. D'Hoker and L. Vinet, Phys.Rev.Lett., 55, 1043 (1985).
[8]
E. D'Hoker and L. Vinet, Lett. Math. Phys., 12, 71 (1986).
CLASSICAL r-MATRICES,
LAX EQUATIONS,
POISSON LIE GROUPS AND DRESSING TRANSFORMATIONS
M.A.
Semenov-Tian-Shansky *) **)
ABSTRACT We discuss the theory of Poisson Lie groups which provides a natural framework for the study of integrable Hamiltonian systems on a lattice and of the dressing transformations in soliton theory.
Classical originally Quantum
r-matrices
introduced
Inverse
of a classical
range
of
classical
To
/2/
describe
integrable
are the semiclassical
R.
Scattering
notion
Drinfel'd
by
Baxter
Method
r-matrix
a
fully
developed
step
was in
the
relevance
of
systems
recall
that
then
a
counterparts
exploited
by L.D.
was originally
r-matrices
as
and
classical they
classified
usually
which seem at the first glance unrelated.
(a)
is
a natural
Poisson
by
bracket
on
the
the
the
and his disciples.
for
The
the
space
/i/. A wide
Be lavin
quantum
and
by
and
V.G.
R-matrices.
study
characterized
phase
frames
Sklyanin
A.A.
of
r-matrices
are
of
by E.K.
classification
properties There
Faddeev
proposed
of quantum R-matrices
within
of
classical
the
following
the
equations
of
motion are Hamiltonian. (b) All
integrals
of motion
arise
are in involution with respect (c)
The
solution
of
the
as spectral
invariants
of a linear operator.
They
to the Poisson bracket.
equations
of
motion
may
be
reduced
to
some
kind
of
Riemann-Hilber t problem. As was first noticed by the present author /3/ the r-matrix ral case
link between when
Poisson
the
these
Poisson
submanifold
properties. brackets
are
The
simplest
linear,
i.e.
approach provides a natu-
theorem of this kind deals with the the
phase
of a dual space to a certain Lie algebra.
monly known in the literature.
space
is realized
This case is most com-
For this case the theorem may be regarded
*) Ecole Normale Sup~rieure - Paris **) On leave of absence from the Steklov Mathematical Talk given at the Paris-Meudon Seminar Series.
as a
Institute,
as a slight
Leningrad - USSR
175
generalization However,
of the Kostant-Adler-Symes
commutativity
theorem /4/.
the r-matrix approach naturally leads to a very interesting
new class of Poisson brackets that are non-linear. Poisson brackets
This class of
(eventually ref@rred to as the Sklyanin brackets,
/I/ ) is particularly
well suited to the study of integrable
on a lattice arising naturally as lattice approximations integrable models, V.G.Drinfel'd
or the semiclassical versions
Poisson Lie group°
+)
of quantum magnetics.
concept,
original
that of a
.
The geometry Of Poisson Lie groups has an interest ticular we are naturally led to generalize reduction,
of its own. In par-
the notions of Hamiltonian
the moment map etc. So far, several applications
concepts were already indicated (a) The description
systems
of continuous
/5/ was the first to notice that Sklyanin's
definition leads to an important new geometric
cf.
of these
:
of the symplectic leaves of Poisson Lie groups
due to Drinfel'd. (b) The geometrical
theory of Lax systems on the lattice,
due to the
present author. (c) The theory of dressing transformations. We shall give a few comments on the latter point. Recall that dressing transformations (However,
were introduced by V.E.Zakharov
they have made no emphasize
ties). In a different
and A.B.Shabat /6/
on their group theoretic proper-
setting they were rediscovered
in a now famous
series of papers of the Kyoto scientists /7/. An excellent geometrical treatment
of the subject is given
of dressing transformations brackets
in
/8/,/9/.
The puzzling property
is that they do not respect the Poisson
on the phase space. It is particularly
interesting
to under-
stand this phenomenon in order to decide whether the dressing transformation group survives quantization. the case if they preserved
Normally this would have been
the Poisson brackets.
so, the situation becomes obscure.
As it happens,
Since they fail to do the theory of Poisson
+) I prefer to rectify the original term introduced was "Hamilton Lie groups". the natural actions
This latter term
by Drinfel'd which
seems ambiguous since
of Such groups are not Hamiltonian.
176
Lie groups provides a clue for a natural characterization
of the beha-
viour of Poisson brackets under the dressing transformations most all the technique available
/10/. Al-
by now is used for the proof, so it
seems to be a good example to introduce the reader to the domain which I believe is still promising. The present lectures are organized as follows. We start with the definition
of classical r-matrices and the proof of
the generalized Kostant-Adler-Symes
theorem. Applications
of this
theorem to particular dynamical systems are now quite numerous /11/, /12/, /13/), so we shall not dwell upon this. Instead, two examples of r-matrices
(see
we display
that are less widely known, the rational
and the elliptic ones. Our exposition
combines ideas borrowed from
/3/ and /4/. It is based on my recent
joint paper with A.G.Reyman
/15/. These examples actually will not be used in the sequel,
so the
reader may skip them. ~n extension to systems depending on a continuous x parameter is briefly indicated. Our second major theme is the definition
of Poisson Lie groups. The
main results here are due to V.G.Drinfel'd study of Lax equations
dressing transformations. /10/. In order to the continuous
of
The latter subject was already exposed in
diminish the overlap I shall consider here mainly
case rather than the discrete one.
Acknowledgements
.
Conversations
years were extremely important sincere gratitude
with V.G.Drinfel'd
during the past
for me. I also wish to express my
to Prof.L.Breen
Ecole Normale Sup~rieure
§I.
/5/. We proceed then to the
on the lattice and the Poisson properties
and Prof.J.L.Verdier
and to the
for their hospitality.
DOUBLE LIE ALGEBRAS AND THE GENERALIZED KOSTANT-ADLER-SYMES THEOREM
1.1.
Generalities.
Throughout
the
paper we shall deal with the
Poisson brackets. Recall that by definition bracket on a smooth manifold ~ C~
(~)
which satisfies
The degenerate
is the structure
brackets are not excluded.
f ; ~-~of
of a Lie algebra on
the Leibnitz rule
lying in the center of C ~ ( ~ ) mapping
(cf. /16J) the Poisson
By definition,
functions
are called Casimir functions.
A
two Poisson manifolds is called a Poisson
177 mapping if
~i.~) t~..i-, I',~ ~,,,. A submanifold V o f ~ { i s
= f~, *~.x,'~
for any
'f' ~
C~(~)
called a Poisson submanifold if there is a
Poisson structure on V such that the natural embedding V c . , ~ is a Poisson mapping. Such a structure is clearly unique~,if it exists.Every point x ~
is contained in a minimal Poisson submanifold which carries
a symplectic structure. These symplectic manifolds form a stratifica ~ tion of
~'1,.
The best known example of a Poisson bracket is the Lie Poisson bracket defined on the dual space to a Lie algebra ~
. By definition,
(1.3) ~ ~, ~ ~ ~) = b ([ ~?(L), a,#(L) ]), I',~F~ C~(~9 *) . (Obviously, d ~ (L) ~ ( ~
)*
= ~
). The symplectic leaves of the
Lie Poisson bracket coincide with the orbits of the adjoint Lie group in the space ~ e . ~-
By an abuse of language we shall simply call them
orbits.
Other examples of Poisson structures will be given in §3 • 1.2.
Definition of double Lie algebras.
Definition 1.1 tor acting o n ~ .
:
Let ~
be a Lie algebra,
We shall say that R
double Lie algebra o n ~
R ~ E~J~a
linear opera-
defines the structure of a
if the bracket
(1.4) [×' Y]R satisfies the Jacobi identity. ~n that case there are two structures of a Lie algebra on the same linear s p a c e ~ .
The Lie algebra
equipped with the Lie bracket[~]Rwill be d e n o t e d ~ R . there are two Poisson brackets on the dual space brackets of the algebras ~
~ ,
Consequently, the Lie Poisson
, ~R
We denote by I(CO~)the space of Casimir functions of 2 " As is well known, I ( ~ )
coincides precisely with the space of
-invariants in C ~ ( ~ ) .
C ~ _
The following theorem serves as a main
motivation of the definition above. Theorem 1.1. (i)
Elements of I ( ~ )
are in involution with respect to
the Lie-Poisson bracket o f ~ R . (ii)
Let h ~ I ( ~ ) .
The equation of motion defined by
with respect to the Lie-Poisson bracket o f ~ R
has the generalized
h
178 Lax form
~k
(~ .5)
~'--E-
=.
M.L
Z(aJ) and put x± = dhi(~) , L ~ ~
Let h 1 , h 2 ~
Proof. (±)
/ ' ~y
definition ,
Z Now, since the functions h l
are a d ~
- invariant both terms vanish.
(ii) By definition, the hamiltonian equation of motion d e f i n e d by a function h @ ~ iV)may be written in the form
(I .6)
~L
~
=
From (1.4) it is clear that
(1.7)
~_~
.
Since h ~ i ( ~ )
_~ ~
=
RX.
t
• ~
the second term vanishes and we arrive to (1.5).
The geometrical meaning of theorem 1.1
is fairly simple. Recall that
in the present case there are two systems of orbits i n ~ and of ~ R
~ , those of
" Theorem 1.1 says essentially that hamiltonian equations
of motion with Hamiltonians h ~ I ( 9 )
respect both. Indeed, formulae
(1.5), (1.6) mes~u that the velocity vector is always tangent to the intersection of a g - o r b i t
and a 9 R - o r b i t
of L ~
. In many cases
of interest these intersections coincide precisely with the Liouville tori for our H ~ i l t o n i a n
(of.
/11/
systems which implies complete integrability
) .
So far we have not specified the conditions on R imposed by demanding that (1.4) satisfy the JacobL identity. We choose to impose the following sufficient condition which is called the (modified) classical ~ang-Baxter identity:
for any X,Y
~
. It is straight forward to check %hat (1.8) implies
the Jacobi identity for (1.4). Indeed for any X, Y, Z 6 ~
= _ [~,[y,~]
] -c.p.
=
o.
The motivation for the choice of condition (1.8) which is of course stronger than merely demanding the Jacobi identity to be valid is provided by the global version of Theorem 1.1. which we are going to
179 formulate. We start with the following simple statement. Proposition 1.1. mapping
Q
(i) Suppose R ~ E n d
(1.8) then the
eg:
..,
:
~satisfies
is a Lie algebra monomorphism. (ii)
Any X ~ a d m i t s
(11oI ~ith
a unique decomposition
X (X+ , X_ ) E
Proof.
We write
(ii).
=
X,
~
~'~ .
- X-
X~(~X~).
In particular,
Clearly , X+ - X_ = X
iR is a monomorphism.
which implies
The f o r m u l a [ [ ~ / ~ ) + ~ I ~ l T ± ]
is a direct consequence of (1.8) . Let G(G R) be a connected simply connected Lie group with the Lie algebra ~ ( ~ )
. Extend iR
Let m : G R _ ~ G
be
Proposition 1.2.
to a Lie group monomorphism iR : G R ~ G X G -I x ~_, (x+ ,x_) ~_~ x+x_
the composition map (i)
The image of m is an open cell in G.
(ii) For x E ~ t h
(1.11~
~=
Z * * -~ --
)
e decomposition
(Z+, ~_) ~ 2~ ~
is unique. We are now able to state the global version of Theorem 1.1. Theorem 1.2.
Let h 6 I ( 9 ) .
For L ~
put
~
= dh(L). Let ~+(~)~(~)
be the solutions to the factorization problem (1.11.) with the lefthand side given by (1.12)
~
(~)
=
~
~ ~CL
The integral curve of the equation (I .5) with the Hamiltonian h starting at
L ~
is given by
We shall outline the proof that will be eventually generalized to the case of Poisson Lie groups considered in
§5.
Let T~G be the cotangent bundle of G . There are two natural actions of G on T~G by left and right translations.
Both actions are Hamilto-
nian with respect to the standard symplectic structure. Hence there are two moment
maps ~ 1 ~ r
:T~G _, ~
. The following statement
is of course well known (cf. /16/). Proposition 1.2. '~
Equip ~ ~ with the Lie Poisson bracket and let
be another copy of ~
equipped with the opposite Poisson bracket
180
Then
~t/
(i.14)
T*Q
~
~F
#
is a dual pair of Poisson mappings. In particular, we may regard~r
as a Hamiltonian reduction map with
respect to the group action which corresponds to ~ Now choose h e I ( ~ )
and vice versa.
and extend it to a function ~ C ~ ( T G )
which is
both right- and left-G-invariant. Lemma I
Fix a trivialization of T~G _~ G x ~ ~
translations.
The Hamiltonian flow on T~G
by means of left
defined by ~ is given by
the formula
(I .15) In a trivialization chosen the moment maps /Q~. ~ . are given by
Proof. (1.16)
#t4~' ( ~ , L ) ~
A~:~.L
(X,L)
l
_L
Now, consider the action of the group GR on G given by
where
h~
h~
are the standard homomorphisms
GR--~G defined above.
This action clearly extends to a Hamiltonian action on T~G ~ G Lepta 2 . (i)
The mapping T~G _ ~
x ~
given by
•
J
defines a cross-section of the action (1.17) over the open cell G+.G_ x 9 ~
T~G
(ii) The quotient Poisson structure on ~ ~ coincides with the Lie Poisson bracket of ~ R " (iii) The reduced Hamiltonian o n ~
~
corresponding to
is h itself. The quotient Hamiltonian fl0w is given by (1.13). The proof of (i), since map
m
(iii) is obvious. To prove (ii) let us notice that
m : G~-~G :
is an immersion it naturally extends to a symplectic
T~GR
the standard
--~T~G. This mapping intertwins the action (1.17) with
action
GR x T G R
bE left translations. Hence (iii) is
corollary of Proposition 1.2 Note .
The reader will eventually see that
most of these arguments
extend to the case of Poisson Lie groups. This does not apply to the last argument, so we shall replace it by a straightforward
computation
181 (which is fairly possible in the just considered case as well). § 2,
EXAMPLES
2.1.
The following class of r-matrices satisfying (I .8) is by
most important. Suppose ~ +
(2.1)
~.
,J_ ~ ~
far the
are Lie subalgebras such that
+ ~_
~.
as a linear space. Let
P±
be the projection operator onto ~ ~
paral-
lel to the complementary subalgebra . Put
(2.2)
R--
?,-
P-
The check of (I .8) is straightforward. I .I.
Tn this particular case Theorem
coincides with the Kostant-Adler-Symes
commutativity theorem.
We now give some examples of decompositions
(2.1) which lead to nume-
rous applications. Let ~
be a simple Lie algebra, and let
~(~)
= ~
~4 ~ [ ~ ~ ~]
be
its loop algebra. Fix a graduation
(2.3)
~ (%)
j6z. "t. ?t 0
Clearly,
(2.4)
c~ (0-~)-~
~
~
(OJ)).l.
3
-'1" o~(¢b~)_
<"o
}
so any graduation gives rise to an T-matrix of the form (2.2). Let us indicate in passing an example of an ~-matrix satisfying (I .8) though not of the form (2.2). Suppose that (2.3) is a principal graduation and put
~_~_ -- ~J~O ~ ± ~ "
algebra in ~ .
Let
P+
j
~--~o
. Clearly, f is a Cartan sub-
' Pc be the projection operators onto k± , f
in the decomposition Proposition 2.1.
(2.5)
Fix
R~
satisfies
=
~End
~ such that
U.,. -P_
det(1 -~)) ~ 0 .
Then
t C4,e}C.~-e)-' P,,
(1.8) .
Note that (2.5) differs from an opera~or of the form (2.2) by a finite dimensional perturbation. A theorem
due to Belavin and Drinfel'd as-
serts that all graded r-matrices on loop algebras are essentially of this type. See /2/, /3/ for more information on the subject. 2.2.
There is an extension of the decomposition
Cherednik /14/. For
~/- ~
C~
=
~
~
~
~
(2.5) suggested by let
~be
the local
182 i.e.
parameter
X~=
X-~ ,
For a fixed finite set ~ C (2.6)
~
, ~ .
~ ~
=
put
e
Put
9 ~ C [Bj~l~r]]
Put 9 v
•
~
and let ~ in ~
(~)
be the Lie algebra
which are regular outside m
of rational functions with values .
Let
i : ~C~)
-~)
the
natural embedding which assigns to each function the set of its Laurent series at points Proposition 2.2.
~r& ~
.
There is a direct sum decomposition
This is merely a reformulation of the well knownMittag-Leffler for ¢ ~
theorem
.
Hence we may associate with (2.8) an ~-matrix of the form (2.2). This leads to the so-called Lax equations with rational spectral parameter. 2.3'
As the construction above suggests, we may try to replace
by analgebraic
~
curve of higher genus. However, in this case there is
an obstruction to a decomposition of the form (2.8), as it follows from the Mittag-Leffler
theorem for curves. We shall briefly indicate how
this obstruction may be avoided for elliptic curves. This leads to the so-called elliptic r-matrices /I/, /2/ . Recall first the so-called Heisenberg representation.
(2.9)
---
I
( ° ' O "'.4°)
For a =(e~.a~) e ~
× Z~
put
I~
defines a projective representation of
(2.1o) Z ~ =
I~E.~
~-
~-~ Z~
Z. The assignement a ~ × Z~
~
:
<~,~>~ ~ - ~
The associated r e p r e s e n t a t i o n of ~ x ~ i n End ~ to the r e g u l a r r e p r e s e n t a t i o n . One has, obviously,
Now, let ~=
C/Zco~
~=
sl(n, C ) .
t Z ~
Let m
i s equivalent
be an elliptic curve,
. We i d e n t i f y ~
of points of order n. Fix a finite s e t ~
X~with
the subgroup ~ C ~
r which does not contain
183 any points
which are equivalent modulo
_['I •
Let 6 (~)be the Lie
algebra of rational functions on ['I which ~re regular outside ~ . and satisfy the automorphy condition
(2.:2)
~ C~,~)
=
Z~
X'(,~) Z~. -~
As above, let
~ - = ~ ~ ~ [~-~,~.]~ ¢~+= ~ ~
Clearly, ~C~)is naturally embebbed into ~ D Proposition 2.~.
(2.12)
Q'3~
Proof .
[~
].
•
There is a direct sum decomposition
=
~ (~))
@ ~'~ *.
By the Mittag-Leffler theorem a rational function on ~ w i t h
the prescribed principal parts
~
at the points
F~ ~ exists
~
if and only if
I~.13)
~
This is immediate
~
×~ ~ = o
from (2.11) since
tr X ~
= 0 .
Hence the decompo-
sition (2.12) exists. The uniqueness follows from the irreducibility of Heisenberg~representation. Let~
be the unique rational function which has only simple poles at , satisfies the functional equation
and is normalized by the condition
~.
~
~
=i
.
Put
o-..~ o
Proposition 2.4.
The projection operator onto ~ is given by the formula
where the linear operator
I_a ~ i a
(. ~))
is acting by
parallel to ~
X __> I_a tr(XI a ).
Proposition 2.4. establishes a link between the formalism which is used in /I/, /2/ and our present approach. Originally the r-matrices were considered as functions with the values in J ~
~
. In our
approach we associate with such functions linear operators. The ellip-
184 tic r-matrix (2.15) was indeed (for n = 2) the first example of a classical r-matrix 6ver studied /I/. 2.4
Let us now indicate how the present formalism may be used to
produce integrable Lax equations. (a)
We proceed in several steps.
The pairing
Y.,-
y > -
(2.,7)
tr,~ D
"~
is non-degenerate and allows to identify ~ )
with its dual. Note also
that (~#3~) ~ _~ ~(~). Another model fer the dual to ~ by ~ -
- ~.~
~
~
[
~
is provided
The two models are related by a map
which assigns to a rational function on ~ satisfying (2.12) the set of its principal parts at ~ 6 ~ . space
~b
The Poisson submanifolds in the
are easy to describe. In particular, we have
Proposition 2.4.
Functions with simple poles at ~-6 ~
submanifold of ~ ( ~ ) a ¢
(~+)~.
The symplectic leaves lying in it
coincide with the eoadjoint orbits of (b)
Let I ( ~ )
form a Poisson
be the algebra of
~ r
I ~
-- S ~
Ad G-invariants on ~
The algebra of Casimir functions of ~
(~I ~ ) = ~(~l~)
is generated by the func-
tionals of the form
%
where
~ r(O..3 ) ,
o~v.-e e [ a¢ -4 , gCJ~, L=C~..,v-~@~6~
By restricting these functionals to the orbits described above we get Hamiltonians in involution giving rise to Lax equations of the form
(2.i9)
d._..~L = [ L,P'I ]
~
[, ~ £ &(£)
/ t'l=P ° (d~,9[[])
They are usually referred to as Lax equations with the spectral parameter on an elliptic curve. By applying Theorems 1.1, 1.2, we may systematically construct such equations and their solutions (cf./15/). 2.5.
The examples considered so far give rise to finite dimensional
systems admitting Lax representations L~
d[/~
= [ [l ~ ]
#
where
are matrices PoSsibly depending on spectral parameter. In many
cases it is natural to assume that Li~ variable
also depend on a spatial
x . Lax equation then takes the form
~
i ~
~ [ Ll M ~
.
There is a natural way to include such equations into the present iormalism.We explain it in brief since it will be of importance in the
185 study of dressing transformations (see § 6. below). Let ~
be a Lie algebra with an invariant scalar product. It will be
convenient to assume that ~ i s
a matrix algebra. We denote by G the
corresponding matrix Lie group. For the time being the reader may assume t h a t ~
is finite dimensional. However, in realistic applications
is always a loop algebra (see below). ~=
C°O(~/Z
~ ~). Suppose R ~
~
C ~C/~/~/
~/
End ~ s a t i s f i e s the Yang-Baxter
identity (1.8) . We extend it t o ~ ~
Put ~
by setting (RX)(x) = R(X(x)). Let
be the corresponding Lie algebra with the Lie bracket (1.4).
There is a 2-cocycle on ~ d e f i n e d
by
Y).
(2.20) Let Put
oen,
,.e
ex,eo
(2.21) ~ C ~ , y )
*oo
ooo o e
¢/
~ L ~(X,y)
~ 4_ ~ C x , R y )
Proposition 2.5. (i) Formula (2.21) defines a 2-cocycle on ~ ( i i ) L e t ~ # be the corresponding central extension o / / ~ Then (~/~l ~ ) is a double Lie algebra. It is particularly nice when the operator R is skew with respect to the inner product o n ~ . sion
~
In that case
c0~ = 0 , so the central exten-
splits. Hence the orbits of %
they are clearly
and
%
coincidei
"continuous products" of orbits o f ~ .
Since in the sequel we shall be dealing almost entirely with this case, it is worth giving a formal definition. Definition 2.1 algebra if
A double Lie algebr: (~, ~
(i) the operator
Baxter identity (1.8) duct on ~
and R
R ~
End ~
) is called a Baxter Lie
satisfies the modified Yang-
; (ii) ~here is a (fixed) invariant inner pro-
is skew with respect to it.
Let us now describe the Casimir functions on ~ Proposition 2.6.
Let us i d e n t i f y ~
.
with its dual by means of the
inner product
so that
--
@ ~. The coadjoint action of
by
(2.22) ~
X.(L
~)
=
(
I X , L]
on
,~X,
is given
o),
186
It integrates to the action of G
(2.23)
A~ ~
Notice that
given by
cI
=
-~
,
(2.23) coincides with the gauge transformations which are
connected with the linear differential equation
(2.24) Let
L~L
be the fundamental solution to (2.24) normalized by the condi-
tion
(2.25)
~ L (O)
=
~
•
(the identity matrix)
By definition, the monodromy matrix
T(L) = ~L(1).
Theorem 2.1. (Floquet).
Two points (L,e), (L',e') (e ~ 0)
same coadjoint orbit in
~#
if and only if
T(L), T(L') are conjugate in Corollary . where ~ Note.
e' = e and the matrices
G'.
The Casimir functions on
~@)is
are of the form L - - ~ ( T ~ L ) )
a central function.
it is clear now that the codimension of orbits in
equal to rank
~ L = ~ o There are also precisely
~
the algebra of Casimir functions on each hyperplane
~
lie on the
~
is
generators of e~= const ~
0 in
. Hence to get sufficiently many integrals of motion provided by
theorem 1.1. we must assume
~ = cw~ . This is the ease when ~
is a
loop algebra. Theorem 2.1.
shows in particular that our geometric approach incorpo-
rates the conventional inverse spectral transform methods which are based on the study of the auxiliary linear problem (2.24) . An extremely important point is the study of Poisson properties of the monodromy map which we now state. For ~ ~
~
~G ( )let
nition ~y l ~ ,
~l ~t
~ ~
be its left and right gradients. By defiand
I Theorem 2.2. tionals [_~ ~
x)=IgJ
'
Let ~ 4 , ~ ~ C "~ ~ G ~ • The Poisson bracket of the func(T(LI)
is given by
C~×,, x;)-,, C~c~,),x-), )<< = ~
(.T(L))
,
(
X<
(
= 'gy,: CTLL}),
187 In particular, the right hand side satisfies the Jacobi identity. We shall give a proof of a more general formula in §6 . For the time being (2.27) will serve us as a motivation for the following definition. Definition 2.3.
Suppose
G
is a Lie group and that there is a struc-
ture of a Baxter Lie algebra on its Lie a l g e b r a ~ . bracket on
~
The Poisson
is defined by
Formula (2.28) will be referred to as the Sklyanin bracket on Corollary . Equip
T;
~
~.
with the bracket (2.28). The monodromy
~.... CT: L--~ T ( L )
is a Poisson mapping.
We shall study the properties of (2.28) directly in §3 o Proposition 2.7.
Multiplication in
~
induces a Poisson mapping
G × G ..~ ~ Although the direct proof is quite simple we shall give a 'physical proof' based on theorem 2.2. Observe that functions on a small patch q-~ C ~ / Z say.
if
~A ~ ~ =
T(L) = T(LI) T(L2) Proposition 2.7
.
~
supported
form a Poisson subspace in
~, then
manifolds. Finally, if
L ~
~ =
I~U~
LI + L ~
= ~4 i
~
~
~" ~
~
~
# e.~ ,,~"4
as Poisson
/
then
Now our claim follows from (2.27)
was indeed the key motivation to introduce the notion
of Poisson Lie groups. We shall return to the study of equation (2.24) in §6
and proceed now to formal definitions.
§3 •
POISSON LiE GROUPS. DEFiNiTiONS AND PROPERTIES.
3.1.
The following definition was already anticipated in the proof
of proposition 2.7. Definition 3.1. manifold
~ x ~
The product of two Poisson manifolds
~ 4 ~
is the
equipped with the Poisson bracket
In other words, (3.1) is the unique Poisson structure on
~1~J~such
188 that
(i)
natural projections
are Poisson mappings
C~
~
(ii)
i ~wq~,
h°~ ~ C ~
Definition 3.2.
: 0~4 .JW~
&: ~ = ~
~t ~ ~
5
=
O
for any
(0%).
Poisson Lie group is a Lie group equipped with a
Poisson bracket such that
(i)
mapping
(ii) inversion
~
~
~
~
--* ~
multiplication in ~
defines a Poisson
x ~-~ x -I changes the sign
of the Poisson bracket. Examples
3.1.
(i)
Any Lie group equipped with a zero Poisson bracket
satisfies the axioms.
( i i ) Let ~#1 be the dual space to a Lie algebra
equipped with the Lie Poisson bracket. We regard it as abelian group. Then the axioms are also satisfied. A much less trivial example is provide~ by (2.28). We shall come up to its study later. Let ~ i
~z
an element
be the left and right translation operator on C ~ ) ~
~
: ~
Multiplication in
~
~
induces a Poisson mapping
T(~) ~ ~C~
)
, ~
~(~) ~x
~-@
by
= T ~ ) @
if
Recall that any Poisson bracket is bilinear in the derivatives of functions. It is convenient to write down a Poisson bracket on a Lie group in the right- or left-invariant frame. Define the left and right
C~:'°~@')
differentials of a function ~
Let us define the Hamiltonian operators
by the formula
~l'~l
: C.--,-~. g .
which correspond to our bracket by setting
t ,e,
=
Proposition 3.1.
<
Suppose @
b y , 09, ) - ~#~t
is a Poisson Lie group.Then functions
satisfy the functional equations
(x.,~)
=
A-,a. z °
(3.5)
_4 •
Proof .
Obviously, !
~.
~t~)
=
I
189
Clearly, (3.2), (3.6), (3.4) imply (3.5) 3.2.
Functionalequations (3.5) imply in particular that
4]1(~ ~ O
~(a) =
, hence the Poisson structure on ~
is
always degenerate at ~he unit element. By linearizing the Poisson bracket at the point
e
be more precise, fix
~4 ~z ~ J
such that
(3.7)
~C
( ~ ) = ~d
[
and choose
~
~
of ~ ~
. To
C ~ )
. Put
=
{
<,
Formula (3.7) defines the structure of a Lie algebra
Proposition 3.2. on ~ M
we get the tangent Lie bialgebra
.
Proof .
Formula~(3.4), (3.7) imply that
hence the definition is unambiguous. The Jacobi identity for (3.7) is obvious. Definition 3.3. is a
Let ~
be a
Lie algebra, ~
its dual. Suppose there
Lie algebra structure on ~ ~ , i.e. a mapplng . i~.~A ~~ ~
satisfying the Jacobi identity. The Lie brackets on ~ a n d
sa~dtobeconsistentifthedual~ap is a 1-cocycle on ~ ,
(9'
~)
~--,
~
are
~ a j = ~
,
i.e.
(3.9) ~ C x , Y])= A pair
~:
~
~C~x').~y)Jv~Cy)°~X' -~y°~ - -
~C×)~
with consistent Lie brackets is called a Lie bialgebra.
Lie bialgebras form a category in which the morphisms are such mappings
p : ~__~
~ that both p and
p~ : ~ __~ ~
are Lie algebra
homomorphisms. Theorem 3.1. /5/
(i)
Formula (3.7) defines t h e structure of a Lie
bialgebra on ( ~ ,
~).
( We shall refer to it as the tangent Lie bi-
algebra of ~
).
(ii) Conversely, let
(~, ~ )
be a Lie bialgebra,
a Lie group corresponding to ~. There is a unique Poisson Lie group structure on ~
such that its tangent Lie bialgebra is ( ~
~)
(iii) The correspondance between the Poisson Lie groups and Lie bialgebrae is functorial.
190 Sketch of a proof.
Let
~ be the Hamiltonian operator which corres-
ponds to the Poisson bracket o n ~
. From (3.8) we get
Now, (3.5) implies (3.9). To prove the converse statement we must integrate the 1-cocycle on aj to a 1-cocycle on is trivial i.e. r ~ Hom ( ~ ,
~)
~ )
=
~
~,Jh
~
-
. Observe that if
~o
~
where
is a fixed element, the corresponding 1-cocycle on C_~
is given by the obvious formula
(~) Operator
r
=
A4 ~ o ~
° A4*
~-4-
"~
is also called classical r-matrix. We shall see below that
the general case may be reduced to this special one (cf. the note following Theorem 3.3. below) 3.3.
As the reader may have already noticed, there is a difference
between the notions of double Lie algebras and Lie bialgebras. Since the former was motivated by applications to integrable systems and the latter by the geometry of Poisson Lie groups, it seems natural to combine the two. An appropriate class lying in the intersection are precisely the Baxter Lie algebras already introduced in §2 (Definition ~.2.
)
.
is a Baxter Lie algebra. The isoProposition 3.3. Suppose ( ~ , ~ ) morphism_~ ~ ~ induced by the invariant scalar product on equips ( ~ , Proof.
~)
with the ~tructure of a Lie bialgebra.
The cocycle
~ is in this case trivial and equals
=
Xo R
-
R o
X.
We now come up to the study of Sklyanin brackets (already defined in :§2) and their generalizations. Let ~
be a Lie group with a Lie algebra
invariant inner product
~ , 2 .~on ~
~
.
Assume there is a fixed
and identify
its means. Accordingly we shall use the notation
~
Proposition 3.4.
be skew and satisfy Formula
defines a Poisson structure on
G.
(1.8) .
~
__~t ----~_ as in
(2.26) for left and right gradients of functions on R, R' ~ End ~
with ~
. Let
by
Igi
Sketch of a proof.
The obstruction for the Jacobi identity to be valid
is a tri-linear form in the gradients. Consider first the left- and right-invariant brackets
The corresponding obstructions are given by ;
(3.~)
,
,
~cyclic permutation.
'
)~
+cyclic
permutation
The sign difference is caused by the fact that the Lie algebras of left- and right-invariant vector fields on G
are anti-isomorphic. By
virtue of the Yang-Baxter identity (1.8) the right hand side in (3.11) simplifies to give
Since the inner product on ~
is invariant, these obstructions cancel.
Specifically,the brackets 1
satisfy the Jacobi identity. The bracket
~ I
3~
i s t h e Sklyanin
bracket~already introduced in (2.28). The Hamiltonian operators which correspond to (3.10) in the rightinvariant (left-invariant) frame are given by
,4 In particular,
~R-~. is obviously a trivial 1-cocycle on ~
I'~' with the
J
values in End ~
. Hence we immediately get
Proposition 3.5 •
The Sklyanin bracket defines the structure of
a
Poisson Lie group on An indirect proof of this statement was already given in §2 . We shall denote the group ~
equipped with the bracket (3.10) by
G(R,R')
.
Let us now study the behaviour of the Poisson bracket (3.10) under the left and right transformations. First, we introduce the following important definition: Definition 3.4
Let
~be
a Poisson Lie group, ~
a Poisson manifold
192 An action
~x
~-~is
called a Poisson group action if it is a
Poisson mapping, the space ~
being equipped with the product
Poisson structure. We now turn to the study of (3.10) . Proposition 3.6.
The natural action of
~
on itself by left (right)
translations defines a left Poisson group action ~al_R)~ ~ i R 0 --~ ~ R I R ~ (correspondingly,
--,. ~(~,~,) ).
a right Poisson group action
Let us prove e.g. the first statement. translation operators by an element
~(RtRg~
Let ~ , ~
x G ~
~_~-,
)
be the left (right)
acting in C-~ ~ 3 .
It
suffices to check that
for any
~, ~ ~
C~CO)
i ~,~
~ (~
. Obviously, one has from (3.13)
(3.,5) Clearly,
(3.15) and (3.6) imply (3.14) •
Note .
We may convert a right Poisson action into a left one by chan-
ging the sign of the Poisson bracket on Proposition 3.6. provides us with the first example of Poisson group actions. Further examples will appear later on. It is useful to have an infinitesimal characteristic actions. Let ~
be a connected Poisson Lie group,
Lie bialgebra. Suppose that ~ C~J~)let ~ the Ist variable
).
generated by an element Proposition 3.7.
(~, ~)
is tangent
~ is acting on a Poisson manifold ~
(x) = dg~(g.X)g=1 g~ ~
of Poisson group
Let
( X
where dg
.For
means differential in
be the vector field on ~/~
X
The action of ~r is a Poisson group action if and
only if ~
for any
~,~
c~(~)
, X~
O"
The direct claim follows immediately from the definitions. if the right hand side is not zero identically in ~ / ~ the vector field
X
is certainly non-Hamiltonian.
~
Notice that C~(J~)
193
3.4.
The squares of Lie algebras.
Let (~, ~ R ) that 9 R
be a Baxter Lie algebra. Recall from proposition 1.1
is embedded into
~
@ ~
via (1.9) . Let
be the diagonal subalgebra. We denote
d = ~@
~
~ ~@
and equip
d with
the inner product
Proposition 3.8.
(it)
(i)
The subspaces
There is a direct sum decomposition
~ ~
' ~R c d
are isotropic with respect to
(3.17). Let
P$~
' P~R~
be the projection operators onto
' ~R
in the
decomposition (3.18). Put
Proposition 3.9.
The operator (3.19) is skew with respect to (3.17)
and satisfies the Yang-Baxter identity (I .8). Hence R d
defines the structure of a Baxter-Lie algebra on d. We shall
refer to (d, dRd) as the square of Note.
(~,
~R ) .
A similar construction works for arbitrary double Lie algebras
(with the effect that R d is no longer skew). This shows that by enlarging our Lie algebras we can always replace an arbitrary R ~ End satisfying (Io8) with an operator of the form (2.2). We leave it to the reader to show that by applying Theorem 1.1. to Casimir functions on d ~
restricted to
R
~
we get the same Lax equations
as in (I .6). Proposition 3.9 is a special case of the following more general result which holds for arbitrary Lie bialgebras. Theorem 3.2 /5/ •
(i)
Let (~, ~ )
be a Lie bialgebra. There is a
unique structure of a Lie algebra in the space (a) ~ ,
9~C-
d
d = ~
~
such that
are Lie subalgebras; (b) The inner product on
d
given by ~ ~ X~,~ ), C X ~ l ~ ) ~--- #~Xz )t~(~)is invariant. (it) Conversely, suppose d is a Lie algebra equipped with an inva~ riant inner product and ~ , ~ are two its Lie subalgebras such that (a) d = ~
~ (b)~,
~C
~
are isotropic subspaces. Then (~, ~)
194
(iii)
is a Lie bialgebra.
Natural embeddings ~ ,
~
d
induce Lie
bialgebra morphisms
also a Lie bialgebra. We shall
refer to ( ~ , 9 )
as the dual Lie bialgebra of (9'
order to obtain the global counterpart of Theorem first
3.2
~).
In
let us quote
the following corollary of Theorem 3.I.
Theorem
3.3.
Let H
be a Poisson Lie group and
K ~ H its subgroup.
Let k, f be the corresponding Lie algebras. Suppose that an ideal. Then
KCH
ki C
h~
is
is a Poisson submanifold and induced Poisson
bracket on K equips it with the structure of a Poisson Lie group. Its tangent Lie bialgebra is (k , ~ / k ~). By applying Theorem 3.3
to our present situation we get the following
result. Let ~ l Gl C-~# be the Poisson Lie groups whose tangent Lie bialgebras are (d, d_Rd),
(~,
~),
Proposition 3.10.
(~,
(i)
~)respectively.
Natural embeddings
~--~ ~ 2
~
C
are Poisson (anti-Poisson) mappings. Note.
Instead of using Theorem 3.1. to prove the above result we may
analyse the restriction of Poisson bracket to This permits actually to prove Theorem 3.1. cle ~
C_rtC~ ~ C
~YJdirectly.
Indeed, on ~
the 1-cocy-
is always trivial. By restricting it to ~ C
~
we get a (not
necessarilytrivial) cocycle which determines the Poisson structure on (cf. Proposition 5.11 below). Since
d = ~
~ ~
,
~r . ~ C ~ s
an open cell
Proposition 3.10
can be refined as follows. Proposition 3.11. the natural mapping
Change the sign of Poisson bracket on ~
@~--@ ~
~ - ~ . Then
: (g, g~)}-~ gg~ is a Poisson
immersion. We shall give a proof in §4 • Examples 3.2.
Poisson Lie groups described in Example 5.1. are dual
to each other. The Lie gromp which corresponds to Ta~
Crx ~
d =~
@ ~
is
equipped with the Lie Poisson bracket depending on a
195
second argument. We now return tD the situation of Proposition 3.9 • Our next
step
will be to describe explicitely the Poisson structure on ~ R which corresponds to the Lie bialgebra (~i, ~] . We shall need the following simple formula. Lemma.
Coadjoint action of the group ~
is given by
(~.2o) where x+
=
~-~
~
I/2(~XzX
: h ~-~h+ are natural
homomorphisms and
) .
Proof. Decomposition (3o18) allows to identify easy to check that A ~ ~¢r~ ~'( ~ ~ ) = ~ which yields (3.20) . Proposition 3.1~.
9~ with ~ 9 " It is ( ~ ~¢'~ ~ & ~ . ' X J J
The Poisson structure on ~ w i t h
the tangent Lie
bialgebra (~R' 9 ) is given by
where
~ (h) @
~
(~9;9~)is
given by I
The proof follows immediately from Proposition 3.10 (we restrict the 1-cocycle on ~9----@×@to G ~
and project it down to get a linear
9 -~- ~ ;into 9 R
mapping acting from
)
Note in particular that
(3.22) which implies that on ~ .
(3.22) defines indeed a Poisson Lie group structure
§4. POISSON REDUCTION.
DEFINITION AND APPLICATIONS
Let ~ b e a Poisson manifold. Suppose that a Lie group ~ is acting o n ~ and that the space of ~ -orbits is a smooth manifold. Definition 4.1. An action ~ . ~ is called admissible if the space of invariants ~ ( ~ ] G C~(j~j is a Lie subalgebra. If the action of
~
is admissible there is a unique Poisson structure
196
on
~\~t
such that the natural projection ~ - : ~ _ ~
Poisson mapping. We shall refer to ~ \ ~
@\~
is a
as the reduced Poisson mani-
fold. The concept of Poisson reduction extends the Poisson approach to the Hamiltonian reduction which was exposed in /16/ (and goes back to S.Lie) . It was originally suggested by V.Drinfel'd. Various examples Of admissible group action are provide~ by the following
result:
Proposition 4.1. Lie subgroup, &C
~ ~
Let ~
~ a ~
be a Poisson Lie group, H C ~
its connected
the corresponding Lie algebra° Suppose that
is a Lie subalgebrao Let ~ - ) ~
tion on a Poisson manifold
~.
be a Poisson group ac-
Then the space C 8 C
Co°(~J
of
-invariants is a Lie subalgebra, hence the restriction of our action to
H
is admissible.
In particular, the action of ~
itself is admissible (The point is that
H C ~
is not necessarily a Poisson Lie group).
Proof.
We use Proposition 3.7.
~
= ~ ~
= 0,
for X ~ ~ and
Let ~ ) ~ ~s
~ ~
C. H ~
~
. Then " By (3.16) we get
A
a
whence
C"
~s usual, the difficult part of reduction lies in the description of symplectic leaves in the quotient space. For Hamiltonian group actions this is done by means of the moment map. The point is to indicate its analog in the present case. We shall not do it in full generality (although the corresponding theorem was recently proved by Karasiov /17/) but rather present a series of examples. Let (~, ~=
~R)
~Cr
be a Baxter Lie algebra, d = ~ @ .
Equip ~
with the Poisson bracket
~
its square , ~r
~l~
given
by (3.12), (3.19) Proposition 4.2. on an open set in Proof.
~
.
The operator ~^( X) defining the bracket is certainly non-dege-
nerate at Note.
The Poisson bracket just described is non-degenerate
x -- identity~r~and hence also on an open set in
The construction described extends straightforwardly ~o the
square of arbitrary Poisson Lie group. In particular in the situation of examples
3.1, 3.2
we get the standard Poisson bracket on ~ - ^ ~
197 Hence proposition 4.1
provides a generalization of cotangent bun~Les.
This observation is due to V.Drinfel'd(unpublished). From ~ropositions 3,6, 3.10, 4.1 we see readily that the action of ~ C~g on ~ by left and right translations is admissible. Recall that ~ ~g is open in ~ m ~ X @ . Hence we may identify ~ C ~ g ) with an open cell in ~ / ~ (respectively~in2-)/C~). Proposition 4.3.
(i)
Natural projections
form dual pairs in the sense of /16/. (ii) The quotient Poisson structure on ~_-v ~/~gcoincides with the Sklyanin bracket (3.12), (2.28) . (iii) The quotient Poisson structure on ~ --~ ~ i ~ coincides with the bracket (3.21), (3.22) . -invariant functions Proof. By definition, (i) means that right- ~ have zero Poisson brackets with left- ~-~ -invariant functions, and similarly for the secend case. Suppose ~ ~(~) is r i g h t - ~ invariant, n~ @ C_~'~'t/ ~D ) is l e f t - ~ a - i n v a r i a n t . Then ~ I ~ ~ ~]-. ~ . Hence
The 2nd
case is considered in a similar way. ~-- ('(" ~ ' ~
>>- << ~ I ~ > ~ 0
We now pass to the calculation of the quotient Poisson structure on
, We extend ~, ~ to
and similarly for ~
right- ~
t
-invariant functions ~ ~
y# I~. on ~ . T h e n
. By definition
(4.2) ~ ~R and~/q~R C d checks easily that Since
~
is is°tr°pic the 2nd term vanishes. One
198 By substituting (4.1), (4.3) into (4.2) and making use of the defini ~ tion (3.17) we get after some easy computation
",/,>- <,£x, y> which coincides with (3.12)
. We leave it to the reader to prove the
last assertion which is done similarly. As a corollary of Proposition 4.3. we get Theorem 4.1.
(i)
action.
(ii) Let us identify the quotient space with ~
Natural action ~
~/~Ais
a Poisson group ~.
Then this action is given by the formula
(4.4)
("t~,~): ~ l'--~ Z'~ ~ ( ~-~ ~-t~ ~
In particular, the subgroup
(4.5)
~:
~
~-*
~
~
~gC~is
= ~ (~-tX-4~ ~) -
acting via
( ~-I ~ - 4
k_ ~ ) ~
This action is a Poisson group action and its orbits coincide with the symplectic leaves in ~ ( e q u i p p e d with{~Sklyanin bracket). We shall call (4.5) the dressing action, it may be regarded as an analogue of the %he coadjoint action° Proof.
Since natural projections
form a dual pair, we are in a position to apply a general theorem from /16/. It asserts that if ( ~I W') is a dual pair of Poisson mappings, then the symplectic leaves are obtained by blowing up points in the double fibering (~s~ ~) , i.e. they are the connected components of r~'CrF -4)
(~.The
projection map ~
~-~
G~\~
•
@
is given by
whence This makes the last assertion obvious. (All the rest is perfectly evident). Note. The result we have quoted is a slightly refined version of a theorem due to V.Drinfel'd. In a dual fashion we may give a description of symplectic leaves in ~
O/@
. Note first of all that
@
serves as another model for
199 ¢T% J__
the quotient spaces 6~/~
, (D'\ c~'. Canonical projections are then given
by
~
(4.6)
[X.,~)
#
Corollary I.
~
#I
-~
9C~ "4
Symplectic leaves in ~
are mapped onto conjugaey clas-
ses in
~
under the canonical mapping
Proof.
Both groups are different models of the same quotient space.
Corollary 2.
Casimir functions on ~
ture described in Proposition 4.4 on
~
m : ~
_, ~
with respect to the Poisson stru~
are precisely the central functions
•
For completeness we give an explicit formula for this Poisson structure Proposition 4.7. (4.7) ~ ~ ~ l ~ u ~
The quotient Poisson structure on 6 i s given by ~ < ~(X)~
y'>t<~(×'),y>_(R(x)/y~-~CX'~y~
×,Y'> < ~~' , y > V, ; - y'=
where ~ = V~ , X'= V~+ ,< y =
We leave the proof ~o the reader (cf. the proof of Proposition 4.3). As another application of the reduction technique we give a proof of Proposition 5.11. Proposition 4.8.
Canonical projections
J# C~, - ~ )
i
:~/~
form a dual pair. The proof is the same as in Proposition 4.3 (Note the sign difference in the Poisson bracket on ~
!)
Corollary ~quip ~ / ~
~
G~ \ B with t~e product Poisson structure. Canonical embedding "~,~Z~,'R~ L ~ I ~ × 6-R\~is a Poisson mapping.
It is easy to check that the quotient Poisson structures on ~ / ~ #
~R\~
are again given by (3.21) - (3.22), (2.28) - (3.12), respectively.Since $~. ~aC ~ is an open subset this finishes the proof of Proposition
(3.~I).
200
§~.
LAX EQUATIONS ON POISSON LIE GROUPS: A GEOMETRIC THEORY
We start with the simplest theorem on the subject which will then be generalized te include Lax equations for lattic~ systems. Throughout this §
we assume that
~is
a Poisson Lie group and that its tangent
Lie bialgebra is a Baxter Lie algebra. Recall from the end of §4 that there are two different Poisson structures on ~ (3.12-),
(4.7).
This suggests ~hat
which are given by
we may use them to construct inte-
grable systems in almost the same way as in Theorems 1.1, 1.2.
As we
shall see now, this is indeed the case. Denote by
I(~)
Theorem 5.1.
the space of Casimir functions for the bracket
(i)
(4.7)
Casimir functions of the Poisson bracket (4.7) are
in involution with respect to the Sklyanin bracket (3.12-). (ii)
Let ~ E I ( ~ )
. The equation of motion defined by ~
with respect
to the Sklyanin bracket has the Lax form
& (iii)
Let
z x+(t)
be the solutions to the factorization problem (1.11)
with the left hand side given by The integral curve of the equation (5.1) starting at L @ ~
is given by
The proof is parallel to the proof of Theorem 1.2. Observe first of all that left and right gradients of a function ~
I(@)
coincide. This
makes (i), (ii) directly obvious from the definition of Sklyanin bracket. Proposition 5. I. projection , ~ ~ tonian
~
(5.4) (
I(Gr), h ~
Recall that
by !
)
W
~ is included into a dual pair (4.6) . Projections
of the integral curve i n ' o n t o
the quotient spaces ~ / £ ~
reduce to points since the reduced~Hamiltonians Since
~
be the standard
• The integral curves of the Hamil-
=~,~
on ~ ~l~)are given
,
Proof.
: (x,y)#.~ xy -I
Let ~ , ~..~
is both right- and l e f t - ~ - i n v a r i a n t
,
are Casimir functions. we have
201
Obviously, V~y = (x' ,x') G d where X' = V~ (xy
) is ti~e-indepen-
Now (5.5) follows immediately.
dent.
Consider the action
~
× ~
~
Notice that the subgroup (~,e)
defined by
~ ~ is a cross section of (5.4) on an
open cell in ~9 . Hence we get a canonical projection
whose fibers coincide with ~ g -orbits in Proposition 5.2.
(i)
The action (5.5) is admissible.
(ii) The
quo-
tient Poisscn space is canonically isomorphic to ~(-R, g). We shall prove a more general statement below (Theorem 5.4). To finish the proof of Theorem 5.1 observe that f o r ~ E i ( G ) ~ = ~ , ~ hence (5.4) defines a quotient flow on
@(-~i~ith
Hamiltonian
Projecting the flow (5.4) down to C_~ gives (5.3). We shall indicate a generalization of Theorem 5.2
which is suited for
the study of lattice systems. Recall from Proposition 3.4
that we may
use more general Poisson brackets given by (3.10), with the left and right R-matrices not necessarily coinciding. This observation is used to twist the Poisson bracket on Let ~ b e
.
an automorphiem of a Baxter Lie algebra (~, R) i.e. an ortho-
gonal operator
"~
automorphism of ~ conjugation
(5.7) Let
~
Aut ~
which commutes with R. It gives rise to an
which we denote by
~rx ~--~
$:
~t'I(gm)
~
k~
g~-~ g
. Define the twisted
by
~L'~
be the space of smooth functions on ~ i n v a r i a n t
with
respect to twisted conjugations. Theorem 5.3.
(i)
Functions
~
~ ~I(G)
pect to the Sklyanin bracket on G by a Hamiltonian ~ ~ ( ( ~ )
(5.8) with
B =
~= ~ R
LA- 6L
have
.
are in involution with res-
( i i ) Equations of motion defined
the following form
,
L(~,
(V~(L)~A=z(Z).(iii) Let
x~ ( t )
, x_ ( t ) be the
202
solutions to the factorization problem (1.11) with the left hand side given by
The integral curve of equation (5.8) defined by ~ ~ ~ ( ~ )
(~.1o) ~he p r o o f
L(~) is
=
~:~ (~).
.L
x± (~)
are given by
based on t h e use o f a t w i s t e d
Poisson structure
on
Extend 9 t o
(~.~)
~c(×,y)=
(X, ~ Y )
and put
We also put
~C'~.~ =
~
()~)
=
t ( ~; ~)(~ ) 2"
Equip ~ w i t h the Poisson bracket (3.10) with Proposition 5.3.
R = Z R d , R' = R d .
(i) The natural action of ~ o n
translations is a Poisson action.
~(~,~]
by left
( i i ) The natural action of
on
~'R~,R~) by rig.t translationsis a right Poisson action This is a corollary of Proposition 3.6 s i n c e
~G
C
~(g4
; -Ra3,
are Poisson subgroups. Proposition 5.4.
Canonical projections
~: :Bc~R~, ~ ) -" ~/~ ~,
$'~ ~B(~R~'~ )
~ ~
\D
are dual to each other. Both quotient spaces are naturally modelled on ~
. Projections ~, ~l
are given by
Proposition 5.5.
Symplectic leaves with respect to the quotient
Poisson structure on ~ a r e
orbits of twisted conjugations (5.7).
Proof. it suffices to compute .~ ( ~ _ 4 ( Clearly,
~,-t(~)
~..
{(..~-4 ~ : ~ )
= [~-~,~-~ Corollary.
))
/ ~ ~ G ~ ( ~Oj-'s [~))'-,,. , ~ G ~ ~
Casimir T'unctions of the quotient Poisson structure on
are invariants of twisted conjugations. A generalization of formula (4.7) for the quotient Poisson structure on G
is given by
(5,14) a ~ v , * ~ , d . = <~ (-c.×), y'>+<~(×'),~.y>-{R(×),Y>- < Rex,), y'> ~ <-c.x, y,> - < ~,, -c.y> Now everything is ready for the proof of Theorem 5.5.
203 Proposition 5.6. Hamiltonian
Let ~ ~ I ~ G )
h T on ~ ( ~ l ~ 3 a r e
,
hT = fo ~ . Integral curves of the
given by (5.4).
We leave the proof to the reader since it is completel~
parallel to
that of Proposition 5.1. Consider the action
k
(5.15)
,~ ~ ,
Theorem 5.4.
G R xb~
~,
(i)
~+
To check
~
,
+ ~
The action (5.15) is admissible.
Poisson bracket on ~ / ~ Proof.
~ g i v e n by
(i)
( i i ) The quotient
coincides with the Sklyanin bracket.
we use Proposition 4.1.
Observe first of all that by combining left and right translations we get a Poisson group action:
C.R~, .R~ x SO~Rd,~ ) x ~ h ~ , ~ )
-->
SO(-~, ~ ) :
We have changed the sign of the ~oisson bracket on the second copy of ~so
as to consider left actions (More generally, if there are two
commuting Poisson group actions ~ x ~ - ~ ;
~ J b ~
] their combi-
nation gives rise to a Poisson group action of ~ x ~ ( w h i c h is equipped with the product structure). Now, ~ g is embedded into ~ X ~ Since the tangent
via
Lie bialgebra of
~
is (d @ d, ~ _ ~ A @ ~ , ~ )
our claim follows, by virtue of Proposition 4.1,
from the following
lemma. Lemma I. ~ C
Proof of the l e n a . nihilates
~
is a Lie subalgebra in ~ -~t~l.
~ • &
An element (X I ' X2 ' ¥I , Y2 ) ~ d @ d
(~,~,
Since there are natural Lie algebra embeddings '
R ~
~1~
an-
if and only if
Equivalently,
'
~
d ~
~ , ,~L3,~
~
' ~R
, ~_~'=~
~ dR d '
it suffices to check that
implies
R_([I.,~,]R
) .
~.~-C~;,~]~)
=o
The last assertion follows immediately from the Yang-Baxter identity.
204
We now come to the proof of the second assertion of Theorem 5.4. Observe that the subgroup (G, e)~-~ is again a cross section of the action (5.15) on an open cell in ~ .
The canonical projection
?: ~__~ C_T is now given by
X' = V~ , Y = V~ of
HT
, Y' = V ~
. It is easy to compute the gradients
. Their restrictions to the surface ( ~, e ) C ~ are given by
v,,~ --
( x, x;.
-~-×_,)
,
v ~ , : (x', x~. - ~ x _ ) .
Similar formulae hold for the gradients of
~r.'~'~,~
=
( x', x'-y"
H~
. Now
+--r y_}
After substituting these expressions into the definition of ~ W, ~ I(~,U~ we get after some remarkable cancellations
s
-- t
4)--
f r,
Note; Unfortunately, T do not know how to extend to the present case the qualitative argument which w~ have used in the proof of Theorem I .2.
This argument is now replaced by a direct computation.
Let us now apply Theorem 5.3
~o the difference Lax equations. Let
(~, R) be a Baxter Lie algebra, ~ t h e Put ~ - "
~
~l~r~N.We shall regard elements of ~
ping~//V~ into ~
(5.16)
corresponding Poisson Lie group.
. Equip ~
( X, x/ >
_---
as functions map~
with the natural inner product
~"
< /k(" , "x/~, >
h
and extend t~ ~ En~ ~
(~)~, -- ~(X~) . ~his makes (~,R) a Baxter Lie algebra.t1Equip ~ w i t h the product Poisson struc-
ture. Clearly, G
is ( ~ ~ )
is a Poisson Lie group and its tangent Lie bialgebra
We shall denote elements of G by S = (el...... S ~ )
Define the mappings
Functions ~
(5~)
~
t o m by setting
~j
T:
G--~
~
by
satisfy the linear difference system
= ~
~,
, %=
while T is the monodromy matrix associated with (5.18)o Obvieusly, one has Proposition 5.7.
The monodromy map T : G - ~
is a Poisson mapping.
205 This property of the Sklyanin bracket has served as a motivation for the whole theory. The quantum version of this statement goes back to R.Baxter. Let ~
Aut ~
(5.19)
~,
be the cyclic permutation
(
×~ . . . .
, x,)
~
( X~,
Clearly, the twisted conjugations
L ~
x~,
g L
×~ . . .
-~'-1 g
X~.~ )
coincide with the
gauge transformations for (5.18) induced by left translations ~
~
in its solution space. The operator (5.19) preserves
the inner product (5.16) and commutes with R .
Hence
applies to the present situation. The space
Theorem 5.3
is described by the
following simple theorem. Theorem 5.5. ("Floque~") gauge orbit in 6
~
.
(ii)
The algebra
, L --~ ~ (~- [ L ) )
As a c o r o l l a r y of Theorem 5.3 Theorem 5.6.
Two elements L, L' g ~ l i e
(i)
Functions
,
h~
,Ig I ( ~ )
tion~of motion with the Hamiltonian
-
(iii)
~
Let (gm) ~ (t)
~_,
mI(~)
T ~
I
is generated by the
(~
•
we ~et
respect to the Sklyanin bracket on ~
~
on the same
if and only if their monodromy matrices ~(L), T'(L')
are conjugate in ~
functions
(i)
-
. h~
are in involution with
( i i ) The Hamiltonian equais given by
~
L~,
be the solutions to the factorization problem
(I .11 ) with the left hand side given by
cLoll) The integral curve::of (5.20) with the origin at is given by
(5.22) Note.
=
(
% . )+_
A completely different approach to the study of difference Lax
equations was described by B.Kupershmidt / ~ / .
These two approaches may
be linked together by a discrete version of the Drinfel'd-Sokolov theory / lq/. However, a detailed analyses of this link goes beyond the scope of the present paper.
206
~6. DRESSING TRANSFOrmATIONS.
In the present paragraph we return to the study of Lax equations on the line described in §2.5.
Our notation will be close to that intro-
duced there, the only difference being that we now drop out the periodicity condition. Thus let (~, ~ R )
be a Baxter Lie algebra. Let ~; (=~
be the corresponding dual Poisson Lie groups.
I. Let~= ~ ( (6.1}
~ , ~ ) / ~=C~,~,~.We define an inner producto,~ by
x, Y>
Clearly,~,~)is
=
<
again a Baxter Lie algebra. The dual space ~ "
is
equipped with the Lie Poisson bracket, i.e.
We shall consider only smooth functionals on ~ C
C ,~)
An element
Y ~.~R,~)
and identify ~ -
with
defines a linear functional on
by (6.1) Let us associate with each function L ~ ~ t h e
linear differe.~tial
V
equation (The charge e will be henceforth chosen to be I, cf. (2.24). Let ~ U be its fundamental solution i.e. a function with values in ~satisfying (6.3). We shall normalize by the condition ~[ (0)= I Now we are in a position to state one of our main results. Let m : ~R-"
G
: h~.~
h+h[ I
be the canonical mapping (cf.Proposition
1.2). Define the "dressed" ~otential formula
~
qoq
L~
, L ~
~,
h ~ ~
by the
~ _i -~
=
Here x + is defined as in (1.11) and the choice of the sign is irrelerant. Theorem 6.1. Formula (6.4)defines a right Poisson action G e x ~ - ~ Note.
In typical applications ~
thoug~ as a ~ie group ~
is a simple Lie algebra. Thus al-
often ~splits <e.g ~ =
G.~G_ if R is given
by (2.2}) it does not contain Poisson Lie subgroups (cf. Theorems 3.1, 3.3 ) . Hence in the statemen~ of Theorem 6.1 the group~acannot be replaced by a smaller subgroup. Before we start a proof of Theorem 6.1 let us discuss the definition
207 (6.4)
and its versions. Notice first of all that (6.4) follows from
the dressing transformation for wave functions
(6.5)
=
(V
Observe indeed that
-~ Jr
is an identity, so the choice of sign is indeed irrelevant. Clearly, q(O]---~@~hCo)_-~hence
g~may
tions to the system % ~ : - -
be regarded as the fundamental solu-
-Lh~ ~ --
whence we get for L h
the de~i~ed
formula (6.4). This argument shows in particular that (6.5) determines a well defined transformation on the phase space ~ -m -.
Starting from
the papers /6/, /7/ people usually consider a different definition
Unlike (6.5), this formula does not preserve the normalization condition ~(o)= ~ and hence defines an action on the space of all solutions to equations (6.3) which is a principal fibering over ~
. Notice
that (6.6) differs from (6.5) only by an apparently inessential constant factor. However, this factor leads to a drastic change in the composition law of dressing transformations. The Poisson properties of transformations
(6.6) are not as nice as those of (6.5)-(6.4).
(cf. Theorem 6.2 below) 6.2.
We proceed now to the proof of Theorem 6.1.
Its main steps are as follows. Since (6.5) is more easy to deal with than (6.4) we start with the description of the Poisson structure on the space ~o
of all normalized wa~e functions. The Poisson structure
on the space
~
of all wave functions will also be of importance.
However, -(6.5) is still too complicated, so it is unreasonable to prove Theorem 6.1 by a direct computation. Instead we shall use the reduction technique and identify ~
with the quotient of a larger
Poisson space. This will enable us to understand the geometrical meaning of (6.5) and make the composition law for dressing transformations completely evident. Let
~ --- ~
By solving
(6.3)
~ ~,(r)
t
~o =
we get a mapping
{~
~ ~/
'g{: ~A6'K~" ~
~(o)=
~
< : Lp,
define a Poisson structure on Vo by demanding that ~
Let us
is a Poisson
mapping. To characterize this structure we compute Poisson brackets of cylindrical functionals on ~o depending on the value of
at one
208 particular point
1'~
x ¢~
each. Let ¢ ~ C~° ( ( ~ ) . Put __~Z[~]= (~ (~'(~:)_)
Vo
. ]~e=ote b:, V
Let Cx, @~ be two cylindrical functionals
Then for sgn x = sgn y
For sgn x ~ sgn y Proof.
l e f t and right ~radients of 16 , 121~< I ~
we have
the Poisson bracket vanishes.
The compt~ta~ion we are going to outline is a version of a
standard one commonly used in the inverse scattering method. Lemma I.
The gradient of the functional
where ~ ( x )
L ~-~¢~(~L] is given by
is a step function, ~(x) = 0, x < 0 ,
0(x)
= I , x #/ 0.
This follows from routine perturbation theory computation applied to (6.3) . Lemma 2.
The gradient (6.8) satisfies the differential equation
(6.9) "~ ~ = [ L 1 ~(J with the boundary conditions
(6.10)
X(O;
=
V~
I
( ~t (Z])
o.< ~
I
.<~
X(Y.,) : ~
(~L(Z})
Let us now use (6.9), (6.10) to compute the l e f t hand side of (6.7). 0b~iously ,
~
[%~
We ~enoted
Xl(Z) = (
j
, X2(s) = (grad T~ , V r L j ) ( Z )
and assumed that IKi 4 1 ~ s g n x = sgn y . 11" the signs of x, y are different the right hand side vanishes since X I , X 2 have disjoint supports. We now use formula (6.7) to define a Poisson bracket we put
(6.11) ~ {~ ~ ~(~)=
x = sign
A %~{,)- o if
sgn x ~ sgn y.
y
,
and
on
V D ~
. Thus
~'~ (~) >-<~(~0~
209 Proposition 6.2. ~ c ~ Proof.
is a Poisson submanifold.
It is sufficient to show that for any
~
~ !~ 6
~
C r ] ~ ," ~ o~ (
z
vanishes on ~o • Obviously, we have
if
~(o}
6.3.
= I.
As our next step we shall realize V as a quotient space of a
larger Poisson manifold. Put ~ z on W
C
We define a Poisson structure
by setting
for any two cylindrical functionals ~J ~-e ~ <'tc)(z;), ~1--~ ~(W(~O; ~I{l~I If sgn x ~ sgn y the Poisson bracket is set to be zero. We are making use of the notation introduced in §3.4 (Proposition ~.8, 3.9). Let us define an action of ~
Lemma ~. Proof.
= ~C~6~on~
by left translations
The action (6.13) is admissible. We must show that fmnctionals on ~
which are invariant under
(6.13) form a Lie subalgebra with respect to (6.12). Observe that if a functional ~ z is left- ~-invariant, then ~ ~ C _ ~ ] is left~-invariant give (6.14) ~ ¢ ~
and hence ~ ~
~]J=-
9R
= ~R
4~(~'z~
" Thus (6.12) simplifies to
)/
L~Ct4(~)
~7~(~(~]]~>
Since right gradients are invariant under left translations the r.h.s. is again a left-~-invariant The diagonal embedding
~ ~
as a subspace of W • Since ~ the quotient ~\k~] with V . Lemma 4.
functional. Q.E.D. ~
: ~(~i~J
allows to consider ~/
G~.~(at least locally) we may identify
The quotient Poisson structure on V ~ ~\Wcoincides
with
(6.~) Proof. Let ~ I ~ W be two cylindrical functionals on V . Assume that Ix~~< ~ y~., sg4a x = sgn y . We extend them to left- ~ -invariant functionals ~ j ~ on W . One checks easily that for ~ ~ ~ C ~ ]
Hence the right gradient is given by
210
Substituting (6.16) into (6.14) we get, after some remarkable cancellations, (6.11). For completeness we reproduce this calculation. We have
.~
<~ ~(~)-'~<_),,
=
-
Aa%)-'
7.)+<(M.,{~-'×,)_,
-'
yL
We ~enoted Y' = ~
X
=- v , ( ~ ( x ) ) . x. = v ) ( ,
(~(y))
~
3.=
(x)). ~ = v4~ ( , p ( y ) ) .
and make use of simple properties o f homomorphisms
I/2(~+I)~.
Define the "diagonal" right action W X ~ - ~ ~
(6.~7)
~.
Lemma 5.
~,c~;~y.> *
~ p~ ,,,,, ~ Equip ~
~
by setting
~_ ~ ~D
with the standard Poisson structure. Then (6.17)
is a Poisson group action. Proof.
Let us rewrite (6.12) using left-invariant frame
where
~,.[~]
=
~.
~i~)
-~ . g ~ o
~ ( ~ ;
-
We have
Since V
(~J(x)g) = Adg-1(
~(~J(x)))_
our assertion immediately
follows from the definition of Poisson group actions. By projecting down the action (6.17) to the quotient space G ~ I W ~ we get the following transformation
Theorem 6.2.
V~©
Formula (6.18) defines a right Poisson group action
...~ V .
This is an immediate corollary of Lemma 5.
211
Indeed the actions diagram,
(6.17), (6.~8) are included into a commutative
W~L9
~W/ V
We point< out the following special cases (i)
The action o f ~ C ~ i s
serves (ii)
~
a Poisson group action. This action pre-
and its restriction to
The action of ~ ~ r C ~ i s
given simply by ~
~o coincides with (6.5)
a Poisson group action. Clearly it is
. The quotient space is isomorphic to The natural projection is given by 7V:
~.~.
(x, x) = ~
(iii)~ The action of subgroup ( e , ~ ) ~
~
on ~ is given by (6.6). Un-
like the preced~n~ cases, this is not a Poisson group action. Indeed, (0, ~ ) & =
(~,
0) is no~ even a Lie subalgebra in dRd
Of course, formula (3.16) provides a precise information on the "nonconservation"
of Poisson brackets even in this case. Now, however,the
right hand side is not defined in intrinsic terms, it depends rather on the Poisson structure on the larger group
~__O(~leJor
on its
tangent Lie bialgebra. 6.4.
Dressing transformations
and dynamics.
For concreteness we shall assume througho.~t this n£ that = sl (2, ~ [ ~ ,
~-J] ~ . The standard decomposition
gives rise to an r-matrix on .~
(6.20)
%
-
which is skew with respect to the inner product
(6.2~)
< w, ",/> =- ¢~,X=o
In physical literature
4=,- ×(~) Y(~,).
(6.20) is often referred to as the (classical)
Yang r-matrix. Let
El
~j
G_
be the corresponding Lie groups (i.e. the loop group
of sl(2) and its subgroups consisting of absolutely convergent Laurent series). The factorization problem associated with (6.20) is the standard matrix Riemann problem. Let ~ c sl(2) be the standard Cartan
212
subalgebra, ~ C G
x~= ~ (6.22) ~(~,~)=
its centralizer in
, t=(~,
.... , ~,,...)
~ ± = H ~ G.~
@j
/
~.x= ~
. Let
~ x~.
V o c ~ ~x = ~c~) ® ( e ~x, ~ . x )
Let ~ ~o be a wave function obtained from ~ by a dressing transformation (6.18). Since ~÷ C ~ centralizes ~o , we may assume without any loss of generality that
(4,
j
c G_
the element g_ being defined uniquely up to a right factor which belongs to H_ . Dressing transformation defined by g_ transforms the "free wave function" ~ ( x , t ) into ~ ~I~ ~
the element
adg~ IX
being defined uniquely. By projecting (6.24) back
to W o ~ V/$(rwe get tk~
Formula (6.25) d e f i n e s an a c t i o n of H+ on the ~ - o r b i t P r o p o s i t i o n 6.3,
Vector fields
of ~o
•
on Ve which correspond to t h i s a c t i o n
are Hamiltonian. By contrast, the apparently more simple action VX ~ - *
V
given by
(6.24) i~ not Hamiltonian. This is easily checked using formula (3.16) to control the non-conservation of Poisson brackets under the transformations (6.24). This observation, however slightly puzzling, does not contradict of course to Proposition 6.3. Indeed, the point is that embedding H+ C_~ ~ given by with some fixed g_ ~ ~_is not a group homomorphism.
213
REFERENCES
I.
Sklyanin E.E. equation.
2.
On complete integrability
Preprint
LOMI. E-3-79, Leningrad:
Belavin A.A., Drinfel'd V.G. Yang-Baxter
of the Landau-Lifshi~z
equation.
LOMI
On the solutions
, 1980.
of the classical
Funct. Anal. and its Appl~
, 16(1982),
159-180. .
Semenov-Tian-Chansky Anal. and its ~ppl.
4.
M.A.
What is the classical r-matrix.
17(1983)
Kostant B. Quantlzation and representation of the Research Symp. on Representations
.
Notes Series,
Drinfel'd V.G.
structures
bialgebras
theory.
- in Proc.
of ~le grou~s,
1977, London Math. Soc.Lect. Hamiltonian
Funct.
, 259-272.
Oxf.,
1979, v.34.
on Lie groups, Lie
and the geometrical meaning of Yang-Baxter
equations.
Sov. Math. Doklady 27 (1983), 68. 6.
Zakharov V.E.
, Shabat A.B.
Integration of nonlinear equations
by the inverse scattering method. 13 (1979) .
8.
, 166-174.
Date E., Jimbo M., Kashiwara M., Miwa T. for soliton equations.
Proc.Japan.
3806-3816;
Physica
(1982), 343-365
18 (1982),
1077-1119.
Segal G., Wilson G. Publ. Math. I.H.E.S.
.
II. Funct. Anal. and its Ap~l.
Wilson G.
Habillage
4D
Transformation
groups
Acad. Sci. 57 A (1981), • Publ.RIMS Kyoto Univ.
Loop groups and the equations
of KDV type,
61 (1985), 4-64. et fonctions ~T , C.R.Acad.Sci.
Paris,
299 (1984), 587-590. 10.
Semenov-Tian-Chansky group actions.
11.
M.A.
Dressing transformations
Publ. RIMS F~voto Univ. 21 (1986)
Reyman A.G., Semenov-Tian-Chansky systems, Math.,
M.A.
Reduction
affine Lie algebras and Lax equations.
54 (1979), 81-100;
63(1981)
, 423-432.
and Poisson
, N6. of Hamiltonian
I, II . Invent.
214
12.
Adler M., Moerbeke P.
Complete integrable systems, Euclidean
Lie algebras and curves. Adv.Math., 38 (1980), 267-317. 13.
Reyman A.G., Semenov-Tian-Chansky M.A.
A new integrable case of
She motion of the 4-dimensional rigid body. Comm. Math.Phys. (lo~j
14.
Cherednik I.V.
Definition of R-functions for generalized
affine Lie algebras. 15.
Funct. Anal. Appl.
Keyman A.G., Semenov-Tian-Chansky M.A.
17 (1983), 243-244. Lie algebras and Lax
equations with the spectral parameter on an elliptic curve, Zapiski Nauchn. Semin. LOMI (in Russian), v.155 (1986). 16.
Weinstein A.,
Local structure of Poisson manifolds, J.Diff.Geom.,
18 (1983), 523-558. 17.
Karasjov M.V., To appear in Sov. Math. izvestija (1986).
18.
Kupershmidt B.A.,
Discrete Lax equations and differential-
difference calculus, 19.
Asterisque 123 (1985).
Drinfel'd V.G., Sokolov V.V., Lie algebras.
Equations of KdV
type and simple
Sov. Math. Doklady, 23 (1981), 457-462.
ON MONTE CARLO SIMULATIONS OF RANDOM LOOPS AND SURFACES
M.Karowski Institut f u r T h e o r i e der E l e m e n t a r t e i l c h e n Freie Universit~t Berlin A r n i m a l l e e 14 D-IO00 B e r l i n 33
1 Introduction
I would l i k e
to report
ration
with
"Freie
Universit~t
concepts
[I-5]
R.Schrader,
Random l o o p s
regions
theories
Symanzik's [7]
F.Rys,
Berlin".
in different
Quantum f i e l d [6].
on some r e s u l t s
W.Helfrich,
of
polymer
can be u n d e r s t o o d
follows.
The " p a r t i t i o n "
description for
the
function
in
and s u r f a c e s
collabo-
at t h e
are u s e f u l
physics.
can be f o r m u l a t e d
theories
obtained
and H . J . T h u n
i n terms
of random w a l k s
of e u c l i d i a n
simple
quantum f i e l d
case o f f r e e
bosons
as
of t h e t h e o r y (l
reads
in regularized
form on a l a t t i c e
Ld (2
where by
~)
is
a Ld-dimensional
vector
and t h e m a t r i x
~
is
given
216 The i n t e g r a t i o n s
in e q . ( 2 )
= l/,Z
can be performed
.t
-
[ - t,-
=
exe
Since the m a t r i x
7_ T
~
each term ~ t r ~
(I- m] tr
"connects"
in e q . ( 4 )
(4)
T~
n e a r e s t neighbours on the l a t t i c e ,
r e p r e s e n t s a sum over a l l
l e n g t h ~ c o n s i s t i n g of simple bonds. Thus the p a r t i t i o n can be w r i t t e n
Z
=
in terms of a s t a t i s t i c a l
loops of function
system
>-- ~
~×p
(5)
~o~.
where the sum extends over a l l
( p o s s i b l y o v e r l a p p i n g ) one-loop con-
figurations.
The c o n f i g u r a t i o n a l
energy - T , 9 - ~ . ~
to the t o t a l
l e n g t h of the loop.
Expanding the e x p o n e n t i o n a l
(5) we get a r e p r e s e n t a t i o n analogous to the w e l l r a t u r e expansion f o r c l a s s i c a l
Z
= 7__ ~×p
in eq.
known high tempe-
spin systems
~s~
I~ Z
I s~
is proportional
<~ Y> I
CO~. where a c o n f i g u r a t i o n
contributing
s e v e r a l disconnected l o o p s .
to the sum now may c o n s i s t
The e n t r o p y f a c t o r s
model. For the I s i n g model One o b t a i n s a f t e r
of
cz depend on the
resummation a sum over
n o n o v e r l a p p i n g loops
_ #~
-#
=T-e
a
:o.i~.
The 0(N) n o n l i n e a r
=tK~
)
~ -model in the l i m i t
"
(7)
N~0 d e s c r i b e s s e l f - a v o i -
ding random walks [ 8 , 9 ] . The polymer f o r m u l a t i o n of fermions leads (due to P a u l i ' s
princip-
217
le)
a l s o to s e l f - a v o i d i n g
The " p a r t i t i o n
loops.
function"
for
T h i s can be seen as f o l l o w s
free
fermions
[I].
is
F
(8)
11"
x
where sgn = s g n ( ~ ) ,
sgn(l+l=)x~(~ )
and
z
I
~=~ (Note t h a t
for
Kogut-Susskind
dependent number.)
(9)
~_m~t " fermions
Any p e r m u t a t i o n
~
~ is
can be taken a product
m u t a t i o n s I ~ .... J~s
which can be r e p r e s e n t e d
intersecting
The f e r m i o n
loops.
partition
as
function
as an x-
of c y c l i c
s
per-
oriented
non-
can be w r i t t e n
as
= where t h e
(I0)
sum e x t e n d s over a l l
avoiding
loops of t o t a l
tistical
system by
configurations
length ~.
Introducing
of o r i e n t e d
self-
an a u x i l i a r y
sta-
polymer : we can w r i t e <
~
k
the f e r m i o n
)<>F -~
(II)
expectation
v a l u e of an o b s e r v a b l e
X
< ~<'s~m" > p o l y m e r < S@~ > p o l y m e r
Whereas normal dom w a l k s , surfaces.
as
quantum f i e l d s
one can d e s c r i b e Wilson's LI0]
(defined gauge f i e l d
lattice
(12)
on p o i n t s ) theories
gauge t h e o r y
are r e l a t e d
to r a n -
in terms of random
formulation
is
g i v e n by
i @#~f. where U I , . . o , U 4 ( G
(eg.G = SU(N))
are d e f i n e d
on t h e
links
around a
218 plaquette.
The h i g h t e m p e r a t u r e
sion
analogously
leads
Z where t h e total limit
>--
:
s.
N->O
where t h e
of t h i s
expres-
(14) over a l l
The e n t r o p y
[9,11]
sum i s
(g-~)
to
~,
sum e x t e n d s
area
expansion
to eq.(6)
two d i m e n s i o n a l
factors
closed
surfaces
c s depend on t h e model.
of
In t h e
one g e t s
now r e s t r i c t e d
to
self-,avoiding
surfaces
and ~( i s
the Euler characteristic. The s t r i n g
quantization
problem
over random s u r f a c e s [ 1 2 ] presented to
by s e l f - a v o i d i n g
self-avoiding
viour
of random s u r f a c e s
Moreover, faces
to
there
should
fermionic
l e a d to
in their
are p o s s s i b l e
strings
a better
state
physics play
phase t r a n s i t i o n s
surfaces in
of e n t r o p y expected to
p o l y m e r s ~ 13]
field
theories
tion
eqs.(ll,12). (loop
involving
gas model)
fermions
in
liquid
surfaces
for
loops
i n [ 2] to
on t h e c r i t i c a l
model
the
on a b a l a n c e
is
which is
be
sheet
a natural useful
to
and i n t e r f a c e s .
simulation of
be u s e f u l relies
two d i m e n s i o n a l
on t h e b a s i s
model
crystals)
sulphur [15 ] . Self-
model [ 1 8 ]
was proposed
the e x c l u d e d volume r e p u l s i o n
of
random s u r f a c e
a Monte C a r l o
A statistical
in the context
L 17J . They can a l s o
of f l e x i b l e
solid-on-solid
we d e v e l o p e d
in
might L I6]
interfaces
properties
of t h e
of
di-
role
whose s t a b i l i t y
t h e r o u g h e n i n g of c r y s t a l
In r e f . [ l ]
sions
dimensions
. The s e l f - a v o i d i n g
generalization describe
three
and e n e r g y of t h e i r describe
(e.g.
polymerisation
of microemulsions
of
and s u r -
i n two and t h r e e
defect-line avoiding
beha-
understanding
an i m p o r t a n t
. They have been s t u d i e d
understanding
may be r e l a t e d
of random w a l k s
polymer p h y s i c s [ 1 3 J mediated
are r e -
continuum limit.
random c h a i n s
L 14J and t h e e q u i l i b r i u m
as summation
particles
of t h e c r i t i c a l
applications
and s o l i d
Self-avoiding
as f e r m i o n i c
An i n v e s t i g a t i o n
theories
statistical
mensions.
walks,
surfaces.
gauge and s t r i n g
has been f o r m u l a t e d
. Similar
method f o r of the
quantum
polymer formula-
in d=2,3,
and 4 dimen-
s t u d y the i n f l u e n c e equilibrium
of
properties
219 of s t a t i s t i c a l
line
systems.
The c r i t i c a l
e v a l u a t e d by means of the " c r i t i c a l loop gas model
i n two d i m e n s i o n s .
diagrams of s e l f - a v o i d i n g mensions w i t h exponents
surface
R,~,~,
In r e f . [ 3 ]
tension
and ~ were
and c u r v a t u r e
intersecting
d i m e n s i o n s were e v a l u a t e d i n r e f . [ 5 ] .
for
the
we e x p l o r e d the phase
random s u r f a c e models i n t h r e e
and ~ f o r
Monte C a r l o s i m u l a t i o n s
exponents ~ , ~ , ~ ,
window" method i n r e f . [ 4 ]
energies.
surface
For o t h e r
and f o u r
di-
The c r i t i c a l
gas models i n t h r e e investigations
of random walks and s u r f a c e s
and
see r e f e r e n c e s
in
El-5J
2 Models The models to be c o n s i d e r e d lattices with
Ld w i t h
periodic
are d e f i n e d
on s q u a r e ,
boundary c o n d i t i o n s
ILdl ~ I 0 4. The p a r t i t i o n
functions
cubic,
hyper c u b i c
in d=2,3,4-dimensions
are d e f i n e d
by
c6(~ where the s e t s of c o n f i g u r a t i o n s E(c)
depend on the s p e c i f i c
gurations
of l i n e s
~i
the l a t t i c e four
links
tained
= ~closed
intersecting
c,~ i
at a common v e r t e x
for
loops
point.
loops
(link)
(17)
(surfaces)}.
(18)
in
of l i n k s c
each v e r t e x Thus t h e l i n e s
(link).
at a v e r t e x ,
two t y p e s of c o n f i -
(surfaces) }
a collection
For C6~sa
(plaquettes).
energies
surfaces):
loops
each v e r t e x
(plaquettes).
are a l l o w e d to touch nected t h i s
comprises
and the c o n f i g u r a t i o n a l We d i s t i n g u i s h
self-avoiding
such t h a t
i n two l i n k s
intersect
model.
(two d i m e n s i o n a l
~sa = { c l o s e d A configuration
~
(link)
The e n e r g i e s may i n c l u d e
con-
may not
distinct
be c o n s i d e r e d
in
in two or
in c is
(surfaces)
But two l o c a l l y
they will
(plaquettes)
is contained
surfaces
as d i s c o n -
t h r e e terms
and
(2o)
220
f o r surfaces. The f i r s t , loop length #
the tension term is p r o p o r t i o n a l to the t o t a l
(surface area
s).
The i n t e r s e c t i o n energy is p r o p o r t i -
onal to the number of i n t e r s e c t i o n points
i
(links ~).
The t h i r d con-
t r i b u t i o n s are curvature energies. They can also be understood as chemical p o t e n t i a l terms of the t o p o l o g i c a l q u a n t i t i e s : number of loops n
and Euler c h a r a c t e r i s t i c X,
r e s p e c t i v e l y . We are i n t e r e s t e d in the
nature of phase t r a n s i t i o n s ( f i r s t and c r i t i c a l
exponents of these models.
3 Monte C a r l o
For s i m p l i c i t y A configuration ratively
of
Method
I shall
describe Starting
change i n
one has t o make s u r e figurations
an o l d
square.
by o c c u p i e d ones and v i c e that
one g e n e r a t e s ci.
In t h e
in
fig.l.
no c r o s s i n g s samoles o f
heat
bath
terms of
random l o o p s .
can be g e n e r a t e d on a c o m p u t e r i t e -
from
a unit
such changes a r e d e p i c t e d
simulation
t h e method i n
(c.f.eqs.(17,18))
as f o l l o w s .
one by a l o c a l empty l i n k s
or second o r d e r ) , phase diagrams
configuration
This
versa.
means t h e The f o u r
one g e t s possible
For t h e
self-avoiding
appear.
By a Monte C a r l o
equilibrium
updating
a new
replacement
ensembles of
of
types
case
(17)
con-
p r o c e a u r e we s e q u e n t i -
"'-- ii
"I-i Figure l .
;;--If
Local changes of loop configurations within a
plaquette a l l y sweep a l l
d ( d - l ) / 2 Ld
plaquettes
new c o n f i g u r a t i o n with p r o b a b i l i t y
of the l a t t i c e and accept the
221
P = Wnew/(Wol d + Wne w) where t h e
w
(21)
are t h e B o l t z m a n n f a c t o r s exp(-E/kT).
T h i s means we t a k e (equally ciple
the
old
be a t t a i n e d
condition"
is
a probability is
stable
initial after
t h e new c o n f i g u r a t i o n
distributed
we r e t a i n
in the unit
one. after
sufficiently Obviously
distribution
under t h i s
A
obtained
is
calculated
after
about f i v e
is
than
reach
up" p e r i o d .
the
P, o t h e r w i s e can i n
prin-
"ergodic
set of c o n f i g u r a t i o n s
to
Moreover,
with
the B o l t z m a n n f a c t o r starting
from
N ~I03
sweeps t h r o u g h
(22)
an a r b i t r a r y
such an e q u i l i b r i u m The thermal
as t h e mean o v e r complete
less
many i t e r a t i o n s a large
we e x p e c t t o "warming
a pseudo-random number
allowed configuration
proportional
procedure.
configuration,
if
interval)
Since every
satisfied.
an a p p r o p r i a t e
riable
(22)
set
average of
a va-
configurations the
each
lattice
(23) The c o m p u t a t i o n s
are u s u a l l y
done i n
"thermal
. . . . T m a x , T m a x - a T , . . . , T m i n where we s t a r t
at
cycles"
Tmi n, Tmin+aT,
low t e m p e r a t u r e
from t h e
empty l a t t i c e .
4 Some R e s u l t s
A) S e l f - a v o i d i n g In r e f . [ 2 ]
l o o p gas
we c o n s i d e r e d
self-avoiding
l o o p gas systems
in d=2,3,
and
4 dimensions (24) C ~ ~$~
We "measured" the fluctuations
the
average
length
<~t> _<~>~
<~>
(proportional
(proportional
to
to the energy) the
specific
and
heat)
in
222 thermal sists
cycles.
For low t e m p e r a t u r e s
of a few small
with temperature gurations
loops.
and < ~ > _
<~>Z
others
loops, on ~
heat decreases a g a i n .
lattice
maximum.
l a r g e ones.
<~> and <~>-<~
is
is filled
displayed in f i g . 2 .
s i z e the peaks of the s p e c i f i c
i
i
i
l
i
The c o n f i -
i
I
loops of
For high
a s y m p t o t i c v a l u e and the
The l a t t i c e
w i t h many
among them a very l a r g e one. The c h a r a c t e r i s t i c of
con-
grows r a p i d l y
some of them c o n t a i n
also comprise r a t h e r
t e m p e r a t u r e the energy approaches i t s specific
configuration
<~>
has a
show s t r o n g f l u c t u a t i o n s :
medium s i z e ,
a typical
Around ~ 0 . 8 6
heat are h i g h e r and s h a r p e r .
i
i
8
dependence
With i n c r e a s i n g
i
i
(IZ)-(1)2
88O~o 80 o 8§ 8§,
2c.
.
o°
8
00
~AAAj~(I)
o
00
oo
oo
o
8o
o88"ti t ~8 ° .1
08
8 o
oo
:t
~888888o
" t -! "t! fill.
I
!
I
I
0
I
F i g u r e 2. The average energy <42>-<~>Z
For
L -->
for
oo
<~>-
I
I
I
05
I
I
10
<~>
and the l e n g t h f l u c t u a t i o n s
a loop gas on a 5 x 5 l a t t i c e .
t h e y approach
<~>~
,-,.,
l # -
~,~t
I -~
(25)
223 a behaviour t y p i c a l the c r i t i c a l
for
lattice tively,
We found
points: .86
in
.50 9
in 3-dimensions in 4 - d i m e n s i o n s .
ti
~ crit In r e f . [ 4 ]
a second order phase t r a n s i t i o n .
=
we used the f a c t
that
2-dimensions
in two dimensions loops d i v i d e the
into "interior" and " e x t e r i o r " p a r t s , to i n t r o d u c e an order parameter
rFt
(26)
Vin
and
Vex, respec-
---
The c r i t i c a l
exponents are d e f i n e d by e q . ( 2 5 )
<~>
~
<~>
~
and
l~-#c"l~ I #->#";~ k=o I#-#~,.~I -r (28
X i s the s u s c e p t i b i l i t y
given by
<~> o~ #-~ <,~> (29 and
h
a magnetic f i e l d
introduced
by the replacement ##-~ # ~ - h ~
eq.(24). We were able to f i n d which i s l i m i t e d , rature side,
a t e m p e r a t u r e regime w i t h i n
close to T c r i t ,
(where d i v e r g i n g thermodynamic q u a n t i t i e s and the end of the c r i t i c a l
rection-to-scaling "critical
window"
slope of the l i n e a r logarithmical
plot
portion
round o f f )
In t h i s
i n c r e a s i n g system
and ~ where c a l c u l a t e d from the
of the corresponding q u a n t i t y
near T c r i t .
on one
(where c o r -
on the o t h e r s i d e .
(whose e x t e n s i o n i n c r e a s e s w i t h exponents ~ , ~ ,
region,
rounding tempe-
regime away from T c r i t
terms become i m p o r t a n t )
s i z e ) the c r i t i c a l
the c r i t i c a l
by the f i n i t e - s i z e
Furthermore,
the exponent
in a doubly ~
was de-
termined from the i s o t h e r m at the c r i t i c a l t e m p e r a t u r e . We found I s i n g - l i k e values f o r a l l exponents c o n s i d e r e d .
224 B) S e l f - a v o i d i n g In r e f . [ 3 ]
surfaces
we i n v e s t i g a t e d
The E u l e r c h a r a c t e r i s t i c 7
= 2
self-avoiding
is defined
C m.o~
-
surface
gas systems
by
~k..Z)
(31)
where ncomp (nhand) i s the number of connected components (hand] e s ) of the s u r f a c e • F i g . 3 shows the average energy <s> and the
I
1.5
I
I
........ <S)/104 cs~/103 1.0 ....... "°•°'.l.o...l°"
1.0
• i,,O,o:~T
°
°'l "%
.5
% ".o~.
.5
%"•"'•..••..,••.
.0
.0
• ..i"
............ -.5
|..
°t:..°.................. ;::~:|ljl|jlIe,..o,•.°..
•
.... ,'':'3<X~/104 i
--•D
.1
g '
"5
.9
i
L
.5
.7
a)
obtained
10 3 and (b)
Euler characteristic lations
for
the s e l f - a v o i d i n g
surface
The average s u r f a c e <s> and E u l e r c h a r a c t e r i s t i c
on (a)
for
configuration
.9
B
b)
F i g u r e 3. Monte C a r l o r e s u l t s gas model.
•
3
10 4 l a t t i c e s .
obtained
the c a s e ~ = O . consists
were
in
"~-cycles"
by Monte C a r l o simu-
In the l o w - t e m p e r a t u r e
predominantly
of small
phase a t y p i c a l
s e p a r a t e d compo-
225 nents.
In t h e h i g h - t e m p e r a t u r e
typical
configuration
many handles
like
phase
consists
a sponge.
Note t h a t point.
= 0.353
dimensions cating
(in
a first
we found
order P tran
with
(in
an energy jump ~ < s>L -4
transitions in t h r e e
are i n
and f o u r
~
phase t r a n s i t i o n
= 0.68
object
(Fig.3a)
with
the data
at
3-dimensions).
(32)
v a n i s h e s at the c r i t i c a l to s c a l e i n v a r i a n c e .
hysterisis
D
connected
dimensions
m i g h t be r e l a t e d
(Fig.3b)
n e g a t i v e and a
phase t r a n s i t i o n
the E u l e r c h a r a c t e r i s t i c
This f l a t n e s s
is
of a s i n g l e
In t h r e e
show e v i d e n c e of a s e c o n d - o r d e r ~crit
~
In f o u r
loops i n the # - c y c l e s
indi-
at
4-dimensions) = 0.45.
agreement w i t h
(33)
The observed d i f f e r e n t
t h o s e of l a t t i c e
t y p e s of
gauge t h e o r i e s
dimensions.
i
i
i
low t e m p e r a t u r e
low temperature 0,5
phase
.0oo,,,,
""i'ii'ii" ",,,
0
/
I \"V
0.5
a)
Figure
./..1 phase/" or°poet w1
/
1 I
I
I
I
0
1
P
t.1
2
b)
4. Phase diagrams f o r
d i m e n s i o n s showing t h r e e order transition lines.
-
model
phases,
(30)
first
in (
(a) t h r e e )and
and
second-
b) f o u r (. . . .
)
226 For nonvanishing chemical p o t e n t i a l
for
the Euler c h a r a c t e r i s t i c
/~>0 we found the phase diagrams d e p i c t e d in f i g . 4 .
For l a r g e ~
new phase appears separated from the others by f i r s t - o r d e r tions.
This " d r o p l e t
sisting
(/~=0)
phase" i s r e l a t e d to a new ground s t a t e con-
of simple cubes, each touching
In r e f . [ 5 ~
the c r i t i c a l
e i g h t o t h e r s at i t s
behaviour of the s e l f - a v o i d i n g
corners.
surface gas
in t h r e e dimensions was i n v e s t i g a t e d by the c r i t i c a l
method. Analogously to the loop gas case,
&,#, ~,
a
transi-
window
I found I s i n g exponents
and &.
C) I n t e r s e c t i n g
surfaces
An i n t e r s e c t i n g
s u r f a c e gas model in t h r e e dimensions
c ~E; p r e v i o u s l y discussed in [ 1 8 ]
was i n v e s t i g a t e d in r e f .
it
approaches the s e l f - a v o i d i n g
i s the I s i n g model and i t
(for/w=O)
in the l i m i t
# ~
order t r a n s i t i o n cal
lines
and t r i c r i t i c a l
It
and mean f i e l d
shows f i r s t -
points.
window" method I obtained I s i n g - l i k e lines
For #~ =0 model
By means of the " c r i -
behaviour along the c r i t i -
behaviour at the t r i c r i t i c a l
p o i n t s which i s
D) H a u s d o r f f dimension A model of a s i n g l e s e l f - a v o i d i n g
random s u r f a c e in t h r e e d i -
mensions w i t h the f i x e d t o p o l o g y of the sphere was considered in ref.[5]
e
At the c r i t i c a l point ~ c r i t = 0.53 gyration diverges like
(35)
[19]
is
and second
expected in t h r e e dimensions.
: Z
(30)
. The phase diagram d e p i c t e d in f i g . 5
symmetric w i t h r e s p e c t to ~ - 7 - # - 2 # £ . tical
[5].
the average radius of
(36)
227
|
I l
\ l
(1)
~
(2) \
o
\ \ \
(3)
o
Figure
05
5.
Phase d i a g r a m o f t h e
showing a d i s o r d e r e d magnetic ( .....
phase
(3)
) transition
points
intersecting
, a ferromagnetic
separated lines.
by f i r s t -
At t h e i r
(
juncture
surface (2)
gas model
and a n a n t i f e r r o ) and s e c o n d - o r d e r
are t r i c r i t i c a l
(a).
The c r i t i c a l of t h e
(I
6
exponent V
is
related
to
the
"Hausdorff
dimension"
surface
~N : 1/,# defined
at ~ = ~ c r i t
~, where s(R)
= ]-~ is
by
<~.~s(~>/~.~P.
the part
of the
w i t h r a d i u s R, such t h a t Carlo result is
the
~k = 2 . 3 0 i n good agreement w i t h in [20~.
(37)
(38)
surface surface
contained passes i s
_+ o.o~
a Flory-type
formula
in
a sphere
centre.
The Monte
(39) dw = 2 1/3 d e r i v e d
228 References I.
M.Karowski, (1985)5
R.Schrader,
and H.J.Thun,
2.
M.Karowski, H.J.Thun, Gen.16(1983)4073
3.
M.Karowski
and H.J.Thun,
4.
M.Karowski
and F.Rys,
5.
M.Karowski,
J.Phys.A:
6.
J . F r ~ h l i c h , in ' P r o g r e s s in Gauge F i e l d T h e o r y ' , ed.G. t ' H o o f t et al (NATO Advanced Study I n s t i t u t e S e r i e s B Ro 115) (Plenum, New York 1984)
7.
K.Symanzik, in 'Local Quantum T h e o r y ' , P r o c . l n t . S c h o o l of Physics ' E n r i c o F e r m i ' , Course XLV, e d . R . J o s t (Academic, New York 1969), p.152
8.
P.de Gennes, Phys. Lett.38AC1972)339
9.
A. Maritan
I0.
K.Wilson,
II.
B.Durhuus, (1983)185
~.Helfrich,
Commun.Math.Phys.97
and F.Rys,
J.Phys.A:Math.
Phys.Rev.Lett.54(1985)2556
J.Phys.A:
Math.Geno19(1986)2599
Math.Gen.(in
and C.Omero,
press)
Phys.Lett.lO9B(1982~51
Phys.Rev. DlO(1974)2445 J.Fr~hlich,
12. A . P o l y a k o v ,
and T.Jonsson,
Nucl.Phys.B225
Phys.Lett.lO3B(1981)207
13. P . J . F l o r y , ' P r i n c i p l e s of Polymer C h e m i s t r y ' C o r n e l l U n i v e r s i t y Press,1969) 14. F.Rys and W . H e l f r i c h , 15. J . C . W h e e l e r , (1980)1748
(Ithaca,
S.J.Kennedy,
and P . P f e u t y ,
Phys.Rev. L e t t . 4 5
Phys. L e t t . l O 2 A [ 1 9 8 4 ) 4 2 0
17. P.de Gennes and C.Taupin,
J.Phys.Chem.86[1982)2294
18. J.D.Weeks, ' O r d e r i n g in S t r o n g l y F l u c t u a t i o n Systems' ed T . R i s t e (Plenum, New York 1979)
20. A . M a r i t a n
and J . G r e e n s i t e , and A . S t e l l a ,
N.Y.:
J.Phys.A15(1982)599
16. T.Hofs~ss and H . K l e i n e r t ,
19. T . S t i r l i n g
FS9
Condensed M a t t e r
Phys. L e t t . 1 2 1 B ( 1 9 8 3 ) 3 4 5
Phys.Rev. L e t t . 5 3 ( 1 9 8 4 ) 1 2 3
FIELD THEORETIC
METHODS
WITH
IN CRITICAL
PHENOMENA
BOUNDARIES
AJvL Nemirovsky
The James Franck Institute
The University of Chicago, Chicago, IL 60637
ABSTRACT
Recent work on field theoretic methods in critical phenomena with boundaries by the author and collaborators is described. The presence of interfaces and boundaries in critical systems produce a much richer set of phenomena than that of infinite sized systems. New universality classes are present and interesting crossover behavior occurs when there is a relative variation of additional length scales associated with either the size of the system or the boundary conditions (BC) satisfied by the order parameter on the limiting surfaces. A recently proprosed crossover renormalization group approach is very well suited to study these rich crossovers. Since functional integrals provide an indefinite integral representation of field theories, Feynman rules in configuration space are independent of geometry and BC. Renormalization of field theories with boundaries is discussed and various geometries and BC are considered. Application of field theoretic techniques are described for studying conformational properties of long polymer chains in dilute solution near interfaces or in confined domains. Also, related problems in quantum field theories with boundaries are presented.
The work I present here was performed in collaboration with ICF. Freed. Also, Z-G. Wang and J.F. Douglas have contributed to some of the work described below.
1, INTRODUCTION Experiments and computer simulations can only probe finite systems with limiting surfaces. On the other hand, theoretical studies of phase transitions (PT) usually consider infinitely extended systems. Although surface effects can, in general, be neglected in large systems, these effects become very relevant near a second order PT point as the correlation length grows unbounded? Critical singularities at a second order PT only occur in the
230
thermodynamic limit as they are rounded off in finite systems. On the other hand, systems which are of infinite extension in two or more dimensions and which are unbounded in the remaining directions, show interesting dimensional crossovers as the transition is approached. 2 Then, it is important to extend theoretical approaches to understand finite systems with limiting surfaces. Phenomenological finite-size scaling methods are widely used to extrapolate computer data to the thermodynamic limit, 2 but there are many aspects of finite size scaling which remain to be described by fundamental theories such as renormalization group (RG) methods. Such a fundamental theory becomes more important as interest extends to the study of particular finite systems with interacting boundaries. This is because universality classes of finite systems are more restricted than those of unbounded systems. The finite systems are characterized not only by the dimensions of the embedding space and of the order parameter but also by the geometry of the system and the boundary conditions (BC) for the order parameter on the limiting surfaces. 2 Here I discuss the application of field theoretic RG techniques to study critical phenomena in the presence of boundaries. The systems may be finite (or semi-infinite) along one (or several) of their dimensions, but they are of infinite extent in the remaining directions. Examples include systems which are finite in all directions, such as a (hyper) cube of size L, and systems which are of infinte size in d' = d - 1 dimensions but are either of finite thickness L along the remaining direction (e.g. a d-dimensional layered geometry) or of semi-infinite extension, etc. The presence of geometrical restrictions on the domain of systems also requires the introduction of BC (periodic, anti-periodic, free surfaces) for the order parameter on the surfaces. Critical systems with boundaries or interfaces display a very rich set of phenomena because the (totally or partially) finite and semi-infinite cases contain several competing lengths and hence have interesting crossover behaviors as these length scales vary with reslx~t to each other. These additional lengths are either associated with the finite size of the system in one or more of their dimensions or to the boundary conditions on the order parameter ~.1.2 Consider, for example, a semi-infinite critical system with a scalar order parameter which satisfies either the Neumann or the Dirichlet BC at the surface. These two cases belong to different universality classes called the special and ordinary transitions, respectively. 1 A surface interaction parameter c is usually introduced as (1/¢)(~/~n~n
= c where (3¢~/~n) stands for the normal derivative o f ~ at the limiting surface 3i2.1 Then, as c
ranges from zero to infinity the system crosses over from the special to the ordinary transition. These transitions are characterized, among other things, by different surface critical exponents. 1 On the other hand, systems that are bounded in one direction but of infinite extent in the remaining ones show a very interesting dimensional crossover as follows: In the critical domain, but away from the critical point, the behavior is dominated by the non-trivial 3d bulk fixed point, while as the transition is approached the 2d fixed point controls the physics. 2
231
Section 2 shows that Feynman rules of field theories in configuration space are independent of geometry and boundary conditions, so they are identical to the well-known rules for unbounded systems. Geometrical constraints and boundary conditions are implemented through the explicit form of the zeroth order two point .corrclati0n function. Semi-infinite critical behavior is briefly discussed in Section 3 where we inla'bduce a model of two coupled semi-infinite critical systems which possess a very rich physics. Section 4 considers the renorrealization of field theories with boundaries and discuss a crossover renormalization group approach that is very well suited to describe interesting multiple crossovers present in these field theories with boundaries. Section 5 deals with other interesting geometries. We begin by briefly discussing curved surfaces and edges, and then pass on to layered geometries with various boundary conditions (such as periodic, anti-periodic, Dirichlet and Neumann), and to cubic and cylindrical geometries. An important conclusion is that the usual eexpansion technique can be utilized to study any geometry and boundary condition as long as the smallest finite system size is not much smaller than the bulk correlation length of the system. Field theoretic methods can also be utilized to study the statistics of long polymer chains in solution near (liquid-liquid, liquid-solid) interfaces or in confined domains (such as a polymer chain in a cylindrical pore). This is the theme of Section 6. Finally, in Section 7 we present some analogies between the statistical mechanical problems of the preceding sections and related problems in quantum field theories.
2. Indefinite Integral Representation of Field Theories Functional integrals provide an indefinite integral representation of the differential equations of a field theory. However, this representation does not contain a complete specification of the boundary conditions. Hence, the same functional integral representation of a field theory applies for various boundary conditions.3 Consider, for example, an O ( N ) N-vector scalar ¢4 field theory in d = 4 - ~ dimensions in a region of the space ~ with a (d-l) dimensional boundary 3f~. The partition function Z[J] is a functional of the external source J given by Z [ J ] = ~D [¢]exp [ - F { ¢ } - ~ddxJ ( x ) ~ x ) ] ,
(2.1)
where F is the free energy functional, D [¢] represents the sum over all configurations of the order parameter ¢(x), x is a d-dimensional position vector inside the region fL to ~ T - To, with Tc the (mean field) bulk critical temperature, and uo are the bare reduced temperature and coupling constant, respectively. It is possible to formally integrate Eq. (2.1) over ¢(x) to obtain
232
.°.pI
t
,,,
where N is a normalization constant such that Z[J = 0] = 1 and G (°) is the bare propagator (two-point correlation function) which is the solution to the usual Klein-Gordan wave equation ( - V 2 + to)G (°) (x, x') = 8(a)(x - x ' ) ,
(2.3)
Eq. (2.3) is satisfied in the region f~ and it must be supplemented with appropriate boundary conditions at ~f~. Equivalently, G(°)(x, x') in (2.2) is only properly defined when boundary conditions are specified. The integral representation (2.2) of the ~p4 field theory is indefinite and applies to arbitrary boundary conditions which are implemented through the properly chosen propagator G (°). Coordinate space Feynman rules follow from (2.2), so they are independent of the explicit form of G(°)(x, x'). position space diagrammatic
rules remam
unchanged from
Hence, the above discussion implies that those o f
an infinite volume
theory, but
that the appropriate zeroth-order propagator G (°) (x, x') must be utilized. Chapter 14 of Ref. 4 contains expressions for the zeroth order two-point Green's function (in the context of the heat conduction problem) for a wealth of geometries and boundary conditions. Translationally invariant systems have G(°)(fx, x'l) but, in general, the presence of interacting surfaces breaks this symmetry making G(°)(x, x') * G(°)(Ix - x'l). The n-point Green function also depends on all n coordinates rather than on n-I coordinate differences as in full space. Diagrammatic expansions for unbounded systems are more conveniently performed in momentum space5 where the translational invariance of the theory is reflected in momentum conservation conditions. The "most" convenient choice for finite systems depends on geometry and BC as discussed in Ref. 3. In the following sections we discuss various geometries and BC.
3. Critical Behavior at Surfaces Semi-infinite critical systems have been studied by several workers using a variety of methods as described by Binder in his comprehensive review on the subject) Renormalizadon group techniques have proven to be one of the most powerful theoretical techniques to study critical phenomena at surfaces. An excellent review by Diehl describes recent advances in this area. 6 Thus, the topics presented below sketch out very recent results which, in general, are not covered in either review. The interested reader may find the details in the reviews of Refs. 1-and 6 and inthe original papers.
3.1. Semi-Infinite Geometry. Two Coupled Semi-Infinite Systems. We begin with the usual Ginzburg-Landau free energy functional of (2.1a) in a semi-infinite geometry. Thus, the region f2 is the positive half-space z>0 bounded by the (d-1)-dimensional flat surface 0 ~ at z = 0.
233
The position vector x of (2.1) is decomposed into its Cartesian components p and z with p a (d-1)-dimensional position vector parallel to the surface ~f2 at z = O. Mean field theory predicts the appearance of four phase transitions depending on the values of the reduced temperature t o and the surface interaction parameter c o (introduced through the boundary conditions satisfied by G (°) at 0f~ as discussed in Sec. 1). 1 These phases are depicted in Fig. la. For co~--~0the system orders at the bulk critical temperature to = 0. When c o is large, or more precisely when Co.~t] a , the transition is called ordinary, while for small values of co, such that co,~tio a , it is known as the special transition. For c0
sFESpP
0
BF
BF
(a) FIGURE
C )
(b)
1
(a) Mean field predicts three phases depending on the values of the reduced temperature t and the surface interaction parameter c for semi-infinite systems: a paramagnetic (P) phase for t>0 and -tll2
234
Recently, we 7 have considered a continuum theory of two coupled inequivalent semi-infinite systems based on suggestions in an alternative form by Diehl et al. g This theory was originally designed to understand the failure of the interesting conjecture of Bray and Moore 9 that the semi-infinite surface crossover exponent Cs equals 1 - v, where v is the usual bulk correlation length exponent. Nevertheless, due to technical difficulties (such as the presence of "quasi-local" couplings), Diehl et al. g failed to construct an explicit realization of the theory, but their analysis of the problem enabled a realization to be devised by alternative means,g Before discussing the continuum model for the two coupled systems, it is convenient to introduce a discrete version as illustrated in Fig. 2a. The model in Fig. 2a consists of two coupled semi-infinite spin systems on a d-dimensional hypercubic lattice. Neighboring spins in each of the semi-inlinite systems A and B interact with bulk coupling constants J~ and J~. Surface spins have different interaction constants Jl~ and J r , respectively. The two semi-infinite systems are coupled by perpendicular bonds with interaction strength J.t. When J.L = 0, the two systems decouple. For J . t ~
the two surface layers "collapse" onto a layer with the
nearest neighbor interaction Jn = J~l + J~l as shown in Fig. 2b.
Jll
B
FIGURE
2
(0)
B
(b)
Figure 2a illustrates a discrete version of the model of two coupled semi-infinite systems of Section 3 whose symmetric limit (J~ = J~t and J~ = Jff) is discussed in detail by Ref. 7. The model consists of two coupled semi-infinite spin systems on a d-dimensional hypercubic lattice. Neighboring spins in each of the semi-infinite systems A and B interact with bulk coupling constants jA and Jff, respectively. Surface spins have interaction constants J~l and J~ that differ from the bulk ones. The two semi-infinite systems are coupled by perpendicular bonds with interactions Jj.. When J± = 0, the two systems decouple. For J.L~
the two surface layers "collapse" onto a single layer with a single surface parameter JII = J~ + J~
as shown in Fig. 2b.
235
The full phase diagram of the model of Fig.2a is very rich. In addition to the (double-bulk and doublesurface) paramagnetic phase, it exhibits four distinct broken symmetry phases as follows: a double-balk, a single-bulk, a double-surface, and a single-surface ferromagnetic phase. An experimental realization of this model can be obtained by inserting a thin layer of a magnetic material X between two magnetic bulk materials A and B. If the two bulks are identical (and consequently J~ = Jl~ and Jba = J~), then the model reduces to that of an infinite critical system with a defect plane. The continuum Ginzburg-Landau free energy of the model is again given by (2.1a), but now f2 contains both the positive and negative half-spaces, i.e., ~ = {x = (p, z) with z>0 and z<0} and ~f2 = {x = (p, z) with z = 0+ and z = 0_}. In general, the reduced temperatures (t/t, tB ) and coupling constants (u/t, uB) of the two half-spaces can be different. To completely define the model, the boundary conditions for the (zeroth order in u/t and uB) two-point correlation function GC°)=-G(°)(Ip-p't,z,z ') are specified on the surfaces at z =0+andz =0_by 7
~G(O> 3z ~+ = c/t:~G¢+°>+ 3°(G+¢°~ - G¢-°)) '
(3.1a)
~G to) Io_ = Ca,oG ~-°)+ 3'o(G ¢-t~- G ¢+°)),
(3.1b)
3o = 3'0, where
the
arguments
of
GO°) have
been
dropped,
(3.1c) and
~G¢°)/3z[o. and
G¢+°) stand
for
3G~°)(Ip - 13'I,z, z')/3zL__,o,and G¢°>(Ip - p'l, z~0+, z'), respectively. The bare surface interaction parameters are denoted by c/t,o,Ce,o and 3o. The latter one couples the two semi-infinite systems. Since the (zeroth order in u/t and UB) two point Green's function satisfies the second order partial differential equation (2.3), boundary conditions are specified by given functions of G~°) , G~-°) , (3G¢°)/~z)~o+and (3GC°)/3zlo on the surfaces atz=0+andz=0o The boundary conditions (3.1a) and (3.1b) are the most general linear and homogeneous ones relating
3GC°)/3z~+, (3GC°)/3z)~o_,GC+°) and GJ°). Eq. (3.1c) is required to obtain a symmetric Green's function (in its arguments) i.e., G ¢ ° ) ( I p - p ' l , z , z ' ) = G ¢ ° ) ( I p ' - p l , z ' , z ) .
The presence of surface magnetic fields
h~ and h~ produces inhomogeneous contributions to (3.1a) and (3.1b). We take h'~ = hin = 0 for simplicity. Finally, the fact that we are considering renormalizable scalar field theories near four dimensions does not permit the use of nonlinear boundary conditions. (Nonlinear terms are associated with irrelevant surface operators heard =4). As discussed above the presence of surfaces breaks the translational invariance of the theory along the direction perpendicular to the surfaces. Of course, momentum is conserved in the direction parallel to the sur-
236
faces and, hence, G (°) =- G(°)(Ip - p'l, z, z'). The function G C°) is, in general, discontinuous at z ( z ' ) = O. (It only becomes continuous for ~o = 0"). This should not come as a surprise since a similar discontinuity is well known to occur in other contexts. For example, the presence of dielectric layers produces a jump in the poten~I, and the temperature is not conunuoas, in general, at the surface of separation of two media of different conducuvlties. After some algebra and using equations (2.3) and (3.1), the zeroth order two-point correlation function G(°)(p, z, z') with p the momentum variable conjugate to p - p' is found to be of the form
G <°~6n,z, z') = G~>O(z)O(z ") + G~°~O(-z)O(-z ")
(3.2)
,- G~O(z)O(-z') + a ~ 0 ( - z ) 0 ( z ' ) ,
The functions G~,~ O, G~,~ ~, G ~ and G ~ are presented in Ref. 7 and some limiting cases are of interest. When go = 0, then G ~ ) = G~°)= 0, while G ~ ) and G~°) become identical m the two-point correlation functions of semi-infinite systems with surface interaction parameters CA.oand ca.0, respectively. 1 When ~o ~ 0, the two semi-infinite regions are coupled. The ~ 0 ~
limit produces the single surface interaction model of Bray and
Moore 9 with c ~ M) = c,A.o -~ CB.O. As can be seen, the model of two coupled semi-infinite systems describes a very rich physical situation. Even the exactly solvable Gaussian theory with uA.o= us,0 = 0 is of interest and is far from trivial.
4, Renormalizatlon of Field Theories with Boundaries As stated in Section 2, Feynman rules of field theories in configuration space are independent of geometry and boundary conditions. These constraints are implemented through the explicit forms of the two-point functions so, for the problem of two coupled semi-infinite systems, the usual flee propagator is replaced by the Fourier inverse of (3.2), and standard Feynman ruless are utilized to evaluate diagr,uns. Nevertheless, the breaking of translational invarlance introduces novel features in these problems such as the presence of one-particle reducible primitively divergent diagrams as shown in Fig. 3 and as discussed in length by Ref. 6. Ren0rmalization of field theories in presence of interacting boundaries has been studied by Symanzik, 1° by Diehl and Dietrich, H and more recently by Diehl 6 and by us.3 In addition to bulk renormalization constants Zo, Zf and Z,, which remain unchanged by the presence of surfaces (as do the I] function and the fixed points {u* }), it is necessary to introduce two additional renormalization functions Zc and Z1 required to renormalize surface interaction parameters and the fields on the surface. In the two coupled semi-infinite systems problem Zc becomes a 2 x 2 non-symmetric real" matrix.7
237
(o) FIGURE
ooo
(b)
c)
3
A new feature associated with the breaking of translational invariance m the existence of one-particle reducible primitively divergent diagrams. For example, the bare two-point functions G, G~ and G~I with 0. I and 2 points on the surface of (a), (b) and (e), respectively, have different singularities, thus requiring different renormatization constants as discussed in Section 4.
With a semi-infinite geometry it is useful to work in a mixed momentum-configuration space representation. For example, we define the bare n-point connected Green's function G } ") (Pi, zl, Co, to, uo) [with Co the column vector (Co, ~0) for the problem of coupled semi-infinite systems] as
Ga('~)(p,,z,, Co,to.
Uo) =
I (2~)"-' " •' exp(-ipvpw)l](an)d-~Sa-1(XPO i=1
(4.1)
Gs(~)(Pi, zi, Co, to, uo)
where Gt}')(pl, zi, ¢o, to, uo) is the bare n-point function in configuration space, and the 8 function reflects the fact that momentum is conserved in the direction parallel to the surfaces. The renormalized Green's function G/¢~)(pi, zi, e, t , u, 1¢,) is then given by
Gfl°(pi, zi, e, t, u, ~) = Z~""/2 [Z1('~)]-l/2G/~")(pi, zi, Zc c,Z, t, S E I ~ Z w u ) ,
(4.2)
where R is a parameter having dimensions of (temperature)1~2 used to define a dimensionless coupling constant u and Sa is lhe area of a sphere of unit radius. Minimal subtraction dimensional regutarization is the most
widely used technique to renormalize these field theories. 6 Minimal subtraction has ZI ") of the form Z~") = [Zi(u)] '~, if zl = 0, i = 1, 2 ..... m, and zi ~ O, i = m ~- 1 ..... n
(4.3)
Although this renormalizauon procedure is very convement to study the physics near a given fixed point [bP] such as the special FP, the ordinary FP, the bulk FP, etc., 6 minimal subwacdon techniques are not well suited to describe the rich crossovers of field theories with boundaries with two or more competing fixed points.12 We have recently proposed a crossover RG approach that is very convenient for studying critical phenomena with several competing lengthsJ 2 Amit and Goldschmidt ~3 utilize mathematically similar techniques to fully describe the bicritical crossover. In our coupled systems problem the surface normalization constants
238
are taken to depend on the extra lengths through the dimensionless combination ~cz, c/w., and 3/~c. This dependence emerges naturally by imposing appropriate normalization conditions on the two- and four-point Green's functions of the theory. In contrast, minimal subtraction dimensional regularization has the normalization constants independent of these lengths but only dependent on u and e. Due to the explicit 1c-dependence of the renormalization constants in addition to the usual implicit ~cdependence through the dimensionless coupling constant u, the renormalization group equations become more involved, but they now describes the full crossover between all fixed points. Consider, for example, a semi° infinite critical system near the special transition. We have evaluated to one-loop approximation ,2 the full zdependence of the surface susceptibility Xlx(z) describing the response of spins in a plane at a distance z away from the surface at z = 0 due to a magnetic field applied on the same plane. At the non-trivial fixed point u ° , the crossover RG equation implies the following scaling form for the renormalized surface susceptibility 7~,n
Xa,n(z, t, u* ,It) = ~-t + 2v(1- n)t*(1 - n)g (x, y ) F ( x ) ,
(4.4)
where v and 11 are the usual bulk exponents, y = Kz and x = ~z(t/K2)v. Ref. 12 presents the functions g(x, y)
and F(x) to O(e), and here we only give some interesting limiting cases. We always consider the asymptotic limit t < ~ ,
but the magnitude of ~
remains at our disposal.
Thus, y =w.z is always larger than
x = (re.)(t/~)" and three regimes exist. The x--coo limit gives 12
g (x, y) --->exp[ex-U2exp(-x)] ,
(4.5a)
F(x) --* 1 + 0 [exp(-x)] ,
(4.5b)
where higher order corrections, varying as x -l exp (-x), have been dropped. Eqs. (4.4) and (4.5) imply that bulk behavior is approached exponentially fast for 0_l, we obtain 12 g ( x , y ) = C ( y ) x -[(N + 2~(N + 8)le
F(x) = 1 + O(x, xlnx) ,
(4.6a)
(4.6b)
where C ( y ) is a finite function of y. The form predicted by (4.4) and (4.6) is in accord with scaling assumptions and previous calculations using minimal subtraction. 6 We stress that this near surface behavior is a
direct donsequence of the full crossover renormalizadon group approach. ,There is no need to utilize operator product expansion techniques, t2 Instead, these techniques are only required in the usual minimal subtraction
239
approach because the standard RG equation does not contain information about the near surface behavior. 6 Finally, as y - t 0 we find 12
g (x, y) = C ( y ) (x/y) -tOy + 2~(N + 8)]~
(4.7a)
F (x ) = 1 + O(x, xlnx)
(4.To)
d'(y) = I - [(N + 2)/(N + 8)]e y ln(y/2).
(4.7c)
Hence, as r.z--~0, 7~.~(z) reduces to 7~.11 = 7~m( z = 0) as expected physically and in contrast to the results of the minimal subtraction renormalization approach. It is interesting to note that 7~.xl(z) is a continuous function of z for 0 ~ <~, but no derivates exist at z = 0. Thus, it cannot be Taylor expanded about z = y = 0. We have also evaluated the surface susceptibilities {7~,1x} for the symmetric version of the model of two-coupled semi-infinite critical system discussed in Sec. 3. 7 Since there are two surface magnetizations m~ and m~ on either side of the interface and since a surface external field h ~ ( h f ) can be applied to the positive (negative) side of the interface, four surface susceptibilities {Zll} can be defined. The symmetric model (ca = cB = c, uA = uB = u) only has the two susceptibilities. These susceptibilties are calculated near the special, ordinary and bulk fixed points. 7 Our results are in agreement with the qualitative discussion of Eisenriegler and Burkhard 14 based upon plausible renormalization group flow properties of the model of Fig. 2a. In particular, we find that the two crossover exponents ~, and 0,' near the special fixed point are the usual semi-infinite crossover exponent 0~ and the special exponent "fi~, respectively. Both of these exponents are given in Ref. 6 to O (e2). Near the bulk fixed point only the surface interaction parameter c is relevant, and its associated surface exponent ~ equals (1 - v) as argued by Bray and Moore. 9 Finally, from oar work it can be inferred that the infinite and semi-infinite ordinary transitions belong to the same universality class as predicted by Ref. 14 since the two limiting cases ( c / r ) ~
with (d/K)= 0 and with ( ~ / ~ ) ~
yield the same surface susceptibilities in
leading order. (Of course, they have different corrections).
5. Other Geometries
5.1. Curved Surfaces and Edges, Curvature effects in critical systems have barely been considered beyond mean field theory. Recently, Eisenriegler L~ has studied 04 field theory inside a d = 4 ~ e dimensional hypersphere near the ordinary ~transition to one-loop order, and Diehl has briefly discussed curvature effects in his recent review. 6 Consider, for example, a critical system bounded by the exterior surface of a sphere of radius R close to the special transition (the outside problem). Then, as the ratio (UR) ranges from zero to infinity, the behavior of the system crosses over from that of a semi-infinite system at the special transition to bulk behavior. We
240
expect this crossover to be described using the crossover renormalization group approach with surface normalization constants Z~ and Zc that depend on rd~ in analogy with problems discussed in Sec. 4. More generally, for an arbitrary curved surface, we believe that the surface constants should depend on K{R }, where [R } stands for the local radius of curvature of the surface. Cardy 16 has studied critical phenomena near an edge where there is an edge interaction parameter 5 (analog to the surface interaction parameter c) that measures how interactions near the edge differ from those deep in the bulk. In addition to the bulk and surface renormalization constants, there are now two additional edge renormalization constants Z8 and ZE (analogs of Z c and ZI) that renormalize the edge interaction constant and the fields on the edge respectively. Cardy has considered the case c = ** and 8 = ** (analogous to the ordinary transition).
He finds a new angular dependent critical edge exponent (even Gaussian edge exponents are
angular dependent!).
5.2. Layered Geometries Before our recent work 17 it was widely believed that e-expansion techniques were inapplicable for treating finite size problems. 2 We considered 17 a layered geometry (infinite in ar = d-1 directions and of thickness L in the remaining one) with periodic boundary conditions (BC), and we employed analogies with mathematically similar problems in finite temperature field theories to illustrate the applicability of the e-expansmn methods away from the critical point. Since then, other geometries and BC have been studied 15'3 with well defined eexpansions. Our studies of field theories in confined reglons have used a variety of BC. We show 3 that as long a (L/~)>I, where ~ is the correlation length, e-expansion techniques can be utilized to describe corrections to bulk quantities due to the finite extent of the system. Similar general results were derived by us for the effects of interacting boundaries where L is a parameter associated with surface interactions (as briefly summarized in Fig. lb). The theory is illustrated for the N-vector model in a layered geometry with periodic, anti-periodic, Dirichlet and Neumann BC where the correlation functions and susceptibilities are evaluated to O (e). Away from the critical point and when ( L / ~ ) ~ ,
we find that first order contributions to scaling functions due to finite size are
exponentially small, proportional to exp(-L/~), for periodic and anti-periodic BC, while these corrections behave as (~L) for free surfaces. This is in accordance with previous numerical calculations and results obtained from various models. 2 As the scaling variable (L/k) approaches unity, we show that first order in e corrections to scaling amplitudes become comparable with zeroth order terms. This marks the beginning of a dimensional crossover where
241
expansion methods break down. The finite size scaling literature 2 usually states that dimensional crossover occurs when the bulk correlation length becomes comparable to the typical system size L. While this is demonstrated by us to be true for a layered geometry with periodic or Neumann BC, it does not hold for example, for a layered geometry with anti-periodic or Dirichlet BC for which the e-expansions are well behaved even at the bulk critical temperature T, .3.1s
Close to the transition a region of dimensionally reduced physics emerges. Layered systems near the shifted critical temperature and semi-infinite geometries near the surface transition have d" = d - l .
Of course,
different geometries, such as an infinite cylinder, a cube, etc., give different ae. We discussts two mechanism for producing dimensional reduction (the emergence of d'-dimensionat physics out of an underlying ddimensional system): a geometrical one (e.g., a layered geometry very close to the shifted critical point), and an interaction drive one (e.g., a semi-infinite system close to the surface transition). An L dependent d" dimensional effective free energy functional for the lowest mode of the order parameter (massless mode) is evaluated by integrating out the higher (heavy) modes. Our approach presents some conceptual difficulties that still remain to be understood to fully describe the dimensional crossover. Can the crossover renormalization group approach be applied to this problem? We are presently investigating this interesting possibility.
5.3. Cubes, Cylinders and Other Geometries. Dynamical Critical Phenomena and First Order Transitions Our recent work 3 and that of Ref. 15 show that the usual c-expansion techniques can be applied to study any geometry and boundary conditions as long as the bulk correlation length of the system is not much larger than the smallest dimension of the system. As the system approaches arbitrarily close to the critical (or pseudocritical) point, the e-expansion break down. Related techniques to our effective free energy functional method have been proposed by Brtzin and Zinn-Justin,19 and by Rudnick et al.20 to investigate the deep critical region for cubic and cylindrical geometries with periodic BC. Their approaches do not present the technical difficulties of ours as discussed above, since these authors only consider systems with no true critical points. Brtzin and Zinn-Jusdn have also proposed a 2 + e expansion to study finite size effects in critical phenomena below Tc.19 Since then, several authorsm have extended the methods of Refs. 19 and 20 to study finite size effects on dynamics and in first order transitions always for systems with no true critical points and with periodic boundary conditions.
6. The Statistics of Polymers in Various Geometries. The study of conformational properties of long, flexible polymer chains near penetrable (liquid-liquid) or impenetrable (liquid-solid) interfaces or in various confined geometries (e.g., polymer chains in cylindrical or
242
spherical pores) has a variety of important practical applications. These applications include cohesion, stabilization of colloidal particles, chromotography reinforcement and floccalation. Also, we note that finite-size effects are present in computer simulations of polymer systems. Simulations generally employ periodic boundary conditions to remove the surface interactions, but the finite size of the computer still affects the computed thermodynamic properties. Therefore, systematic extrapolation of the simulation data is required in order to describe properties of the infinite system. It is, therefore, of theoretical interest to understand how the thermodynamic limit is approached as the size of the system is increased. The statistics of long flexible polymer chains with excluded volume in dilute solutions is well known to belong to the same universal class as that of the O(N) ¢4 field theory with N = 0. 99 This holds not only for unbounded systems but also for systems with interacting interfaces and those in confined geome~ies. Thus, most of the results for critical systems discussed in the previous sections can be transcribed to corresponding polymer problems. We have used powerful field theoretic techniques to study the conformational properties of polymers near interacting impenelrablez3 and penetrable z4 interfaces and polymer chains in confined geometriesz~ such as polymers between two parallel plates with various polymer-surface interactions on the limiting surfaces or polymers near the outside surface of a repulsive sphere. Some of the rich array of situations that can now be treated using renormalization group methods are illustrated in Fig. 4.
(a)
(b)
(d)
(c)
(e)
These figures illustrate some interesting systems involving a single polymer chain with excluded volume and interacting boundaries in several geometries. These geometries can now be studied by employing the RG methods discussed in Section 6 [as long as the radius of gyration of the polymer chain is not longer than the smallest dimension of the system]:
(a)
A polymer attached to a sphere with an interacting surface.
243
(b)
A polymer in the shell formed by two concentric cylinders.
(c)
A polymer near a sphere formed by two different solvents, e.g. oil and water. The quality of these two solvents is, in general, different. Furthermore, the interfacial region can be such that one side of the interface attracts the polymer whereas the .other side repells it.
(d)
A polymer in an edge where the power law exponent for some property(ies) can depend on the edge angle.
(e)
A polymer in a cone.
7. Quantum Field Theories with Boundaries
It is well known that there are many analogies between statistical mechanics and quantum field theories (Qk-'T) for unbounded systems.5 For example, the Green's functions of the QFI"s are the analogues of the correlation functions in statistical mechanics, and Z[J] of (2.1) can be viewed as the generating functional of Euclidean self-interacting scalar QFT. Successive derivatives of Z[J] respect to the external source J produce all the Green's functions of the theory. These analogies, of course, also hold when boundaries are present.
The Casimir effect, the attraction of two neutral and parallel plates in a vacuum environment, predicted and experimentally confirmed several years ago, is the earliest example of boundary effects in QFT. 26 An interesting example of the scalar "Casimir effect" in statistical mechanics, as discussed by Diehl,6 is provided by fluctuation-induced force between two plates with a binary fluid mixture at its consolute point held in between. Systems that are of infinite extent in two or more of their dimensions and finite in the remaining directions such as a layered geometry in d=3 dimensions, display 3d physics away from the shifted critical temperature (but inside the critical domain) but ae=2-dimensional physics in the deep critical region.2 Dimensional reduction, the emergence of a quasi ar dimensional physics out of an underlying d dimensional system, is one of the main ingredients of the Kaluza-Klein theories. 27
In fact, Kaluza-Klein masses are the analog of
experimentally observed 2s shifts in critical temperatures of finite size systems from those of the bulk.
We have used the analogy between finite size problems in a periodic layered geometry and similar problems in finite temperature field theories to demonstrate how e-expansion techniques can be employed to study finite systems away from the shifted critical point as described in Sec. 5.2. At finite temperatures 1]-1 (where = (kT) -1, k is Boltzmann's constant and T is the absolute temperature) the causal boundary conditions of field
theories in real time are replaced by periodic boundary conditions with period !3 in Euclidean time. 29 Thus, a
244
finite temperature field theory is identical to one contained between two-parallel (hyper) plates with periodic boundary, conditions. The (hyper) planes are perpendicular to the Euclidean time direction, and the periodicity is
8. ACKNOWLEDGEMENT I am grateful to H.J, de Vega for his kind hospitality at paris VI and to K. Binder. H,W. Diehl and E. Eisenricgler for useful discussions. This research is supported, in part, by NSF grant DMR 83-18560.
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2.
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H.B.G. Casimir, Proc, [{on. Ned. Akad. Wetenschap., BS1, 793 (1948). Experimental evidence is discussed by M J . Sparnaay; Physica, 24, 751 (1958).
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For a description of Kaluza-Klein theories, see, for example, E. Witten, Nucl. Phys. B186, 412 (1981); A. Salam and J. Strathdee, Ann. of Phys. 141, 316 (1982).
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