Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, Mijnchen, K. Hepp, Zijrich R. Kippenhahn, Mijnchen, H. A. Weidenmiiller, Heidelberg and J. Zittartz, KGln Managing Editor: W. Beiglbijck
246 Field Theory, Quantum Gravity and Strings Proceedings of a Seminar Series Held at DAPHE, Observatoire de Meudon, and LPTHE, Universit6 Pierre et Marie Curie, Paris, Between October 1984 and October 1985
Edited by H. J. de Vega and N. S6nchez
Springer-Verlag Berlin Heidelberg
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Editors H. J. de Vega Universite Pierre et Marie Curie, L.P.T.H.E. Tour 16, ler Stage, 4, place Jussieu, F-75230
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N. Sanchez Observatoire de Paris, Section d’Astrophysique de Meudon 5, place Jules Janssen, F-92195 Meudon Principal Cedex, France
ISBN 3-540-16452-g ISBN O-387-16452-9
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PREFACE
Perhaps the main challenge in t h e o r e t i c a l physics today is the quantum u n i f i c a t i o n of a l l i n t e r a c t i o n s , including g r a v i t y . Such a u n i f i c a t i o n is strongly suggested by the b e a u t i f u l non-Abelian gauge theory of strong, electromagnetic and weak i n t e r a c t i o n s , and, in addition, is required for a conceptual u n i f i c a t i o n of general r e l a t i v i t y
and
quantum theory. The r e v i v a l of i n t e r e s t in s t r i n g theory since 1984 has arisen in t h i s context. Superstring models appear to be candidates f o r the achievement of such u n i f i c a t i o n . A consistent description of primordial cosmology ( t ~ t Planck) r e q u i r e s a quantum theory of g r a v i t y . Since a f u l l quantum theory of g r a v i t y is not yet available, d i f f e r e n t types of approximations and models are used, in p a r t i c u l a r , the wave function of the Universe approach and semiclassical treatments of g r a v i t y . A nice p o s s i b i l i t y for a geometrical u n i f i c a t i o n of g r a v i t y and gauge theories arises from higher-dimensional theories through dimensional reduction f o l l o w i n g Kaluza and K1ein's proposal. Perturbat i v e schemes are not s u f f i c i e n t to elucidate the physical content of d i f f e r e n t f i e l d theories of i n t e r e s t in d i f f e r e n t contexts. Exactly solvable theories can be helpful for understanding more r e a l i s t i c models; they can be important in four (or more) dimensions or else as models in the two-dimensional sheet of a s t r i n g . In addition, the development of powerful methods f o r solving non-linear problems is of conceptual and p r a c t i c a l importance. A seminar series "Seminaires sur les ~quations non-lin~aires en th~orie des champs" intended to f o l l o w current developments in mathematical physics, p a r t i c u l a r l y in the above-mentioned areas, was started in the Paris region in October 1983. The seminars take place a l t e r n a t e l y at DAPHE-Observatoire de Meudon and LPTHE-Universit~ Pierre et Marie Curie (Paris Vl),and they encourage regular meetings between t h e o r e t i c a l physic i s t s of d i f f e r e n t d i s c i p l i n e s and a number of mathematicians. Participants come from Paris VI and VII, IHP, ENS, Coll~ge de France, CPT-Marseille, DAPHE-Meudon, IHES and LPTHE-Orsay. The f i r s t
volume "Non-Linear Equations in Classical and Quantum Field
Theory", comprising the twenty-two lectures delivered in t h i s series up to October 1984, has already been published by Springer-Verlag as Lecture Notes in Physics, Voi.226. The present volume "Field Theory, Quantum Gravity and Strings" accounts flor the next twenty-two lectures delivered up to October 1985. I t is a pleasure to thank a l l the speakers f o r accepting our i n v i t a t i o n s and f o r their
i n t e r e s t i n g c o n t r i b u t i o n s . We thank a l l the p a r t i c i p a n t s f o r t h e i r i n t e r e s t and
f o r t h e i r s t i m u l a t i n g discussions. We also thank M. Dubois-Violette at Orsay and J.L. Richard at Marseille, and B. Carter and B. Whiting at Meudon for t h e i r cooperation and encouragement. We acknowledge Mrs. C. Rosolen and Mrs. D. Lopes for t h e i r typing of part of these proceedings.
JV
We p a r t i c u l a r l y thank the S c i e n t i f i c Direction "Math6matiques-Physique de Base" of C.N.R.S. and the "Observatoire de Paris-Meudon" f o r the f i n a n c i a l support which has made t h i s series possible. We extend our appreciation to Springer-Verlag f o r t h e i r cooperation and e f f i c i e n c y in publishing these proceedings and hope that the p o s s i b i l i t y of making our seminars more widely available in t h i s way w i l l continue in the f u t u r e .
Paris-Meudon
H.J. de Vega
December 1985
N. S~nchez
TABLE
OF
CONTENTS
LECTURES ON QUANTUM COSMOLOGY S.W. Hawking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SOLITONS AND BLACK HOLES IN 4, 5 DIMENSIONS 46
G.W. Gibbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TRUNCATIONS IN KALUZA-KLEIN THEORIES C.N. Pope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
CANONICAL QUANTIZATION AND COSMIC CENSORSHIP P. H a j i c e k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
QUANTUM EFFECTS IN NON-INERTIAL FRAMES AND QUANTUM COVARIANCE D. Bernard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
STOCHASTIC DE SITTER (INFLATIONARY)
82
STAGE IN THE EARLY UNIVERSE
A.A. S t a r o b i n s k y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
SOME MATHEMATICAL ASPECTS OF STOCHASTIC QUANTIZATION G. J o n a - L a s i n i o ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
: ......
127
SUPERSTRINGS AND THE UNIFICATION OF FORCES AND PARTICLES M.B. Green . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
CONFORMALLY INVARIANT FIELD THEORIES IN TWO DIMENSIONS CRITICAL SYSTEMS AND STRINGS J.-L.
Gervais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
LIOUVILLE MODEL ON THE LATTICE L.D. Faddeev ( * )
and L.A. T a k h t a j a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
166
EXACT SOLVABILITY OF SEMICLASSICAL QUANTUM GRAVITY IN TWO DIMENSIONS AND LIOUVILLE THEORY N. S~nchez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOME FEATURES OF COMPLETE INTEGRABILITY
~80
IN SUPERSYMMETRIC GAUGE THEORIES
D. Devchand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190
MONOPOLES AND RECIPROCITY E. C o r r i g a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
206
Vl
NON-LOCAL CONSERVATION LAWS FOR NON-LINEAR SIGMA MODELS WITH FERMIONS 221
M. Forger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INVERSE SCATTERING TRANSFORM IN ANGULAR MOMENTUMAND APPLICATIONS TO NON-LOCAL EFFECTIVE ACTIONS
242
J. Avan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GENERAL STRUCTUREAND PROPERTIES OF THE INTEGRABLE NON-LINEAR EVOLUTION EQUATIONS IN I+I AND 2+I DIMENSIONS
267
B.G. Konopelchenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HIERARCHIES OF POISSON BRACKETS FOR ELEMENTS OF THE SCATTERING'MATRICES
284
B.G. Konopelchenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MULTIDIMENSIONAL INVERSE SCATTERING AND NON-LINEAR EQUATIONS A . I . Nachman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
298
AN SL(3)-SYMMETRICAL F-GORDON EQUATION Z B = ~ ( e Z - e -2Z) B. Gaffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
301
THE SOLUTION OF THE CARTAN EQUIVALENCE PROBLEM FOR d2y = F(x,y, dy) UNDER THE PSEUDO-GROUP~ = ~(X), y = ~ ( x , y )
~
dx
N. Kamran(*) and W.F. Shadwick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
320
QUANTUM R MATRIX RELATED TO THE GENERALIZED TODA SYSTEM: AN ALGEBRAIC APPROACH M. Jimbo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
335
SOLUTION OF THE MULTICHANNEL KONDO-PROBLEM N. Andrei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
362
THE DIRECTED ANIMALS AND RELATED PROBLEMS Deepak Dhar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
368
INCOMMENSURATE STRUCTURESAND BREAKING OF ANALYTICITY S. Aubry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L i s t of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (*) Lecture given by t h i s author
373 377
Lectures
on Quantum
Cosmology
S, W. Hawking
Department of Applied Mathematics & Theoretical Physics. Silver Street, Cambridge CB3 9EW.
1.
IntroduoUon,
The aim of cosmology Is to describe the Universe and to explain why it should be the way it is.
For this purpose one constructs a mathematical model of
the universe and a set of rules which relate elements of the model to observable quantities,
This model normally consists of two parts:
[11
Local Laws which govern the physical fields in the model, physics,
these
Laws
are
normally
expressed
which can be derived from an action can
be
obtained
from
a
path
I.
as
tn classical
differential
equations
In quantum physics the Laws
integral
over
all
field
configurations
weighted with e x p ( i I ),
[2]
Boundary Conditions which pick out one particular state from among the set of those allowed by the Local Laws. specified some
by the
initial
asymptotic
boundary
time
and
conditions
conditions
the on
quantum
the
class
for
The classical state can the
state C of
can field
differential be
equations
determined
configurations
be at
by the that
are
summed over in the path integral,
Many were
not
a
people would
question
for
say that
science
the
but for
boundary conditions
metaphysics
or
for
religion.
the
universe
However.
in
classical
general
b e c a u s e there
relativity
are
one
a number
cannot
avoid
of t h e o r e m s
the
problem
['1] which
of
boundary
show that the
conditions
universe
must
have started out with a s p a c e t l m e singularity of infinite density and s p a c e t i m e c u r v a ture.
At this
singularity
all the
Laws of
physics
would
break down.
Thus
one
could not predict how the universe would e m e r g e from the Big Bang singularity but would
have to impose
it as a b o u n d a r y condition.
the singularity t h e o r e m s in a
different way:
namely,
One can,
however,
Interpret
that they indicate that the g r a v i -
tational field was so strong in the very early universe that classical g e n e r a l relativity breaks down and that quantum
gravitational effects
have to be taken
There does not seem to be any necessity for singularities as I shall show,
into account.
in quantum
gravity and.
one can avoid the problem of b o u n d a r y conditions.
I shall a d o p t what Is called the Euclidean a p p r o a c h to quantum In this
one
performs
a path
integral
over
Euclidean
rather than over metrics with
Lorentzian signature
continues
Lorentzian
the
result
Euclidean a p p r o a c h
to
the
iI
+ + +) The
basic
and then
metrics
analytically
assumption
of
the
g/zv and
is proportional to
exp(= -
positive definite
is that the "probability" of a positive definite 4 - m e t r i c
matter field configuration •
where I
regime.
(-
i.e.
gravity.
(1.1)
I[g~v,O])
is the Euclidean action.
°/ 2
i[g/Lv,~] = .--Pz6rr
-
f(R
-
2A
M
where h i j fundamental
Is the 3 - m e t r i c form
-
_
~ 2KhZ/2d3x aM
(1.2)
m(g~v,~'))gl/2d4x P
on the b o u n d a r y aM and K Is the trace of the second
of the boundary.
The surface
term
in the action
is n e c e s s a r y
because physics
the
curvature
of the
scalar
universe
R contains
is g o v e r n e d
second
derivatives of the
by probabilities
metrics g/zv and matter field configurations
of the form
b e l o n g i n g to a certain
metric.
The
above for
all 4 -
class
C.
The
specification of this class d e t e r m i n e s the quantum state of the universe.
There seem to be two and only two natural c h o i c e s of the class C:
a)
C o m p a c t Metrics
b)
Non-compact metry,
metrics
i.e.
Boundary conditions
which
are asymptotic to metrics
of maximal s y m -
flat Euclidean s p a c e or Euclidean a n t i - d e Sitter s p a c e
of type b)
define the usual vacuum
state.
In this state the
expectation values of most quantities a r e defined to be zero so the vacuum state is not o£ as much Interest as the quantum
state of the
universe.
In particle
scattering
calculations one starts with the vacuum state and one c h a n g e s the state by creating particles by the action of field o p e r a t o r s at infinity in the infinite past. particles
interact
and
then
annihilates
field o p e r a t o r s at future infinity. s u p p o s e d that the quantum state, one
the
resultant p a r t i c l e s b y t h e
One lets the action
of other
This gets one back to the vacuum state.
If one
state of the universe was s o m e such
particle scattering
one one would loose all ability to p r e d i c t the state of the universe b e c a u s e would
have
no
idea what was
matter
In the universe would
would
decrease
to
zero
at
coming
become large
in.
One would
concentrated
distances
also e x p e c t that the
in a certain
instead
of
the
region and that it
roughly
homogeneous
universe that we observe,
In particle scattering p r o b l e m s , ity.
one is interested in o b s e r v a b l e s at infin-
One is therefore c o n c e r n e d only with metrics which are c o n n e c t e d to Infinity:
any d i s c o n n e c t e d c o m p a c t parts of the metric would not contribute to the scattering of particles from infinity. o b s e r v a b l e s In a finite whether
this
region
the class C which
In c o s m o l o g y , region
In the
is c o n n e c t e d
middle
to an
defines the quantum
on the other hand, of the
infinite
one is c o n c e r n e d with
s p a c e and
asymptotic
It does
region.
not matter
Suppose that
state of the universe consists of metrics
of
4 type b ) ,
The expectation value of an o b s e r v a b l e In a finite region will be given by
a path integral which contains contributions from two kinds of metric.
I)
Connected asymptotically Euclidean or a n t i - d e Sitter metrics
ii)
Disconnected metrics which consist of a c o m p a c t part which contains the region
of observation
and
an
asymptotically
Euclidean
or a n t i - d e
Sitter
part
One cannot exclude d i s c o n n e c t e d be a p p r o x i m a t e d by c o n n e c t e d thin tubes.
metrics
metrics
from
the path
in which
the different
b e c a u s e they can
parts were joined
The tubes could be chosen to have n e g l i g i b l e action.
logically non-trivial
metrics
by t o p o l o g i c a l l y trivial
cannot
metrics.
be excluded
It turns
path integral c o m e s from d i s c o n n e c t e d
defines
the
quantum
state
to
Similarly,
because they can
out that the d o m i n a n t
be
contribution
more
metrics.
This
natural to c h o o s e would
without any singularities edges
at
which
emphasised, universe.
mean
at which
boundary
however,
that
non-compact
universe
metrics
would
of
type
b)
would
is only
a
to the as far
would
be
It would t h e r e -
be c o m p l e t e l y
the laws of physics
conditions this
Thus,
C to be the class of all c o m p a c t
that the
topo-
the result of c h o o s i n g the class C
almost the same as c h o o s i n g it to be c o m p a c t metrics of type a ) . fore seem
by
be a p p r o x i m a t e d
metrics of the second kind.
as observations in a finite region are c o n c e r n e d , that
integral
non-singular self-contained
break clown and without any
have
to
be
orocosal
for
the
set.
It
quantum
should state
be
of the
One cannot derive It from some other principle but merely show that It
Is a natural choice, but whether
The ultimate test is not whether
it e n a b l e s
one to
make
predictions
It Is aesthetically a p p e a l i n g
that a g r e e with
observations.
I
shall e n d e a v o u r to do this for a simple model.
2. The Wavefunctlon In entire
of the Universe
practice,
4-metric,
one
is
but of a more
normally restricted
interested
in
the
probability,
set of o b s e r v a b l e s .
can be derived from the basic probability ( ] . ] )
Such
not
of
the
a probability
by Integrating over the unobserved
quantities.
A particularly Important
case
Is the probability P [ h i j , ~ o ]
of finding a
closed c o m p a c t 3 - s u b m a n l f o l d S which divides the 4 - m a n i f o l d M Into two parts M± and on which the induced 3 - m e t r i c is h i j
and the matter field configuration is ¢ o
is
(2. 1)
P[hij,d>0] = fd[g#v]d[d>]exp(-~[g/zv,~] )
where the Integral Is taken over all 4 - m e t r i c s and matter field configurations b e l o n g Ing to the class C which contain the submanlfold S on which the Induced 3 - m e t r i c is h i j into
and the matter field configuration Is Do, the
product
of
two
amplitudes
P [ h i j , ¢ ~ O] = ~ ' + [ h i j , C ~ o ] ~ _ [ h i j , ~ o ]
@±[hij,~o]
=
This probability can be factorized
or
wave
functions
~'± [ h i j
,¢~0].
where
(2.2)
fd[g#v]d[~]exp(-~[g#v,~])
C±
The path integral Is over the classes C+ of metrics on the compact manifolds M+ with boundary S. are real.
With the choice of c o m p a c t metrics for C,
I shall therefore drop the subscripts
+ and -
~z+ = ~_ and both
and refer to ~z as the
"Wavefunction of the Universe'.
In a neighbourhood of S in M, one can introduce a time coordinate t , which is zero on S,
and three space coordinates x
i
and one can write the metric
in the 3 + 1 form
ds 2 = _ (N 2 - NiNi)dt2 + 2Nidxidt + hijdxidxJ
(2.3)
A Lorentzian metric corresponds to the lapse N being real and a Euclidean metric corresponds to N negative imaginary.
The shift vector N i
In the Lorentzian case the classical action is
is real in both cases.
6
I =
I(Lg
+
Lm)d3xdt
(2, 4)
where
Lg
=
mD ijklw h%/2 16nN(G ~.ijKkl + 3R)
I
(2.5)
J
(2.6)
Kij = ~N - -at + 2N( ilJ )
is the second fundamental form of S and
G ijkl
= -1/2 h~%(hikh jl +
hilh jk - 2h ijhkz)
(2.7)
In the case of a massive scalar field
f I -2 a~
-[hiJ-
_ 2N ia~a~ N2
NiN___3_' _a_~_N 2 J]axiax jS~
In the Hamiltonian treatment of General ponents h i j
of the 3 - m e t r l c
(2.8)
m2~21
Relativity one r e g a r d s the c o m -
and the field ¢, as the c a n o n i c a l
coordinates•
The
c a n o n i c a l l y c o n j u g a t e momenta are
•,
a~
7r13 = ---- = at%ij
an
TP4~ = _ _ m 8+
-_
-
hh ~"-m m 2 16 167/P ( K i j
N-lhlh I~
-
hiJK)
- t"i--a+-] axZJ
(2.9)
(2.10)
The H a m l l t o n l a n Is
H = ~(~iJF*ij + ?r~ ~ - Lg - Lm)d3x
(2.11)
= I(NH 0 + NiHi)d3x
where
2
HO " 16Xrmp2Gijkl TrijTrkl - 167r mph%/~ 3R
k
+ ~,2h]/z 7r + hiJ a~. a¢. + m2~2 axZax 3
(2.12)
1
H i = _ 2 ijl j + hiJ a~.
(2.18)
ax 3
and
Gijkl = ~/2h-~/~(hikhjl + hilhjk - hijhkl)
From its path integral definition, the 3 - m e t r i c of t ,
hij
(2.14)
the w a v e f u n c t i o n ~, is a function only of
and the m a t t e r field c o n f i g u r a t i o n D 0 on S but it is not a function
which is m e r e l y a c o o r d i n a t e that can be given any value.
lows that ~I, will be u n c h a n g e d
It t h e r e f o r e fol-
if the surface S is displaced a d i s t a n c e N a l o n g the
n o r m a l s and shifted an a m o u n t N i
a l o n g itself.
The c h a n g e
in •
p l a c e m e n t will be the q u a n t u m H a m t l t o n i a n o p e r a t o r acting on "#'.
u n d e r that d i s Thus ~ will o b e y
the zero e n e r g y S c h r o e d l n g e r e q u a t i o n .
H~
=
0
(2.15)
8 where
the
Hamlltonian
operator
is obtained
from
the
classical
Hamiltonian
by the
replacements
TriJ(x) ~ -- i ~ i O(j x ) '
7r#(x) --', -- i---~ 5 ~ 0( x )
(2, ] 6 )
3 Quantlzatlon
The wavefunction ~" can be r e g a r d e d as a function on the infinite d i m e n sional manifold W of all 3 - m e t r i c s h i j to W change
is a pair of fields
(Tij,P,)
of the metric h i j
and matter fields •
on S where ~ / i j
can
on S.
A t a n g e n t vector
be r e g a r d e d
as a small
and /~ can be r e g a r d e d as a small c h a n g e of ~.
each c h o i c e of N on S there is a natural metric F ( N ) on W
ds2 = J
[321;
~ij~kl + I/2hlh/~2
For
2
(3. "1)
The zero e n e r g y S c h r o d i n g e r equation
H~' = 0
(3.2)
can be d e c o m p o s e d into the m o m e n t u m constraint
H ~' -= fNiHid3x ~'
= $hv'~ i
This Implies
2
that ~" is the
~-; =j same
equation,
c o r r e s p o n d i n g to
axJ 8~(x)j
on 3 - m e t r i c s
are related by c o o r d i n a t e transformations
(3.3)
In S.
and
matter field configurations
that
The other part of the S c h r o e d l n g e r
HI~ = o
where
H I = "j N H o d 3 X
Wheeler-DeWitt system
in
called
the
equation for each
of s e c o n d
ambiguity
is
the
o r d e r partial choice
of
equation.
There
of N on S.
One can
differential
equations
for ~I, on W.
operator
ordering
in these
is
one
regard them There
equations
as a
is some
but this
will
not
We shall assume that II I has the form 2
( -
Laplacian
Wheeler-DeWitt
choice
affect the results of this paper.
where v 2 is the
(3.4)
+ ~RE + v)~" = o
z/zv 2
in the
metric
F(N).
(3.5)
RE is the curvature
scalar
of this
metric and the potential V Is
2 V = j.hl/ZN
where U
2
T OO
~'2n¢,.
_ mp 3R + E + U d3x 167r
The c o n s t a n t
the c o s m o l o g i c a l c o n s t a n t A.
/ (3.6)
E can be r e g a r d e d as a renormalization
We shall assume that the r e n o r m a l i z e d A is zero.
shall also assume that the coefficient ~ of the s c a l a r curvature
Any Wheeler-DeWitt quantum which
wavefunctton equation
for
~I, which each
state of the Universe.
represents
the
quantum
metrics without boundary.
satisfies
choice
regard
constraint
on S d e s c r i b e s
the
We shall be c o n c e r n e d with the p a r t i c u l a r solution state
defined
by a
path
integral
over
compact
4-
In this case
-
I(g~v,~))
as a b o u n d a r y condition on the Wheeler-DeWItt
that tI, tends to a constant,
and
a possible
(3.7)
is the Euclidean action obtained by setting N negative imaginary. (3.7)
We
RE of W is zero.
momentum
of N and N i
= Id[g~v]d[~]exp(
where I
the
of
which can be normalized to one.
equations. as h i 3
One can It implies
goes to zero.
10 4 Unperturbed
Friedman
References
Model
3,4,5
considered
the
Minisuperspace
m o d e l which
consisted
of a F r i e d m a n m o d e l with m e t r i c
ds 2 = 02( - N2dt 2 + a2dN~)
w h e r e dn~ is the metric of the unit 3 - s p h e r e .
(4.])
The n o r m a l i z a t i o n factor 0 2 =
2 2
3Trmp has been included for convenience,
The model contains a scalar field (21/2T/O)-I~
with mass u-lm which is constant on surfaces of constant t.
One can easily gen-
eralize this to the case of a s c a l a r field with a potential V ( ~ ) .
Such g e n e r a l i z a t i o n s
include m o d e l s with h i g h e r derivative q u a n t u m c o r r e c t i o n s 6.
a2
The classical
The action is
N 2 tdtJ
+ m2~2
(4.2)
H a m l l t o n l a n Is
H = ~2N(
-
a-l~ a 2 + a - 3 n~2 - a + a3m2~ 2)
(4.3)
where
ada Ndt
7Ta
7T#
The classical H a m i l t o n i a n c o n s t r a i n t is H = o.
a t tN
Nd
f!
+ a d--£ d t
da
N2am2~2
a3d# = N dt
(4.4)
The classical field e q u a t i o n s are
+
=
o
(4.5)
11
The W h e e l e r - D e W i t t
e q u a t i o n is
]/zNe-3a[
+ 2Vl~(a,#) =
a2 a,2
(35
0
(4.7)
where
V =
and ~x = t,n a.
zAz(eeam2¢2
One can r e g a r d e q u a t i o n
the flat s p a c e with c o o r d i n a t e s
(~z,~)
-
e 4=)
(4.7)
(4.8)
as a h y p e r b o l i c e q u a t i o n for ~' In
with a as the time c o o r d i n a t e .
The b o u n -
dary c o n d i t i o n that gives the q u a n t u m state defined by a path Integral over c o m p a c t 4-metrics dary V >0,
is ~ -* 1 as o~ -. - ~o
condition, I#1
one
> 1 (this
finds
that
If o n e i n t e g r a t e s e q u a t i o n the
wavefunction
has been c o n f i r m e d
starts
numerically
5).
(4.7)
with this b o u n -
oscillating
in
One can
the
region
i n t e r p r e t the
o s c i l l a t o r y c o m p o n e n t of the w a v e f u n c t i o n by the WKB a p p r o x i m a t i o n :
= Re ( C e iS
where
C is
a
slowly
varying
amplitude
and
S
)
Is a
(4.9)
raplclly varying
phase.
One
c h o o s e s S to satisfy the classical H a m i t t o n - J a c o b i e q u a t i o n :
H(Yra,rr#,a,#)
= o
(4. lO)
where
s ~'a = aa-~'
~~ = as a-~
(4,11)
One can write (4. "10) in the form
I/zfab as as + e-3~'v = o aqaaq b
(4.12)
12 where fab is the inverse to the metric F(1):
fab = e-3~diag(-i,i)
(4. ]3)
The wavefunetlon (4, 9) will then satisfy the Wheeler-DeWltt equation If
v2c + 2ifab aC a S + iCV2S = 0 aga~q b
where V 2 is the Laplacian in the metric l a b '
(4. ]4)
One can ignore the first term in
equation ( 4 . 1 4 )
and can integrate the equation along the trajectories of the vector
field X a = d r~-
= l a b a.__S and so determine the amplitude C.
These trajectories
aq b
correspond to classical solutions of the field equations.
They are parameterized by
the coordinate time t of the classical solutions. The solutions that correspond to the oscillating part of the wavefunction of
the
Minisuperspaee
model
start
out
at
V = O,
I~J
> 1
with
~da
= d_~ dt =
o.
They expand exponentially with
S = - ~el 3=m ~1(1 - m - 2 e - 2 = ~ - 2 )
~
dt
After a time of order 3 m - ] ' ( l # . l l
= ml~l
-
dl~l
'
1),
starts to oscillate with frequency m.
dt
=
"
-
- ~e3=ml~l
1
z-m
(4, 15)
(4.16)
where ~1 is the initial value of ~. the field The solution then becomes matter dominated
and expands with e a proportional to t 2/3.
If there were other fields present,
the
massive scalar particles would decay Into light particles and then the solution would expand with e ~z proportional to t z/z,
9~
Eventually the solution would reach a maximum
2
radius of order e x 9 ( - ' ~ - ) or e x p ( 9 ~ l ) depending on whether it is radiation or matter dominated for similar manner.
most of the expansion.
The solution would then
recollapse in a
13 5 The Perturbed
Friedman
Model
We assume that the metric is of the form ( 2 . 3 ) side has been multiplied by a normalization factor o
2
except the right hand
The 3-metric h i 3
has the
form
2
hij = a (nij + Eij)
where Nij
(5.])
Is the metric on the unit 3-sphere and Eij
Is a perturbation on this
metric and may be expanded in harmonics:
z3
E
[61/2
• ' = n,l,m
+ 2%/2 c e
~
_n
an~m 3 ij~Jim +
e
n
n£m (Sij)Im + 2
The coefficients a . m , b
d°
n
bn~m (Pij )~m +
0
n
n£m (Gij)Im + 2
2]/2
0
S° " n
CnEm ( z 3 )~m
de Ge n ] nero ( ij)~m I
(5.2)
d° de n~m' n£m' n~m are functions of the time c o o r i dlnate t but not the three spatial coordinates x .
n£
The Q ( x z)
. ,c °.
6%/2
nLm
n£m
,c e
are the standard scalar
harmonics on the 3-sphere.
P i j ( x 1) are given by (suppressing all but the i , j
indices)
1
Pij
They are traoeless, P i
i
= 0.
(n 2 1- I) Qlij + 3-~ijQ
The S i j
Sij
where
Si
are
the
transverse
transverse traceless tensor harmonics.
(5.3)
are defined by
= Sil j
vector
The
+ Sjl i
harmonics,
(5.4)
sill-o.
Gi i = Gij I j
= 0.
The
Gij
are
the
Further details about
the harmonics and their normalization can be found in appendix A.
14 shift and the scalar field ~(xi,t) can be expanded in terms
The lapse, of harmonics:
{
n)
N = N O i + 6- ~
(5.5)
F. gn£m Q£m n, £,m
n + 2 ~ Jn£m (Si);m ] N i . e (= Y. {6-]/2 kn£m (Pi)£m n, £,m
= o-1
1
where P i
Qli"
#(t) +
1
Hereafter.
nl
(5.6)
(5, 7)
F. fn£m Q£m n, £,m
the labels n , 9 . , m , o and e will be denoted
(n 2 - l) simply by n. ground"
One can then expand the action to all orders in terms of the "back-
quantities
a,#,N 0
a n , b n , On, t i n , f n , g n , k n ,
but
only
to
second
order
in
the
"perturbations"
j n :
I =
I o ( a , # , N O) +
(5.8)
F.I n n
where I O is the action of the unperturbed model ( 4 . 2 )
and In is quadratic in the
perturbations and is given in appendix B. One can define conjugate momenta
in the usual manner.
~a = - NLle3a& + quadratic terms
~ = NLIe3~ ~ + quadratic terms
= - NLle3a[~ n + &(a n - gn ) + !e-a k ] 3 nJ
77
an
They are:
(5, 9)
(5.10)
(5.11)
2
=
~bn
NLIe3U iD__=_~I [~n + 4&bn - ~l e - a k n,] (n 2 - 1)
(5.12)
15 /;c = N; le3'~ (n2 - 4) [~n + 4&c n - e-aJn ] n
(5. ]3)
(5. ] 4 )
I
l
~rf = Nole3(~ fn + ~(3an - gn ) n The q u a d r a t i c terms In e q u a t i o n s
(5.9)
and
(5.]0)
(5. 15)
are given in a p p e n d i x B.
The
H a m i l t o n l a n can then be expressed in terms of t h e s e m o m e n t a and the o t h e r q u a n t i ties:
.-.o
0,1,2
The subscripts perturbations
.,o÷ ~.?~+ ~n Hn,~I ÷nE{knSH~I
I
on the "1
and
H_
d e n o t e the o r d e r s
and S and V d e n o t e the s c a l a r
the H a m i l t o n i a n .
HIO is the H a m i l t o n i a n
"g0
The s e c o n d
order
a
Hamlitonian
is given
(5. ]6)
+ Jn VHn_lj}
of the quantities
and v e c t o r parts of the shift part of
of the u n p e r t u r b e d m o d e l with N = 1 :
~
+
-
by H i 2 = E H / 2 n -
(5. 17) S n
= F.( HI2 13
+
Vn
Hi2
+
where
+
_ ~2
an
-
+ L_~_:!/
2
(n2-4) ~bn
2)an +
+ ~f2
in the
n,.
~ n + 2an~an
(n-~--i) n
+ 8bn~ b n ~
- 6an~ f n ~#
Tn
HI2)
16
+
e'=m
[ n + 6anfn~)l +
[2
n
-
(n2_l) nJj
(5.
]8)
VHI2n = Z/2e-3aI(n2_4)c2[lOTr2+ 6~] + i_.__(n2_4) Tr2Cn+ 8Cn/TCnTr + (n2-4)C2n[2e 4(z - 6e6am2~2]]
(5. "19)
TH,2n -~'2e" -3~, lan f.2 [1OTr2 + 6/T~] + TrC~n + 8dnTrdnTr
d2n[(n2+l)e 4(z - 6e6(Zm2#2]]
+
(5.20)
The first order Hamlltonlans are
H[1
1/'ze- 3a =
an
n
+ m2e6(Z[2fn. + 3an.21 - 2e~a[(n2-4)bn + (n2+~'~)anll
(5.21)
The shift parts of the Hamlltonlan are
_1 = "~e
- nan
n
(n2_1)
nj
~z
VnH_/ = e -(z{n,cn + 4(n2-4) Cn~ a]
(5.22)
(5.23)
The classical field equations are given in appendix B. Because the Lagrange
multipliers
No,gn,k n , j n
are Independent.
the
zero energy Sohroedlnger equation
H~!" = 0
(5.24)
17
can be d e c o m p o s e d as before into m o m e n t u m constraints and Wheeler-DeWitt tions.
As the m o m e n t u m
constraints
guity in the o p e r a t o r o r d e r i n g .
are linear in the m o m e n t a ,
I
a ab n
The
first
order
addition o f terms by multiplying probabilities
I
an
+ 4(n 2 (n 2
4) bn 1)
-
1
Ba
( 5, 25)
3fn ~l~!, = 0
-°I ~ <°
+ 4(n 2 -
Hamiltonians
second o r d e r differential e q u a t i o n s , that we are using,
there is no a m b i -
One therefore has
S n i -3~ a _ H_.I~, = - ~e a~n
H..,].~" = e
equa-
n Hll
4) c n
give
~" = 0
a
one for each n.
series
of
(5.26)
finite
dimensional
In the o r d e r of a p p r o x i m a t i o n
the ambiguity in the o p e r a t o r o r d e r i n g will consist of the possible a linear in ~-~. The effect of such terms can be c o m p e n s a t e d for
the wavefunction of different
by powers of e a.
observations
at
a
given
This will not affect the value
of
a.
We
shall
relative therefore
ignore such ambiguities and terms.
2
Finally,
4¢e
2
(5.27)
one has an Infinite d i m e n s i o n a l s e c o n d o r d e r differential equation
I.,o + ~.,s.?2 +. where HIo
Is the
operator in
v"?2+ T.~2,1. = o
the Wheeler-DeWitt
equatlon
(5.28)
of the
unperturbed
18 Friedman Mlnlsuperspace model:
= ]/,ze_3(z[ (92
l~
,o
82 + e6(Zm2¢,2 _ e4~Z]
(5.29)
a~2
and
HI2 = ]/2e -3(z I-I =a n +
!0_/._n_2-4) b21 (92 (n2_l) nJaa2 ---
a2
(n.2-1__). a 2
a2
aan 2
(n2-4) (gbn2
afn 2
[15a2 + 6--(n2-4) b21 a 2 [2 n (n2_l) nJ a¢,2 --
a a
8b -=(9 a__
2anSana~
a a
nabna~ + 6an~fna~
- e4al'l[~(n 2..~)5a 2n + (n2-7_.1 (n2-!). b 2 + 2(n2 4 ) anbn _ (n2_l)f2n]j 3 (n 2 -1 ) 17.
(5.30)
+ e6amZ[fZn + 6anfn# ] + e6(Xm2"2[3a2, (~ n - 6(n2-4)b2]](n2_l) njj
VHI2n
= ]/2e-3(~
t
a2
t (ga
1 j__2_2_ (n2-4) (9cn
aa
8Cna ~-(gu *
(n2-4)CZn[2e 4a - 6e6am2~2]]
T l-If2 n = ]he-3a
-
d 2 [10 a2 + 6
n( aa2
ad 2 n
d2[(n2+l)e 4u - 6e6Um2#2]]
n
(5.31)
a a 8d na--~-~na~ +
(5.82)
We shall call equation (5.28) the master equation. It is not hyperbolic a2 because, as well as the positive second derivatives -a~ - 2 in HIO, there are the posi-
19 a2
tlve second
c o n s t r a i n t (5, 25) solve
the
cn
and
differential
constraint
then
solve
(5.26) on
momentum
equation
on
one
a n = o.
to substitute for the
c n = O.
hyper bolic for small fn" use the
However.
can
use
the m o m e n t u m
to substitute for the partial derivatives with r e s p e c t to a n and then
resultant
momentum
S n Hi2.
derlvatlves---aa 2 in e a c h n
One
thus
Similarly,
one
can
partial derivatives with
obtains
a
modified
to c a l c u l a t e the wav e f u n ct io n
the
r e s p e c t to
equation
If one knows the w a v e f u n c t i o n on a n = 0 = c n ,
constraints
use
which
is
one ca n
at o t h e r values
of a
n
and c n .
6 The WavefuncUon
Because
the
perturbation
modes
are
not
co u p le d
to
each
o t h e r,
the
wavefunctlon can be expressed as a sum of t e r m s of the f o rm
(6. l )
= Re (~I,0(¢z,@)I-[tI,(n)(cz,#,an,bn,Cn,dn,fn)) n
= Re ( Ce is )
w h e r e S Is a rapidly varying function of a and
@ and C Is a s l o w l y ' v a r y i n g
of all the variables,
Into the
by ~,
If one substitutes
(6. ] )
m a s t e r equation
function
and divides
one obtains
(6.2)
n~m
2~!'(n)~! "(m)
+ E n
t,
+ e _ 3 a V(~,~)
= o
20 where v 2 is the Laplaclan in the Minisuperspace metric f a b
= e3=diag(-l'l)
2
and
the dot product Is with respect to this metric. An individual perturbation mode does not contribute of the sums in the third and fourth terms in equation ( 6 . 2 ) .
a significant fraction Thus these terms can
be replaced by
(V2~I') _(v2~I'(n))
~" "~n
~(n)
I v2~I'(n)}2
VZlEn--~((n'l~'-
+
(v2~I,(n)) "
-
In order that the ansatz an,bn, Cn,dn,fn
I v "#'(n)]2 ,fE In1,"
n
(6.])
be valid,
have to cancel out.
(6.3)
the terms
" tJ
in
(6.2)
that depend
Thts implies
(v2~I').(v2~F(n)) + V2v22~I,(n)= ~2-~,(n) II,
( -
where J
(6.4)
~I, -
I/2V2 +
on
e -3e
V
+
(6.5)
~2J,J)~Z 0 = 0
V ~,(n) 2
=
~..--:-:--:-
~ ~,(n)
In regions In which the phase S is a rapidly varying function of ~z and ~,
one can neglect the second term in ( 6 . 4 )
In comparison with the first term.
One can also replace the T/'CZ and Tt# which appear in H~2 by _~S and ~ tively.
The
vector X a = f a b 8_S_ obtained aq b
inverse minisuperspace metric f a b
can
by
raising
the
a be regarded as ~
covector v2S where t
respecby the
is the time
21 parameter
of
approximation.
the
classical
Friedman
metric
that
corresponds
to
1/, by the
WKB
One then obtains a time dependent Schroedtnger equation for each
mode along a trajectory of the vector field x a :
a~.( n )
i-
Equation ( 6 . 5 )
H:L~(n)n iz
(6.6)
can be Interpreted as the Wheeter-DeWitt equation for a
two dimensional minlsuperspace perturbations,
at
model with an extra term ] / z J . J
arising from
in o r d e r to make J finite, one will have to make subtractions.
the Sub-
n tracting out the ground state energies of the HI2 corresponds to a renormalization of the cosmological constant A.
There Is a second subtraction which corresponds
to a renormallzatlon of the Planck mass mp and a third one which corresponds to a curvature squared counterterm,
The effect of such
action has been considered elsewhere
One can write
higher derivative terms
~,n as
~n = S~(n)(a'#'an'bn'fn) V~(n)(a'#'Cn) T~z(n)(a'#'dn) S~,(n),V~I,(n) and T~,(n) S.n V.n Tn where HI2,
HI2 and
in the
6
HI2 respectively.
obey
independent
Schroedinger
equations
(6.7)
with
22 7 The B o u n d a r y
Conditions
We want to find the solution of the master equation that c o r r e s p o n d s to
• [hij,~] -- fd[g~v]d[~ ] exp(-I)
where the integral is taken over all c o m p a c t 4 - m e t r i c s bounded by the 8 - s u r f a c e S.
and matter fields which
are
If one takes the scale p a r a m e t e r u to be very n e g a -
tive but keeps the other p a r a m e t e r s fixed, e 2a.
(7.1)
the Euclidean action
I tends to zero like
Thus one would expect ~ to tend to one as ~z tends to minus infinity.
One
can
S~(n),V~'(n),T~ t(n)
estimate of the
the
form
perturbation
of ~(n)
the
scalar,
from
vector
the
path
and
tensor
parts
integral
(7. "1).
One
takes the 4 - m e t r i c g/.Lv and the scalar field 4) to be of the b a c k g r o u n d form
ds 2 = 02( - N2dt 2 + ~_2~z( t )=~2 u*=3)
and
#(£)
(an,bn, fn)
respectively
plus
a
small
perturbation
, c n and d n as functions of t ,
to be c o m p a c t ,
described
it has to be Euclidean when o~ = -
is Lorentzian.
where /z = 0 at t at t
= 0,
= 0.
an,bn, Cn,dn,fn
variables
In regions in which
In o r d e r to allow a smooth
we shall take N to be of the form
In o r d e r that the 4 - m e t r i c have
the
,= ie N has to be purely n e g a -
N will be real and positive.
transition from Euclidean to Lorentzlan,
by
In o r d e r for the b a c k g r o u n d 4 - m e t r i c
tive imaginary at u = - =o, which we shall take to be £ = 0. the metric
(7.2)
-
i e i/~
and the scalar field be regular
to vanish there,
The tensor perturbations d n have the Euclidean action
T~n = ~/zfdt d n TD dn + boundary
where
term
(7.3)
23
(7.4)
+ Vze-2a
+ 4iN0 e3a
The
last term
In
field e q u a t i o n s .
.2 _ _..!__#.
3 2,2
- ~-m ~
(7.4)
vanishes
If the
1 df }l
3&2
2(iN O)2
2(iN O)2
background
metric
s a t i s f i e s the
background
The a c t i o n Is e x t r e m l z e d w h e n d n s a t i s f i e s the e q u a t i o n
TD d
F o r a d n that satisfies ( 7 . 5 ) ,
n
= o
(7,5)
the action is just the b o u n d a r y term
]
T~el l e3a[ d (~ n = 2iN 0 [ n n
4&d21 nj
+
(7.6)
The path Integral o v e r d n will be
Id[dn] exp(- Tin)
(detTD) -~/2 exp(- T~cl =
One
now
has
wavefunction ground
to
integrate
T~(n)
One
(7.7)
over
expects
the
different dominant
background contribution
m e t r i c s that a r e n e a r a s o l u t i o n of the c l a s s i c a l
F o r such
metrics
b o u n d a r y c o n d i t i o n d n = 0 at t
metrics to
v = e-U(n2-1)
~'z and
f i e l d s which
are
to
come
obtain from
the
back-
b a o k g r o u n d field e q u a t i o n s .
Then the s o l u t i o n of ( 7 . 5 )
one regards
which o b e y s the
= o is
d
background
(7.7)
)
o n e can e m p l o y the a d i a b a t i c a p p r o x i m a t i o n in which
a to be a slowly v a r y i n g f u n c t i o n of t .
where
n
n
= A(e vr
-
e -vT)
~- = f i N 0 d t .
This
near a solution
of the
(7.8)
approximation background
will
be
valid
for
field e q u a t i o n s a n d
24 for which
I~001 ~
(7.9)
ne -a
°
For
a
regular
Euclidean
Euclidean
solution
of the
adiabatic a p p r o x l m a t l o n the
solution
metric,
of the
I~ol
background
should
field
near
t
equations,
= o.
then
hold for l a r g e values of n
background
a p p r o x i m a t i o n can be used.
= e -a
field
equations
becomes
If the
metric
I~ool<e-a.
is
Thus
the
Into the r e g i o n
in which
Lorentzian
the
and
WKB
The w a v e f u n c t i o n T,~,(n) will then be
(7.
in
the
Euclidean
co'ch(vT)
"
1.
region, In the
~" will Lorentzian
be
real
region
and
positive.
For
large
values
w h e r e the WKB a p p r o x i m a t i o n
will be c o m p l e x but It will still have a positive real part and c o t h ( v T ) a p p r o x i m a t e l y 1 for l a r g e n.
I0)
of
applies,
n, r
will still be
Thus
T~(n) = B exp[ - 2i ~aSd2n _]/zne2ad2n]
The n o r m a l i z a t i o n constant B can factor,
a
be c h o s e n
to be 1.
Thus,
(7.11)
a p a r t from
a phase
the g r a v i t a t i o n a l wave m o d e s e n t e r the WKB r e g i o n in t h e i r g r o u n d state. We
now c o n s i d e r
the v e c t o r
part V~,(n)
of the w a v e f u n c t i o n .
This
is
pure g a u g e as the quantities c n can be given any values by g a u g e t r a n s f o r m a t i o n s p a r a m e t e r l z e d by the J n '
The f r e e d o m
to m a k e g a u g e t r a n s f o r m a t i o n s
Is reflected
q u a n t u m m e c h a n i c a l l y in the c o n s t r a i n t
e
+4(
~'=
o
(7. 12)
25 One can integrate ( 7 . 1 2 )
to give
• (a,{Cn}) = ~(a - 2 E(nZ-4)C2n,O) n
(7. 13)
where the d e p e n d e n c e on the other variables has been suppressed, replace ~
by
aS i~F.
One can then solve for
One can also
V~,(n).
exp[2i (r12-4.)C2n~]@S
(7. 14)
The scalar perturbation modes a n , b n and f n
involve a combination of
V~(n)
=
the behavlour of the tensor and vector perturbations. is given in appendix B.
The scalar part of the action
The action is extremized by solutions of the classical e q u a -
tions
NO d( e 3aan. + l(n2 - 4)No2ea(an+ bn) + 3e3a(~n No )
-
No2m2@fn) =
d le2(zknl N2{3e3am2@2 - ~(n2+2)ealgn + e3a~gn - 3/N° ~[ N0J
N O
d e3a~n 1 2 dr( ~0 ) - ~n
_
+
1 )No2e~(a n + bn)
3!NO
d
[
=
~n 2
_
(7. 15)
1 )N2e~gn
2akn]
(7.18)
°
NO ~t (e 3afn NO )
+
• n 3e3~a
tWo[m2 e3 a
.2[ +
+
(n2-l)e~Z]fn
e3~/- 2N2m2~gn + ~gn- e-a~kn I
(7."17}
26 There is a three parameter family of solutions to ( 7 . ] 5 ) obey the boundary condition a n = b n = f n
= 0 at t
= o.
to ( 7 . ] 7 )
which
There are however,
two constraint equations:
an + Ln2--416
+ 3fn~
(n2_1) n
3an ( _ &2 +
"
e-=
= agn
(n2_l)
k
(7.18)
n
.2 ¢ ) + 2(¢~ n _ &~n)
2 2 -20~ [(n 2 -4)b n + (n 2 + ]/2)an ] + N O2 m2(2fn @ + 3an~2 ) _ ~N0e
.2
=-2&e-Uk3 n + 2gn( _ &2 +
@ )
( 7 19)
These correspond to the two gauge degrees of freedom parameterized by k n and g n respectively.
The Euclidean action for a solution to equations (7. "15) to (7. ]9)
S~cl 1 30~I n = ~00 e - anan
is
.(_n_2-4lb ]~ + fnfn
+ (n2_l) n n
+ &I -a2 + 4(n2-4)b2] n (n 2-z) nj
* 3~anfn
(7.20)
(nZ_z) njj
where the background field equations have been used. In many ways the simplest gauge to work in is that with g n = k n = o. However,
this
gauge
does
not allow one
to find
a compact 4 - m e t r i c
bounded by a 3-surface with arbitrary values of a n , b n and f n tion of the equations (7. ]5)
to (7. ]7)
which
Is
and which is a solu-
and the constraint equations.
Instead, we
shall use the gauge a n = b n = 0 and shall solve the constraint equations ( 7 . 1 8 )
27 a n d (7. 19) to find g n and k n :
(7,21)
(n2-4)& ~ + s ~2
I
°°"
k
I
-
= 3 ( n 2 - 1 ) e a~. . . . . .
( 7 . 17)
With t h e s e s u b s t i t u t e d .
NO dt~
(7.22)
becomes
No
a second order equation for f
n
+ N.2[ 0[m 2e 3a + (n2_l)e=]fnj =
(7.23)
For
large
n we
can
again
the s o l u t i o n of ( 7 . 2 3 )
when
I#1
> 1:
use the
adiabatic
approximation
fn = Asinh(vr)
where v 2 = e-2a(n2-1).
S~(n)(u,~,
This S~.(n)
Is of the
ground
at n o n - z e r o
equations (5.25)
to e s t i m a t e
(7, 24)
Thus f o r t h e s e m o d e s
0 , 0 ,fn)
state f o r m
*, e x 9 -
apart
from
v a l u e s of a n and b n c a n
and ( 5 . 2 7 ) .
. 2o~.2 ~2ne rn -
a smal l
. aS ]/2Z~gnf
phase
factor.
n
]
(7.25)
The val ue
of
be f o u n d by i n t e g r a t i n g the c o n s t r a i n t
28 The tensor and scalar modes start off In thetr ground states, sibly from
the modes at low n.
The
neglected.
Thus the total e n e r g y E = n ~ - - ~ ( - ~ -
when
ground
apart p o s -
vector modes are pure g a u g e and can be H(n)t,(n)
the
state
energies
are
of the perturbations will be small
subtracted.
But
E = i(vzS).J
where
V2~( n ) J = F. t,(n" ~ n
Thus J
is
small.
This means that the wavefunction ~0 will obey
the Wheeler-DeWitt equation of the unperturbed m i n i s u p e r s p a c e model and the phase factor
S will
be
mode
# will
not start
first,
regularity
require # = o.
approximately -
at
t
out = o
Second,
in
i l n ~ O.
its ground
requires
However state.
the
There
homogeneous are two
an = b n = cn = d n = fn
scalar
reasons =o.
but
field
for
this:
does
not
the classical field equation for ~ i s o f the form f o r a damped
h a r m o n i c oscillator with a c o n s t a n t f r e q u e n c y m rather than a d e c r e a s i n g frequency -5 e n. This means that the a d i a b a t i c approximation is not valid at small t and that the solution of the classical field equation is # a p p r o x i m a t e l y constant. of such solutions is small, the other variables.
so large values of I$1
They will c o r r e s p o n d to classical solutions which have a
long inflationary period and then model which
the oscillations thermal
of the
spectrum.
are not d a m p e d as they are for
Thus the WKB trajectories which start out from large values of
I$1 have high probability.
realistic
The action
go over to a matter d o m i n a t e d
included other fields of low rest mass, massive The
scalar field
model
would
expand
as
In a
the matter e n e r g y in
would d e c a y into
then
expansion.
light particles
a
radiation
with
a
dominated
universe.
8 Growth
of Perturbations The tensor modes will obey the S c h r o e d l n g e r equation
iaT~ (n) at
n [ [~J
THI2T~(n )
adn 2
(8. I)
8dn i ~
a ad n
29 d2n{(n2+l)e4a - 6e6¢Xm2#2}}
(8,
2)
One can write
T~(n)
exp(-2a) exp{-2i @S d2} T,(on)
=
(8.3)
then
.aT$1n) I at
I
~ze Sa
a2
_ ___
ad2
The WKB a p p r o x i m a t i o n in deriving oscillator
(8.4). with
a
j
+ d2(n2_l)e4~
to the
T~(n)
background
Then
(8.4)
time
dependent
v
(8.4)
Wheeler-DeWitt
has the form
equation
of the S c h r o e d i n g e r
frequency
v
=
has
been used
e q u a t i o n for an
(n2-1)VZe -a,
Initially
the
w a v e f u n c t i o n T,~'(on) will be in the g r o u n d state ( a p a r t from a n o r m a l i z a t i o n factor ) and the f r e q u e n c y
v will be l a r g e c o m p a r e d
to (~.
In this case o n e can
use the
a d i a b a t i c a p p r o x i m a t i o n to show that T~(on) r e m a i n s in the g r o u n d state
T~(n), exp[ - ~ne2=d2n}
The
adiabatic
approximation
will
break
(8.5)
down
when
v -
ie
the
w a v e l e n g t h of the g r a v i t a t i o n a l m o d e b e c o m e s e q u a l to the horizon scale in the infl a t i o n a r y period.
The w a v e f u n c t i o n T~/,(on) will then " freeze " :
T~o(n) ,
where
a,
is the value
exp
of
{
-
the
w a v e f u n c t i o n T~'(on) will r e m a i n of the form
~ne
mode (8.6)
1 goes
(8.6)
outside
the
horizon.
The
until the m o d e r e - e n t e r s the h o r -
izon in the matter or radiation d o m i n a t e d era at the much g r e a t e r v a l u e ~ze of a.
30
One can then
apply the adiabatic a p p r o x i m a t i o n again to
l o n g e r be in the ground excited
states,
This
state;
is the
it will
phenomenon
(8.4)
be a superposition of the
T~n)r
but
of a number
amplification
of the
of highly
ground
is more
behaviour of the scalar modes is rather similar but their description
complicated
because
of the
gauge degrees
section we evaluated the wavefunction S ~ ( n ) prescription.
state
7,8,9
fluctuations in the gravitational wave modes that was discussed in r e f e r e n c e s
The
will no
The ground state form
( in f n
of freedom.
In the
previous
on a n = ]on = o by the path integral ) that we found will be valid until the
adiabatic approximation breaks down ie until the wavelength of the m o d e e x c e d e s the horizon distance during the inflationary period.
In o r d e r to discuss the s u b s e q u e n t
behaviour of the wavefunction,
to
constraint ( 5 . 2 7 )
it is c o n v e n i e n t
use the first
to evaluate S~.(n) on a n ~ o , b n = f n
= o.
order
Hamlltonlan
One finds that
S~(n)(u,#,an, o, o )= B exp{iCa2] S£,(n)(=,#,an)
(8, 7)
The normalization and phase factors B and C d e p e n d on ~z and ~ but not a n .
C = l[aS] 2 [ a a J -if [8S} Lao~J 2 - l(n2-4)e4(Z]
(8.8)
At the time the wavelength of the mode equals the horizon d i s t a n c e during the inflationary period,
the wavefunction S~'(on) has the form
I
S~(n) = exp - I/2 n
where y, y
Y,
= 3#,. =
is
the
of
More g e n e r a l l y ,
- r _~_Y3- z
6V[a~j
value
.
y;2
y = ~r~l.~.~a[a.~j-z when
e
2=, a the
mode
(8.9)
leaves
the
horizon,
in the case of a scalar field with a potential
V(~),
31
One bn = fn
can
obtain
a
Schroedinger
= o in the scalar Hamlltonlan
the momentum constraint respectively.
aS (n) tO z at
(5.25)
S HI2 n
equation
for
S~,(n) -0
by
putting
a a and substituting for ~--~-- and ~ - - from n n
and the first o r d e r Hamlltonian constraint
(5.27)
This gives
•
3a
]/ze
_ y2 6a 2
e4a(n2_4)[12- 1 4a[asI-2] a2 I St(on)
(8. 10)
~y
where terms of o r d e r -12 have been neglected, The term e4a[aS] [8oe.I -2 will be small n 1 compared to ~ except near the time of maximum radius of the background solution. Y The S c h r o e d i n g e r equation for S t ( o n ) ( a n ) is very similar to the equation for Tt(on)(dn),-
(8.4),
potential term wavelengths exp(-
is
except that the kinetic term is multiplied by a factor y 2 and the divided
within
.. -2 2aa2n ) %,2ny e
the
by a
factor
horizon,
2
y .
St(on)
One
would
would
have
and this is bourne out by ( 8 . 9 ) .
therefore the
expect that
ground
for
state
form
On the other hand.
when
the wavelength becomes larger than the horizon,
t h e S c h r o e d i n g e r equation (8. ]0)
indicates that Tt(on) will freeze in the form ( 8 . 9 )
until the mode r e - e n t e r s the h o r -
izon In the matter dominated era.
Even if the equation of state of the Universe
changes to radiation dominated during the period that the wavelength of the mode is greater than the horizon size. (8.9).
it will still be true that St(on) is frozen in the form
The ground state fluctuations in the scalar modes will therefore be amplified
in a similar m a n n e r to the tensor modes. the rms fluctuation in the scalar modes, be greater by the factor y , same wavelength.
At the time of r e - e n t r y of the horizon
in the gauge in which b
than the rms fluctuation
n
= f
n
= o.
will
in the tensor modes of the
32 9 Comparison
with Observation
a knowledge of T~(on) and S~(on) one can calculate the relative
From
probabilttes of observing different values of d n and a n at a given point on a t r a j e c tory of the v e c t o r field X i
ie at a given value of <x and ~ in a b a c k g r o u n d
which is a solution of the classical
field e q u a t i o n s .
will be u n i m p o r t a n t and we shall n e g l e c t it. ties of o b s e r v i n g
different
amounts
In fact,
metric
the d e p e n d e n c e on
One can then c a l c u l a t e the p r o b a b i l i -
of a n l s o t r o p y
In the m i c r o w a v e b a c k g r o u n d
and
can c o m p a r e these p r e d i c t i o n s with the u p p e r limits set by o b s e r v a t i o n . The t e n s o r and s c a l a r p e r t u r b a t i o n m o d e s will be in highly excited states at
large
values
of
,v.
This
means
that
we
e n s e m b l e evolving a c c o r d i n g to the classical
can
in
(in
and
=4,n
will
be
their
development
as
an
equations of m o t i o n with initial d i s t r i b u -
tions in d n and a n p r o p o r t i o n a l to IT~(on)12 and distributions
treat
IS~(on)12 respectively.
proportional
T ~rO , , , ( n ) ," d
to
The initial TPno I
and
re-enter
the
n IS~("n)n'a,, S~'(n)lv n horizon,
respectively.
In fact.
at the time
the distributions will be c o n c e n t r a t e d at d The
surfaces
with
b n = fn
= 0
be
surfaces
density in the classical solution during the Inflationary period. of e n e r g y ,
modes
= a. = O. n
n
will
that the
of
constant
energy
By local c o n s e r v a t i o n
they will r e m a i n surfaces of c o n s t a n t e n e r g y density in the e r a after the
inflationary period when the e n e r g y is d o m i n a t e d by the c o h e r e n t oscillations of the homogeneous
background
scalar
field
particles and heat up the universe, of c o n s t a n t ground be
temperature.
will be such
considered
to
The
a surface have
¢.
If the
scalar
particles
the surfaces with b n = f n
surface
of
last
scattering
with t e m p e r a t u r e T s.
propagated
freely
to
us
decay
Into
light
= 0 will be surfaces
of the
microwave
back-
The m i c r o w a v e radiation can from
this
surface.
Thus
the
o b s e r v e d t e m p e r a t u r e will be
T
T
o
s I + z
(9. "I)
33 where
z
ts
the
redshlft
of
the
surface
of
last
scattering,
Variations
In
the
o b s e r v e d t e m p e r a t u r e will arise from v a r i a t i o n s In z in different d i r e c t i o n s of o b s e r vation.
These a r e given by
1 + z = .t/Zn /z
e v a l u a t e d at the surface of last s c a t t e r i n g faces surface
of c o n s t a n t
t
in the
of last s~atterlng
and
gauge
(9.2)
w h e r e n/L ts the unit n o r m a l to the s u r -
gn = kn
= J n = o and
b n = fn
= o on t h e
p/L Is the p a r a l l e l l y p r o p a g a t e d t a n g e n t v e c t o r to the
null g e o d e s i c from the o b s e r v e r n o r m a l i z e d by
£/~n/z=
1 at the p r e s e n t time,
One
can c a l c u l a t e the evolution of t./Zn/z down the past light c o n e of the o b s e r v e r :
d fl;Ln/zJ] d-i[ where
k is the afflna
parameter
on the
= n ~ ; v t'p't'v
null
(9.3)
geodesic,
The
only n o n - z e r o
com-
p o n e n t s of n # . ; v are
nis j = e2=I&nlj + F.(A n n + &an~ljQ
+ F.(5 " n n + (xbn)Pij
(9.4)
+ E(dnn + &dn)Gijl
In the g a u g e that we are using, on the scale of the horlzon,
the d o m i n a n t a n l s o t r o p l a terms In ( 9 , 4 )
will be t h o s e involving & a n and &dn.
These will give
t e m p e r a t u r e a n i s o t r o p i e s of the form
((AT/T)2>
'=
or
-
(9.5)
The n u m b e r of m o d e s that c o n t r i b u t e to a n i s o t r o p i e s on the scale of the horizon Is of the o r d e r of n 3.
From the results of the last section
34
= y2n-le -20~*
(9.6)
= n-Ze-2~*
(9, 7)
The dominant contribution comes from the scalar modes which give
<(AT/T)2> ~, y,n 2 2 e -2~,
(9.8)
-a. But ne
*, & , ,
the value of the
Hubble constant at the time that the
horizon size left the horizon during the inflationary period. limit of about 10- 8
on
about 5.10-5m
(Ref. P less than 1014 GeV .
10 Conclusion
< ( A T / T ) 2> ]0)
restricts
this
present
The observational upper
Hubble constant to be less than
which in turn restricts the mass of the scalar field to be
and Summary
We started from the proposal that the quantum state of the Universe is defined
by a path integral over compact 4-metrics.
boundary Universe metrics
condition on
and
the
for
infinite
matter
field
the
Wheeler-DeWltt
dimensional
equation
manifold,
configurations
on
This can for
the
superspace,
a 3-surface
S.
be regarded as a wavefunctlon
the
space
of
of
the
all
3-
Previous papers
had
considered finite dimensional approximations to superspace and had shown that the boundary condition led to a wavefunctlon which could be interpreted as c o r r e s p o n d Ing to a family of classical
solutions which
were homogeneous and isotropic and
which had a period of exponential or inflationary expansion.
In the
we extended this work to the full superspace without restrictions, two
basic
homogeneous and
degrees of freedom
Isotroplc
to second order.
degrees of freedom
present paper We treated the
exactly and
the
other
We justified this approximation by showing
that the inhomogeneous or anisotropic modes started out in their ground states.
35
We derived time d e p e n d e n t S c h r o e d l n g e r equations for each mode. showed
that they r e m a i n e d
In the
ground
state
horizon size during the Inflationary period. state fluctuations
got frozen
until
their wavelength e x c e d e d the
In the s u b s e q u e n t expansion the ground
until the wavelength r e - e n t e r e d
radiation or matter d o m i n a t e d era,
We
the horizon during the
This part of the calculation is similar to e a r l i e r 7
work on the d e v e l o p m e n t of gravitational waves
and density perturbations
ll, 12 .
m
the inflationary universe but It has the a d v a n t a g e that the assumptions of a period of exponential expansion and of an initial ground fied.
state for the perturbations are justi-
The perturbations would be c o m p a t i b l e with the upper limits set by o b s e r v a -
tions of the microwave b a c k g r o u n d
if the s c a l a r field that drives the Inflation has a
mass of ].014 GeV or less.
In section 8 we c a l c u l a t e d the s c a l a r perturbations
in a g a u g e In which
the surfaces of c o n s t a n t time are surfaces of c o n s t a n t density. density fluctuations
in this gauge.
g a u g e in which a n = b n = o.
However.
one can
There are thus no
make a transformation
to a
In this g a u g e the density fluctuation at the time that
the wavelength c o m e s within the horizon is
<(Ap/p)2>
.2
==
Pe
},2
.2
.2 (2,
(lO.l)
2
aeO e Because y tions,
and & ,
d e p e n d only logarithmically
on the wavelength of the p e r t u r b a -
this gives an almost scale free s p e c t r u m of density fluctuations.
These f l u c -
tuations can evolve a c c o r d i n g to the classical field equations to give rise to the f o r mation
of
galaxies
and
all
the
other
structure
that
we
observe
in
the
Universe.
Thus all the c o m p l e x i t i e s of the present state of the Universe have their origin In the ground
state
fluctuations
In
the
H e l s e n b e r g Uncertainty Principle.
tnhomogeneous
modes
and
so
arise
from
the
36 Referenoes
1
S.W. Hawking Time".
2
& G . F . R . Ellis.
S.W, Hawking and D, N, Page.
"Relativity.
Session XL,
Amsterdam.
of S p a c e -
,1973).
"Operator Ordering and the Flatness of
DAMTP preprlnt (,1985)
S,W. Hawklng In: ]983,
L a r g e - S c a l e Structure
(Cambridge University Press,
the Universe', 3
"The
edited
Groups and Topology I1", Los Houches
by B.S. DeWitt & R. Stora
(North
Holland
]984)
4
S.W. Hawking. Nucl. Phys. B239 257 ( 1 9 8 4 ) ,
5
S.W, Hawktng and Z.C, Wu,
6
S,W, H a w k t n g & J.C, Luttrelh
NueI, Phys, B247 250 (1984)
7
V.A. Rubakov.
&
Phys, Lett ,151B ,15 ( ] 9 8 5 )
M.V. Sazhin
A, V. Veryaskin.
Phys. Lett. 1,15B
,189
(]982)
8
L.P. Grischuk,
Zh,
40 409 (1975)]"
Eksp. Ann.
Teor, N.Y.
Fiz 6_7 825 ( ] 9 7 4 )
Phys,
JETP
Sci 30__? 439 (1977)
9
A . A , Staroblnsky. PIs'ma Zh.
.10
S.W. Hawktng, Phys. Lett. 1508 339 (1985)
11
S.W. Hawking, Phys. Lett.
12
A . H . Guth
& S.Y. Pi,
[Sov,
Eksp.
Teor.
FIz 30 719 ( ] 9 7 9 )
"115B 295 (,1982)
Phys. Rev. Lett. 49
1110
(,1982)
[JETP
682 (,1979)] 13
E,M. Llfshitz and I.M. Khalatnlkov, Adv. Phys.
]2 ,185 (.11963)
,14
U . H , G e r l a c h and U.K, Sengupta. Phys. Rev D,18 .11773 (,1978)
Lett.
30
37
Appendix A:
Harmonics
on the 3 - s p h e r e
In this a p p e n d i x we d e s c r i b e the p r o p e r t i e s of the s c a l a r , tensor harmonics
on the 8 - s p h e r e
S 3.
v e c t o r and
The m e t r i c on S 3 is G i j
and so
the line e l e m e n t is
d£ 2 = f~..dxZdx 3 = dx 2 + sin2x(d82 z3
A vertical
stroke
metric R i j ' (1)
Indices
Scalar
The s c a l a r
will
denote i,j,k
covariant
+ sin28d# 2)
differentiation
with
(A])
respect
to
the
are raised and lowered using R i j "
Harmonics
spherical
harmonics
O/_m n ( x,O,#)
the Laplacian o p e r a t o r on S 3 ' Thus,
Q(n)
Ik
Ik
I) Q(n)
eigenfunctions
n = 1,2,3...
for given n,
n-i ~
Q(n)(x,O,~ )
scalar
of
they satisfy the e i g e n v a l u e e q u a t i o n
(n 2 . . . .
The most g e n e r a l solution to ( A 2 ) .
are
J ~
An
= £=o m = - £
(A2)
Is a sum of solutions
n
(A3)
Im Q ~ m (x'°'~)
w h e r e A nl m are a set of a r b i t r a r y constants.
n The QP.m are given explicitly
by
n
Qjm(X,e,¢)
where [I ( X )
Ylrll ( 8 , # ) are
constitute on S 3.
the
are
the
usual
Fock
harmonics
a complete
orthogonal
=
n;(x)Ytm(e,#)
harmonics 13, )4
on The
the
( A4 )
2-sphere,
spherical
S 2,
and
harmonics
n Qim
set for the e x p a n s i o n of any s c a l a r field
38 (2)
Vector
Harmonics
The t r a n s v e r s e v e c t o r of
the
Laplaclan
n (Si)jm(x,e,#)
harmonics
operator
on
S3
which
are
are vector eigenfunctions
transverse.
That
is,
they
satisfy the e i g e n v a l u e e q u a t i o n
s(n) Ik l Ik
= _ (n 2 .
2 ) S~ n ) l
n = 2,3,4...
(AS)
and the t r a n s v e r s e c o n d i t i o n
Shn ' '"i " 1
The most g e n e r a l s o l u t i o n to ( A 5 )
s(n)(x,8,#)
n-i E ~=I
=
= o
(A6)
and ( A 6 )
is a sum of s o l u t i o n s
£ E Bn m=-£ ~m
n (Si)~m(X'e'#)
w h e r e B n£m a r e a set of a r b i t r a r y
constants.
n (Si)£m
w h e r e it is a l s o e x p l a i n e d how they a r e
a r e given in r e f e r e n c e 14
c l a s s i f i e d as odd have
two
(o)
linearly
or e v e n
independent
(e)
Explicit
(A7)
e x p r e s s i o n s f o r the
using a parity t r a n s f o r m a t i o n .
transverse
vector
harmonics
We thus
S°i
and
sei
( n , £, m s u p p r e s s e d )
Using harmonic
the
scalar
n
harmonics
Q£m
we
may
construct
a third
vector
n
(Pi)£m. defined by ( n , £ . m suppressed)
1
-
Pi
(n 2 -
1)
Qli
n
(AS)
= 2,3,4...
It may be shown to satisfy
Pilk
Ik
=
-
( n 2 - 3) P. z
and
p. I i = x
- Q
(A9)
39 The three vector h a r m o n i c s S °i ' S ei and P i
constitute a c o m p l e t e o r t h o g o n a l
set for the expansion of any vector field on S 3,
(3)
The
Tensor
Harmonics
transverse
elgenfunctions traceless.
traceless of the
That is.
tensor
Laplacian
n (Gi3)9.m (x,e,~)
harmonics
o p e r a t o r on S 3 which
are
tensor
are transverse and
they satisfy the e i g e n v a l u e equation
G(n) Ik G(.n) ij Ik = - (n2 - 3) 13
(AI0)
n = 3,4,5...
and the transverse and traceless conditions
c~n )li
l3
= o
The most g e n e r a l solution to ( A l l )
n-i G(.n)(x,8,#) 13
G~ n)i = 0 '
=
(All)
1
and (A12)
is a sum of solutions
£
)'1 ~" C n (Gij); 9=2 m=- P. Em m( x ' e ' # )
where Cn£m are a set of arbitrary constants. may be classified as odd or even.
(A12)
As in the vector case they
13 and Explicit expressions for ( GO i j )£m
n ( Gei j )9,m are given In r e f e r e n c e 14
Using the transverse vector harmonics construct traceless tensor harmonics for odd and even,
i
we m a y
o n e n (Sij)£ m and (Sij)£ m defined,
= Sil j ~ Sjl i
= 0 since S . is transverse. 1
shown to satisfy
n and (s ei)£m,
both
by ( n , £, m s u p p r e s s e d )
Sij
and thus S , 1
( S io )n E m
In addition,
(A'I3)
the S i j
may be
40
S..
13
Ij
=
._ ( n 2 _ 4 )
S,,.i~l i 13
Ik
Sijlk
Using n
(Qij)Jtm
the
scalar
and ( P i j
n
)~m defined by ( n , £ . m
1
Pij
are
traceless
i
"Pi
= 0,
6)
(AI6)
Si3
may
construct
two
tensors
suppressed)
n = 1,2,3
1 (n 2 - i) Qlij
and Pij
(A]5)
we
Q£m"
Qij = 3-flijQ
The
_
n
harmonics
(A]4)
1
= o
(n 2
= -
S,
(A17)
÷ 3~ij Q
and
in
n ~ 2,3,4
addition,
may
be
(A]8)
shown
to
satisfy
Pij
Ij
2
= - ~n
Pijlk Ik =
•. P13
2
- 4) Pi
-- ( n 2 - 7) Pij
]ij = ~2( n 2 -
The six t e n s o r h a r m o n i c s
(A19)
Qij,Pij,
S °i j '
4)
(A20)
(A2I)
Q
S ei j ' G°i j
and G ei j
constitute
a c o m p l e t e o r t h o g o n a l set for the e x p a n s i o n of any s y m m e t r i c s e c o n d rank tensor field on S 3,
(4)
Orthogonality
The n o r m a l i z a t i o n
and
Normalization
of the scalar,
the o r t h o g o n a l i t y relations, dp..
Thus
v e c t o r and t e n s o r
harmonics
We d e n o t e the i n t e g r a t i o n
Is fixed by
m e a s u r e on S 3 by
41 d/z = d3x (dehflij)]/2 = sin2X sine dxded~
The
n _Otto
(A22)
are normalized so that
fd# Qim n Q~'m' n' = onn' 8£~, 8mm,
(A23)
This implies
n
i
n'
fd/A (Pi)~m (P)~'m'
=
_
1
(n 2 - I)
6nn'
8mm,
~i'
(A24)
and
n ij n' = 2(n 2 - 4) 6nn' 8mm, fd~ (Pij)im (P )£'m' 3(n 2 i) 6££,
The ( S i ) £nm .
both odd and even,
(A25)
are normalized so that
n (S z )£'m' n' = 8nn' 8££, 8mm, ;d# (Si)£m
(A26)
~d/L (Sij)Imn (siJ)£n',m, = 2(n2 -. 4) 8 nn' 8£i , 6ram '
(A27)
This implies
Finally,
the ( G i 3 )£m' n
both odd and even,
are normalized so that
fd/~ (Oij n (dijon ' = 8nn' )£m " "£'m' 8££,
The information
mm'
given in this a p p e n d i x about the spherical
(A28)
harmonics
is all that is n e e d e d to perform the derivations presented in the main text, Further details may be found in r e f e r e n c e s
13 & 14
42
Appendix
B:
Action
and Field
Equations
of the Infinite
Dimensional
Model
The action
I =
where
lo((Z,~,N0)
+ EI n
(Bl) rl
I 0 Is the action of the unperturbed model:
Io = -
%fdt
. SOe
3a{& 2 /~-
e
-2a
.2
-
[N o
{n
_~_ + m z c z
j
(B 2 )
2
NO
is quadratic in the p e r t u r b a t i o n s and may be written
In = Idt(L3 + L mn )
(B3)
where
Ln = //ae°~o g
~i n 2-25_)an2 + (n2-7_/~ 4 / b 2 _ 2(n2_4)C2n _ (n2÷l)d2n ÷ ~(n2-4)anbn 2 3 (n2_l) n
gn['32-(n2-4)bn 4 32-(n2÷I/2)anl+ ~ I
3(n2-i!)k2 I + (n2 n
2
3(z
+ I~ 0
4)j2nl
.2
_ ~2 ÷ LD/-4)62 ~ (n2 4)62 + d n (n2_l) n n
÷ & I - 2an&n + 81D~-n~/b (n2 i) n 6n + e(n2- 4)CnCn + Sdndnl an + 6 + 21 - ~32 02+ (n21) n
6(n2 - 4)C2n +
+ gn[2 o+ 23an Onl
60:I
43
and
! L n = VaNo e3a
f n +
1
m
-I- 3 ~ - m 2 , ~ 2 2 -~-2-No
j
2
_ m 2 .2
+
-
4(n2-4-)b 2 - 4(n2-4)C2n - 4,d (n2_].) n
an-
I
(
-
)n
+ - . ~ g 2n 0
fn~ "21
- gn 2m2fn@ + 3m2an ¢2 + 2--~-2- + 3-~D'--~e--a NO N 2 J - ~knfn¢
"
(B5)
The full expressions for n a and nl~ are
7/'a
++3+, -~-0 [ - &
+ F.. n
n
I_ anon 2 n
+~=.-
(n2_l)
1
n
t &(3a n
(
~-(~ _ +__..: _ +
field
n
equations
-
gn) + 3 e
kn]
(B7)
F. gn{~n n
n
The classical
2
+ 4(n -4)b 6 + 4(n2-4)Cn6 n + 4rindn (n2 I) n n
+ 3an~- e-+kn~olJ
may be obtained
from
varying with respect to each of the fields in turn.
the action
(B7)
(B1)
by
Variation with respect
44 to = and # gives two field equations, similar to those obtained In section II. 4. but modified by terms quadratic in the perturbations:
°I o"l
"° :'
l(j
NO d E &
_
_ (~2 +
#
÷ N~m2~ = quadratic
.2
+ 3 #
- INOe
-
terms
(88)
N2e -2a
+ N m2#2
Variation with respect to the perturbations an,bn,Cn,d n and fn leads to five field equations:
d. 3dan. 1 2 ~ ; ) + ~ ( n - 4)N20en'(an -f b n ) + 3 e 3 e ( ~nf , - No2m2#fn ) -
NO ~E(e
N°213e3am2#2 - ~(n2+2)ea}gn
d e3Ul~n N O dt(
1 2
- ~n
N;)
2 a
- l)Noe
~oJ
+ e3a&gn - ~N° ~dl f e ~ J d
(an
+ ?o dt!
d. 3c~Cn.
+ bn ) =
1 n2 ~(
(8,o,
- 1 )N2eagn
"ol
d [ 2aJn}
(8ll)
~(e Eo,= ~{e EoI
(812)
d e3adn NO dt( ~00 ) + (n2 - l)N2e~dn = 0
(B'I3)
" n • [~o[m . 2[ 2 e 3a + (n2-1 ) e a l f n = N O d~tt(e3afN-~) + 3e3a~a 0
45
e3={ - 2N~m2#g n + ~C3n - e-O~#knJ
In
obtaining (Bl0)
(B14).
-
(814)
the field equations (B8) and (89)have
been
used and terms cubic in the perturbations have been droppped. Variation with respect to the Lagrange multipliers kn, J n , g n and NO leads to a set of constraints.
Variation with respect to k n and 3n leads
to the momentum constraints;
&n 4. U~J-415 4- 3~n~
=
(n2_l) n
"
e-=
~gn
(n2_l)
k
(815)
n
6 n = e-aJn
(816)
Variation with respect to gn gives the linear Hamiltonian constraint:
.2
+ NoZ ma(2fn ~ + San@2) - 3-~0 2e-Zaf[(n 2-4)bn 4. (n2 + 1/Z)anJ
.2
"-~e-~Zk3 n + Zgn( _ &2 4-
Finally.
variation with
@ )
(817)
respect to NO yields the Hamiltonian
constraint.
which we write as
~e3a
- - - ~~2 .+ - - -
NO
NO
e -2~ 4. ~ $ z
= quadratic
terms
(818)
SOLITONS
AND B L A C K HOLES
IN 4,5 D I M E N S I O N S
G.W. Gibbons of A p p l i e d M a t h e m a t i c s and T h e o r e t i c a l of Cambridge, Silver Street, C a m b r i d g e U.K.
Department University
Physics, CB3 9EW
Contents
I) 2)
Topology
3)
The Black Hole
4)
Solitons
5)
Pyrgon-Monopole
I.
Introduction
Introduction
This February [1,2]
and Initial
as Soliton
in 5-dimensions duality
is the w r i t t e n 1985.
Since
I have d e c i d e d
to comment on the
version
importance the
or otherwise
p o i n t will be that while
particular
that one cannot
of topology quantum
there
the a p p a r e n t l y
situation analogue NUT
with that
whose
by Gross,
occurrence
alters
theory.
aspects
monopoles importance
Perry
also be described.
and Sorkin. In section
Their
that the
in section
value
2 I will
problem.
case,
In
black holes and the
contrast
the
argue that the true
theory
for K a l u z a - K l e i n
data
- will be similarly
4 I will
and I will
singularities
of the importance
feel one can regard
in Y a n g - M i l l s
In
of initial
likely
sense
In section
My m a i n
(Cosmic Censorship)
Reissner-Nordstrom
in 5 - d i m e n s i o n s
topology
w i t h the s i t u a t i o n
ones views
initial
in p a r t i c u l a r
of spacetime
is as follows:
of the
but rather
differences.
It is highly
why I don't
in the extreme
of m a g n e t i c
similarities
in
elsewhere
dimensions.
by event h o r i z o n s
of this to supergravity.
solutions
stresse d will
inevitable
of the article
3 I will d e s c r i b e except
and spacetime
it make m a t h e m a t i c a l
section
as solitons
verbatim
assume that the time e v o l u t i o n
some t o p o l o g i c a l
in Paris
in gravity,
are s i g n if i c a n t
substantially
discuss
relation
are many
shielding
- should
The plan
of spatial
there
in the c l a s s i c a l
theory
affected.
This
the lectures of solitons
in 4 and in 5 spacetime
theory
and their c o n j e c t u r e d
is continuous.
problem
given
as given has now a p p e a r e d
not to r e p e a t
situation
in Y a n g - M i l l s - H i g g s
of two lectures
the m a t e r i a l
on the general
contrasting
means
Data
are the m u l t i - T a u b
theory was
relation
5 I will d e s c r i b e
first
to black holes a duality
47
conjecture
analogous
to that of O l i v e
& Montonen
in the Y a n g - M i l l s
case.
2.
Topology
and the Initial
It is an a t t r a c t i v e topological data
set
features
where
Riemannian
metric
and second
fundamental
region,
~ ,
Z
is k n o w n
Relativity
E
and o t h e r an initial
manifold,
gij
a
K.. the second f u n d a m e n t a l form. The m e t r i c 13 form just satisfy c e r t a i n c o n s t r a i n t s and be imagine
is in the S c h w a r z s c h i l d
may be i m a g i n e d
topological
more than one a s y m p t o t i c vacuum
to
E
with
k
classification
[3] that for o r i e n t a b l e
solution.
to be c o m p a c t i f i e d
being diffeomorphic
uniquely
solitons
is to start with
is a 3 - d i m e n s i o n a l
Indeed one could
is no complete
expressed E
flat.
regions
manifold There
and
just as there
asymptotic
idea that the way to study
in General
{E,gij,Kij}
asymptotically
Data
manifolds
as the c o n n e c t e d
The
to give
points
removed.
of 3 m a n i f o l d s
and factors
sum of a number
of
k
a compact
E
but it
may be
"prime m a n i f o l d s "
l
~ ZI # Z2 A complete
list of prime
for instance d iscret e data
S2 x S I
subgroup
satisfying
Schoen
"'" # Zn
and Y a u
(I) manifolds
is not k n o w n
and e l l i p t i c
of SO(4)
with
free
the c o n s t r a i n £ s [4] p r o b a b l y
spaces
where
action on
which
limited
but it is k n o w n that
S3/F
S3
are o r i e n t a b l e
to a sum of
F
is a suitable
are prime. are,
S 2 × S1's
Initial
according
to
and e l l i p t i c
spaces. The
existence
argue that there there were
of a u n i q u e
factorization
are no solitons
one w o u l d
expect
has
led W i t t e n
in 4 - d i m e n s i o n a l
an a n t i s o l i t o n
gravity
3-metric
[5] to
because
if
such that one
could write:
S3 : Z
# ~ s
where
Zs
soliton.
(2) s
is the soliton If
E
s
now seems to have soliton
The way out of this
pair
to give
spacetime
topology
number".
any t o p o l o g i c a l l y singularities
and
~s
that of the anti-
is ruled out by the u n i q u e n e s s
cannot have
particular
is not a "good q u a n t u m that
this
a p r o b l e m with CPT
antisoliton
appears
3-space
is prime
since
difficulty This
(2)
implies
the q u a n t u m would
seems
non-trivial
of the vacuum.
seem to be that
reasonable
in its future
that the
numbers
initial [6].
data
One
because
topology it
set m u s t evolve
48 A c c o r d i n g to the widely believed but still as yet u n p r o v e d Cosmic C e n s o r s h i p Hypothesis event horizons. state
[7] these singularities will be shielded inside
Furthermore
it is also widely believed that the final
(in the classical theory) will consist of one or more time
independent black holes.
These black holes will have the m e t r i c of
the Kerr solution. The consequences of this are spatial topology is concerned.
rather d i s a p p o i n t i n g as far as
Suppose one started with for instance
one of Sorkin's n o n - o r i e n t a b l e wormholes non-orientable
S2
bundle over
initial d a t a w i t h this t o p o l o g y topological proper£ies
[3].
[8].
That is
E = P
the
SI
It is not d i f f i c u l t to construct
[9].
This has a number of fascinating
For instance,
topologically:
P # (S 2 × S I) ~ P # P
(3)
w h i c h one m i g h t i n t e r p r e t as saying that two n o n - o r i e n t a b l e w o r m h o l e s could turn into a n o n - o r i e n t a b l e w o r m h o l e and a c o n v e n t i o n a l o r i e n t a b l e wormhole.
All of this however will be invisible from infinity since
p r e s u m a b l y each or maybe both will be s u r r o u n d e d by event horizons and the fact that they are t o p o l o g i c a l l y n o n - t r i v i a l will play no role in the e x t e r i o r dynamics.
The final black hole s o l u t i o n will be a
S c h w a r z s c h i l d or Kerr metric and no hint of the interior t o p o l o g y will show up in that. Very much the same applies to the s i g n i f i c a n c e of the 8-vacuum structure of the initial data. Q
One might v i e w the c o n f i g u r a t i o n space
for gravity as the space of R i e m a n n i a n metrics on
the set of d i f f e o m o r p h i s m s at infinity)
Diff,(E)
and its tangent space invariant.
c o n n e c t e d the c o n f i g u r a t i o n space
Q
E
having a point on If
factored by E
(the point
Diff,(~)
is not
will not be simply connected and
% - v a c u u m analogous to those in Yang-Mills theory are possible
[10].
A p a r t i c u l a r instance of this is the beautiful work of Sorkin and Friedman
[11] on spin ½ from gravity.
Because
Q
is not simply connected
a r o t a t i o n of the spacetime relative to infinity may result in one moving around a closed loop in path.
such a rotation. where
Q
which is not h o m o t o p i c to the constant
The q u a n t u m wave function could in principle change sign under
F
As an example consider as they do
~
to be
S3/F
is the 8 element group c o n s i s t i n g of the q u a t e r n i o n s and their
negatives t o g e t h e r w i t h
±I .
It is quite easy to construct time
symmetric initial data c o r r e s p o n d i n g £o this space.
The r e s u l t i n g space
can be thought of as c o n t a i n i n g 7 black holes s u i t a b l y i d e n t i f i e d
[9].
Despite the exotic t o p o l o g y it seems rather likely that the end result
49
will be just one external
metric
Finally t opolog y
solitons
without
goo
of the initial
let me remind Pauli
other
b l a c k hole.
as a final
Einstein,
regular
large
argument
horizonts" static
= v2'
with
V ~ 0
t h e o r e m of Serini,
states
The a r g u m e n t
depends
V ÷ I
as
"No
that there are no
of the v a c u u m E i n s t e i n
and
of 3-space
I like to p a r a p h r a s e
The t h e o r e m
~
be no sign in the
the s i g n i f i c a n c e
of the well k n o w n which
solutions
on
there will
topology.
against
the r e a d e r
than the flat one.
imply
exotic
and L i c h n e r o w i c z
globally
Again
on the
at i n f i n i t y
equations
fact that
if
the field e q u a t i o n s
that
V.VIv = 0
(4)
1
w here
Vi
metric
is c o v a r i a n t
gij"
The m a x i m u m
The r e m a i n i n g
R
w here
gij
is the Ricci and hence
The remarks
tensor
of the
regard
property
The s i t u a t i o n
from the work
even
should
"intermediate
We are still
theory
By central
of the f u n d a m e n t a l
of the
in E i n s t e i n
fields
theory
of the theory.
but
since we against
final outcome
of this
t h e o r y but m a y require guess
If this
theory
conserved
Indeed
are u n s t a b l e
is that the
is true
the
as an u n s t a b l e
than a stable p a r t i c l e - l i k e
completely
They have no
theory.
A plausible
in the q u a n t u m
as opposed
area is a n y t h i n g
in the q u a n t u m
ignorant
the reader
not m e a n that one can
Far from it.
to this w o u l d be if the hole I mean
of the t h e o r y
in the c l a s s i c a l
of gravity.
rather
to c o n v i n c e
This does
in a puff of radiation.
be r e g a r d e d
state",
The e x c e p t i o n
shows
be flat.
[12] that black holes
evaporation.
quantum
d y na m i c s
is even w o r s e
of H a w k i n g
simply d i s a p p e a r s
been i n t e n d e d
of the e v e n t h o r i z o n
w h i c h may not be c a l c u l a b l e
charge.
V = I.
in ~ - d i m e n s i o n s
m e t r i c must
as solitons.
thermal
b lack hole
which
data.
process
a consistent
gij
4-dimensional initial
or angular m o m e n t u m
solitonic.
of
2 have
black holes
the n o n - d e c r e a s i n g
hole
that
as S o l i t o n
in s e c t i o n
importance
necessarily
know
shows
now reads
the 4 - d i m e n s i o n a l
to that of 3 - d i m e n s i o n a l
fixed mass
immediately
to the spatial
(5)
The B l a c k Hole
of the
principle
field e q u a t i o n
with respect
0
Rij
that
3.
=
l]
differentiation
carried
state. a "central"
and not c a r r i e d
For e x a m p l e
in
N=2
by any
ungauged
50 extended s u p e r g r a ~ i t y N=2
[13] there is a Maxwell field.
supergravity m u l t i p l e t s are the graviton,
gravitino.
The fields of the
the photon and the
These are all e l e c t r i c a l l y neutral with respect to the
Maxwell field - that is why the theory is "ungauged".
It is quite
possible for black holes to carry this charge - e s s e n t i a l l y because the lines of flux are "trapped in the topology" in the days of "Geometrodynamics". rotating) the mass
is that of Reissner and Nordstrom. M
and charge
Q .
as people used to say
The m e t r i c of such holes
(if non-
It is p a r a m e t e r i z e d by
Because of the duality,
invariance of the
theory of any m a g n e t i c charge may be rotated to zero by a suitable d u a l i t y rotation.
The s i n g u l a r i t y is clothed by an event horizon if
M ~ IQI/K where
K 2 = 4~G
detail e l s e w h e r e
(6) and
G
[I,2]
[see also 14,15,16,17].
is N e w t o n ' s constant. how one m a y v i e w
I have d e s c r i b e d in more
(6) as a B o g o m o l n y type i n e q u a l i t y
The electric charge
Q
is truly central in
the sense of the s u p e r s y m m e t r y algebra and the inequality in saturated by extreme black holes which are "supersymmetric" they possess
"Killing spinors".
black hole m e t r i c s M a j u m d a r metrics
(6) is in that
There exist a whole family of multi-
[17] satisfying
(6).
These are the Papapetrou-
[18] which are included in the general class of
I s r a e l - W i l s o n metrics [19].
Tod
[20] has shown that the I s r a e l - W i l s o n
metrics e x h a u s t all the metrics with Killing spinors in
N=2
supergravity.
It has been known for some time that the throat of the extreme ReissnerN o r d s t r o m metric has the g e o m e t r y of the R o b i n s o n - B e r t o t t i i.e. the product metric on anti-de Sitter space.
S 2 x (ADS)2
when
(ADS) 2
solution,
is 2 - d i m e n s i o n a l
The R o b i n s o n - B e r t o t t i metric shares with flat
space the p r o p e r t y of being m a x i m a l l y s u p e r s y m m e t r i c - i.e. of having the largest possible number of K i l l i n g spinors. R e i s s n e r - N o r d s t r o m metrics "vacua" of
N=2
Thus the extreme
spatially interpolate b e t w e e n the 2 possible
u n g a u g e d supergravitY.
The p o s s i b l e relevance of this
remark for spontaneous c o m p a c t i f i c a t i o n is intriguing.
For the present
let me remark that this is typically s o l i t o n - l i k e behaviour. Since the charge is central it cannot be lost during H a w k i n g e v a p o r a t i o n and so a hole with an initial charge must settle down to the lowest mass state with that charge. temperature)
state.
This is the extreme
(zero
This extreme R e i s s n e r - N o r d s t r o m holes seem to
behave just like solitons.
The hole with the opposite charge is clearly
the a n t i s o l i t o n and it seems e x t r e m e l y p l a u s i b l e that a s o l i t o n - a n t i soliton might completely annihilate one another.
They cannot do this
51
classically
if C o s m i c
Censorship
holds
since by H a w k i n g ' s
the final event h o r i z o n m u s t have n o n - v a n i s h i n g Schwarzschild
black hole
The m a i n way that
there
Since
from e q u i l i b r i u m
parameters
specifying
argue
in
N ~3
permutation space.
.
were
case
is
- i.e.
identical
the m o t i o n
that
should
change
doesn't
one w o u l d N
slowly.
However
smaller
equation
could
than
the
is the theory
solution the positions though
If the holes,
of
I will having
by the a c t i o n of the
Thus we k n o w the m o d u l i if one makes
all the others
in the
order
This
coincide
happen.
factor
should move
in Y a n g - M i l l s
by giving
positions.
is not known.
In the slow m o t i o n
they
departures
is to lowest
successfully
the points
of the small hole
motion
order
space",
is s p e c i f i e d
on the
is very m u c h
the same mass),
the P a p a p e t r o u - M a j u m d a r
that this
g i v e n by the s t a n d a r d
geodesic
solution
In p r i n c i p a l
SN
The m e t r i c
mass x K).
from solitons or charge
to c o n s i d e r
To lowest
"moduli
b l a c k holes)
group
that one hole that
the
in a short w h i l e
equal masses,
their mass
(which need not all have
as has been used
In the p r e s e n t
N points
holes d i f f e r
it is r e a s o n a b l e
perturbatively.
approximation
[21,22].
holes
on a suitable
(representing
thermally.
rule.
in e q u i l i b r i u m
on g e o d e s i c s
the extreme
to be no w a y of fixing
the e x t r e m e
can remain
same
in w h i c h
seems
no q u a n t i z a t i o n
can then e v a p o r a t e
area theorem
area but the r e s u l t a n t
the a p p r o x i m a t i o n one can anticipate
field of the others
for a charged g e o d e s i c limit this does
should
(with charge
indeed give
be =
non-relativistic
in the m e t r i c
ds 2 = U3dx 2
(7)
w here i=N-1 i=I This m e t r i c would
take
space.
respect
possible
(8)
Ix -
an infinite
non-relativistic
with
l
is complete
The q u a n t u m
moduli
GM.
Z
U = I +
on
~ 3 _ {xi} "
time
to merge
scattering
to the m e t r i c
that p o t e n t i a l
holes
correspond
on the m o d u l i
terms m i g h t
space,
all had equal mass The moduli
The wave
function
could
equation
to the though
appear due to one
the p e r m u t a t i o n S N.
could be studied
at the S c h r 6 d i n g e r
presumably
the case that the holes group.
approximation
the holes
or coalesce.
of e x t r e m e
limit by looking
This w o u l d
In this
one
should
space w o u l d have
in p r i n c i p l e
in the on the
scalar L a p l a c i a n it is also loop effects. divide
out by
fundamental
group
then be even or odd u n d e r
In
52
permutation.
Thus one could imagine
"fermionic"
black holes!
This is
the analogue of the e f f e c t of S o r k i n and F r i e d m a n I d e s c r i b e d above. It is p o s s i b l e to find e x t r e m e b l a c k holes in the e x t e n d e d s u p e r g r a v i t y t h e o r y as w e l l be t h o u g h t of as solitons.
[23].
N=4
ungauged
T h e y s h o u l d also p r o b a b l y
Like the e x t r e m e holes
in
N=2
they also
have no natural mass q u a n t i z a t i o n - at least as far as c l a s s i c a l or semi-classical considerations
are concerned.
To get a s a t i s f a c t o r y
q u a n t i z a t i o n rule one seems f o r c e d to turn to K a l u z a - K l e i n theory.
4.
Solitons
in 5 - d i m e n s i o n s
M u c h of the d i s c u s s i o n about the r e l e v a n c e of t o p o l o g y 2 could be r e p e a t e d here w i t h
4 replacing
3.
in s e c t i o n
The d e t a i l s of the
t o p o l o g i c a l d i s c u s s i o n w o u l d d i f f e r and we c e r t a i n l y d o n ' t have d e t a i l e d singularity theorems
and black h o l e u n i q u e n e s s
theorems
in h i g h e r
d i m e n s i o n s - i n d e e d we k n o w v e r y little about b l a c k holes in h i g h e r dimensions.
However
in h i g h e r d i m e n s i o n s
gravity
is e v e n m o r e a t t r a c t i v e
(having a force
i n v e r s e l y as d i s t a n c e to the p o w e r of the d i m e n s i o n of
spacetime minus
2) than in 4-dimensions.
distance
In 5 - d i m e n s i o n s
in the same way as the r e p u l s i v e
centrifugal
i n v e r s e l y as d i s t a n c e cubed in all d i m e n s i o n s ) .
force
(which is
In h i g h e r d i m e n s i o n s
it rises e v e n m o r e r a p i d l y t h a n the c e n t r i f u g a l
repulsion.
seem to m a k e g r a v i t a t i o n a l c o l l a p s e and s p a c e t i m e more
it d e p e n d s on
This would
singularities even
l i k e l y in h i g h e r d i m e n s i o n s . H o w e v e r there
is an i m p o r t a n t d i f f e r e n c e .
nor w o u l d we w i s h to c o n f i n e o u r s e l v e s asymptotically Euclidean.
We are no l o n g e r o b l i g e d
to i n i t i a l d a t a w h i c h
are
If we do so the a r g u m e n t that the v a n i s h i n g
of the Ricci t e n s o r implies that the 4 - s p a c e
is flat still goes t h r o u g h
a c c o r d i n g to S c h o e n and Yau's P o s i t i v e A c t i o n T h e o r e m
[25].
If we
d o n ' t r e q u i r e that the 4 - m e t r i c be a s y m p t o t i c a l l y E u c l i d e a n there are m a n y c o m p l e t e Ricci
flat 4-metrics,
s u r f a c e - w h i c h is compact.
static 5 - m e t r i c w i t h no horizons. are still f o r c e d to have on
~ x M , where
M
i n c l u d i n g one - t h a t on the K3
Any gravitational Note
i n s t a n t o n will give a
that if we have no h o r i z o n we
V = I, that is the m e t r i c m u s t be a p r o d u c t
is the 4-manifold.
s p a c e t i m e w o u l d be said to be
In the o l d e r l a n g u a g e
N o t all of these o b j e c t s w i l l be c l a s s i c a l l y will be g o v e r n e d by s p e c t r u m of the L i c h n e r o w i c z s y m m e t r i c t e n s o r s on
M .
If
M
stable.
The s t a b i l i t y
L a p l a c i a n a c t i n g on
has a s e l f - d u a l m e t r i c this is k n o w n to
be p o s i t i v e
and h e n c e the c o r r e s p o n d i n g
stable.
M
If
the
"ultrastatic".
static lump w i l l be c l a s s i c a l l y
has a m e t r i c w h i c h is not s e l f - d u a l the s p e c t r u m is not
58
likely to have unstable. and the
a positive
Examples
"Taub-Bolt
of these
The E u c l i d e a n
objects
tational
and attempts
would
involve
hole will
~3 × S I
a spatial
the T a u b - B o l t
that
these
metric Again
black holes in w h i c h
in
[28].
[24]
and
to evolve
The b o u n d a r y
i.e.
that
or that latter
appear
the
is the self-dual
S1
The t o p o l o g y bundle
over
a black
apply
to
It is p o s s i b l e
from a 4 - d i m e n s i o n a l those d e s c r i b e d
cause these black holes
context
metric
flat.
metric,
w i t h Hopf
but
this
down to the Taub-
likely.
for K a l u z a - K l e i n
in this
flat p r o d u c t
Taub-NUT
since
solution.
of i n t e r e s t
at infinity
S2
to settle
seems
effect m a y then
locally
is
to gravi-
likely that
should be i n c l u d e d w i t h
be w h a t has been called it a p p r o a c h
energy
The same remarks
regular w h e n ~ i e w e d
The H a w k i n g
conditions
It seems
tries
formation
case they
it be a s y m p t o t i c a l l y
centres. the
presumably
theory
down to the flat m e t r i c
change.
to the flat or the T a u b - N U T
the m e t r i c
[26]
has the same a s y m p t o t i c s
it loses
some sort of s i n g u l a r i t y
is not known.
black hole
stand-point
to settle
topology
which
solution
so p r e s u m a b l y
forming
be formed but this
NUT metric.
solution
in the full n o n - l i n e a r
Schwarzschild
on
it can't do this w i t h o u t
lump will be
Schwarzschild
[27].
as the flat m e t r i c radiation
and the c o r r e s p o n d i n g
are the E u c l i d e a n
solution"
The e v o l u t i o n unclear.
spectrum
of this
on
theory
R3 x SI
The typical
invariant
is
~×
N - i.e.
flat -
at infinity example
or m u l t i - T a u b - N U T
in this case
is that
asymptotically
BN
the
of the
with
N
when
BN
lens
is
space
L(N,I) . Gross, solution Perry
Perry
plays
and S o r k i n
[29] have p o i n t e d
the role of a m a g n e t i c
and m y s e l f
asymptotically
[30] have
locally
out that the T a u b - N U T
monopole
in K a l u z a - K l e i n
shown that the m o n o p o l e
flat
solution
should
moment
satisfy
P
theory.
of any
the B o g o m o l n y
type
inequality
IPl < M 2<
with
[9)
--
equality
in the s u p e r s y m m e t r i c
to note that the g r a v i t a t i o n a l [22]
is s e l f - d u a l
but has
it has the topology dihedral
group.
solutions
at i n f i n i t y
of the W i t t e n
The m u l t i coincident
Taub-NUT
points
in
instanton
a negative
The crucial
of
solutions
case.
solution
mass.
This
~ x (S3/F)
point here
equation
~3.
self-dual
It is i n t e r e s t i n g
of A t i y a h
& Hitchin
is p r e s u m a b l y
where
is w h e t h e r
F
because
is the b i n a r y
or not
suitable
exist. are s p e c i f i e d
Permutating
by giving
the points
gives
N
non-
the same m e t r i c
54
so the moduli where and
A
space is the well k n o w n c o n f i g u r a t i o n
is the points
SN
is as before
on the m o d u l i
space
the p o s s i b i l i t y monopoles
(jR3)N
P
study.
of m u l t i v a l u e d
property
satisfies
where
the p e r m u t a t i o n
is under
space
{ (jR3) N - A } / S N
two or more points
group on Again
wave
can r e a l l y be thought
An i m p o r t a n t charge
in
N
coincide
symbols.
The m e t r i c
the q u a n t u m m e c h a n i c s
functions
though whether
of as fermions
of the T a u b - N U T
remains
solutions
the Dirac q u a n t i z a t i o n
offers
these
at p r e s e n t
unclear.
is that the m a g n e t i c
condition:
eP = 2~
where
e
(10)
is the basic unit of charge
This
in turn implies
that
in K a l u z a - K l e i n
(using the e q u a l i t y
theory.
in
(9)) the mass
M
is quantized:
M
Given
-
their
charge
(11)
1
4~<e stability
it seems
representing
and the q u a n t i z a t i o n
reasonable
solitons
some of the t o p o l o g i c a l conserved. Hirzebruch
the T a u b - N U T
this does require,
numbers
In the p r e s e n t signature
to regard
though
of the mass
associated
case
with
two such numbers
and the Euler
number.
Hirzebruch
signature.
magnetic
charge.
Since
can be read off
boundary
conditions
number
infinity
Hawking,
invariant
in Y a n g - M i l l s
accept
resemblances
theory
typically
by an amount
one does
is the
This means
and
this
8~2/g 2
action
that
the m a s s i v e
of the next
Another
in General
topologically
to
The Euler
there
Relativity
to another. where
is not
is the coupling
action
is unlike
the a c t i o n
different
due to
little
This
from seems
argument,
arbitrarily
one sees a number
modes
The
monopole
it c o r r e s p o n d s
cannot be d e t e r m i n e d
in 4-dimensions
section.
are not
are of interest.
of s i n g u l a r i t i e s
configuration
g
the o b j e c t
2, that
from the asymptotic
it may cost
where
as
to be conserved.
This
occurrence
them as solitons
with
subject
Roughly
for it to be conserved.
from one t o p o l o g i c a l
the case
differ
expect however.
is that the E u c l i d e a n
invariant.
to pass
matter
and g i v e n the likely
to be no good r e a s o n
scale
one might
is a d i f f e r e n t
solutions
as in section
The m u l t i p l e
has n o n - v a n i s h i n g
this
and m a g n e t i c
is scale
configurations constant.
If
of s t r i k i n g
of the K a l u z a - K l e i n
theory.
This
55
5.
Pyrgon-Monopole The p h y s i c a l
viewed
duality
content
of the
5-dimensional
Kaluza-Klein
theory when
from the p o i n t of v i e w of 4 - d i m e n s i o n s
I)
A set of m a s s l e s s
states,
the graviton,
graviphoton
and
dilaton 2)
A tower of m a s s i v e m
and charge
states
decay
linearized into
level
account
to d e c a y
into a neutral
into
all the m a s s i v e interactions
lower mass
state.
Thus
be a b s o l u t e l y
stable
except
state.
stable
lowest mass
These
Thus the p e r t u r b a t i v e pyrgons
physical
and antipyrgons.
the r e l a t i o n
(12) c o r r e s p o n d s to avoid
[32] the m a x i m u m
supersymmetries. zero modes
analogy,
If these
case we
[30]
in the
they
has been
an e f f e c t i v e
it m u s t
be the o r i g i n a l
because
of the s u p e r m u l t i p l e t s
massless
fields
antimonopole
This
suggested
create
field theory
pyrgon
++
massless
÷+
anti-pyrgon.
fields
fit
for example
charge
allowed.
2.
spinors
is a rather between
suggested
in Y a n g - M i l l s exist
create
model
of
and hence when
close the m o n o p o l e s
duality theory
is [33].
in the full monopole
states.
the m a s s l e s s
states.
it is e s s e n t i a l l y
of the pyrgons.
structure.
++
N=8
supergravity
and a n n i h i l a t e
which
field t h e o r y
states,
fit into s u p e r m u l t i p l e t s There
dualities:
monopole
N=8
[31].
of m a s s l e s s
than
of K i l l i n g
that there m i g h t
which
antiparticle
the pyrgons
In
central
to say a duality,
there will be o p e r a t o r s
satisfy
theory
charge.
number
into account.
suggest
theory operators
In a d d i t i o n
in
cannot
n=1 , should
w i t h their
spin greater
and the p y r g o n s .
to that w h i c h
In the p r e s e n t quantum
theory
central
possess
state
states,
space consists
with
stable.
the h i g h e r
have been called Pyrgons
to the m a x i m a l
one is tempted
of K a l u z a - K l e i n analogous
with
permitted
As shown are taken
indeed
Hilbert
are t r i v i a l l y expect
but a charged
annihilation
states
states
N o w the G-P-S m o n o p o l e s Cremmer
states
In a s u p e r s y m m e t r i c
supermultiplets
is n e c e s s a r y
states
one m i g h t
the lowest mass
against
into m a s s i v e
the
1 and 2 each with mass
n = 1,2,3,... At the
This
of spin 0,
by
(12)
W h e n one takes mass
states
given
lel 2K
m = n
w here
e
This
Thus we have
unique
-
is e s s e n t i a l l y
the c o n j e c t u r e d
56
It is d i f f i c u l t to see with present day techniques how such a conjecture could be verified.
In the Yang-Mills case some partial
e v i d e n c e has come from a study of m a g n e t i c and e l e c t r i c dipole moments. It has been v e r i f i e d that the g y r o m a g n e t i c ratio of the o r d i n a r y YangMills particles equals the g y r o e l e c t r i c ratio of the m o n o p o l e s plus fermionic zero-modes
[34].
It is known that the g y r o m a g n e t i c ratios
of the pyrgons are anomalous and equal unity, value of 2 [35].
rather than the Dirac
It w o u l d be i n t e r e s t i n g to calculate the electric
dipole moments of G-P-S m o n o p o l e s with their fermionic zero-modes. Further insight into this c o n j e c t u r e d d u a l i t y m i g h t come from a study of m o n o p o l e - p y r g o n
interactions.
A number of authors
pointed out that there is no "Callan-Rubakov" effect catalyze the decay of pyrgons.
[36] have
[37] which would
This is m o s t easily seen from the fact
that scalar modes on Taub-NUT are well defined and u s i n g the c o v a r i a n t l y constant spinor fields on T a u b - N U T one can obtain all solutions of the Dirac equation. Thus if
g
is a c o v a r i a n t l y constant spinor or T a u b - N U T and
a solution of the wave e q u a t i o n w i t h energy ~
~±
= [~
± ~i
then:
(~)]~
are solutions of the Dirac e q u a t i o n with the same energy. A s t r i k i n g fact about the scalar modes on the T a u b - N U T b a c k g r o u n d is that the m a s s i v e scalar Pyrgon wave e q u a t i o n separates in 2 d i f f e r e n t c o o r d i n a t e systems.
One system is the standard radial v a r i a b l e s
in
w h i c h the m e t r i c is ds 2 : (I + ~ ) - 1 4 N 2 ( d ~ + c o s @ d * ) 2 + 1 1
where
0 ! ~ ! 47 .
Thus
K a l u z a - K l e i n circle. ~
where
+~)
8 ~ N = 2~R K
(dp2+p2(d@2+sin2@d*2))
where
RK
(13)
is the radius of the
The scalar field has the form
,n = e -i~t elY~n Yim(0)
eim~f
n
(r)
(14)
~Yzm(8)eim~ is a spin w e i g h t e d spherical harmonic and where z ls a n o n - r e l a t i v i s t i c Coulomb wave f u n c t i o n w i t h angular m o m e n t u m
f (r) n but where the Coulomb potential is energy dependent, u p o n ~ , that is 12 d (p2 df) p dp dp
f
i.e. depends
satisfies i(i+1)f 2 + (2N 2 - n 2 )f + ( 2 _ n 2 ) f = 0 p N P 4N 2
(15)
57 There are no bound states, just scattering equation
better described using parabolic
This
states.
Since the radial
(15) is a Coulomb one one might anticipate
=
(I + cose)
q =
(I - cose)
scattering.
in the
Using them one can give a simple description orbits are especially
They are, when projected
p, 8 and ~, the intersection the intersection
of 2 different
separates
t,~,~,q
of the
simple being conic
into the 3-space spanned by
of a cone centred at
being a hyperbola
The existence equation
defined by
The wave equation also separates
The classical
sections.
is
(16)
is in fact true.
coordinates.
coordinates,
that scattering
p=0
with a plane,
in general. coordinate
systems
is often taken as the indication
and indeed of a "spectrum generating
algebra".
in which the wave of hidden symmetries
The precise nature of
this algebra
in the present case has not been worked out.
It is tempting
to speculate
that it may be related to the known existence
of Kac-Moody
algebras
in K a l u z a - K l e i n
Ano%her tempting
theory
[38].
speculation
full expression
in string theory.
if one considers
10-dimensional
spacelike obtains
dimensions
is that these ideas will find their Mike Green
[39] has remarked
string theory on where
form a torus,
string states with masses
that
10-D of the
each of whose radii equals
R
one
satisfying
M~ R2N~ (mass)2 = i=0[ (--!lR 2 + ___~i)~,2 + _~2~, (NO + No ) No
and
No
integers
are occupation
{M i}
numbers for higher
are K a l u z a - K l e i n
R ÷ 0
i'th compact dimension. and
~' -R-
constant
The resulting D - d i m e n s i o n a l
from the periodicity
{N i }
are topological
This theory is apparently N=2
(10-D) dimensions
Consider
the limit
= I
field theory has an infinite number of
spin 2 supermultiplets
with 10-dimensional with
The integers
The
associated with the number of times a closed string winds
round the
massive
string states.
charges resulting
in the 10-D compact dimensions. charges
(17)
whose masses
identical
are determined
by 1
to the theory obtained by starting
supergravity
and c o m p a c t i f y i n g
having finite radii
R =I
on a hypertorus
58 Now set Cremmer's from
D=5 .
N=8
The reduction of
N=2
d=10 to 5 dimensions
D=5 model, with fts pyrgon states.
(17) we see that the states corresponding
charge but non-vanishing this limit.
topological
These presumably
gives
On the other hand
to zero K a l u z a - K l e i n
winding numbers will survive in
correspond
to the magnetic monopole
states.
References [I]
[2] [3] [4] [5] [6]
[7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
G.W. Gibbons in "Supersymmetry, Supergravity and Related Topics" ed. F. del Aguila, J.A. de Azcarraga and L.E. Ibanez, World Scientific 1985. G.W. Gibbons in "Non Linear Phenomena in Physics" ed. F. Claro, Springer Proceedings in Physics #3. Springer Verlag 1985. J. Hempel "Topology of 3-Manifolds", Princeton University Press (1976). R. Schoen and S.T. Yau, Phys. Rev. Lett. 43 1457 (1979). E. Witten, Commun. Math. Phys. 100 197 (1985). D. Gannon, J. Math. Phys. 2364 (1975). D. Gannon, G.R.G. 7 219 (1976). C.W. Lee, Commun. Math. Phys. 51 157 (1976). R. Penrose, Ann. N.Y. Acad. Sci. 224 125 (1973). R. Sorkin, J. Phys. At0 717 (1977). G.W. Gibbons, unpublished. C.J. Isham, Phys. Lett. I06B 188 (1981). C.J. Isham, in "Quantum Structure of Space and Time", eds. C.J. Isham & M.J. Duff. Cambridge University Press (1982). J. Friedman & R. Sorkin, Phys. Rev. Lett. 44 1100 (1980); see also B. Witt: Milwaukee preprint. S.W. Hawking, Nature (Lond.) 248 30 (1974). S.W. Hawking, Commun. Math. Phys. 43 199 (1975). S. Ferrara & P. van Nieuwenhuizen, Phys. Rev. Lett. 37 1669 (1976). G.W. Gibbons, in Heisenberg Memorial Symposium, ed. P. B r e i t e n l o h n e r and H.P. Durr, Springer Lecture Notes in Physics #160. G.W. Gibbons & C.M. Hull, Phys. Letts. I09B 190 (1982). G.W. Gibbons, in Proc. 4th Silarg Symposium, ed. C. Aragone, World Scientific. J.B. Hartle and S.W. Hawking, Commun. Math. Phys. 26 87 (1982). A. Papapetrou, Proc. Roy. Irish. Acad. A51 191 (1947). S.D. Majumdar, Phys. Rev. 72 390 (1947). W. Israel and G.A. Wilson, J. Math. Phys. 1 3 865 (1972). P. Tod, Phys. Lett. B121 241 (1983). N. Manton, Phys. Rev. N. M a n t o n in "Monopoles in Quantum Field Theory" ed. N. Craigie, P. Goddard and W. Nahm, World Scientific (1982). M.F. Atiyah and N. Hitchin, Phys. Lett. I07A 21 (1985). G.W. Gibbons, Nucl. Phys. B207 337 (1982). W. Simon, G.R.G. 17 761 (1985). R. Schoen and S.T__Yan, Phys. Rev. Lett. 42 547 (1979). D. Page, Phys. Rev. B. Allen, Phys. Rev. D30 1153 (1984). R.E. Young, Phys. Rev. D28 2420 (1983). G.~. Gibbons and D. Wiltshire, Annals of Phys., in press. D. Gross and M.J. Perry, Nucl. Phys. B226 29 (1983). R. Sorkin, Phys. Rev. Lett. 51 87 (1983). G.W. Gibbons and M.J. Perry, Nucl. Phys. B248 629 (1984). E.W. Kolb and R. Slansky, Phys. Lett. 135B 378 (1984). E. Cremmer in "Superspace and Supergravity", ed. S.W. Hawking and S.W. Hawking & M. Rocek, Cambridge University Press 1981.
59
[33] [34] [35] [36]
[37]
[38] [39]
C. Montonen and D.I. Olive, Phys. Lett. 72B 117 (1977). H. Osborn, Phys. Lett. 115B 226 (1982). Bo-Yu. Hou, Phys. Lett. 125B 389 (1983). A. Hoysoyer et al., Phys. Lett. 134B (1984). P.C. Nelson, Nucl. Phys. 238B 638 (1984). H. Ezawa and A. Iwasaki, Phys. Lett. 138B 81 (1984). M. Kobayashi and A. Sugamoto, Prog. Theor. Phys. 72 122 (1984). F.A. Bais and P. Batenburg, Nucl. Phys. B245 469 (1984). V. Rubakov., Pisma Zh. Eksp. Teor. Fiz. 33 658 (1981); Nucl. Phys. 203B 311 (1982). C.G. Callan, Phys. Rev. D25 2141 (1981). A. Salam and J. Strathdee, Annals of Phys. 141 316 (1982). L. Dolan and M.J. Duff, Phys. Rev. Lett. 52 14 (1984). M. Green "The Status of Superstrings" undated Queen Mary College preprint.
TRUNCATIONS
C.N.
IN
KALUZA-KLEIN
THEORIES
Pope
Blackett Laboratory, SW7 2BZ, UK.
Imperial
College,
Prince
Consort
Road,
London
Certain mathematical aspects of Kaluza-Klein theories are discussed, concerned with the ability to truncate the four-dimensional spectrum of states to a finite subsector, including the graviton and Yang-Mills gauge bosons. This yields a criterion by means of which certain exceptional theories are singled out from the generic case.
i.
INTRODUCTION Kaluza-Klein
of
gravity
local
and
gauge
gauge
in
all
the
theory
to
first
sight
freedom
be
are
freedom to
order
be
used
to
tend
those
based
principles
theory
admit
Minkowski
on
on
requirement
a
GeV
based
or
on
by
about
a
~urther
light,
any
17
orders
in
particle
of
is
from be
of
perhaps
able
the to
outset. explain
of
40
Of the
hand
orders to
observed
the
of
theory. that
mind
that
the
the
Planck
is
mentioned energy
magnitude.
be
imposing
phenomena,
natural scale, are lower involve
Seen these
in
it
this
physical
theory but
a
seem
scales
'correct'
the
and
above
comparisons
of
the
the
fermions,
at
are
mathematical
requirements
gravity
such
phenomenological
such in
the which
there
requirement
cosmological
course
all
are
principles
premature
from
chiral
accelerators and
about
rather
borne
physical
magnitude,
one
from
the
with
theory
the
extrapolation it
quantum
whilst
observations
requirements ultimately
of cm,
be
what
has
speaking,
the
Superficially
should
choose also
Broadly
there
be
group
at
usual
criteria
self-consistency
would
gauge
it
theory
on
hand
of
and one
seem the
powerful
derived
other
state.
but
i0
the
realistic
sca~
19
categories;
principles
ground
reasonable,
I0
the
'right' rather
principles
while
physical
space
unification
two
unifying
dimension!
the some
have
pick but
possibilities.
into
physical
based of
the
to
unification the
theories one
and
coordinate
like
does
favorite out
requires
divide
only
Lagrangian,
one's
single
restrict
on
Examples
choose to
would
Kaluza-Klein
Not
the
invariance
general
a successful
theories,
in
one
to
considerations,
very
to try
of
one
respect
badly.
unification
coordinate
in
nature,
this
included
theories,
criteria
in
geometrical
subsectors
However, in
rather
and
general
as
four-dimensional
be
In candidate
and
with
which
arise
forces
fare
to
natural
dimension.
unique, to
a
in
both
a higher
additional
can
fields,
fundamental
, as
fields
provide
invariance
invariance of
theories
should may
well
61
be
that
they
collective The
it
be
in
very
metals,
development
of
at
the
when
as
a
time
be
look
provided
able
I
Wheeler
idea
should
flat,
be
topologically
cosmological observed
A
at
magnitude.
not
illustrate
of
be
point
low-energy will
by
theory
recent such
of
hope.
The
answerable
at
consistency This
is
paper
classical
2.
be
surge a
is
that
present, one
primarily
INCONSISTENCIES
is
admits
four-dimensional a
Kaluza-Klein
to
at
of
gravity
the
same
with and
should
a
should serve
to our
by
physical
be
that
may
in
of
more
such
that
a The
part
on
to
be
mathematical
elementary
consistency these
be
the
theory.
complicated
principle
to
guided
than
based
too
many
be
would
unified is
far
certain
theory
of be
imposing
this
unique
are
KALUZA-KLEIN
interpretation
we
now
that
a
is
does
of
theories
it
orders
scale.
general
concerned
state
it
the
even
model
wary
requirement
at
with
120
simple
it
effective
might
but be
for
to
here
spacetime
This
possible
lead
an
least
be
scale
compared
at
is by
spacetime
constant
Planck
is
applied
ground
seen
Planck
with
units,
self-consistency
be
GENERIC
be
scales
that
by
that
superstring
theories,
IN
it
suggested
the
present,
obvious
it
(4+k)-dimensional
compactifications M4
can
fruitful demanded
macroscopic
perhaps
the
involved but
which
at
view
would in
issues
the
have
theory
at
smaller
of
and
interest
that
order
theory!
at
an
it
of
very
originally
Planck
should
the
finite,
Kaluza-Klein
Any
is
principles
criterion
the
after
to
'foamlike',
in
seriously
Ultimately,
should
powerful
and
cosmological
one
adopt
mathematical
considerations.
and -i
prejudices
therefore
of
non-trivial
been
quantum
In
fundamental
too
a
suggested
which
that
analogy;
theory
long
have
foam',
2
studies
order
the
the
developed
microscopic
negative
taken
the
everyday
more
huge
fundamental
highly
theory
which
Hawking.
scales
a
feature
probably
We
large
Thus
desirable
of
the
superconductivity. in
complex
constant
not
microscopic
on
these
of
being
low-energy
framework
to
understood
would
as
a good
the
applied
consequence
'spacetime by
in
possibly
within
It
way
of
developed
macroscopically
is
was
explain the
different
the
and
to
way
manifest
only
successful
of
very by
a
mechanics
a
illustration
might
is were
mechanics.
of
to
it
quantum
prerequisite
means
equation
which
quantum
no
satisfactorily
but
effects
outset
by
non-trivial
superconductivity
SchrSdinger
collective
An
highly
are
of
understood
electrons
a
which
phenomenon
non-relativistic
the
in
phenomena
theory. can
emerge
levels.
questions we
now
in
turn.
THEORIES which
admits
solutions
of
and
a compact
M k is in
which
the
spontaneous form
M4x internal
fluctuations
Mk,
where space, of
the
82
(4+k)-dimensional
fields
4-dimensional massless these
interpretation and
is
truncate
higher
to
just
theory
because
in
source
terms
these
states
To
see
to
in
cosmological
use
the
-
decomposed
as
is
a
M4
with a
is
The just
are
retained discarded,
with
the
where
generic
as
of
arise
inhomogeneous hence of
pure
field
space
to
Kaluza-
These
equations
with
of
tries
setting motion.
gravity
with
equation
(i)
over ~ runs
state
Einstein
one
and
case
(4+k)-dimensional
ground
a rewriting
a
act
their
of all
0,
running
etc.,
admits
compact
fields 4+k
are
values.
over
M4and
solutions
'hatted', These
m runs
of
the
and
will
over
form
be
M k.
M4
x
Mk,
satisfying
gmn'
(2)
4-dimensional
with
isometry
the
just
in
a
towers
if
then
are
consider
=
is
have
Retaining
inconsistencies.
dimensions,
%N
indices
cosmological
space
us
(4+k)
A
that
2A =-k+2
a
let
+
%N
which
inconsistent
in
M=(~,m), (I)
Rmn
and
R
2
world
Equation Mk
--
notation
are
where
is
detail,
it
However,
states, into
which
states
constant
~N We
states
the
zero
this
M,N,...
the
run
infinite
ref.3).
since
of
to
values
of
example,
theory.
number
liable
general for
for
consistent,
finite
is
ground-state
excitations
(see,
dimensional
a
one
their
as
states
necessarily
original
Klein
a
massive
states
the
around
spacetime
constant.
continuous
We
will
satisfying assume
symmetries,
such
the
Einstein
equation
A is
positive,
so
the
k-sphere
with
as
Mk
can
be
SO(k+l)
group. standard
the
Kaluza-Klein
gravity
corresponds
and
to
gMN
gauge
setting ^ A eM
=
ansatz,
the ^ B eN
boson
which degrees
is of
(4+k)-dimensional
designed freedom
metric
to in
to
4
truncate
to
dimensions,
be
~AB'
(3)
where
e
(x,y)
= e
~(x)
= e a m(y)
~a(x,y) where
A,B ....
are
into
~, ~,...
running
dx
,
(4)
d y m - K1a(y) "
(4+k)-dimensional over
M4
A i ~(x) local
and
a,b,..,
dx
Lorentz in
Mk,
,
(5) indices x
and
decomposed m y are the
63 coordinates
on
generating Yang-Mills on
Mk,
(2),
M4and
its gauge which
M k respectively,
isometry
and
shows
K i are (i=l
The
k-bein
unchanged
from
the
ea eb m n 6ab= straightforward
(5)
G
potentials.
is
by A
group
that
gmn" calculation the
R~
= R~
^ Rab
= Rab
^
Ricci
the is
Killing
) is
to
and
Riemannian
on
A i ~(x)
related
state
given
vectors
G) , and
are
the
still
Mk the
metric
satisfies
curvature
for
(4)
by
I Ki a Kj Fi Fj Y 2 a ~y ~ '
(6)
i Ki K3 b Fi Fje~ + --~ a e8 ,
(7)
1
•
F i
2 K1b DI3
R~b -
eam(y ground
of
tensor
the
, ...dim
~
(8)
~ '
where
F i a~ = 2 V[ a Ai~]+ the
structure
constants
Cijk A j
c.. k are i3
Ak~,
defined
(9)
by
[K i , K j ] = cij k K k , K i = K im
where and
Rab
and
Mkrespectively. In
(I)
are the
5m,
the
Rieci
local
D~
is
tensors
Lorentz
the for
basis
Yang-Mills the
eAM,
the
covariant
vielbeins
derivative.
e ~ (x)
and
(4+k)-dimensional
eam(y field
Re~ ) on
M4
equation
reads
--
RAB
2
Substituting
~,
R
1 8 - - -2R
where (12)
~b and
R
ab
~8
= R~
depend
Killing general
R
"lAB
(6)
inconsistencies. =
in
and
(io)
vectors one
+
A
,
(7)
There
are
in
(ii).
~
is
only
on
K i,
=
O.
and
(8)
three
cases
first
of
(ii)
into to
II)
now
consider,
these
the
which
the
4-dimensional spacetime depend
upon
Ricci
the
corresponding
~
to
coordinates
K ia K j
)
scalar.
coordinates the
reveals
AB
yields
= ~1 ( Fi ~y Fj ~ y ---~I Fi y8 Fjy8
+ A ~8
has
The
'lAB
All
x ~, ym
the
a'
(12)
terms
in
except
for
the
on
In
fact
M k.
64
K ia K j = 8ij + yij(y) a
(13)
where y13(y) is symmetric and tracefree in i and j. Thus defining T 13 by = F(i~Y Fj ) ~y - ~I Fi 78 Fj 78 ~ 8 '
TiJ~
•
(14)
'
4
(12) implies that the tracefree part of T13 ~ must vanish Tij~ while
the
- (dim G) -I 6ij T kke~ = 0,
y-independent
part
of
(12)
(15) gives
the
equation with Yang-Mills source term. However, (15)
on
Einstein
the algebraic constraint
inconsistent
Yang-Mills fields means that the ansatz (4), (e~ (x), A i~(x) ) being an arbitrary solution with
4-dimensional
Einstein-Yang-Mills
The (4)
the
standard
only
and
way
of
overcoming
is
to
restrict
(5)
generate
some
subgroup
K i'a
Since the
(16)
implies
subgroup
vanishes Even is
G'
can
X =
the
one
were
(4),
group
then
be
G
by
G'
U(1)
balancing general
by
object
retained,
that
Killing
then
k=4n+l.
to
with
equation
truncation
in
those
the
a lot
smaller then
On
constant
if fields
G'
is
a group
include
on
which
Euler
(unit)
length,
number
X of
must
vanish
than
G.
SU(2)
For
H,
Mk
somewhere).
when
manifold
restored
for
2
fields,
which
left-hand-side spin
these equations
of
4-dimensional
be
spin
the
massive
their
more
could
massive
setting,
components
~b
has
vector
k-sphere,
truncation.
The
vector only
all
infinitely
the
just
which
usually
when
coupling of
the
to
(16)
consistency
inconsistent
Yang-Mills
is
y-dependence
gravity
for
of
vectors
example
if
k=4n+3, with
G
and = HL x
HLOr HR .
including
grounds
inconsistency
Killin~
non-trivial
prepared
(5),
G
X ~ 6 then
can be either
ansatz
of
each
be
the
of
8 i'j'
can if
this
SO(k+l)-invariant
only
HR, G' If
O,
=
that
G'
(since
if
Mk
the
KJ' a
is
equations.
the
G'
(5)
(ii)
2 can many If
such the
of
the
entire
would
(12).
But
consistently 5 fields , which
~pin
isometry 2
fields
in
the
isometry
introduce we
be
entire
massive
present
of
only
fields
a
know
coupled would
to
defeat
group to
on
zero
G
is is
motion no
difficulties,
yielding
the
65
D
However
F
the
=
ab
(17)
components
K i a KJb" another
0.
of
F i a8
unacceptable
Fj~
=
which
this
inconsistency
one
must
include
scalar
~i'j'~ Q,i'j'~
to
K i' Fi~
subgroup
G'
fields
does
The pure
for
which resolve
described
Kaluza-Klein finite
number
bosons,
one
must
the
subgroup
G'
the
Kaluza-Klein
the
adjoint
(4),
In
scalars G'
with a
(5) isometry
centralized
by
non-trivial,
this
K
are
need
be
invariant of
in
G
means
and
'
Of
that
Mk
and
(5), as
the
course, must
any
graviton
and
also
from
G
to
certainly
be
to
G'
of
can
be
general
of
order
G
include
acting
in
gauge
product
In
subgroup
a
group
and
view:
generic
extracts
isometry
transitively a
scalar
(12).
symmetric
of
the
(4+k)-dimensional
(16),
is
the
in
of
the
form to
of
which
restrict
a G'
general
restrict
the
point under
~k,
(4)
fluctuate
of
the
in
to
theoretical
group
in
the
satisfy
that
to
ansatz
from
G
is
(x)
problem
including
of
here
gmn
also
of
an
group
subgroup
introduction
case
write
The
no
representative
gauge
itself.
must
the
the
Killing vectors i'j' ~ , which
group
is
still
the
fields
the
whose
in
to
4-dimensional
(18)
inconsistency
reasonably
order
restrict
from
ansatz of
of
of
understood
K
is
theory.
now must
since
above
equation
i'j'
components
we
previous
is
resolution
freedom
that
holds,
the
there The
metric
course
(16)
time
of
means Of
not
This avoided.
the
which
situation
(18)
degrees
n.' F3'e~
Einstein
0,
is
allowing
K j' a~
yield
constraint.
for
corresponding
(ii)
subgroup G
which
for an
the
G'
is
to
be
homogeneous
space. The from
need
to
insisting
field
from ansatz
equations.
taken
that
one
action,
and
action.
In
over
Mk,
and
In
in
never
Einstein-Yang-Mills a
prescription field
the
y,
arise). action for equations
to
Thus with satisfying and
the
view
the
an
equivalent term
in
from
one
certain violating
(13)
obtain G.
averaging would
and
( action
a
(12)
disappear,
ansatz
so
the would
4-dimensional
However
components others,
sometimes dimensional
4-dimensional
to
effective
could group
is
higher
the
arose
dimensional
effective
4-dimensional
gauge
scalars,
higher
be
oi'j'
the
into
the
the
obtain
would
y-dependent
in
include
satisfy
ansatz
scalars
them
and
approach,
the
over
the
varying
G',
should
this
case the
of
to
alternative
out
which
G
substitute
dicussion,
of
dimensional
an
integrate our
omitting
course
the
should
possibility
is
restrict that
this of such
the an
procedure higherapproach,
66
which the
would
have
philosophy
prejudices
at
between
to
the
of
'consistent'
most
is
the
we
ELEVEN
we
that
also
runs
lowlose
action
counter
energy
the
and
to
physical
correspondence extrema
now
comprises
the
the
equations
reason
in
certain can
a
of
as
very
of
for
the
made
general
very
cannot
be
of
this
discussion which
special.
It
the
theories,
which
criterion
being
adopting
exceptional
be
the
mathematical
theories
singles is
to
out
one
such
turn.
SUPERGRAVITY.
examine
field
would
compelling
framework
DIMENSIONAL
will
grounds,
imposing
truncations
have
Kaluza-Klein that
One
be
4-dimensional
the
approach
Thus
certain
physical to
action.
within
Section.
We
scale.
consistent
understood
on
premature
the
perhaps
remarkable
theory
is
Planck
extrema
However,
justified
it
higher-dimensional
3.
be
that
the
metric
8
bosonic
tensor
I ~ RAB . 2 . ~AB .
^VA 9ABCD _
sector
^ZMN
and
of
d=ll
supergravity ^ potential AMNP,
a 3-form
1 ^ ~BCD E . ~ .( FACDE
i
576
E
I ^~ 8 F ~ ~AB
BCDEI...E8 ?E
?E5
)
, which satisfying
(19)
'
...E8 ,
I...E4
(20)
A
where These one
=
admit
ground
sets
~e~y6 in
FABCD
=
3m~e~y6; can
necessary ~ABCD
~[A
ABCD]'
and
state
M7
is
we
are
using
local 9 solutions on
Freund-Rubin
components
all
spacetime One
4
A
of an
FABCD t o
Einstein
zero
space
except
Lorentz
indices.
M 4 x M7,
in
in spacetime,
satisfying
Rab
= 6m 2
which
where
6ab , and
M4, R ~ = - 1 2 m 2 ~ . easily
to
show
augment
including
the
that
the
already
elfbein
gauge
at
ansatz
bosons,
in
M 4.
correct
order
the
linearized
(4) to
and
(5)
extract
level
by
an
it
is
ansatz
on
the
massless
FABCD
turns
spin
A
1 degrees be
of
freedom
in
The
ansatz
for
out
to
'
~y6 Substituting
into
1
R8 where
- -- R 2 ~ TiJ~
^
= 3m g ~ y 6 ' (19)
-12m 2
is
defined
and
choosing 1
~
=
F~cd
AB
•
= -- T i3 e~ 2 by
(14).
This
!_
2m E ~ Y 6 = K ia
~8, Kj
F i~6
this
a
equation
Vc
Ki
d"
(21)
yields
+ -1 - 2 YaK i b yaK jb ) 2m should
be
compared
(22)
with
87 (12).
In
whose
this
case,
Killing
the
vectors
equation
Ki'a KJ'a + 1 2 2m It
is
(23),
easy
so
to
the
satisfy seems
to
than
be
no,
consistent
for
a
subgroup
G'
of
G
that
if
in
satisfies
as
For
just
'a
Ki
arises
round
yaK j'b = 6i'j'
to
a one
case
7-sphere,
more
Killing
Killing
has
only
is
satisfies
vectors
space
situation
28
i Ki, a -2
then
Einstein
the
all
(16)
whether
generic
(23)
M7
the
different.
vectors
of
can answer On
S0(8)
the
satisfy
. Of
course
the
above
components
of
(20)
certainly
would
the
the
ansatze a
~MN
that
could
isometry
G
in are
there
N=8
~
be all
7
on
S
, can
graviton,
28
gauge
scalars.
The
complexities
attempts
to
perform
a
the
gauge
since
all
of
be
of
spin
complete
the
whose 35
check
a 35 of scalars
truncation.
is needed
Generically,
one
of
of
and in
already
a
non-abelian
equation. d=ll
of
to
just
sector
scalars
have
so
yield
supergravity
truncated bosonic
d=ll
theories
the
each
and
far
the
comprises pseudo-
defeated
all
the consistency, but partial ii 12 ' . Note that the fact that
is another would
(19)
Kaluza-Klein
known
full
~
scalars
of
Einstein
0 sector
results in this direction are encouraging only
bosons other
the
has
known
consistently
and
the
other
that
multiplet, bosons,
of
Kaluza-Klein calculation
no
indications
supergravity
with
compactification
unique;
components
strong
compactified
massless the
the
the
concerned consistency
of
7-sphere
4-dimensions,
inconsistency
Full
However,
the
to
been
inclusion
AMN P.
yield
in
(19).
the
of
seems
In theory,
and
possibly
group
fact
equation
require
property
supergravity theory
calculation
Einstein
for
exhibited
an
Va Ki'b
(16).
but
SO(8~-invariant (23)
see
question
(23)
is
satisfy
remarkable
have
expected
property to
need
of
this
all
the
r e p r e s e n t a t i o n s occuring in the symmetric product of 28 with itself. The
full N=8 truncation
complications
has been intractable
due to the spin 0 fields.
to date because
A simpler problem,
e x h i b i t s some remarkable properties of the theory, to
N=3
fields)
supergravity and
then
(for
discard
which the
system,
point
consistency
fields
is
atall.
described
that The
in detail
with
calculations, in ref.
supermultiplet
fermions.
Einstein-Yang-Mills here
the
13.
SU(2)
One
should
be
which
is
contains
still
no
left with
group.
achievable
Restricting
of the still
is to first truncate
is then
gauge
which
The with
quite
to the
spin
just
0
the
remarkable no
scalar
involved,
appropriate
is
SU(2)
subgroup of S O ( 8 ) , it t u r n s out t h a t the a n s a t z (21) for ~ A B C D is correct to all orders, and one finds that (4), (5) and (21) yield an
68
exact
solution
arbitrary
of
(19)
solution
with
SU(2)
kind
of
of
gauge
and
the
group.
non-trivial
(20),
where
4-dimensional
There
embedding
(
g~v(x),
A
i
~(x))
Einstein-Yang-Mills
is
no
of
solutions
other
theory of
known
the
is
an
equations to
admit
this
Einstein-Yang-Mills
equations. A the
remarkable purely
feature
bosonic
of
this
subsector
SU(2)
truncation
depends
M..
eleven
dimensional
the
Lagrangian,
with
symmetry.
Thus
related,
although
4.
of
the
precisely
consistency the
"
the
and
precise
S
~
A
FM...N
Fp...Q
which
AR...
S
demanded
this
of
presence
in
the
term
in
A
coefficient
in
consistency
the
^
supersymmetry
way
that
upon
by
seem
to
works
remains
be
super-
intimately unclear.
CONCLUSION We
is
theory
is
crucially
have
seen
often
in
not
massless
section
possible
equations
consistency
is
example
to
of
that
in
make
four-dimensional
dimensional
the
2 to
fields of
section
2,
generic
ansatz and
motion.
reinstate
a
an
In
which
such
some
of
this
would
Kaluza-Klein which
the
previously
the
only
way
the
higher-
to
truncated
include
it
just
satisfies
cases
the
theory
extracts
restore
fields.
infinitely
many
In
massive
spin 2. Suppose, that
however,
one
should
inconsistent) thereby the
and
massless
that are
to
the be
of
the y(1)
of
make
(y)
to
relevant
mass
quadratic of
correct
course
in
fields
a
of
drastically
view it
is
action, just
is
merely the
to
make
a
alter
the
since
it
could ansatz
modification
the the
will add
a
not term
~(x,y)=¢(x) mode
will
Lagrangian
interaction
of
non-linear
a non-zero
four-dimensional
state
respects any
one
massless
ykI)~" is
a statement ground
harmonics one
wishes,
the
non-uniqueness.
around
example,
and such
of
that
one
For
zero-mode
effective
of
though
describing
zero-mode
free
that
Although
the
one
ansatz
times
is
right-hand-side is
is
Provided
masslessness.
operator.
terms
will
the yt0)" "
point
wrong?
M k. one
the (even
action
fluctuations
prescription of
take
higher-dimensional
a massless
on
this
the
problem
the
system,
criterion
where
go
spacetime
the
into
the
level
as
to
ansatz
four-dimensional
would
should
prepared massless
it
ref.3,
mass-operators
y(0)(y),"
In
one
expanded
modification affect 2 ¢ (x)
What
in
linearized
symmetries
were the
effective
fields.
that
at
relevant
it
an
discussed
Specifying
one take
substitute
obtaining
As
that
simply
of
leave
the the
unchanged,
terms.
Which
is
the
choice? the
truncation,
case
of
the
answer
a
Kaluza-Klein is
unambiguous:
theory the
admitting correct
choice
a
consistent is
the
one
69
that
ensures
course the
one
are
ansatz
has
the
fields.)
there the
the
still
massless
then of
that
simply
arbitrary,
freedom
However
is
interaction
no
in
i.e.
their
ansatz
one
chooses.
it
sense
to
the
to if
unique
terms
massless make
satisfies make
is of
Thus the
no
motion.
redefinitions
for
ansatz,
upon
amongst
and
non-linear
structure
so
most
Lagrangian
which
a consistent
(Of
truncation,
four-dimensional depend
only
of
consistent
massless
effective
coefficients
study
field
there
choice the
equations
particular
truncation of
a
does
Kaluza-Klein
theory.
ACKNOWLEDGEMENT I
would
K.S.
like
Stelle
to
for
thank
many
M.J.
helpful
Duff,
G.W.
Gibbons,
B.E.W.
Nilsson
and
discussions.
REFERENCES I)
J.A. Wheeler, in: Relativity, and C.M. deWitt (Gordon and
2)
S.W.
Hawking,
3)
M.J.
Duff,
4)
M.J. 149B
Duff, (1984)
5)
D.
Boulware
6)
N.
Manton,
7)
M.J.
8)
E.
9)
P.G.O.
i0)
M.J. Duff and C.N. Pope, S. Ferrara, J.G. Taylor Singapore, 1983).
ii)
B.E.W.
12)
B.
13)
C.N.
Duff
Nucl. B.E.W.
and
and B.
Freund
C.N.
and
Pope,
H.
Nieolai
Class.
B.S.
deWitt
349.
Pope, Pope
Phys.
and
Phys.
81
Report,
N.P.
in
Warner,
(1975)
print.
Phys.
Lett.
193.
NSF-ITP-83-04. Nucl.
and
M.A.
Phys.
C.N.
Ann.
Pope,
Julia
(1978)
C.N.
Deser,
preprint
Nilsson, Wit,
and
Nilsson,
S.
UCSB
B144
Nilsson
B.E.W. 90.
Cremmer,
de
Phys.
groups and topology, eds Breach, New York, 1964).
J.
Phys.
Scherk,
Rubin,
Phys.
B255
(1985)
Phys.
Lett.
Lett.
97B
in: Supersymmetry and and P. van Nieuwenhuizen
Lett. and
Quantum
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29.
CANONICAL QUANTIZATION AND COSMIC CENSORSHIP
P. Hajicek I n s t i t u t e for Theoretical Physics University of Bern Sidlerstrasse 5, CH-3012 Bern, Switzerland
I . Introduction
One of the most d i f f i c u l t
problems of quantum g r a v i t y originates from the w e l l -
known feature of the theory that the causal structure of spacetime - the system of l i g h t cones - is a function of the dynamical f i e l d i t s e l f .
Moreover, r e a l i s t i c models
of matter f i e l d s and g r a v i t y w i l l be unstable with respect to the gravitational collapse and formation of black holes. This means, in this context, that the deviations of causal structures of possible dynamical developments from each other, or from some standard structure l i k e e.g. that of the Minkowski spacetime, can be d r a s t i c a l l y large. (We have assumed here that black-hole-like objects e x i s t in nature, but this is very plausible due to cumulating observations - see, e.g.
[I]
.) By the way, such an as-
sumption could lead to some l i m i t s on r e a l i s t i c models for g r a v i t y : i t seems that the classical l i m i t of such models had to y i e l d more than j u s t the perturbation series for general r e l a t i v i t y around the f l a t background. Then, a l l such models, even with high d e r i v a t i v e s , with a compound graviton, supergravity, Kaluza-Klein, string theory, etc. w i l l suffer from the above problem. Most of the today investigations are based on the expansion in the number of loops. This means that the true l i g h t cones are approximated by the l i g h t cones of a given, fixed classical solution (corresponding, say, to the ground state). However, such an approximation is dangerous even in the purely classical theory - t h e series leading to divergent integrals in higher order contributions
12] .
Another way to avoid the problem seems to be offered by the Euclidean regime: the dynamical equations become e l l i p t i c
and there is no e x p l i c i t causal structure. How-
ever, the d i f f i c u l t y seems to reappear at a d i f f e r e n t l e v e l : the corresponding quantum theory becomes acausal
[4
, non-unitary
and leads to the loss of quantum coherence
[4] , not asymptotically complete
[5] ,
[4] .
Within the canonical quantization, the above problem takes on the following form. For any canonical formalism to work, we have to f o l i a t e the spacetime by Cauchy hyper-
71 surfaces. However, the existence of such a f o l i a t i o n , the so-called global hyperboli c i t y is a very special property of the causal structure. As the causal structure i t self is a function of the dynamical f i e l d , we do not know whether or not we are in conflict with the quantum dynamics, i f we require the global hyperbolicity from the outset. Let us explain by an example what is meant by a "conflict with the dynamics". First, i t is possible to weaken the global hyperbolicity somewhat. Let us consider only asymptotically f l a t spacetimes. Then, for the scattering problems to be welldefined, i t is sufficient that the part
[ I - ( ~ +) n
I + ( ~ ' - ) ] of each element M' from
a "large" class of such spacetimes is globally hyperbolic. This means, roughly speaking, that in each spacetime M' satisfying the requirement, there are no singularities which are visible from
3 + and at the same time influenceable from ~ - . Large class
means that the exceptions form a set of measure zero. I t is convenient to put the measure on the space of regular Cauchy data for asymptotically f l a t spacetimes. Then, the requirement is clearly closely related to the so-called Weak Cosmic Censorship [6]. In the classical form of this hypothesis, one assumes that the whole system of the classical f i e l d equations is satisfied by all spacetimes M'. Thus, a violation of the Weak Cosmic Censorship could be considered as a sort of conflict between the classical dynamics and the
weakened global hyperbolicity. Now, i t is well known that this form
of Cosmic Censorship is very l i k e l y to be violated
[7] .
However, the dynamics we consider is the quantum dynamics. The corresponding Quantum Cosmic Censorship Hypothesis (Q.C.C.H.) is not equivalent to the classical one. In fact, some people believe that the singularities of the classical general r e l a t i v i t y w i l l be avoided in the corresponding quantum theory. I f this is true, then the Q.C.C.H. w i l l be more l i k e l y to be satisfied than the classical one. We shall touch this problem in more detail later on. Another, more subjective d i f f i c u l t y with the existence of Cauchy hypersurfaces is that the globally hyperbolic spacetimes are "topologically dull" (as Hawking puts i t ) : no change of the space topology is possible (this i s , roughly, the content of a classical Geroch theorem
[ 6 ] ) . However, as f a r as I know, there is no proof that the
corresponding quantum dynamics w i l l also p r o h i b i t any change of space topology and s p l i t , in t h i s way, into the corresponding superselection sectors. Sometimes, one compares the general r e l a t i v i t y with the string theory. Any classical ( i . e . non-quantized) string i s , on one hand, a two-dimensional spacetime with well-defined dynamics. On the other hand, changes in topology of the time-constant folii
of the classical string manifold are possible (and even necessary in order that
there is any interaction between s t r i n g s ) . Why, so one asks, are these two facts compatible in the s t r i n g theory, and not compatible in the general r e l a t i v i t y ? The
72 answer is simple: the strings are able to join t h e i r ends to form a regular internal string point or to be torn into pieces with regular end points. No such discontinui t i e s are allowed for classical spacetimes.
2. The method of canonical reduction
Suppose we have some self-consistent f i e l d - t h e o r e t i c a l
model containing gravity
in the form of spacetime metric. Then, we can t r y to quantize i t by the so-called canonical reduction method (see, e.g.
[ 8 ] ). This method consists roughly in the
following steps. F i r s t , one adds some gauge conditions to the dynamical equations. The gauge conditions have the form of equations (mostly d i f f e r e n t i a l
equations) con-
taining the dynamical variables of the model. Then, the variables are divided into the following classes: the true dynamical variables, the dependent variables, the gauge variables and the Lagrange m u l t i p l i e r s .
The dynamical equations are divided into
gauge conditions, gauge propagating equations, constraints and the true dynamical equations, The gauge conditions, gauge propagating equations and constraints are solved for the dependent variables, gauge variables and Lagrange m u l t i p l i e r s ,
and the true
dynamical equations are expressed through the true dynamical variables only. In this way, the constraints and gauge freedom disappear and we obtain only mutually independent dynamical variables. Such a reduction can be performed within a Lagrangian (second order) or Hamiltonian ( f i r s t order) formulation and the reduced theory can be quantized in the standard way. In most cases, i t is impossible to perform this program e x p l i c i t l y solve a system of d i f f e r e n t i a l
(one has to
equations with arbitrary c o e f f i c i e n t functions). How-
ever, for our purposes, the abstract existence of solutions to these equations is sufficient.
Indeed, one can transform the resulting quantum theory to the form, which
is independent of a p a r t i c u l a r gauge condition, and which enables calculation of relevant physical quantities ( l i k e the S-matrix) without an e x p l i c i t reduction
[9, I0] .
For some theories, there is no gauge condition which works for the whole spacetime manifold and for the t o t a l i t y of possible f i e l d s (Gribov ambiguity, see, e.g. [II I ). However, what we r e a l l y want to do is the deparametrization of the system, that i s , only a partial reduction so that the gauge condition fixes j u s t the spacelike foliation.
This should always be possible, or else no reasonable dynamics would
exist. Within the reduction method, the problems with causal structure become even more numerous and involved. F i r s t , the family of hypersurfaces defined by the gauge condition can become degenerate (containing, e.g. intersection of the hypersurfaces)
73
or non-spacelike for some values of the dynamical f i e l d .
Even i f the f o l i a t i o n is
l o c a l l y regular and spacelike, i t need not represent, globally, a f o l i a t i o n by Cauchy hypersurfaces, i r r e s p e c t i v e l y whether or not the spacetime to be f o l i a t e d is globally hyperbolic. And, f i n a l l y ,
the spacetime need not be globally hyperbolic. However, we
have also more conditions on kinematically possible metrics: the gauge conditions, gauge propagating equations and constraints must hold before any dynamics is set up. We call these "predynamical equations". Let us, now, formulate all such predynamical assumptions more carefully. a) Let the spacetime mandifold (M~g) be I) smooth (C2), and 2) asymptotically Minkowskian, 3) causal, orientable and time-orientable. In many proofs, the requirement al) can be weakened. However, in this i n i t i a l
state
of investigation of these problems, i t is very comfortable. The assumption a2) means that (M',g) has a complete
~
[12] ; scattering problems can be formulated.
b) Let ~ = 0 be a gauge condition in the form of a ( d i f f e r e n t i a l )
equation for the
metric which (possibly supplied with a boundary condition) defines a regular spacel i k e f o l i a t i o n of some neighbourhood N of i ° in M' by asymptotically f l a t hypersurfaces. Let the corresponding time parameter t coincide asymptQtically with a proper time and has the bounds t ~ (- ~, ~). A f o l i a t i o n t ( x ) = const is regular and spacelike in N, i f the vector f i e l d t , i (normal to the hypersurfaces) is well-defined, continuous and timelike everywhere in N. c) Let there be some C> 0 such that the hypersurfaces t = t o for all t o < - C can be extended, as solutions of ~ = 0 in M', to form a regular spacelike f o l i a t i o n of some part Nl of M' by hypersurfaces, each of which is complete with respect to the positive definite metric induced on i t by the metric of (M',g). Thus, in the "remote past", the f o l i i represent a regular, i n f i n i t e , asymptotically f l a t space. I t can have a n o n - t r i v i a l topology and contain incoming extremal black holes (Hawking temperatureT = 0). This is the "assumption of regular i n i t i a l data". Let us define M to be the maximal connected neighbourhood of i ° in M' to which the hypersurfaces t = const can be extended, as solutions of ~ = O, to form a regular spacelike f o l i a t i o n . Thus, M contains N and NI . d) We assume that each hypersurface t = const in M is either complete or has a boundary in M' which coincides with an apparent horizon (AH). The content of assumption d) is twofold: l ) I t states that the kinematically possible fields are regular. Hence, the maximal extension of the f o l i a t i o n ~ = 0 leads either to complete hypersurfaces, or ends at points, which are regular points of M'. Such points can only be singular with respect to the f o l i a t i o n : points of intersection of
74 different t : const hypersurfaces, points, where the hypersurfaces cease to be spacel i k e , etc. 2) I t is a requirement on the gauge condition ~ = O: the singular points of the corresponding f o l i a t i o n must coincide with the AH of the spacetime (M',g). We shall see in the next section that this can, at least for some models, easily be done. This condition developed during the investigation from the attempt to f o l i a t e only the region which was outside of black holes. The AH is chosen because i t is locally well-defined. The assumption of regularity of kinematically possible f i e l d s , as contained in d) i s , on one hand, a very weak analog of assumptions which are usually done, i f one constructs the quantum dynamics of some f i e l d . For example, for the linear fields ~131 , one considers f i r s t some space of Cauchy data which can serve as test fields - they are C~ and, say, of compact support. Then, one finds some norm, or scalar product with respect to which the spaces can be completed to Banach, or Hilbert spaces. The n:orm, or at least the corresponding topology should be preserved by the dynamics in order that the dynamics of the more singular elements be well-defined. Thus, for a construction of quantum dynamics, i t seems necessary that the kinematicall y possible fields are dense in some suitable functional space. On the other hand, the assumptions c) and d) together remind us strongly on the weak cosmic censorship hypothesis. Of course, there are differences: for example, we require that the singul a r i t i e s are hidden beyond AH instead of beyond event horizons. The r e a l l y important difference i s , however, that we do not require the v a l i d i t y of the f u l l system of the classical dynamical equations. We require, f i r s t , just the predynamical equations and, second, the p o s s i b i l i t y to construct a reasonable quantum dynamics. Let us call the points c) and d) together with the assumption that one can construct a reasonable quantum dynamics with them, a Quantum Weak Cosmic Censorship Hypothesis. I t is clear that the quantum censorship could be true even i f the classical one is invalid. e) We assume, f i n a l l y , that the whole system of the predynamical equations holds in M. We shall call our f i e l d theoretical model for gravity to be completely foliable, i f the assumptions a) - e) imply the following properties of M: (i)
M is asymptotically Minkowskian,
( i i ) each hypersurface ~ = 0 in M is a Cauchy hypersurface for M. The gauge condition ~ can, then, be called a "complete f o l i a t i o n " . Thus, at the end, the singularities beyond the apparent horizon are not v i s i b l e from
75 3. BCMNmodel
The most simple known f i e l d theoretical model in which the dynamics can lead to a t r u l y nontrivial causal structure (namely the ~ormation of black hole horizons) is the Berger-Chitre-Moncrief-Nutku model [14~. We show in this section that the model is completely foliable. Let us f i r s t b r i e f l y introduce the model. I t results from the Einstein-Maxwell system to which an uncharged scalar f i e l d is minimally coupled. All dynamical degrees of freedom are frozen except for the spherically symmetrical ones. We have the following variables: a metric gab on a two-dimensional manifold M' (t-r-surface of the original four-dimensional spacetime), a real scalar f i e l d @on M' (r-coordinate), and the real scalar f i e l d ~ on M' (the original scalar). The action has the form:
(see [151 ). Here, G is the Newton constant, Q and P are the electric and the magnetic charges of the possible (incoming) black hole, g is the determinant of gab and R is the curvature scalar corresponding to gab" Let us choose the gauge condition
where na is a normal vector to the t = const surfaces, and supplement i t by the boundary condition at i n f i n i t y
(3)
}oo : The condition (3) specifies the time parameter t up to an additive constant. In
an asymptotically Minko~skian spacetime M',the f o l i a t i o n (2), (3) w i l l be spacelike near i ° and the assumption b) w i l l be satisfied. The t = const hypersurfaces in the maximal extension M with a regular spacelike f o l i a t i o n can only have AH as boundaries in M'. This is clear from the following considerations. The f o l i a t i o n is regular at a l l points, where the direction of na is uniquely determined by the condition (2), that i s , where @a@is a non-zero vector. Any c r i t i c a l point p of @satisfies
78
where ~a and ka are the two independent null directions at p. Thus, p is a future and past AH simultaneoulsy, or "double" AH (DAH). The f o l i a t i o n ceases to be spacelike, i f ~a~ becomes null, that is either at a future AH (FAH) or at a past AH (PAH). In
[161 , the following theorem has been shown.
Theorem l : I f the conditions a) - e) are satisfied, then M is asymptotically Minkowskian Let us consider the predynamical equations. I f we reduce the theory in i t s Lagrangian form, then the equations read as follows: the Hamiltonian constraint:
f~
the momentum constraint:
and the gauge propagating equations:
± ~x )_ (~,) = ~ I ( ~ C6) w'X (~-
0
•
,
Here, we have chosen the x coordinate to be
The following relations hold:
- ~
,
-
~
C
=
-9°°
÷
~£~
77 so ~ is the lapse, B the s h i f t function and ma i s the unit tangential vector to t = const hypersurfaces. I t has been shown in
[17!
that the three equations (4), (5) and (6) are equi-
valent to the following tensorial equation:
SI Thus, i t is e a s i l y transformable to any coordinate system in M; i f written out in a double null coordinates, eq. (7) implies the following Theorem 2: Let H be the future (past) h a l f of an outgoing (incoming) null hypersurface though a future (past) AH p. Let p l i e on the boundary of M and H inside of M. Then, the divergence of the null geodetic generators of H is non-positive (non-negative). Using the theorems 1 and 2, one can show that the FAH cannot be v i s i b l e from ~+ the the PAH cannot be influenced from ~ - , as well as that the world tube of an apparent horizon at the boundary of M is not timelike. These properties and the assumptions a) - e) imply the following Theorem 3: All hypersurfaces t = const in M are complete with respect to the induced n~tri c. From theorem 3 and the r e g u l a r i t y of the f o l i a t i o n in M, i t e a s i l y follows that the t = const are Cauchy hypersurfaces for M. Hence, the BCMNmodel is completely f o l i a b l e and the conditions (2), (3) represent a complete f o l i a t i o n .
4. Hawking e f f e c t , p o s i t i v i t y problem and u n i t a r i t y The picture which the complete f o l i a b i l i t y
of the BCMNmodel gives concerning
the kinematically possible t r a j e c t o r i e s is quite d i f f e r e n t from the current ideas about the expectation value of the n~tric in a spherically symmetric spacetime with a collapse. According to these ideas, a black hole horizon w i l l appear and the most of the information about the collapsing object w i l l be l o s t inside of i t . There w i l l also be a radiation going from the collapse region out to the i n f i n i t y - the so-called Hawking radiation. The o r i g i n of t h i s radiation w i l l be localized to a neighbourhood of the black hole horizon. The energy necessary for the Hawking radiation w i l l be taken d i r e c t l y from the black hole: a l o c a l l y defined current of negative energy w i l ] be pouring through the horizon from outside. Due to t h i s negative current, an apparent horizon can form outside the black hole horizon. Such an AH w i l l be v i s i b l e from ~ +, and i t s world tube w i l l be t i m e l i k e .
78 The crucial question in this respect is which properties of the kinematically possible trajectories can survive the quantization and can, in this way, appear as properties of the expectation value of the metric. Nali;ve]~ i t could seem that all such properties must survive, because the expectation value of the metric can be calf culated as a path integral average over the kinematically possible trajectories. However, this is not true in general. For example, the energy density of, say, KleinGordon field is everywhere non-negative for any kinematically possible trajectory but its expectation value can be negative at some points. In
[16] , this question has been discussed at some length. One possibility, due
to Ashtekar and Horowitz
[18] , is the following: i f a given general property of all
kinematically possible trajectories can be considered as a property of the configuration space of the system, then i t will survive the quantization. Such properties are, for example, the absence of AH in M (this is the absence of critical points of along t = const surfaces), or the fact that M is asymptotically Minkowskian. Another, more obvious possibility, is to look at the operators which represent the components of the metric in the quantum theory to see which sort of spacetime they are likely to yield. In our gauge, the component gll = I/y of the metric gives information whether the t = const surfaces are spacelike (gll > O) or null (gll = 0). The l a t t e r case would mean an AH. After the reduction, gll is a dependent variable, given by
[19] :
i b
where =
4-
G
is a positive function, and =
& with ~(y) being the canonical momentum and ~(y) the canonical coordinate (true dynamical variables), gll(x) is clearly positive classically, but can i t be made to an operator with positive spectrum ? Some subtraction procedure is necessary to define g l l ( x ) ; for example, i f this procedure can be applied directly to T(x) - T(y) (so that the exponential of i t is already well-defined), then gll(x) will be positive even i f T(x) - T(y) i t s e l f is not. I t is interesting to notice that the problem with positivity of gll(x) is closely related to the well-known "positivity problem" in the canonical quantization [ 8 ] :
79 A canonical coordinate Q which describes gravity (in two-dimensional theories, t h i s can be g l l ( x ) indeed) has a l i m i t e d range of i t s values. I f the corresponding operator Q s a t i s f i e s the canonical commutation rules (CCR) together with i t s conjugated momentum ~Q, then the spectrum of Q cannot be l i m i t e d . Two solutions to the problem have been proposed in the l i t e r a t u r e : I ) Choose other variable ( l i k e log Q) and i t s conjugate
[20] , or 2) use the pair
{Q~(Q~Q)} as the pair of variables whose CCR determine the operator algebra
[21] . A
t h i r d solution, possfble only within the reduction method, is to choose such a quantity as a dependent one. (This is analogous to what one does in ordinary quantum mechanics with the canonically conjugated pair {time, energy}). Another important point to discuss is the u n i t a r i t y of the resulting quantum theory. We have shown that the canonical qunatization is applicable to the BCMNmodel, because the relevant part of each kinematically possible spacetime can be f o l i a t e d by Cauchy hypersurfaces. The dynamical development of the quantum states from one such surface to another w i l l be unitary. However, t h i s relevant part M is not the whole spacetime in general. Thus, the f i n a l , t = ~, Cauchy hypersurface can contain a part of the boundary of M in M', e.g. an event horizon, say. I t i s , however, impossible to perform measurements along the event horizon, and one w i l l be interested to know the state only along that part of the t = ~ hypersurface which does not contain the horizons. Such a state w i l l ,
in general, be mixed, and we seem to lose the quantum co-
herence, and u n i t a r i t y , again. Here, the Hawking e f f e c t could, in f a c t , help. I f a l l energy of the collapsing object is radiated away again, then we can end up with a "clean" horizon, that i s , no horizon at a l l , or that one which has been present before the collapse (incoming hole). For t h i s to work, i t is necessary that the Hawking radiation carries away a l l information about the collapsing object. Such a transfer of information seems to be impossible according to the current ideas which l o c a l i z e the o r i g i n of Hawking radiation to a neighbourhood of the event horizon. There was another school of thinking about the Hawking e f f e c t
[22] , l e t us call
i t Boulware school, which localized the o r i g i n of Hawking radiation to the inside of the collapsing object. The energy of the radiation was taken d i r e c t l y from the object so that the horizon could never form. The calculations of the Boulware school, however, did not reveal any better information transfer than Hawking's. This was due to the assumption that a fixed classical background gravitational f i e l d was well-defined everywhere and that i t was the only source of the radiation; such a f i e l d had certain propert i e s which did not depend on the d e t a i l s of the collapse. The struggle about where the o r i g i n of Hawking radiation is to be localized has been won by the Hawking school a f t e r the consens has been achieved about the regularization and renormalization of the
80 stress-energy tensor: the expectation value, , of this tensor gives a l o c a l l y well-defined c-number energy current (see, e.g.
[23]
and the references given
there). Let me very b r i e f l y c r i t i z e these theories. F i r s t , such fine l o c a l i z a t i o n of the origin of the Hawking radiation that i t can distinguish between the inside of the collapsing object and a neighbourhood of the corresponding event horizon need not make any sense at a l l , in spite of the results about . Indeed, the a r b i t r a r i l y detailed l o c a l l y defined c-number energy current given by could have some physical r e a l i t y U~
only i f the corresponding mean squared deviation, <~T2 >, would be negligible with respect to . As far as I know, this was never shown. Second, i t is a n o n - t r i v i a l problem, whether the whole perturbation scheme used to calculate the Hawking e f f e c t is applicable. That is the semiclassical aproximation: f i r s t ,
one calculates the classical solution for the collapsing object and
the surrounding gravitational f i e l d ; second, one investigates the behaviour of small quantum disturbances around t h i s fixed classical background; f i n a l l y , one couples the classical metric to the expectation value of the stress-energy tensor of these disturbances. In our case, the classical background is unstable and the e f f e c t i t produces, is large. In t h i s way, my speculations about the information transfer in the Hawking effect need not be completely wrong and u n i t a r i t y could be saved. F i n a l l y , I would l i k e to stress that my c r i t i c s concern only the so-called dynamical Hawking e f f e c t and not the existence and properties of the thermal quantum states on the s t a t i c black hole background.
Acknowledgement: I am very indepted to R. Penrose and J. Hartle for important c r i t i c a l remarks.
81 References
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QUANTUM EFFECTS IN NON INERTIAL FRAMES AND QUANTUM COVARIANCE
Denis BERNARD Groupe d'Astrophysique Relativiste C.N.R.S. - Observatoire de Paris-Meudon 92195 Meudon Principal Cedex - France.
Abstract
:
We review recent results in non-inertial quantum field theory. By formulating Q.F.T. in a large class of accelerated frames, the classical and the quantum aspects of the theory are unified. We describe the thermal effects, their asymptotic character and the role of the P.C.T. symmetry. A discussion of quantum covariance and detection processes is also given.
0. INTRODUCTION
Quantum field theory in accelerated frames is a possible approach to the understanding of gravitational effects. Most of the known results about quantum field theory in curved space-time,
such as the Hawking radiation,
with non-inertial effects in flat space-time.
can be described by analogies
Therefore, non-inertial quantum field
theory makes possible the setting up of a "laboratory" for studying field quantization in curved space-time. However~
it is also a possible way to discuss non-iner-
tial detection process or a possible way to search for a quantum covariance principle. This paper reviews recent results about non-inertial quantum field theory and presents some new ones, too. In particular, we analyse thermal effects and their asymptotic character and we relate them to proposed quantum covariance laws. A critical discussion of detection processes and their link with quantum field theory in non-inertial frames is presented. The content of this paper is : I. Q.F.T.
in Rindler frame : the role of the P.C.T. symmetry
II. Q.F.T. in analytic accelerated frames III. The asymptotic character of thermal effects from a local principle IV. A hamiltonian formulation V. Vacuum fluctuation in accelerated frames
88
VI. Discussion.
I. QUANTUM FIELD THEORY IN RINDLER FRAME : THE ROLE OF THE P.C.T.
SYMMETRY.
We shall begin with the Davies-Unruh's
III result about the quantification of a sca-
lar field ~ in a uniformly accelerated
frame (Rindler frame).
There are many ways
based on Bogoliubov transformations, Green's functions, and others
121, to obtain
this famous result . But here, we want to present a global description where the role of the P.C.T.
symmetry is illustrated
13l. In particular this symmetry becomes
crucial for the analysis of the state identification proposed by t'Hooft. flat space-time,
law of the tetrad carrying by the observers. transport equation
--+
~.f_ + ~'n(r)
This equation is the Fermi-Walker
141 which reads :
Jet,r)
(1.1) where
First in
accelerated trajectories are completely describe by the transport
._._7 A n e. w
+
is the tetrad and
re(r)
O°
m~r)
-
0
the generator of this transport.
@~m
is a generator of Lorentz transformation and can be written as
@ = 4__ e
(1.2) where
L ~
L~#
w~h
E ~# +
= o
are the generators of Lorentz transformations.
duce the acceleration--~
and the rotation
~
It is useful to intro-
as the "electric" and "magnetic"
part of the antisymmetric tensor
(1.3)
-~
We are looking for accelerating trajectories the operator
~
the inertial ones
for which
~'E -~-
£ ~ > 0
so that
is a "good" generator of temporel evolution"
for any time-like vector . y ~ < ~ ting to introduce the non-inertial
where
"-" _.4 * E ~ # E ~# = -d._Cl 4
and
~
are the translation generators). ~t is interescoordinates
( -~p ~
by
J dr'
In terms of these coordinates,
=
~' the metric take the form
) defined with respect to
84
Then t h e h a m i l t o n i a n .A
(1.~)
Ht
-
"~ ~,__~_ ~
-~-
2
becomes simply the ~-time evolution generator
(1.~)
ILl _- _ i
It is generally supposed that these generators represent the observer's hamiltonian. For the sake of simplicity (and because we can always take it as an approximation at least during a small duration ~ ) , parallel,
we choose
~
~-independant.
For ~
and
the hamiltonian becomes
(1.9)
L. where ~
and
~
are respectively the boost-generator and the angular-momentum
in the direction i. Moreover, the previous non-inertial coordinates become the rotating Rindler coordinates
(I.i0)
p=fJ in cylindrical coordinate The accelerated coordinates
(-~ ~
~
Minkowski space. (see figure)
The region
~
=
~
E~
~I ~)
,
d
cover only a s u b m a n i f o l d ~ 7
~+
)X
>0
cation of "Rindler accelerated observers".
of the
, is the field of communiR-and
~*J
are the past and futur event-
horizons of these regions. The quantum particle states for this "observers" are chosen to be eigen-functions of
85
the hamiltonian
and we shall require that these wave functions vanish on ~ E or on ~ . Because m is the generator of the Lorentz transformation (1.5), the wave functions ¢£1"" satisfy the following transformation law
where
/~(~)is
the Lorentz
transformation
That property characterizes t h e f u n c t i o n plane-wave decomposition(l.13) 16,
~.l~,~) Use of (1.12), yields (i.14)
/(I~
[0 E~ ~
--~Ep--' b u t i t
~ C,2~)~ o ~
+
( ~ k + ~ -+ )m ' o -"~ M
~
i~- ~
~1
Therefore, and
+-
c a n be c h o s e n a s we c a n b u i l t ,
6~qcm
,~(~)
] G£, (~)
0
to i n t r o d u c e
a
; Ek= f d ~
~fter
~
=
-;
c
6~o(~)
),
ei°~
where m is the angular momentum ; ~ = and
simplest
differential equation for
(where ~ is the cylindrical angle of t whose solutions are (i.15)
is
~{~" ~) 6E,,
= a
(1.5).
!
q +m
= va~+ ~{ ,
normalization,
a wave f u n c t i o n
basis,
¢
6,1, m
,. which can be used to construct the Fock-space of the quantum field:
(1.16)
[in the discrete notation] + The operators of creation-annihilation,,__ OG~,rm and ~l~q Ir~ , define the vacuum state I0> : Q~,mlO~'- O Because the ~ , q , m have positive minkowskian-energy, this vacuum is the Minkowski one. Now, the region R~ is outside the field of communication of the accelerator "observers" inside R I. Therefore we would like to diagonalize the hamiltonian separatly inside the region R I and R~. Thanks to the P.C.T. symmetry, we can link i
86
the value of the wave function inside R I to that inside R]I. From (1.13) and (1.15) we get : :
~c(~,m
(1.17)
-[o,-
e
(In this region, the logarithm in the equation (1.15) has been defined on the halfupper complex plane). Since,
(1.18)
we have
and a similar relation for
,I
~ ~O
.
Therefore, the states
vanish
I
~'
~ae÷mD],"
.
-7~ &÷m£l..~----£±-x
'~,~,~
1
J-6,'l,-,~ J
in the region R ~ and are eigenfunctions of H.
Similarly, we define
(1.21)
Z
~,9,~
zl 6%-
which is the P.C.T. symmetric image of
T¢ ~j~
. The [ ~ vanish in the re-
gion R I and are eigenfunctions of H, too. The normalized wave functions i~ and ~ and their complex conjugates make up a wave function basis which defines the Rindler mode. The quantum field ~ reads
(1.23)
and from
"-- ~
(,q,m
(1 32),
I
3~C6j~119 16,q,m 4- ~[ £,q,m
"(J)~ = (~)-I(~1--/@
The creation-annihilation operators Rindler vacuum:
IO~
~ IC I O ~
where
@
6,~,m
]
is the antiunitary
~C6~c]jm and = ~CIO~> = O
-~hC = ~-~z C ~
P,C.T.operator.
define the
Because, the definition (1.20) mixes positive and negative frequencies, the Rindler vacuum is not equivalent to the minkowski-one.
The different creation-annihilation
operators are related by the Bogoliubov transformation
87
I
and similarly Therefore, modes
for
~C
.
the Minkowski vacuum
I05 contains Rindler modes.
The density of Rindler
:
d e s c r i b e s a P l a n e k i a n spectrum.
T=o/~'~
The a c c e l e r a t i o n
plays the r o l e
o f the t e m p e r a t u r e
and the rotation velocity appears as a chemical potential.
The unitary transformation
linking the Rindler mode to the Minkowski-one
can be
written as :
Io5 -- 1110
>
(1.26)
The pure Minkowski vacuum state contains pairs of Rindler modes.
(like the B.C.S.
state).
R I and another crea-
Each pair contains one "particle"
created in the region
ted outside the horizons ~
o But, if we restrict
whose support is restricted
to the region RI, it is better to introduce a density
matrix
~
ourselves
to observable,
~
say
by :
(1.27)
IO>
that is, by taking the trace over the states built from ~
tion of the observable
• Then, the expecta-
~+
, in the Minkowski vacuum takes the form :
A
And the density matrix
~
, describing a thermal mixed state, is
I
(1.29)
=
I n
where
I~)~,~,~>
=
(~!)~
(~q~m)I0~
This thermal character persists integral approach,
in the presence of interactions.
W. Unruh and N. Weiss
theory in a Rindler frame coincides, clidean Q.F.T.
are the n-Rindler mode states.
in an inertial frame.
By using a path
i51 have shown that a thermal quantum field
for the Hawking-Unruh
temperature,
with the eu-
88
Remark on electromagnetic
The description
of the accelerated
(4_~) illustrates, gravitationals
once more,
I.
effectsJ
electromagnetic
tensor.
trajectories
in terms of Lorentz generators
the analogie between classical electromagnetic
The tensor
E ~
becomes the analog
In particular all stationary
of
trajectories
(~)
like
and
times the
(such that 6 w ~
] can be found directly from the study of trajectories
is ~ - i n d e p e n d a n t electromagnetic
analogies.
in constant
(see ref.(6 bis) and ref. (25) for another derivation of
fields.
these trajectories). These analogies persist at the quantum level. Indeed, the Schwinger Lagrangian presence of an electric
in
field E (B = 0)
8-ir '~
~
<,.~E
6
-
can be written as
where
=
It is quite natural production and excitations.
to interpret 2ms as the excitation energy du to virtual particle
~'(S) as the spectral function describing
Then, we may define the temperature
the density of virtual
describing
the average excitation of
the ground state in presence of the field E by
T
-
.lm
_
eft
~]Crn This value is the analog
2. Q.F.T.
of the Unruh-Hawking
temperature
IN ANALYTIC ACCELERATED FRAMES.
The thermal effects analyzed ler rotating coordinate
in the previous
section are net restricted
(i.i0), but persist for a large class of accelerated
nates. Following on from the work of Sanchez j6j we generalize quantum field theories based on analytical mappings to the massive case in four dimensions We describe
or accelerated
/:z._
= F '
space-times
by including possible rotation or drifting.
the sub-manifold by the curvilinear
=
coordi-
17] the formulation
in two-dimensional
, defined by the transformation
(A. l)
to the Rind-
(B.1)
coordinates,
law : t
=
-~
Iz = z'+
-
of
89
where
~ , ~ ) E (~)~/ ~, ~ )
monotonic
are Minkowskian
function defined on •
parameters.
~
completely
an angular velocity and@ ~
their domain of communication.
(A and B.2)
U±---
The horizons
and~
drifting
coordinates
cover
~
:
but if
~
are finite the accelerated (Minkowski
coordina-
space-time)
the metric takes the form
~s~: _ A{~')C~'~J~ ~) . ~ ( J ~ ' ) ~ ÷ I
t
and the whole cinematic
or ~ i ~
a strictly
~(~oO)
tes only cover a limited region of total space-time
~)
~
associated with such a region
of F, the inverse mapping of
U+_ =-+ ~O , there is no horizon,
In such coordinates,
coordinates)
ensures that the accelerated
are defined by the singularities
If
~--
Such a transformation
cylindrical
I~
~p'~
i~
of the "observers"
inside
the static regions
:~')-~#~ A(~,~')
A(~I ~' ) is determined by the mapping.
However at the quantum level, the two statements it is necessary accelerated
to make clear the behaviour
coordinates.
To illustrate what precedes we limit ourselves
manifold with only one horizon the asymptotic
behaviour
Su=h a class of mappings
U~=+co
and to a mapping which has
) O
includes
describe uniformly-accelerated of wave funcions
and
: U_= O
the Rindler mappings
observers.
for global space-time
:
n --~(X~,)=
e
So as to be able to define a complet set
by (see figure)
s3~(~) L, : -~(u') and
2
--S'
which
from the wave functions defined in the sub-
manifold we must also prolong the transformation
(A.5)
to a sub-
: l ~(_oo)=
(A and B.4)
(i) and (3) are not sufficient and / of ~{ioo) so as to define a Q.F.T. in
(B.5)
90 Figure
The Fock space associated with the quantization
of a massive scalar field
built up from a bas~s of wave functions which are solutions
and which have a positive
Relative
"charge" defined by scalar product
, the Fock space is built upon the cyl~n-
to the global coordinates ~ff, ~)
drical waves of positive energy
and
~
is
of
:
is a Bessel function.
or upon the plane wave functions
e The creation-annihilation
With accelerated
operators
coordinates,
ted with accelerated
O~ I O~
define the global vacuum
we must define the quantum states which can be associa-
"observers"
in the region
~
. These wave functions do not
make up a complete basis for global space and thus are not sufficient Fock space. metry.
I0> :
to build a
In order to form a complete basis from these states we use the PCT sym-
The wave functions ~ ~
Cauchy data on se conditions,
-~" ~
relative to the region ~ T
whose support is included in are always null on
~-~-
il.
associated with a state
~
defined as
--~I~
are defined by certain --~----n~-~
. Under the-
(but not on F and P). Each
~ --
is
91
The
~
are null throughout the region R I.
Consequently, for ~ # ~ sufficient
for
~A
~
to constitute a complete basis for global space, it is
to be a complete basis for the class of wave functions which
possess null Cauchy data on l~j
~% ~
~
~ -- ~
ri~R~-
. This can be shown by decomposing
on the basis of the "Rindler states" defined in the previous section of
this paper. The Fock space is thus built upon the creation-annihilition operators and
t~t..~,
O~
and
relative to
~
and
j
~
C~
t C~
we have
[ ~1_, (l),] : 0
The operators
C~, Cll~ define the accelerated vacuum I0'2 c~:lo'> :
d~ t o / >
= 0
The PCT construction ensures that the theory in accelerated coordinates is completely determined by its formulation in the region R I. Indeed~we have
@J
where ~
is the anti-unitary PCT operator. The Bogoliubov transformation between
the two representations of the Fock space is written as
(l and B. 7 )
It is desirable to note that the canonical quantization is achieved first of all in the global space-time ~ .
Otherwise the operator PCT could not be built up. The
Bogoliubov transformation is simply the unitary transformation linking two choices of possible base states for the Fock space. In coordinates
('~--#j I~#)
the wave equation takes the form :
[-'~/+ "~,< wit,,
t~<~
Ac~',~')~ ~ ] ~A,c~',~') : o
<x'~,,c~,~),~ ~'),(s-~$
"< +
92
and
~@=
~+
~@
~-'[~'=~"~'~-4-~: +
and
~:
The asymptotic condition (5) implies that the effective mass
A[X~I~/) N 4
is null at
the horizons and that it is infinite at infinity so that no particle can escape. So we can choose as the base functions, the functions
i +: -
)
which satisfy =
V.
,~
UI--> +
u)
where explicitly :
The functions
~
are orthonormalized with respect to the scalar product o n R I.
Moreover, OO_-~j_~'~-~ in the first case and
6 0 : ~ _ ~ _ ~
in the second one
are interpreted as the asymptotic frequencies on the past-horizon. Another choice of the base states is possible, imposing asymptotic conditions on the future horizon so that :
V~+ee In the first case, the vacuum is denoted I0'
; in > whereas in the second the vacu-
um will be called I0' ; out >. In general, the non-stationary character makes the two vacuums inequivalent (only for the Rindler mapping is 10';in> = 0';out>). From here on, we write I0'> for 10';in> unless explicitly stated. With respect to the region RI, we note that, by construction, the states defined by d~
are not observable. The commutator,
[ ~Ej
~
~= O
expresses the absence of
a causal relationship between R I and R~. So, relative to the region RI, the pure state I0> which corresponds to the global vacuum is described by the density matrix obtained by tracing-out the states A
~
:
This matrix is completely determined by the population functions :
93
An e x p l i c i t
calculation
gives
=
Eb4k, oZ~:lu e
a
"~
+~'
_D,,V'iu) - -~ ,~'+ I,-, u
'AX' -o
CU - h~t+ ; E )e _ i~,,v (u)_.il,~v~L,9 e
.o.
a
~&&'
with
So the Bogoliubov sive case but
coefficients
~(~j~l)
~g$
and ~ ( ~ i )
and ~ ) ~
are not the same as in the non-mas-
are not dependent
on the mass as the asympto-
tic condition imposes a total redshift on the past horizon Thus it is the asymptotic behaviour which determines
(see dispersion relation).
the thermal properties.
Indeed
the results already obtained by N. Sanchez can be extended. p
i) The relation between the mapping
and
~(~, ~')
is reciprocal
and we can
invert the relation
du/L
a
where N l is defined by
¢A.9)
ii) The above relation makes it possible --4~(UI)=eX~(tltl/),-
~y~) we obtain
is the population :
to show that the Rindler mapping,
is the only one which satisfies
the global thermal balance
function for a unity of volume and, in the Rindler c a s e ,
94
(A. I0)
~¥(~)=
~
and (B. I0)
~/'¢(~) --
~
where'~=-~/~ and .~L~ ~ ) ~
/i@;~--(6~-~j2")
-- ~']
~"~
appears as the temperature play the role of chemical potentials.
iii) The thermic properties are defined by the asymptotic behaviour of the mapping. For an asymptotic Rindler mapping,
~(u')=e×~(~_U p)
when
LI/----'~ --4"
the population function behaves according to the law
Wil-~
=
andthere is a simple analogous expression for the case B. Here, the asymptotic temperature "~+
(A and B. II)
--~+ -
~
X ~-
can be written as
ILn~(~l)]I
f
Contrary to the previous case, there is no global thermal equilibrium but only an asymptotic thermal equilibrium in the region where the coordinates
and
tend
towards infinity. Moreover, in order to extend the analogy between the examination of the thermal properties linked to these mappings (but in flat space-time) and those that can exist in curved space-time,
it is useful to introduce the surface gravity'. ~ can be
defined by the ratio of the proper acceleration, a', to the temporal compenent,'1) ~j of the speed of the observers that follow the flux lines defined by the normals to the hypersurfaees, t' = constant.
H'I
Then the asymptotic temperatures are
='
I Vl= *
This relation can also be interpreted as a generalisation of the Unruh-Hawking temperature
T=o/~
for uniformly and linearly accelerated observers.
The asymptotic
character of the thermal effect, and the link between flat space-time and curved space-time effects are clearly shown. In particular, near the horizon of a Kerr black hole the transformation between the Kruskal coordinates coordinates
( II /
r~.-/S )
f
~)
(JI~ Vk. )
and the "tortoise"
95
is basically of type (i) :
with~L=~Li~
the angular velocity of the horizon of the black hole a n d ~
the sur-
face gravity of the Kerr-black-hole:
The Hawking temperature follows from this analogy. further.
In particular,
But the analogy cannot be pursued
the supperradiance effect cannot be reproduced as is shown
by the expression (~o~0)o~ ~{~l~.
If one wished to show schematically such an effect
with another mapping, better reflecting the properties of the Kerr metric~ tionary character would be lost ; the vacua equivalent.
the sta-
10';in> and 10'~out> are then no longer
In that case, it is no longer possible to distinguish the effects of
non-stationarity
from the effects of superradiance due to a difference between asym-
ptotic frequencies.
The same problems would present themselves if one wished to re-
establish the isotropy
: the stationary character is destroyed.
This previous study can he extended to mappings with non-constant rotation or drifting unless they becomes constant at the horizons.
Remark i. In a thermal equilibrium situation at a temperature T, we typically define the thermal average of an observable ~
, by computing the expectation of ~
rature T and by substracting its value at - ~ = O
. i.e.
at the tempe-
:
In this spirit, the natural definition of the average in an accelerated frame seems to be
In particular,
if
~
is the stress tensor in a two dimensional massless case, this
definition gives a renormalized stress-tensor which takes into account the energy carried by the "created particles" due to the acceleration.
[The meaning of this de-
finition is to give a "physical reality" to the created particles).
Namely,
lerated frames (u~v~ :
the stress tensor reads
T.,.,.
181
%;,=
v'J
for acce-
96
(fY 'f is
where
This stress-tensor mation.
Indeed,
the schwarzian derivative.
definition explicitely breaks covariance by coordinate
the choice of the renormalization
riant one because the accelerated vacuum can either abandon the definition
(~-~
prescription(~o~i)is
I0'> is frame dependent. and find a covariant
not a cova-
At this stage, we
one or, find a law
which tells us how must transform the vacuum by a frame transformation. sscial equation of the back reaction problem
gives us this transformation
transfor-
The semi-cla-
:
law. Explicitely,
this equation breaks up 191, in the
two dimensional
case, into a geometrical
the accelerated
frames to the vacuum states. This relation tell us how to transform
the vacuum by frame transformation ter of the renormalization
equation and into a set of equations
in order to compensate
the non-covariant
linking
charac-
scheme.
Remark 2. It will be observed
that our study yields a temperature T = o / ~
case, and not - ~ = O / ~
as t'Hooft suggested recently
in the Rindler
II01. This ambiguity
to the procedure adopted by t'Hooft for the definition of the associated the region R I. In order to define a quantum covariance to-one correspondance
between the global space ~
is due
states in
principle and to secure a one-
and the region RI, he identifies
the physics of the left region R I with that of the right region and, he defines a linear relation between a quantum state in ~
and a density matrix in R I. In order
to describes his proposal, we introduce the P.C.T. ce W E
associated
to the operators
a by,
to the Fock space ~
0 where
Then, to the state is associated
O
IV>
=
~---~ ~
I~
'
Recently,
and J O ~
Fock-
10>, the new density matrix
:
twice the standard one. But the hermitici-
for the density matrix restrict
re, we must restrict ourselves
invariant.
I~>
k><.;
is now, a thermal state with a temperature
~
~'--~ ~
of the choice of the basis in ~
""
density matrix
to the
operator.
~ I ~ > ~
:
It follows that for the Minkowski vacuum state
ty condition
associated
0
is the P.C.T. antiunitary
the density matrix
This relation is independant spaces.
C~
symmetry and we link the Fock-spa-
the~/~
by reality conditions.
to a real quantum mechanics.
More accurately,
is a hermitian operator only if the state I ~ >
Therefothe
is P.C.T.
We do ~ot know if such a real formalism has a physical meaning. an approach to this problem, based on the construction
functions has been given
Iiii. Even if after identification,
of symmetric wave
the period in the imagi-
97
nary time appears to be half the standard one, the resulting Q.F.T. does not have a finite temperature at all.
3. THE ASYMPTOTIC CHARACTER OF THERMAL EFFECTS FROM A LOCAL PRINCIPLE.
The previous study has shown the asymptotic character of the thermal effects. The temperature (8AI) depends only on the behavior of the mapping at the asymptotic regions of the space-time, in particular at the horizon. Recently, Haag and co-workers 1121 have deduced the Hawking temperature from a local principle, implemented outside and on the horizon. Contrary to the commutator ~(~j~l)= [ ~ ) p ~ t ~
which is
state-independant 1131 (in a globally hyperbolic space-time), the anticommutator function
G
(%~)=
(x))
(~q) is state-dependent. This function can be used in
order to implement a local criterion satisfied by the "physically allowed" states. Haag and co-workers have chosen to define their local principle on the tangent-space of the s p a c e - t i m e ~ Let ~
.
a map from the tangent-space - ~
at ~
to~
such that
The l o c a l
principle
becomes
:
lit can be shown to be independent of the choice of the m a p ~ . ~ This local critirion requires just that the singular part of the
G~ ) function
looks like the Minkowski-one. The aim of this principle is to show that the only thermal-equilibrium-state
(with
respect to the Rindler time ~ ) satisfying this critirion on the horizon has the Hawking temperature T = Q / ~
Indeed,zfor all observables A and B say, a thermal equi-
librium state with temperature ~ = ~ / ~
r
o
satisfies K.M.S. condition 1141.
=
t
where the ~
N~)=U(r)n U(r)
~ and L l ( r )
i s the evolution-operator with respect to
-time independent hamiltonian.
It is desirable to introduce the commutator IA,BI, and therefore to write the K.M.S. condition as
98
e~
In particular
, ~/[i~(-e),
e
for A and B being the field operator at different
In order to analyse the singular part of the
~{I)
Joa dr' points we get
function, we express the commuta-
tor function G in terms of the solutions of the Klein-Gordon equation in theRindler frame
(3.6)
K i._.q2.(ll~"1) QX~(x- ~l.~ l ) o.
;
~+ ~'~'
and we use :
4 ~=
(3.8)
~J:= ~ -+ Sz4"
It
~i+
~Z
o
We take :
is easy to show that the equation ( ~ )
outside the horizon allows any value
with the local principle
of the temperature,
part of the r~h.s, of (3.5) is independent
of ~
(3.2) imposed
because the Singular
. But on the horizon ~ ~ I _ _ ~ O
j
the behavior of the modified Hankel function becomes
(3.9)
Q
Then, the K.M.S. equation
(3.5) yields _ ~ s ~-~k~,~ -~
° 4-vc'~p (3.101
ko(N e) a
>
x
Thus the comparison with (3.2) on the horizon shows that the local principle tisfied only for the Hawking temperature
is sa-
99
For quantum fields in accelerated
frames,
the event horizon plays the role of a ther-
mostat and fixes the temperature.
Remark This local principle
is in fact based on the most singular term of the Hadamard
development
I15,161
which postulates
C311)
G (~,~')-
~
the expression
li(%~,)
V(~,~')~
at least in a small normal neighbourhood. geodesic distance between previous principle
~K
and
Where
1171. The currently
O'(~i~-l)
in
However, ~{~I)
#)
singulari-
)the unitarity
isll81
of the stress-tenso~
I
si.~ahr
operator~
only if the
G (4)
function possess a Hadamard development
it seems quite reasonalbe
and to require the asymptotic Hadamard With this asymption,
up to the
to extend this locality critirion
form up to all orders for the anticommutator
we are able to renormalize
cally defined by an expressions
il71 all quantities
classi-
like :
' ~...~q
•~
with an arbitrary number of derivatives
4. A HAMILTONIAN
But the
between diffe-
0)
t~)
x---~ ~ J
order two. Therefore,
A hamiltonian
F(9C~
favored methods of renormalisation
a differential
are applicable
function.
is one-half the square
if we impose the Hadamard and
I 0(.,I//~@
w,"(~,~'
~_i ) and U,V,W some smooth functions.
this critirion.
ty up to the order one in ~J/~
with
function as
is too local to ensure the unitarity equivalence
rents vacua satisfying
ensured
of the
and an arbitrary number of products
of fields.
FORMULATION
formulation
of these effects is crucial for the study of the Wheeler-
De Witt equation for quantum gravitational
fields.
The equation governing
the evolu-
tion of scalar field is built on from the action
by standard procedure.
ds
Since,
..-
the canonical m o m e n t u m S ,
in arbitrary coordinate
systems,
the metric reads
+
defined with respect to the time t
is :
100
3T'-
(4.~)
~ ~()~) _-_ JRi f°
and the hamiltonian density is
In particular,
the Legendre
The submanifold hypersurfaces
transformation
in which the Hamiltonian
( 4 - ~ ) becomes singular if N vanishes.
formulation
remains well defined is bounded by
locally defined by N = 0. They are always null-hypersurfaces
ping with causal nature of the field propagation. wave function ~
The SehrSdinger
in kee-
equation for the
is
(4.5)
~L
We shall compare the hamiltonian particular,
formulation
in Minkowski and Rindler frames and in
we analyse the ground states of the differents hamiltonians.
ke of simplicity,
we restrict ourselves
to the two dimensional massless
other cases are similar and can be found in the reference nate system is chosen to be ( ~ )
For the sacase. The
1191. The Rindler coordi-
:
op :
e
(4.6)
OZ
In a Minkowski
coordinate
~-+-~o function
: e Oy
s,~auj
COSk(Qr)
~e
:
e°Y
system (t,x), with Diricklet boundary conditions)~--Oat
(the case with Newmann boundary conditions ~
O~ ~=
is similar), we expand the wave
as
fi Taking
q~
as canonical variable,
the Schrodinger
equation
.,1,oo
The ground state of the Minkowski hamiltonian O
(4.,)
~. (%)=
For the Rindler case, we expand
t~
as
~rl
iS
can be written as :
101
(4.1o) Taking
Y = l a l ) Q~ i1"~ Q~
as canonical variable, the Schr~dinger equation
clz-
-~ 1'
~(%) = ~
%)
is
Sq~ Sq,
The ground state of the Rindler hamiltonian
i4.1.)
(~-
ex?
~
is
_$ dI) i~lqr%
These two ground states can be compared on the hypersurface ~ = ~ = O .
Inversing the
mode decomposition (4.7) and (4.10), yields the Bogoliubov transformaiton between the canonical variable I~
and 2 ~
:
% =
We way now substitute this relation into the expression (49) for the Minkowski ground state and we obtain
in terms of Rindler modes : -~OO
The two ground state appear clearly different. For high p.momentum, the structure of ~
approaches that of
~7~° . The difference between
~o
and
~
o
is signi-
ficant only for small p-momentum, reflecting the infrared (large distance) nature of the Hawking-Unruh effect. The average of the Rindler number operator for the p-th momentum mode, ~I~)
say, can be calculated directly by analogy with the simple har-
monic oscillator problem :
(4.16)
d e ~I?I/a
_
102
which is the familiar result. We thus see that the Rindler modes in the Minkowski ground state are populated in a thermal distribution.
But, by construction,
coherent state density matrix
the Minkowski state is described by a
1201 and not by mixed one.
Remark In a non-stationary metric,
the choice of the vacuum state as the ground state of
the hamiltonian becomes delicate. ciple (911).
In particular,
it does not satisfy the local prin-
Therefore it does not satisfy the unitary condition and it does not
possess a well defined renormalized stress-tensor normalization schemes
(at least within the standard re-
Ii81).
5. VACUUM FLUCTUATIONS IN ACCELERATED FRAMES.
Hawking radiation is currently interpreted as due to "creation of partlcles' ". T h i s interpretation, which appears naturally in the formulation based on a field decomposition on the Rindler modes,
is supported by a study of particle detection process.
On the other hand, Sciama and co-workers
121,221 have pointed out that this thermal
bath has its origin in the zero-point fluctuations of the quantum fields. The spectrum of the zero-point field energy appears to be distorted by the acceleration.
He-
re, following ref.(23), we define the particle density and the energy-density from conserved currents. For a massless scalar field, the standard density current reads as
Moreover,
if
~
is a Killing vector for the background space-time, ~
generates a
transformation which leaves the action invariant. The conserved Noether current associated to that invariance is
The orbits of the Killing vector can be identified with world lines, ~ ) some observers.
(5.3)
The normalized velocity vector is
~
We opt for the following definition of the density of particles density (e) seen by those observers
(5.4)
say, of
, (n) and the energy
103 where < > stands for the vacuum expectation value. It is convenient functions,
to express the vacuum expectation values in terms of the Wightman
W(~,~#=~¢{~)~(~}>and
to introduce the Fourier transform defined with res-
pect to the proper time along these world lines
---l-
(5.5)
/"
w
:
ioas
:] Is
Then, simple calculations
(5.7)
~f.~(~)
e--
Now, interpreting
give :
t~,~
~J
&O _ _
~/{~).I.W(~/
as the frequency
measured
(with respect to the proper time
) by a detector moving along these trajectories, the particle and energy densities, For inertial observers,
~-j~)
and
become
respectively.
we get :
(5.8)
-
On the otherhand,
for Rindler observers,
we have
:
+
(5.9)
where
~-~
O~o ~
is the (local) acceleration
The particles densities
of these trajectories.
are equal in both cases and express that there is "one parti-
cle" in each phase-space additional
cell. But the zero-point
Planckian term. This is the distortion
acceleration
~v~/~o~ e _7
in agreement with the interpretation
energy has, in the Rindler case, an of the zero-point
energy due to the
of Boyer, Sciama and others.
DISCUSSION
It is quite surprising
to note that, despite of its asymptOtic
effect admits a local description via the detector models
character,
has its origin in the high degree of symmetry of the Rindler accelerated lines, ~ i =
constant,
coincide with the world-lines
ted observers and furthermore, locally the Fermi-Walker ries.
For this reason,
the Unruh
1241. This special feature frame. The
of a system of uniformly accelera-
every where in the region RI, this coordinate
coordinate
system associated with these hyperbolic
system is trajecto-
the Rindler frame appears as the most adapted one to these
104 trajectories.
But in general,
te system which, every where,
for a given f~ow of trajectories
there is no coordina-
is locally the Fermi-Walker system associated with them.
(The Rindler frame is the unique one which has this property).
Therefore,
there does
not exist a coordinate system "naturally" adapted to the flow of trajectories. feature has some consequences in non-inertial quantum field theory. J. Letaw and J. Pfautsch
This
In particular,
125,261 have studied the link between the canonical formu-
lation of Q.F.T. in accelerated frames and the models of quantum detection processes by non-inertial observers responds
: A rotating detector plunged into the Minkowskian vacuum
(the spectrum of the excitations has no simple expression but does not va-
nish) whereas the rotating vacuum defined by a
mode decomposition in a rotating fra-
me is equivalent to the Minkowski one. Here, the choice of the non-inertial rotating frame has been criticized as being highly non-adapted
1271. But, except in the Rindler
case, even if it is possible to consider a single point-like detector by the use of the Fermi-Walker coordinate system associated with it, it is impossible to take into account the finite size of the detector.
( In particular, we do not know the effect
of the acceleration on the internal hamiltonian of this detector). now, the link between models of detection process and Q.F.T. is not really etablished.
Therefore, up to
in accelerated frames
The study of non-inertial quantum field theory seems better
adapted to analyse the link between the formulations of quantum theory in a global manifold and in a submanifold respectively, with direct consequences on quantum gravitational effects (section Z and vacuum covariance).
The problem of quantum detec-
tion is an old one. We would like to point-out some new problems which appear from the non-inertial character.
First, if we would like to represent the quantum measu-
rement by an observable, without describing the detection process, we must find a) which element of the observable algebra represents an ideal detection porcess ? b) A transformation law which tells us how this observable is modified when the same detector is forced to move along some other world-line
?
Up to now, there is no "quantum covariance principle" that answers these questions. Therefore, people looked for models of detection processes.
Thus to ensure that the
internal hamiltonian of the detector is the same along any trajectories, ly studied detectors were based on point-like monopoles. tion of the detectors,
the current-
By studing the reponse func-
these models of detectors propose to analyse the "effective
particle content" of the quatnum state seen along the detector trajectory. Recently, Hinton
128[ and Davies
1291 have made important remarks concerning these models.
i) How to normalize the function reponse,
in particular in curved space-time
ii) Do different models of detectors give the same "effective particle content" to the same quantum state. Their studies indicates that the "effective particle content" is detector model dependant. Therefore,
these questions are still open : What is really measured by these
detection ? or to which measurements do the detection process correspond ? It would be interesting to discuss the problem of anisotropy in the detection of the acceleration radiation in connection with these questions
1301. A way to avoid these problems
105
is to go back to the first point of view, and to search to describe measurement intrinsic quantities
like
,
~
case, the hyperbolic
trajectories
or other vector densities.
are generated by a Killing vector and therefore
Noether current defined in section V is "naturally" measurement.
But in general,
for an arbitrary
and therefore no current is intrinsically b). Formulation
by
In the Kindler
adapted to represent
trajectory,
the
the quantum
such symmetry does not exist
defined that answers the questions a) and
of a quantum covariance principle as expressed by the previous
ques-
tions is still an open problem.
ACKNOWLEDGMENTS
I am grateful
to Norma Sanchez
I acknowledge
Brandon Carter for numerous
for numerous
discussions,
stimulating
advice and encouragments.
discussions
and for a critical
reading of the draft manuscript.
REFERENCES
I.
S.W. Hawking, Com.Math. Phys., 43, 199, (1975). P.C.W. Davies, J.Phys., A8, 609 (1975). W.G. Unruh, Phys.Rev. DI4, 870 (1976). S.A. Fulling, PHys.Rev. D7, 2850 (1973). 2. H. Rumpf, Phys. Rev. D28, 2946 (1983). S.M. Christensen and M.J. Duff, Nucl. Phys. B146, ii (1978). P. Candelas and D. Deutsch, ProC.Roy. Soc. A362, 251 (1978) and Proc. Roy. Soe. A354, 79 (1977). 3. R.J. Hughes, Preprint CERN T.H. 3670 (1983). G.L. Sewell, Ann. Phys.N.Y., 141, 201, (1982). 4. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation,(Freeman, SanFrancisco 1973). 5. W.G. Unruh and N. Weiss, Phys.Rev. D29, 1656 (1984). 6.6. C. Itzykson and J.B. Zuber, Quantum field Theory, (Mc Graw Hill, N.Y., 1980). 6. N. S~nchez, Phys.Rev. D24, 2100 (1981). 7. D. Bernard and N. Sdnchez, preprint in preparation. 8. ~ W . Davies, Proc. Roy. Soc. Lond, A354, 529 (1977). 9. R. Balbinot and R. Horeanini~ Phys.Lett., 151B, 401 (1985). N. Sa~chez, to appear in Nucl. Phys. B. i0. G.t'Hooft, J. Geometry and PHys., i, 45 (1984). G. t'Hooft, Utrecht Preprint, December (1984). Ii. G. Gibbons, in Carg~se Lectures 1985, to appear. N. S~nchez and B. Whiting, in preparation. 12. K. Haag, H. Narnhofer and U. Stein, Com.Math. Phys., 94, 219 (1984). 13. Lichnerowitz, in Les Houches, 1963, edited by C. de Witt and B.S. de Witt, Gordon and Beach. 14. R. Kubo, J.Phys.Soc. Japan, 12, 570 (1957). P.C. Martin and J. Schwinger, Phys.Rev. 115, 1342 (1969). 15. See, for example, F.G. Friedlander, The wave equation on a curved spacetime, (Cambridge University Press, Cambridge, 1975). 16. S.A. Pulling, F.J. Narcowich, R.M. Wald, Ann. Phys. N.Y., 136, 243 (1981). 17. D. Bernard, Meudon Preprint, 1985. 18. N.D. Birrel and P.C.W. Davies, Quantum fields in curved space (Cambridge University Press, Cambridge, 1982).
106
19. 20. 21. 22. 23. 24.
25. 26. 27. 28. 29. 30.
M.R. Brown and A.C. Ottewill, Proc.Roy. Soc.Lond. A389, 379 (1983). K. Freese, C.T. Hill and R. Mueller, Nucl. Phys. B , (1985). T.D. Lee, Columbia University Preprint (1985). D.W. Sciama, P. Candelas and D. Deutsch, Adv. Phys. 30, 327 (1981). T.H. Boyer, Phys.Rev. D2!, 2137 (1980). S. Hacyan et al., Phys. Rev. D32, 914 (1985). W.G. Unruh, Phys.Rev. DI4, 870 (1976). B.S. de Witt, in General Relativity : an Einstein centenary survey, edited by S.W. Hawking and W. Israel, (Cambridge University Press, Cambridge, 19 ). W.G. Unruh and R.M. Wald, Phys.Rev. D29, 1043 (1984). J.R. Letaw, Phys.Rev. D23, 1709 (1981). J.R. Letaw and J.D. Pfautsch, Phys.Rev. D24, 1491 (1982). N. Myhrwold, Phys. Lett., 100A, 345 ( 1 9 8 ~ . K.J. Hinton, J. Phys. A : Math.Gen., 16, 1937 (1983). K.J. Hinton, Class. Quantum Gray., i, 27, (1984). P.C.W. Davies, in Essays in Honor of the Sixtieth Birthday of B.S. deWitt ; edi ted by S. Christensen. (Adam Hilger, Bristol~ 1984). N. S~nchez, to appear in Phys.Lett.A K. Hinton, P.C.W. Davies, J. Pfautsch, Phys. Lett. 120B, 88 (1983). W. Israel, J.M. Nester, Phys.Lett., 98A, 329 (1983).
STOCHASTIC DE SITTER (INFLATIONARY) STAGE IN THE EARLYUNIVERSE
A.A. STAROBINSKY Landau Institute for Theoretical Thysics, Moscow, 117334, U.S.S.R. and ER 176 C.N.R.S. "D~partement d'Astrophysique Fondamentale" Observatoire de Meudon 92195 Meudon Principal Cedex FRANCE
Abstract The dynamics of a large-scale quasi-homogeneous scalar f i e l d producing the de Sitter ( i n f l a t i o n a r y ) stage in the early universe is strongly affected by small-scale quantum fluctuations of the same scalar f i e l d and, in this way, becomes stochastic. The evolution of the corresponding large-scale space-time metric follows that of the scalar f i e l d and is stochastic also. The Fokker-Planck equation for the evolution of the large-scale scalar f i e l d is obtained and solved for an arbitrary scalar f i e l d potential. The average duration of the de-Sitter stage in the new inflationary scenario is calculated (only partial results on this problem were known e a r l i e r ) . Applications of the developed formalism to the chaotic inflationary scenario and to quantum i n f l a t i o n are considered. In these cases, the main unsolved problem lies in i n i t i a l pre-inflationary conditions.
1. Introduction In the models of the early universe with an i n i t i a l or intermediate metastable de S i t t e r ( i n f l a t i o n a r y ) stage with an effective cosmological constant produced both by quantum gravitational corrections to the Einstein equations 111 and by a scalar field
12-4 , of extreme importance is the e x i t from this stage that depends on the
way of decay of the effective cosmological constant because i t determines the spec-
108
trum and amplitude of metric perturbations for the subsequent evolution. These perturbations break the homogeneity and isotropy achieved earlier at the inflationary stage and can, in the worst case, destroy all the advantages of i n f l a t i o n . Two ways of decay of the effective cosmological constant are possible : via (quasi) homogeneous classical i n s t a b i l i t y and via inhomogeneous quantum fluctuations. In the f i r s t case, the amplitude of perturbations of the de Sitter space-time in the modes which preserve (exactly or approximately) the isotropy and homogeneity of the 3-space in some frame of reference is much more than the amplitude of other, inhomogeneous perturbations. Thus, we have a classical (quasi)-homogeneous perturbation from the very beginning and the subsequent evolution is deterministic ; the duration of the de S i t t e r stage is t o t a l l y determined by the i n i t i a l amplitude of this perturbation. This type of decay takes place, for example, in the author's model i l t for the case of the closed 3-space section i f the spatial dimension of this section was of the order of
H "4 at the beginning of the de Sitter stage (in the paper, we put
= c = 1 ; a(t) is the scale factor of the Friedmann-Robertson-Walker isotropic cosmological model ;
~ = J/a).
The existence of a quasi-homogeneous classical scalar f i e l d is also assumed in the "chaotic" inflationary scenario 151 (for the inclusion of the R2 term where R is the Ricci scalar, see 161). Here, the term "chaotic" simply means the unspecified dependence of the metric and the scalar f i e l d on space coordinates though this dependence is weak enough, so that the spatial derivatives of all variables are much less than the temporal ones. In the second case, we have no large (quasi) homogeneous perturbation at the beginning of the de Sitter stage. This p o s s i b i l i t y was f i r s t pointed in 171 in connection with the model i l i .
But, in fact, this situation is more typical for the models whe-
re the de Sitter stage arises from the i n i t i a l l y radiation-dominated, "hot" universe in the course of a non-equilibrium, close to the ~ order phase transition (for example, the "new" inflationary scenario). Here, nevertheless, a large quasi-homogeneous "classical" perturbation with characteristic wavelengths >>H"1 can arise during the de Sitter stage from small-scale quantum perturbations. In
other words, "classi-
cal order" appears from "quantum chaos". In spite of being effectively classical, the evolution of this large-scale perturbation and the space-time metric as a whole is essentially stochastic. The duration of the de Sitter stage also becomes a stochastic quantity in this case. This is just the process we are interested in. I t belongs to the class of the so-called "synenergetic" problems which arise in different branches of science and attract much interest at the present time. We shall consider the new inflationary scenario where the role of the abovementioned perturbation is played by the non-zero largescale scalar f i e l d
~) . I t is assumed that
~
~
0 (or s u f f i c i e n t l y small) at the
beginning of the de Sitter stage. We shall obtain the Fokker-Planck equation for the evolution of the probability distribution of
~
(Sec.2) and calculate the average
duration of the de S i t t e r stage in the new inflationary scenario in Sec.3 (only par-
109 tial
results on t h i s problem or order-of-magnitude estimates were obtained e a r l i e r
18-101). A f t e r t h a t , we shall turn to the chaotic i n f l a t i o n a r y scenario (Sec.4) and discuss the modern state of the Problem of the "creation" of the universe b r i e f l y
(Sec.5).
2. Evolution of a scalar f i e l d in the new i n f l a t i o n a r y scenario. the new i n f l a t i o n a r y scenario is assumed to be produced by
The de S i t t e r stage in
the vacuum energy of some scalar f i e l d with the Lagrangian density (i)
2, where the vacuum e f f e c t i v e potential
v(@.) :
V~)
has the f o l l o w i n g properties :
o I M,~ ~ _±v
M2 can have both signes. ~_.- g o
4 3- ~
(2)
~
is the flat space-time (true vacuum). ~ - 0
the f a l s e vacuum. We include the term in
is
to describe the case of the s o - c a l l e d
"primordial" i n f l a t i o n I l l l simultaneously. At the non-zero temperature T, the pot e n t i a l V acquires the additional thermal term which is either small or, with the
sufficient accuracy, has the form ~-B'I "L ~ ' ,
B(41, T:~a -I.
At the de Sitter stage, H = Ho = const, a = ao exp (Hot), where ~o~= ~a~C~Vo/.~ (the spatial curvature is negligable). In order to have enough long de Sitter stage and enough small perturbations at the subsequent stages, the following conditions should be f u l f i l l e d : ?.
IMP! ~ H°/Zo
;
v/t.t. ~
to"
;
~
~o'"
(3)
The Coleman-Weinberg potential does not e v i d e n t l y meet these requirements, so i t is usually assumed now that
~
is some weakly i n t e r a c t i n g scalar f i e l d ,
in particulaG
i t should be the s i n g l e t with respect to SU(5) or any other grand u n i f i c a t i o n group. In such a way, the s p i r i t ,
though not the l e t t e r , of the "new" i n f l a t i o n a r y scenario
is maintained. The de S i t t e r stage begins when T4~V o, I t can be divided into two successive periods: "hot" and "cold" (vacuum). During the hot period, the temperature T ~H o and quantumg r a v i t a t i o n a l effects caused by the space-time curvature are unimportant. The duration of t h i s period is rather short ; in dimensionless u n i t s ,
~.
,, ~
,,,
z,, ( v~l, 1 H. ) .,. ~
( c, "~ V." '/~ ),
( ~)
110 that
is of the order of 10 t y p i c a l l y .
After that, the cold (vacuum) period begins
where T << Hoand, in fact, temperature effects can be neglected (except only for the calculation of the i n i t i a l dispersion of ~ ; see Eq.(13) below). This period is the most interesting because quantum-gravitational effects connected with the spacetime curvature play the decisive role here (we denote its beginning by to). To obtain q u a n t i t a t i v e l y
(not only
qualitatively)
correct
results one should not
use such quantities as < ~l~ > or < ~ 2 > (the approaches based on these quantities have been correctly c r i t i c i z e d in [12, tum scalar f i e l d
k
Here, ~ ' ( t ,
~)
Instead of t h i s , we represent the quan-
(]~ (the Heisenberg operator) in the form :
oct)
,
=
131),
:
t
't
,
~)o
&:o~Y:.
contains only long wavelength modes with k<< Hoa(t),
4< :£ .
~(~
is the
small correction that can be neglected in the leading order in small parameters NM21/Ho2,
~/H o, ~ and the second integral term in Eq. (5) satisfies the free
massless scalar wave equation in the de S i t t e r background : [ ~ =
O. Thus,
(6) and ~
and a% are the usual creation and annihilation Bose-operators. The a u x i l i a r y
small parameter 6~ is introduced to refine the derivation, i t w i l l not appear in all final equations. ~n fact, i t cannot be a r b i t r a r i l y
small ; the immediate comparison
of d i f f e r e n t terms in Eq. (5) suggests that ~ >> IMI/H o but more refined treatment consisting in the substitution of the solution (6) by the solution of the free massive wave equation O ~
+ M2~
= 0 in the de S i t t e r background (that does not
change Eq. (8) below in the leading approximation in [M21/Ho2) shows that the signif i c a n t l y weaker condition i -~n& I << max (Ho2/M2 , Ho/P , ~ - i ) is s u f f i c i e n t . I t can be also seen immediately that the account of the abovementioned thermal correction to V ( ~ ) results in the substitution
(7) in Eq. (6). This gives an effective infrared c u t - o f f that can be important in some problems. The scalar f i e l d
~
satisfies the operator equation of motion [ ~ ' +
dV /d~ = 0
111 exactly. Using (5, 6) and the conditions of "slow r o l l i n g " (3), one obtains the following equation f o r ~
in the leading order :
a Wo
oL
(zn')#"
(8)
,,
+
That is the main point : the large-scale scalar f i e l d ~l~ changes not only due to the classical force d V ( ~ ) / d ~
but also due to the flow of i n i t i a l l y
small-scale
quantum fluctuations across the de S i t t e r horizon k = a(t)H o in the process of expansion. Moreover, the evolution of inhomogeneous fluctuations is l i n e a r inside the de S i t t e r horizon and even in some region outside i t of ~
; on the other hand, the evolution
is non-linear but here the spatial and second time derivatives o f ~
are small.
Below, we shall omit the bar above ---~ , so ~ w i l l mean the large-scale f i e l d only. Two important consequences follow from Eq. (8). F i r s t l y , there are no spatial derivatives in Eq (8) at a l l . This means that the evolution of ~ can be studied l o c a l l y , in the "point" ( t h i s "point" has, in f a c t , spatial dimension ~ temporal evolution o f ~ ) i s
Ho" I ) .
The
slow as compared to Ho-I ( i f the i n f l a t i o n exists at a l l ) ,
so our time " d i f f e r e n t i a l " dt can be also chosen .,, Ho-1 ; only the processes with characteristic times ~>> Ho-i w i l l be considered. Secondly, though
~
and f have
a complicated operator structure, i t can be immediately seen that a l l terms in Eq. (8) commute with each other because ~k and ~k+ appear only in one combination for each possible ~ !
Thus, we can consider ~)and f as c l a s s i c a l , c-number quantities.
But they are c e r t a i n l y stochastic, simply because we can not ascribe any d e f i n i t e numerical value to the c o m b i n a t i o n [ ~ a r e s u l t , the
~:C.~(-~
~ ) - ( ~ ~ ) ] .
As
peculiar properties of the de S i t t e r space-time - t h e existence of the
horizon and the appearance of the large " f r i c t i o n " term 3Ho~
in the wave equation-
s i m p l i f y the problem of a non-equilibrium phase t r a n s i t i o n greatly and make i t s solution possible, in contrast to the case of the f l a t space-time. I t is clear now that Eq. (8) can be considered as the Langevin equation f o r ~ b ( t ) with the stochastic force f ( t ) .
The calculation of the correlation function for f ( t )
is straighforward and gives ( ~ i s
<
Thus f ( t )
=
the same throughout) :
Ho3
(9)
has the properties of white noise. This appears to be the case because
d i f f e r e n t moments of time correspond to d i f f e r e n t k because of the ~ -function
112
in the definition of f, and ~k and ak+ with different'~commute. separated points,
<
For spatially
> :
e.,~., 14.0 I~-~1
"(10)
We are interested in the average values where F is an arbitrary function. For that case, one can introduce the normalized probability distribution ~ ( ~ for the classical stochastic quantity
=
•
)
,t)
(~
so that (too
~o,O
By the standard procedure, the Fokker-Planck (or, better to say, EinsteinSmoluchowski) equation for ~ follows from (8) and (9) :
This equation has to be supplemented by some initial condition for ~ at t = t o. It should be noted also that Eq. (12) is applicable at the stage of "slow rolling" (I ~ I << Ha ~ ) only. When this condition ceases to be valid ~hat takes place at ~min (Ho ~ -1/2 , Ha2-}) -1)), the second time derivative o f ~ comes into play (though spatial derivatives are s t i l l unimportant), the de Sitter stage ends and ~ reaches its f l a t space-time equilibrium value ~ o during the time interval less than Ho-1. After that, a number of oscillations around ~o is possible. Thus, strictly speaking, we can use Eq. (12) i f only i ~l<<min (Ho ~-1/2,Ho2 ~-1). But just because we are not interested in time intervals ~ t A_sHo-I when calculating such quantities, as e.g., the average duration of the de Sitter stage, we can safely substitute V ( ( ~ ) in Eqs. (8,12) by its expansion for I~LT~I << ~ o (the second line in Eq. (2))and use Eqs. (8, 12) for arbitrary ~ . Then the (stochastic) moment of time t s when the de Sitter stage ends coincides with the sufficient accuracy ( ~ t ~ Ha-l) with the moment when i ~ l reaches infinity according to Eq. (8) (the stochastic force f ( t ) becomes unimportant at the last stage of evolution). A note should be added about time reversibility. The microscopic evolution of the total scalar field operator ~) is, certainly, time-reversible, so the apparent, diffusion-like irreversibility of the evolution of ~ ( ~ ,t) is due to, as usually, "coarse-graining" that takes place continuously in the process of neglecting more and more information contained in separate modes with different k.
113 3.
Average duration of the de Sitter stage in the new inflationary scenario.
Now we have to introduce the i n i t i a l
condition for
part of the i n f l a t i o n : ~ : I o ( ~ a t
t = t O. The simplest possible choice would be
~ )
~
at the beginning of the "cold"
te(~)
:
• In f a c t , the situation is more complicated and depends on the
initial
conditions at the Planckian moment tip = G½. I f one assumes thermal e q u i l i -
brium before the de S i t t e r stage, then the contribution of thermal quanta of the scalar f i e l d
with the rest mass m2(T) << T2 (B<<1) to
~
~ e ( ~ ) is gaussian with
the dispersion
<
: L
el,)
IT~
o
At T <
a r t ~"
(.k'+
i ~ : ) :'
This expression is valid i f the modes with k~, ko are inside the horizon at the beginning of the de S i t t e r stage that requires
B>>GV~vHo/Mp , where M@= G-½ is
the
Planck mass. In the opposite case, the modes with k < kI = Ho a(t=Ho-1) are never inside the horizon. For these modes, ~0~., const, and, in fact, nothing definite can be said about their occupation numbers. The probability distribution needs not be gaussian either, but i t is independent of time (we do not include the term R~2/12 2 into the Lagrangian (1) because then the fine-tuning between M2 and Ho is necessary for the i n f l a t i o n to occur). In this case, the reasonablelower l i m i t on the i n i t i a l dispersion can be obtained by integrating from kI to ~
< <~(t:~.)>
>~
in Eqs. (13,14) that gives
HoV. V~ ,,, Ho'/* M~/z >>Ho z
. (15)
i f thermal equilibrium is assumed in the whole region inside the horizon at the beginning of the de S i t t e r stage. Thus, the i n i t i a l
dispersion of ~
, in general, exceeds Ho2 s i g n i f i c a n t l y . Never-
theless, i t appears (see below) that i f
then the i n i t i a l dispersion can be neglected because its effect on the average duration of the de Sitter stage proves to be small. Therefore, there exists a set of poss i b l e (though not necessary) i n i t i a l conditions at t = t~,for which we can use the i n i t i a l condition
~)o(~)=~(~)at t = to"
Note that, i f the last term in Eq.(12) can be neglected (that takes place in the be-
114
ning of the "cold" period of i n f l a t i o n ) , then Eq. (12)is the usual diffusion equation. Thus, the i n i t i a l l y gaussian distribution ~C(~) remains gaussian in the course of time evolution and its dispersion changes as
~rr { This is just the result obtained in [9,10,15J. In the presence of the quadratic potential V = M2~2/2, the distribution remains gaussian and the dispersion can be obtained from the "one-loop" equation 1101
In this case, Eq.(20) below reduces to that of the harmonic o s c i l l a t o r and can be
solved analytically. In the general case, the solution of Eq.(12) is : 3
a# ~ / where ¥ ~ ( ~ } i s
the complete orthonormal set of eigenfunctions of the Schrodinger
equation
i
-
2,
I t was explained at the end of Sec.2 that we may set V(eo) : -JV(-~)I = - ~ . Therefore, W(%m) = ~ and Eq.(20) has the discrete spectrum of eigenvalues only. For V(~) given in Eq.(2), i t is the equation of the
anharmonic (or doubly anharmonic) oscil-
lator. The coefficients cn are obtained from the i n i t i a l condition for ~)(!T~, ~ ) at t = t o :
115 The behaviour of J C ~ , I : )
at large times i s , as usually, determined by the lowest
energy level Eo. Eo is s t r i c t l y positive that follows from the "supersymmetric" form of the potential W(~). In practice, we are more interested not in ~ ( ~ , i : ) i t s e l f but in w(t s) - the probabil i t y d i s t r i b u t i o n for the stochastic moment t s when the de S i t t e r stage ends'.w(ts) can be obtained from ,~(~i:) by the following way. ~et the r o l l i n g of the scalar f i e l d to both sides is possible : V ~ ) ] ~ ) ~ - ~ that means that [ ~ ) J I* evolution of ~
The integral .~ d~)~.--~)"I converges at
becomes deterministic ; both the stochastic force in Eq.(8) and the
second d e r i v a t i v e with respect to ~ of Eq.(12) for
= -~.
approaches i n f i n i t y in f i n i t e time.~~V'For l~l-," ==, the
~-~±~is,
in Eq.(12) can be neglected. Then the solution
correspondingly,
where g is some unknown function that has to be determined from the previous evolution. The form of the solution represents the fact that the p r o b a b i l i t y is transported without changing along the classical paths
Therefore, one can introduce w(ts)cK.g(ts). The exact c o e f f i c i e n t of p r o p o r t i o n a l i t y is determined by the condition of p r o b a b i l i t y conservation
along the path (23). I f we do not make difference between r o l l i n g down to the l e f t and to the r i g h t sides, then the resulting expression for w(ts) is
_
3Uo
I f the r o l l i n g of the scalar f i e l d is possible to the r i g h t side only (V(-m) = ~ , V(~) = - m ; e.g., when~ = 0 in Eq.(2)), the second l i m i t in Eq.(25)hastobeomitted. The d i s t r i b u t i o n w(ts) is c e r t a i n l y non-gaussian. I t s behaviour for large t s is exponential and is determined by the lowest energy level Eo. Though w(ts) cannot be computed a n a l i t i c a l l y , i t is remarkable that the closed e x p l i c i t expressions for a l l moments <(Ho(ts-to))n> with integer n can be obtained in the form of successive integrals. The approach used here is s i m i l a r to the Stratonovich's " f i r s t time passage" method. Let us consider a set of the functions
116
(26) Then
(27)
Integrating both sides of Eq.(12) over t from t = t o to t = ~ , we obtain the ordinary d i f f e r e n t i a l equation
H? Q~"+
~ ( av Q.), __y.(~)
I t s solution, subjected to the boundary conditions Qo(~.~) = 0 (becauseS(m•,t)=O),is
C
~,
•
~
~
a,: I f the r o l l i n g is possible to the r i g h t ( l e f t ) side only, then C=O (C=1). For the symmetric case V(-~) = V(~) and ~ ( - ~ ) = ~ ( ~ ) , C : ½. Now,
~U--,
~.®
½_,..~
a--¢- ~° ( ]} )
= O - ~ ) - c _- ~ = . ( ~ d t ( ~ o l eo
Thus, the p r o b a b i l i t y w(ts) introduced according to Eq.(25) is properly normalized. By multiplying both sides of Eq.(12) by ( t - t o ) n and integrating over t from t o to t = ~ , the recurrence relation between Qn can be found. I t has the form (n>zl) :
ga"
+
3 ..'o
: - ~ ~?-.-.
The boundary conditions are Qn(~*:) = 0 for a l l n. Then
:~:I
117
Con£~".
(32) Using Eq.(27), we obtain
~/Ho
-
In p a r t i c u l a r , the average dimensionless duration of the de S i t t e r stage is equal to :
where~is
(34)
given in Eq.(4) and Qo is presented in Eq.(29).
Let us now consider several p a r t i c u l a r cases. Let ~ = 0 in Eq.(2) (that corresponds to the original picture of the "new" i n f l a t i o n ) and #o(~ = ~ C ~
• Then Eq.(34)
s i m p l i f i e s (C = ½) :
(the constant term in the potential may be omitted because i t cancels in Eq.(35)). After some manipulation, the expression (35) can be represented in the form contai-
ning only one integration :
4
£'
o
V~ where ~
'~-'
-i-
]
(3,) o~- ~_n" M ~"
is the confluent hypergeometric function.
Three more p a r t i c u l a r cases are of special interest. I)
M2 < 0 ; ~½ Ho2 << IM21 << Ho2 ; 141 >> i .
Then
ZlM'I
-'IG rr 7" I'l 4 ] S~, 14o4 "I-Y
(37)
118
where *'6"= 0.577 ... is the Euler constant. In this case, one-loop approximation which consists in the substitution of <~4> by 3(<~2>)2 in the equation for <~2> gives the result which is correct with the logarithmic accuracy :
4~,,A~'>
o.n,...-Lo,,f
= No ae~ +
H~
3 ~ I m~'l
R 71 ,qo ~-
(38)
However, more accurate approach was developed in 1101 for this case which gave the right answer. I t consists in the observation that in this case the stochastic force f ( t ) in Eq.(8) is important then and only then when the classical force (-dV(~)/d~) can be neglected and vice versa. Thus, Eq.(8) can be integrated directly that gives the following result for the stochastic quantity t s i t s e l f 1101 :
14o(~-~o) = where ~ I
~°~
~
IM'I,
;
(39)
is a gaussian stochastic quantity with zero average and the dispersion
(40)
:,
<~12>
(the thermal contribution to is neglected here for simplicity). After averaging~4 in Eq.(39) over the gaussian distribution, just the correct result (37) appears. 2) IM21 ~ ½ Ho2 ; I~I ~ I. For this case, only one-loop 1101 or order-of-magnitude 191 estimates were known earlier. It follows from Eq.(36) that
+ One-loop approximation gives the numerical coefficient in the second term equal to ~2 / ~ ' ~ 6 . 9 8 that is 2.56 times less. It is intructive to consider the case of a many-component scalar field ~a with the symmetry group O(N) and see how the one-loop approximation becomes exact in the limit N - ~ . Let ~ = (~.a~a) ½. The strightforward application of the developed approach shows that the corresponding generalization of Eq.(12) to the N{I case is :
.~ = ,~..,. ~,~-,
~.~ ~ W~
o
.~)(42) /
119 where SN is the area of the N-dimensional sphere (O(N)-symetrical initial condition for ~ is also assumed). If ~°(~,t) =~(~) at t = t o, then, instead of Eq.(35), the following expression for the average duration of the de Sitter stage results :
~ t ~ ~"-~ For V((): Vo- ~
< H. C~.-~.) >
~'/"~ =
< ,. c~,.~.~ >,.~--
Z)
q~w
p.,F
,-,-'-~
( 4h
•
NI~)
(44)
.,rc=l~"/~)P.,¢~
Thus, both expressions tend to the same limit at N..~aO(but from different sides). Now we return to the N = I case and calculate the dispersion of the quantity Ho(ts-
i ~--Eqs'(32'33)' n ~ I I-"~( J~°°~IbY~ : we : ) ~ ~,IA.d~°°~ s~ - < " : ( ~ . ~ . ) ~ > - ( 4 , .
~/~(~f%/~/Z_~Y~;tp.
c~,-~.) > ) ~ =
_
F ~ (e~
(~')
'I¢~)~t
~f,
where F(~,k) is the elliptic integral of the f i r s t kind. Also interesting is to calculate the change in the result (41) due to the spreading of the initial condition at t = t o (the "thermal" correction). If .~C~}is the gaussian distribution with the zero average and the dispersion ~)~- (see Eqs.(14,15), then by applying Eq.(29) with C = ½ the following result can be found : z
_ ~n-~- @ T
-Z
Ho ;(46)
q'~'~, << Ho
Thus, i f the condition (16) is satisfied, then the thermal correction is small ; in the opposite case, the inflationary stage is very short. 3) ~½ Ho2 ~ M2 ~ Ho2 ; IX~ i. In this case, the result (36) simplifies to the form :
C.&rr= M ~ )
~EM"
e ~ , . z, 3 Ho ~
(47)
120 The exponent j u s t coincides with the r e s u l t obtained by Hawking and Moss 181 with the help of the de S i t t e r instanton. Thus, our approach reproduces the instanton results without using instantons at a l l . Moreover, we have obtained a l i t t l e
more -
the c o e f f i c i e n t of the exponential, that corresponds to the summation of a l l one-loop diagrams on the instanton background in the standard functional integral approach. The corresponding p r o b a b i l i t y d i s t r i b u t i o n w(ts) is determined by the lowest energy level Eo of Eq.(20) with the excellent accuracy and, thus, is purely exponential :
~oI4;
_
4~'~ ~M"
(
_ 2.
4It ~ ~ M ,)
"/ , ' ~ , / , ( 4 8 )
I t is clear in our approach that the transition of the scalar f i e l d through the potential barrier takes place only locally, that is, in the volume.~.Ho"3 (in fact, somewhat larger), but not in the whole 3-space. This fact can be also understood in the functional integral approach i f one rewrites the de Sitter instanton in the static, "thermal" form :
_
~. where
+ (t_
{.,,
M
+
"B""I`
"~ is periodic with the period 2 ~ H o - l . Then the instanton t e l l s us that
has reached the top of the potential b a r r i e r inside the horizon (r < Ho-1) but gives us no information about the behaviour of ~
outside the horizon.
That is enough for the case of the "new" i n f l a t i o n . Now we shall turn to the so-called "primordial" i n f l a t i o n i l l 1 where i t is assumed that P ~ O, ~ = 0 and present the most i n t e r e s t i n g results b r i e f l y .
In t h i s case, the average duration of the de
S i t t e r stage is given by Eqs.(34,29) with C = O. Two l i m i t i n g cases are the most important and representative. i ) IM21 ~ Ho4/3 V 2 / 3 . Then
/ ,~ ,.o ( " % , )
F'("/s)(_g.)i4"o -/s
(5o)
v, .
2) Ho4/3 ~ 2/3 ~ M2 ~ Ho2. In t h i s case,
~,," U2 M~
q Hoe "v'~
(51)
121
Again, the exponent is j u s t the action for the Hawking-Moss instanton which is equal to the difference between the actions for the de S i t t e r instantons (49) with ~ - m ~ x : M~/p and
~:
~m{~=O.
The third case M2
fact, to the second one a f t e r s h i f t i n g the scalar f i e l d : ~ . = ~ E I - |M~Jj/~ ) • The quantitative results presented in the Sec. 2,3 were f i r s t published by the author in the shorter form in Russian in i16,171. Two points should be emphasized, however. Firstly,
though the quantity
Y,~(a(ts)/a(to) ) = H o ( t s - t o )
has the well-defined probabi-
l i t y d i s t r i b u t i o n w(ts), the quantity a ( t s ) / a ( t o ) does not, because EoHo2~l in all cases. Thus, i t seems that the quantity ~na(t) is more suitable for the description of the stochastic i n f l a t i o n than the scale factor a(t) i t s e l f . Secondly, the calculated duration of the de S i t t e r stage gives us the typical size of causally connected regions. However, only a minor last part of this i n f l a t i o n produces regions those remain approximately homogeneous and isotropic in the course of subsequent evolution. This follows from the fact that a f t e r the inflation~the spacetime metric at scales much larger than the cosmological p o s t - i n f l a t i o n a r y particl~ol,~ horizon has the following simple structure in the proper ("ultra-synchronous") gauge. ds 2 = d t 2 - e x p ( h ( ~ ) ) a 2 ( t ) ( d x 2+dy2+dz 2) ;
(52)
#I
h(~) = 2 ~(a(ts(3m))/a(to)), where h(~) is not assumed to be small and a(t) is the scale factor for the s t r i c t l y isotropic and homogeneous solution. The quantity h(~~) is essentially stochastic, its rms value is of the order of its average (see, e.g., Eq.(45)). Thus, the metric (52) becomes anisotropic and inhomogeneous in the course of the a f t e r - i n f l a t i o n a r y expansion when spatial gradients of h ( ~ (omitted in Eq.(52) in the leading approximation) come into play. This situation i l l u s t r a t e s
the well-known fact that "general" i n f l a -
tion produces neither isotropy nor homogeneity of the present-day universe and, therefore, cannot "explain" them without further assumptions. Nevertheless, i f the conditions (3) are f u l f i l l e d , then the last, "useful" part of i n f l a t i o n does produce sufficiently
large regions with the degree of homogeneity and isotropy that matches the
observations. I t is important that during this part of i n f l a t i o n the stochastic force f ( t ) in Eq.(8) becomes small as compared to the classical force ( - d V ( ~ ) / ( d ~ . Then, for regions those are not too large, h(~) can be represented in the form which was used in 110,18-211 : h(~) = const +~h(r~ ; ~h(~) = - 2 H o ~ E ( t , ~ ) /
~
,
(53)
where ~ is the small fluctuation of ~ ( t ) produced by f ( t , ~ ) . Here ~h(~) is r e a l l y small. The duration ~ of this "useful" part of i n f l a t i o n (when i ~ h I < l ) is easily e s t i mated using the expression for perturbations (53) :
122
I M'I /_..~ '~4/~ t-I0~ , ,)=o ;
t4o ~,~:, ",, ~-'H
IMP IM'I
IM'I >> ~,'/~ No= ~ ~=o.
Ho(' (54)
M tl • ~
1 M'!
>) p'~Ho31~ , ~ = o
Ho ~ t I contains no exponentially large multipliers. I f
~ or "~ are fixed, then
A t I is maximal and the amplitude of perturbations at the given present-day scale is minimal when IM2[ ~ ~ 1/3Ho2 or iM2[ ~
½ Ho3/2 ; the upper limits on ~ and
presented in Eq.(3), s t r i c t l y speaking, refer just to these cases. I f M does not satisfy these conditions, the duration of the "useful" part of inflation diminishes ; however, the numerical restrictions on }
and V remain practically unchanged due
to the f i r s t condition in Eq.(3). I t should be pointed also that the case M2>O presents no more advantages than the case M2
4. Evolution of the scalar f i e l d in the chaotic inflationary scenario.
In the chaotic i n f l a t i o n a r y scenario, i t is assumed that the i n i t i a l quasi-homogeneous scalar f i e l d I~I>M~
at t=t~
~is
value of the
non-zero and, in fact, large ; t y p i c a l l y ,
. The potential V(!~) can be a rather arbitrary function ; the only
condition is that i t should grow less faster than exp(const, l~i) for J ~ i - - ~ . Typical examples are V(~) = ~ 4 / 4 15I and even V(~)=M2 ~ 2 / 2 with M2>O (the dynamics of the l a t t e r model was studied in 122-261). Here, the quantity H =~/a cannot be constant in general, but i f IH] ~H2~then the expansion of the universe is quasi-exponent i a l . Thus, the notion of the quasi-de S i t t e r stage with the slow varying H arises. The scalar f i e l d should also change slowly during this stage : I~I ~ H ~ .
Then,
H2 : 81~ GV(~). We can now repeat the derivation of Eqs.(8,12) (Sec.2) for this case. Because of the dependence of H on t , the quantity
-~A~a(t)= j H ( t ) d t appears to be more proper and
fundamental independent variable than the time t. Eq.(6) retains its form with the change : Ho..~H. I t is straightforward to obtain the following equation for the large-scale scalar f i e l d
~A~
:
3H ~ ~
~I
(55)
123
Then the corresponding Fokker-Planck equation takes the form (H2can be expressed through V(~)) :
-
?)
4 {
(5e)
I t is worthwhile to note that this equation has just the form one would expect to follow from quantum cosmology because i t is no longer depends on such classical quantities as t or H, but contains only fundamental variables ~ a and ~ which remain in quantum case.
Now, the problem of the initial condition for j O ( ~ a ) of classical chaotic i n f l a t i o n ,
i t is usually assumed that
arises In the studies = ~Po at t=tp that
corresponds to ---'%eC~) °(. ~ C ~ - ~ ) f o r some . ~ 0 ~ . But such a condition contradicts the whole s p i r i t of quantum cosmology. A natural idea is to consider stationary solutions (e.g., independent of ~ v ~ ) of Eq.(56). They can be thought of as being in "equilibrium with space-time foam" which may arise at planckian curvatures. At f i r s t ,
we introduce the notion of the probability f l u x j ( ~ j ~
) by rewriting
Eq.(56) in the form
"a~o.
S
(57)
Then, two types of stationary solutions arise : with no f l u x and with a constant f l u x
Jo :~9 = const. V- l e x p ( 3 / e G 2 v ) - ~ 3 ~Jo(GV)-I exp(3/8 G2 V) J d ~ l exp(-3/8 G2 V(~l)).
(58)
- -
The f i r s t
solution (with j = O) is just the envelope of the Hartle-Hawking time-sym-
metric wave function 1271 in the c l a s s i c a l l y permitted region (a2~ (83~GV)-1) ; the exponent is the action for the de S i t t e r instanton with ~ = const (with the correct sign). Moreover, we have obtained the c o e f f i c i e n t of the exponent, so the solution appears to be normalizable. I t iseasy to v e r i f y that the average value of ~ ted with th~ use of this solution p r a c t i c a l l y coincides_L..with ~ $
calcula-
--the value of
for which IHI~H2 and the de S i t t e r stage ends ( q~s,~l~pif V(.~ = "~h~_n/n). This does not mean that the dimension of the universe a f t e r i n f l a t i o n is small (because all ~ are equally probable for stationary solutions) but suggests that the "usef u l " part of i n f l a t i o n is t y p i c a l l y very small ( i f exists at a l l ) in this case. I t is possible to obtain the "useful" part of i n f l a t i o n that is long enough, but with the very small probability ~ exp(-3/eG2V(~s))~ exp(-lolO).
124
I t is interesting that the second solution with j { 0 does not, in fact, contain any exponential at a l l . For G2V(~)~I that corresponds to curvatures much less than the planckian one, its form for Jo
In this case, the stochastic force is unimportant. Thus, we have only two possibil i t i e s : e i t h e r the stationary solution contains the instanton contribution exp(-S) (where S is the action for the instanton, S
5. Conclusions and discussion. We introduced and elaborated the approach consisting in taking into account the change in a large-scale scalar f i e l d due to the continuous flow of small-scale quantum perturbations of the same scalar f i e l d across the de Sitter horizon during the de Sitter (inflationary) stage. That gave us the p o s s i b i l i t y to find the e x p l i c i t expressions for the average duration of the de Sitter stage (and for any higher moment i f necessary) in the case when the i n i t i a l probability distribution of the scalar f i e l d before the beginning of de Sitter stage was known. Certainly, the method used in the paper (as any other mathematical method) cannot solve the problem of i n i t i a l pre-inflationary conditions ; new physical hypothesises (or "principles") are necessary for this purpose. What can be said now about the p o s s i b i l i t y of "spontaneous quantum creation of the universe" which was so extensively discussed in 132-351 ? To make the terminology more precise, the author proposed some time ago 1361 (see also 130i) to call the "quantum creation of the universe" the situation when we have a solution for the wave function of the universe with a non-zero probability flux emerging from the region of small values of a (or, equivalently, large values of space-time curvature). This proposal can be used in our stochastic approach also. Then the f i r s t stationary solution of Eq.(56) (the f i r s t term in Eq.(58)) corresponds to the time-symmetric universe which has no beginning and was not created. This coincides with the Hawking's interpretation of the Hartle-Hawking wave function in quantum cosmology. In the case of our f i r s t solution, we encounter the serious d i f f i c u l t y connected (as was explai-
125 ned in Sec. 4) with the very small p r o b a b i l i t y of having the large duration of the "useful" part of i n f l a t i o n . The second s t a t i o n a r y solution with the non-zero p r o b a b i l i t y f l u x does correspond to the "creation" of the universe but t h i s creation has very l i t t l e
in common with the
picture that was introduced in 132-35J. In p a r t i c u l a r , no quantum tunneling takes place, and the evolution of the metric and the scalar
f i e l d remains classical up
to the planckian curvatures. This type of creation was called the "classical creat i o n " in ]28J but i t should be clear that the "classical creation" is not a new concept but simply the paraphrase of the standard classical picture of a s i n g u l a r i t y as a boundary of the space-time through which the space-time cannot be continued ; the only difference is that now t h i s boundary is assumed to have a f i n i t e thickness o The d i f f i c u l t i e s
with the second solution are connected with our i m p o s s i b i l i t y at
the present time to prove the very existence of such a solution (in other words, to prove the p o s s i b i l i t y of the quantum change of topology) and to say something d e f i n i t e about the value of Jo, i f i t is non-zero. Thus, the problem of the p o s s i b i l i t y of the quantum creation of the universe remains open. The author would l i k e to thank Prof. Norma Sanchez for the h o s p i t a l i t y in the Groupe d'Astrophysique R e l a t i v i s t e de l'Observatoire de Paris-Meudon where t h i s paper was completed and the Centre National de la Recherche S c i e n t i f i q u e for f i n a n c i a l support.
References 1. A.A. Starobinsky, Phys. Lett. 91B (1980) 99. 2. A.H. Guth, Phys. Rev. D23 (1981) 347. 3. AoD. Linde, Pys. Lett. 108B (1982) 389. 4. A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. 5. A.D. Linde, PismavZhETF 38(1983) 149 ; P h y s . Lett. 129B (1983) 177. 6. L.A. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Lett. 157B (1985) 361. 7. V.F. Mukhanov, G.V. Chibisov, PismavZhETF 33 (1981) 549. 8. S.W. Hawking, I.G. Moss, Phys. Lett. 110B (1982) 35. 9. A.D. Linde, Phys. Lett. 116B (1982) 335. i0. A.A. Starobinsky, Phys.Lett. II7B (1982) 175. 11. J. E l l i s , D.V. Nanopoulos, K.A. Olive, K.Tamvakis, Phys. Lett. 120B (1983) 331. 12. G.F. Mazenko, W.G. Unruh, R.M. Wald, Phys.Rev. D31 (1985) 273. 13. M. Evans, J.G. Mc Carthy, Phys. Rev. D31 (1985) 1799. 14. A. V i l e n k i n , Phys. Lett. II5B (1982) 91. 15. A. V i l e n k i n , L.H. ford, Phys. Rev. D26 (1982) 1231. 16. A. Starobinsky, in : Proc. 6th Sov.Gravit.conf., Moscow 3-5 j u l y 1984 (MGPI Press, Moscow, 1984), vol. 2, p. 39. 17. A. Starobinsky, in : Fundamental I n t e r a c t i o n s , ed. V.N. Ponomarev (MGPI Press, Moscow, 1984) p. 54.
126 18. S.W. Hawking, Phys. Lett. 115B (1982) 295. 19. A.H. Guth, S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. 20. J. Bardeen, P.I. Steinhardt, M.S. Turner, Phys. Rev. D28 (1983) 679. 21. A.A. Starobinsky, Pis'ma Astron. Zh. 9 (1983) 579 ISov.Astron. Lett. 9(1983) 302J 22. L. Parker, S.A. Fulling, Phys. Rev. D__77(1973) 2357. 23. A.A. Starobinsky. Pis'ma Astron. Zh. 4 (1978) 155 ISov.Astron. Lett 4(1978) 82J. 24. S.W. Hawking, Nucl.Phys. B239 (1984) 257. 25. D.N. Page, Class.Quantum Grav. i (1984) 417. 26. V.A. Belinsky, L.P. Grishchuk, I.M. Khalatnikov, Ya.B. Zeldovich, Phys. Lett. 155B (1985) 232. 27. J.B. Hartle, S.W. Hawking, Phys. Rev. D28 (1983) 2960. 28. Ya.B. Zeldovich, A.A. Starobinsky, Pis'ma Astron. Zh. 10 (1984) 323 ISov.Astron. Lett. 10 (1984) 135J. 29. A.D. Linde, Zh.Eksp.Teor. Fiz. 8_7_7(1984) 369 ; L e t t . 30. A. Vilenkin, Phys. Rev. D30 (1984) 509.
NuovoCimento 39(1984) 401.
31. V.A. Rubakov, Pis'ma ZhETF, 39 (1984) 89 ;Phys. Lett. 148 (1984) 280. 32. E.P. Tryon, Nature 246 (1973) 396. 33. P.I. Fomin, Dokl.Akad.Nauk Ukrain. SSR (1975) 831. 34. L.P. Grishchuk, Ya.B. Zeldovich, in : Quantum structure of space-time, eds. M. Duff and Co Isham (Cambridge U.P., Cambridge, 1982) p. 409. 35. A. Vilenkin, Phys. Lett. 117B (1982) 25 ;Phys. Rev. D27 (1983) 2848. 36. A.A. Starobinsky, talk at the seminar in the P.K. Sternberg State Astronomical Institute (Moscow, November 1983), unpublished.
SOME
MATHEMATICAL
STOCHASTIC
ASPECTS
OF
QUANTIZATION
G.Jona-Lasinlo Dipartimento
dl
Fislca GNSM
In
this
thematical program
report
I would
problems
of
and
INFN
like
encountered
stochastic
Yong-Shl/I/
- Universit~
and
to in
quantization
then
developed
-
briefly a
its
Sapienza",
outline
rigorous
first
in
"La
Rome
of
the
ma-
implementation
some
of
the
and
Wu
proposed
formal
by
Parisi
aspects
by
several
au-
thors/2/. We sists
recall
in
that
mechanical
system
stationary
state
The to
the
basic
the
with of
standard
solve
the
considering
finite
some
or
for
F_c(])
stochastic
quantlzatlon
associated
degrees
of
to
a
freedom,
con-
quantum as
the
process.
the
stochastic
of
measure
infinite
stochastic
proposal
following
idea
Euclidean
construction differential
of
such
a process
is
equation
"a (i)
/ where in
the
S(~)
is
Wiener
E (W ( x , t )
the
Euclidean
process
W (x l j
t'))
action
characterized
= min
(t,t')
describing by
the
~x
the
system
and
W(x,t)
covarlance
-
x')
(1.2)
128
The
typical
where
form
V(~)
dimension
is of
of the
local
a
the
functional
S(~) I
polynomial
space.
is
in ¢
Introducing
of
(1.3)
even
in
degree
(i.I) we
V
and
the
is
obtain
(I .4)
is the L a p l a c i a n . As non
it
is
eq.(l.4)
dlfferentiable.
case
of
come
by
equation The integral that
ordinary
only is
stochastic
integrating into
is
This
with
an i n t e g r a l
natural
well
because
known
differential
respect
to
the
Wiener
difficulty equations
the
time
and
case
of
(1.4)
process
already
where
it
in is
is the
over-
transforming
the
equation/3/.
thing
to
by
using
equation
formal
a
do
in
the
the Green
function
of
is the
to
obtain
linear
is
~0
where
G satisfies
~&
qt
~
G + G
and Z is the G a u s s l a n
= ~(t'-t)
~(_x*- x)
(1.6)
process
z(t,_~) =~ dV~/jOat' a(t,t',~,~')
~W(t',x') ~ t'
+
~o(t,x )
an
part,
(1.7)
129 i
w h e r e ~= next
is
step
= on
I.
a
-
the
solution
depends
In
this
case
circumstance are
continuous
solved
for
each
theory number
of
It
interesting
is
quantum
degrees
in
the
typical in
case
note
can
that
been
be
is
however
applications. has
of
In
useful
in
of case
recent
of
Quantum
years
entirely
process then A
in
a
=I
finite
Mechanics. goes
beyond
the
theory
fact
different
be
rather
The ~
with
relevance,
depends
can
ref./4/.
of
The
the
Z(t,x).
systems
its
(1.4).
This
(1.5)
input
found
the
in
equation.
the
that
of
.
variables.
quantization
freedom,
part ~
trajectories
both
realization
this
to
homogeneous
dimensionality
a meaningful
functions
of
/4/
is
stochastic
mechanical
developed
the
(1.5)
of
the
linear
on
continuous
treatment covers
the
that
Z(t,x)
complete
of
crucially
domains/5/-
/6/
=
2.
- At ~
appear. but
a
The
=2
free
distribution.
value
Ew
tion
with
(Z2(t,~)) respect has
this
however,
way
to
equation this
ing
the
make cient the
and
spite this
of
stochastic us
first
present term
fact we
us that
the
situation and
this
then
by
introducing
means
expecta-
counterterms.
and
become
In
infinite,
The w a y
consists
in d e v e l o p -
then
adjusting of the
the
out
renor-
form
~ (t n, xl~))
in
equation
treatment
does
a
not
The
a cut-off
~
is
(1.5)
product
To
follows. terms.
of
the
in
solve
solution
as
counter
suffi-
Therefore
of
a meaning. a weak
idea
in
not
approach.
have
the
the W i c k
is
"solution"
call
basic
introducing taking
this
perturbative
defining
prohabilists
introducing
expectation
meaning.
literature
expectations
to a n o n
means
Ew
counterterms
mathematical
by
the
Theory
function
(1.2).
theory
equation.
(1.5)
Field
continuous
example
a mathematical
that
to w h a t
differential
for
the
physical
consists
appeal
regularize
V'(~)
have
(t 2, x 2 ) - -
to r e s o r t
facing
Quantum a
logarithmically.
end
a way
a rigorous
one has
the
not
of
anymore
process
the
in the
in such
For
problem
diverges
in
does
I, X l ) ~
not
evaluates
in p e r t u r b a t i o n
terms
problem
one
is
to be m o d i f i e d
since
taken
solution
sense.
If this
itself
E w (~(t
difficulties
Z(t,x)
therefore
problem
malization
typical
to the W i e n e r
Eq.(l.5)
the
the
field
In
of a Let the
nonlinear
evaluating
it.
That
l
is
the
nonlinear
term
will
be:
V' ( ~ ) : /
field.
The
Wick
product
can
be
taken
where
~
is
the
cut-off
-
with
respect
to
the
covarianee
130 of
the
free
: V'(/):
field
=
~
C (x,y)
(/3
-3
= ( - ~ + I ) -I. C
(x,x)~
).
If for With
example
these
V(~)
= ~--
modifications
(1.5)
becomes
= - C *: V (
and
as
for
each
long
solutions (in
the
that
as ~
imput of
(1.8)
The
of
has
If
weak
calculus
ron-Martin
formula,
}.L~
when
the
the
}
a
can
be
solved
in
we
removed.
formula,
on the
limiting
as~->~
cut-off
the
Radon-Nikodym
. This
~oT(: V / ( Z ~ ) : , d W ) -
to
perturbatlvely
known
to/~
which
a measure ~
measures)
non
provides
respect
induce
converges
is a well
which
equation
now
of
implementing
there
the m e a s u r e / ~ w i t h
= exp I-
now
solution
in
stochastic
Z will
convergence
a weak
consists
is a m e a n i n g f u l
process
(1.8).
sense
therefore
this Z.
(t.8)
): + Z
this
measure shall Our
say goal
idea.
In
Girsanov-Camederivative
of
is/3/
V/(Z.): I/ 2} (1"9)
I' :
~Tdt'
where
(: V/(Z~):,
dW)
=
S
d2x:
V/(Z/~(t'x)):
dW
(t, x)
A is
a scalar
induced gral dered
by
product
in
it.
We
work
appearing
in
(1.9)
as a limit
SL~.li= °
of
the in
the
space
variables
finite
is a Ito
and
volume A
integral,
[[
• The
that
is
"[I i s
the
stochastic it must
be
norm inte-
consi-
sums
(:v/(z~(ti)):,
w(ti+ 1) - w (tf))
(1.1o)
131
where
the
ti represent
definition
the
the
Markov
property
ral
probabilistically,
rules
(1.9)
basic
is,
the
T]
Wiener not
process.
obey
as
. Notice
with
the
This
it
that
with
integrand
integral,
is well
known
now
consists
in showing
stochastic
variable
and
that
problem
special
the
very
natu-
to the
usual
that
when
~
---~ ~
in p a r t i c u l a r
normalized.
expectation
now
we
to
(i.li)
reminds
lagranglan
explicit
this
due
calculus/3/.
the m e a s u r e ~ i s
The
of [ o ,
uncorrelated
(d d - - ~ ) = 1
Z o means
rather
is
does
problem
is a good
EZo
more
of
of d i f f e r e n t i a l Our
that
a partition
increment
notice
of
is
is
taken
constructive
involved,
that
with
by
the
To
Z (o, _~x) = Zo(X). field
make
rules
the
of
theory,
only
connection
the
Ito
a
even
calculus
it
follows
f(:V/(Z
):,
dW)
= ~
(T,~):
I : Z4
d2x
- ~ ~
z%
:Z4(o,~):
d2x +
A (1.12)
+
0Tdt
Using At of
(I.12),
this
neither variable.
by
in
For
and
the
square
a similar
reason
is
the
the
that
thing
the
the
The
first
(1.97
this
type
operates structure
in
to
exotic
apply
the
the
rather
is
more
difficult.
a well
defined
of
the
second
we
just
higher
(1.9)
mentioned
expansion order
which
is
of
the
methods
realizes In
fact
stochastic
term does
form.
straight
One
is
the
with of
ready
of d i v e r g e n c e
divergence term
a less
constructed/7/.
problem of
takes
particular was
expectation
is that
mechanism special
our
72
(1.9)
to be
in
(~
exponent
theory! of
P
in
seems
theory,
which
example
remarkable
in p e r t u r b a t i o n by
everything
however
term
exponential
field
methods
immediately
the
point
constructive
forward
The
(Z 3
diverges.
not
show
is c a n c e l l e d exponential
contributions. such
up
as
to
The
insure
132
in
any
of
the
case
the
normalization
cancellation
divergences
in
constitute
a
however
E ' ~ _d/~¢ _
condition
mechanisms
) = I. This
supersymme~r~c difficulty
reminds
theories(*).
in
a
non
These
perturbative
approach. At
this
specific must to
that
its
methods
/
)2 seem
way
The
other the
equilibrium and
d ~(t,x)
with
In/9/
it
which
(t', £'))
was
shown
(C-$~
to
that
prove
the
same
of
for
to
the
stochastic 3 " in
such
represents
a
its
by J o n a - L a s i n i o
stochastic
equilibrium
we
expansion led
eq.(l.8)
still
of
look
cell
in
recently
family
(t,x)
+ C I-/
measure.
:V/(/(t,x)))
previous
to e q u i l i b r i u m conclude analysis methods
with
min
(t,t')
for ~ L T ~
the
methods
existence
of
the
approach
(*) has
followed
the
sufficient
which
encountered
theory
a whole
= C I- & (~,E')
The
the
the w a y fact
admit
- ~
(1.13).
push
in
space
methods
modifying
Euclidean
not
a case
difficulties
those in
are
such
phase
on
diffeThe
one
is
W
sufficient
powerful
)2
was
is
as
in
insist
quantization
dt
(1.13)
I and
E (W(t,x)
We
the
above
consists
( ~
This
= dW (t,x)
O ~
In/9/
P
There
equations
considered
nature
must
group
the
If we
stochastic
for P ( ~ ) 2
We
example
fact
possibility
state.
for
devised
for
. In
usual
Mitter/9/.
rential
~$
possibilities. basis
renormalization
of a s i m i l a r
that
two the
counterpart.
like
the
of
as
the m e t h o d s
generally
construction P (
are
stochastic
powerful
more
there
of e q . ( l . 8 )
conclude
treat
more or
stage
form
equation is some
to treat mentioned
(1.8)
slower
for
comments. the
ease ~
an
(1.14)
used
ergodie
corresponds
for
P
weak to
(
j
)2
solution ~
=
i.
are of The
(1.13). It
would
= I i.e.
eertainly eq.(l.8)
be with
worth
to
the m o r e
before.
The c o n n e c t i o n b e t w e e n s t o c h a s t i c c a l c u l u s been c o n s i d e r e d by m a n y a u t h o r s / 8 / -
and
supersymmetry
133
In/9/
only
interesting
to
the take
ultraviolet the
problem
was
limit A--> 6~ . In
that the formalism of the cluster
this
studied.
It
connection
expansion applies
would we
be
remark
also to the study
of (1.13). In Mitter.
conclusion My
I
would
understanding
our pleasant
and fruitful
of
like the
to
express
subject
my
discussed
gratitude here
owes
to
P.K.
much
to
collaboration.
References i)
G.Parisi,
2)
For a review see for example B.Saklta, 7th Johns Hopkins Workshop, ed. G.Domokos, S . K o v e s i - D o m o k o s (World Scientific, Singapore 1983).
3)
See e.g. I.I. Gihman, A.V.Skorohod, Equations", Springer 1972.
4)
W.Faris,
G.Jona-Laslnio,
5)
R.Benzi,
A.Sutera,
6)
M.Cassandro,
7)
E.Nelson, in "Constructive Quantum Field Theory" Lecture Notes in Phys. Vol. 25, Springer 1973; B.Simon, "The P ( / ) 2 Euclidean (Quantum) Field Theory" P r i n c e t o n NJ, Princeton University Press 1974; J.Gllmm, A.Jaffe, "Quantum Physics" Springer 1981.
8)
S.Cecotti, L.Girardello, Phys.Lett. IIOB, 39 (1982); G.Parlsi, N.Sourlas, Nuel.Phys. 206B, 321 (1982); E.Gozzi, Phys.Lett. 129Bn 432 ( i - ~ ) ; V.de Alfaro, S.Fubinl, G.Furlan, G.Veneziano, Phys.Lett. 399 (1984).
9)
Wu Yong-Shi,
Sci. Sin. 24,
G.Jona-Lasinio,
J.Phys.A, 15,
J.Phys.A, 18,
E.Olivieri,
483 (1981).
P.Picco,
P.K.Mitter,
2239
"Stochastic
3025
Differential
(1982).
(1985).
Ann.Inst.
Comm.Math.Phys.
H.Poinear~,
I01, 409
in Press.
(1985).
142B,
S U P E R S T R I N G S A N D T H E UNIFICATION O F F O R C E S A N D P A R T I C L E S
Michael B. G r e e n , P h y s i c s D e p a r t m e n t , Queen Mary College, U n i v e r s i t y o f L o n d o n , U.K.
The q u e s t i o n of how to r e c o n c i l e t h e c l a s s i c a l d e s c r i p t i o n o f t h e g r a v i t a t i o n a l force
embodied
in
Einstein's
general
theory
of
relativity
q u a n t u m t h e o r y is a c e n t r a l i s s u e in t h e o r e t i c a l p h y s i c s .
with
the
principles
of
A simple a p p l i c a t i o n of t h e
u n c e r t a i n t y p r i n c i p l e s h o w s t h a t a t a d i s t a n c e , Ax, a r o u n d t h e P l a n c k scale, i.e.
(i)
Ax ~ /Gh/c3 ~ 10 -35 m e t e r s
(where G is the gravitational constant) space-time
must
perturbative
be
considered
calculations
space-time
is
small
in
on
to
q u a n t u m f l u c t u a t i o n s become so l a r g e t h a t
contain
quantum
all
a
sea
gravity
length
scales
of
virtual
assume they
that
are
black the
invalid
holes.
Since
curvature and
of
lead
to
non-renormalizable infinities. Non-perturbative methods have not led to calculable
consequences. I t a p p e a r s likely t h a t s u p e r s t r i n g in a c o n s i s t e n t m a n n e r .
theories unite gravity and quantum mechanics
This is a c h e i v e d b y a m o d i f i c a t i o n of g e n e r a l r e l a t i v i t y a t
s h o r t d i s t a n c e s so t h a t E i n s t e i n ' s t h e o r y e m e r g e s a s a l o n g d i s t a n c e a p p r o x i m a t i o n . Furthermore,
the
quantum
consistency
of
superstring
theories
provides
s t r i n g e n t r e s t r i c t i o n s o n t h e p o s s i b l e u n i f y i n g Yang-Mills g a u g e g r o u p s .
very
As a r e s u l t
g r a v i t y is u n i f i e d w i t h t h e o t h e r f o r c e s a n d p a r t i c l e s in a n almost u n i q u e m a n n e r . The o n l y p o s s i b l e u n i f y i n g g r o u p s a r e
S0(32) o r E8 x E8
(2)
[$0(32) is a l a r g e o r t h o g o n a l g r o u p while E 8 is t h e l a r g e s t e x c e p t i o n a l Lie g r o u p . ] The d i m e n s i o n a l i t y of s p a c e - t i m e is a l s o r e q u i r e d
to t a k e
a special
(or
"critical")
value
D = 10
in o r d e r have
any
(3)
to o b t a i n a c o n s i s t e n t s u p e r s t r i n g chance
four-dimensional
of world,
describing six
the
quantum theory.
observed
dimensions
must
physics turn
out
Clearly, in o r d e r
of to
our be
to
(approximately) curled-up
(or
135
" c o m p a c t i f i e d " ) t o a v e r y small size. The i d e a o f h i g h e r
dimensions arose
in m o d e r n p h y s i c s
in t h e
proposal
by
Kaluza a n d Klein 1 in t h e 1920's to u n i f y e l e c t r o m a g n e t i s m w i t h g r a v i t y b y a s s u m i n g t h e e x i s t e n c e of a f i f t h d i m e n s i o n w h i c h f o r m s a v e r y b e e n r e v i v e d in the c o n t e x t of s u p e r g r a v i t y the
interactions
in
this
particularly popular.]
manner.
[A
theory
in
eleven
In this respect
ten-dimensional theory
arise
superstring
from t h e
since the possible gauge groups
very
has
been
symmetries of the
s y m m e t r i e s of t h e
theories are
t h e r e is more g a u g e
This idea has
dimensions
In t h e s e Kaluza-Klein schemes the gauge
effectively four-dimensional theory space.
small c i r c l e .
t h e o r i e s w h i c h h a v e t r i e d to u n i f y all
different.
compactified
A l r e a d y in t h e
symmetry than anyone could wish for
(in (2)) a r e so l a r g e .
The e o m p a c t i f i c a t i o n o f t h e
e x t r a d i m e n s i o n s is h e r e e x p e c t e d to r e d u c e t h e g a u g e s y m m e t r y d o w n to a s m a l l e r symmetry group. the
T h i s s h o u l d lead to s o m e t h i n g like a " G r a n d U n i f i e d " s y m m e t r y in
effective four-dimensional theory
observed standard
accelerator
physics,
this
at
high
energies.
symmetry
model w i t h s y m m e t r y g r o u p s
must
in
Furthermorey turn
break
to
down
explain to
the
SU(3) (for c o l o u r ) a n d SU(2) x U(1) ( f o r t h e
electro-weak forces). A l t h o u g h a c o m p l e t e l y r e a l i s t i c w a y in understood
it
is a l r e a d y
clear
that
making contact with observed physics. theories
contain
logically have
no f r e e
free
input
might happen
theories
have
a
is n o t
good
yet
chance
of
The p r o g r a m m e is v e r y a m b i t i o u s s i n c e t h e s e
parameters
parameters).
which this
superstring
(although
The t e c h n i q u e s
the
space
required
phenomenological predictions and the theoretical structure
of
solutions
for analysing
may
both the
of t h e s e t h e o r i e s i n v o l v e
t h e u s e o f m a n y i d e a s in m o d e r n m a t h e m a t i c s t h a t h a v e n o t b e e n u s e d b y p a r t i c l e p h y s i c i s t s u n t i l now.
Conversely, many a s p e c t s of s u p e r s t r i n g
theory raise issues
of i n t e r e s t in p u r e mathematics.
CHIRALITY A key chirality
constraint
(i.e.
interactions. indicated that
parity
on any
theory
violation)
of
is t h a t
the
it m u s t
give rise
four-dimensional
world
to t h e o b s e r v e d due
to
the
weak
O v e r t h e l a s t f e w y e a r s t h e s t u d y o f t h e Kaluza-Klein m e c h a n i s m h a s chiral physics
can p r o b a b l y only emerge from a h i g h e r - d i m e n s i o n a l
t h e o r y if two c o n d i t i o n s a r e s a t i s f i e d 2 : (a) t h e
higher-dimensional theory
is c h i r a l
(which excludes odd-dimensional
t h e o r i e s , s i n c e c h i r a l i t y o n l y e x i s t s in e v e n d i m e n s i o n s ) a n d (b) t h e r e is a g a u g e group~ G, i n t h e h i g h e r - d i m e n s i o n a l t h e o r y . be
necessary
to
compactification. configuration
avoid The
(such
as
losing
gauge a
fields
magnetic
the can
chirality twist
monopole)
up in
property into the
distinguishes the different four-dimensional chiralities.
a
in
This seems to
the
process
topologically
internal
space
-
of
non-trivial this
then
136
CHIRAL ANOMALIES
A n y c h i r a l t h e o r y is l i k e l y to b e p l a g u e d b y i n c o n s i s t e n c i e s k n o w n a s c h i r a l gauge
"anomalies".
These
represent
the
s a c r o s a n c t c o n s e r v a t i o n laws t h a t w e r e may
in
general
arise
in
the
breakdown
in
the
quantum
theory
built into the classical theory.
conservation
of
Yang-Mills
currents
of
Anomalies and
in
the
c o n s e r v a t i o n of g r a v i t a t i o n a l c u r r e n t s i.e. t h e e n e r g y - m o m e n t u m t e n s o r (as well a s in the supersymmetry current). contains
a n t i - f e r m i o n s of
complex r e p r e s e n t a t i o n
I n f o u r d i m e n s i o n s a t h e o r y w i t h Weyl f e r m i o n s also
the
of
a
opposite gauge
chirality.
group
(so
Only that
if
the
complex c o n j u g a t e r e p r e s e n t a t i o n ) is t h e t h e o r y c h i r a l . Yang-Mills a n o m a l i e s b u t
the
fermions
anti-fermions
lie
in
be
both
Yang-Mills
anomalies
(with
fermions
the
is i n s e n s i t i v e to
H o w e v e r , in t e n d i m e n s i o n s ( a n d
g e n e r a l l y in
4n+2 d i m e n s i o n s ) a f e r m i o n a n d i t s a n t i - p a r t i c l e h a v e t h e same c h i r a l i t y a n d can
a
I n t h a t c a s e t h e r e may b e
no g r a v i t a t i o n a l a n o m a l i e s s i n c e g r a v i t y
the gauge group quantum numbers.
lie in
in
any
there
representation)
and
g r a v i t a t i o n a l anomalies. The e x i s t e n c e of anomalies r e n d e r s a t h e o r y i n c o n s i s t e n t b e c a u s e t h e y lead to a violation o f u n i t a r i t y d u e to t h e c o u p l i n g of u n p h y s i c a l l o n g i t u d i n a l m o d e s of g a u g e p a r t i c l e s to t h e p h y s i c a l t r a n s v e r s e that
there
were
dimensions.
no
modes.
anomaly-free
chiral
I t was t h e n d i s c o v e r e d 3
Up to l a s t s u m m e r it h a d b e e n t h o u g h t theories
with
gauge
groups
in
ten
t h a t a n o m a l i e s may be a b s e n t f r o m t h e o r i e s
with the gauge groups mentioned earlier. Superstring
t h e o r i e s with t h e s e g a u g e g r o u p s a r e b o t h f r e e f r o m a n o m a l i e s a s
well a s t h e i n f i n i t i e s t h a t p l a g u e q u a n t u m t h e o r i e s of g r a v i t y checked).
(as f a r a s h a s b e e n
T h e s e s u c c e s s e s a r e u n p r e c e d e n t e d in a n y q u a n t u m t h e o r y o f g r a v i t y .
WHAT ARE SUPERSTRINGS?
In
contrast
constituents theory between
have
are
to
extension
string
usual
relativistic
structurless
field
in
point
one
theory
field
dimension. and
Yang-Mills o r g e n e r a l r e l a t i v i t y .
theories,
particles~ t h e This
conventional
in
which
constituents leads
to
"point"
the
fundamental
of a n y
string
significant
field
theories
field
differences such
as
A s i n g l e c l a s s i c a l r e l a t i v i s t i c s t r i n g c a n v i b r a t e in
a n i n f i n i t e s e t of normal m o d e s with u n l i m i t e d f r e q u e n c i e s .
The s e p a r a t i o n b e t w e e n
t h e f r e q u e n c i e s of t h e s e m o d e s is d e t e r m i n e d b y t h e r e s t t e n s i o n o f t h e s t r i n g , T. The
modes can
be
quantized
so
that
the
quantum
mechanics of a
single
string
d e s c r i b e s a n i n f i n i t e s e t of s t a t e s w i t h m a s s e s w h i c h i n c r e a s e w i t h o u t b o u n d , t h e i r separation given by A ( m a s s ) 2 = 2~T
(4)
137
These
states
also
have
spins
which
straight-line Regge trajectories
increase
without
bound
(with s l o p e c~' - 1/2zyT).
since
they
lie
on
This is n o t a n a c c i d e n t -
s t r i n g t h e o r y o r i g i n a t e d in t h e l a t e 1960's w i t h t h e d u a l r e s o n a n c e model 4 w h i c h was developed to explain h a d r o n i c phenomena. string
The e a r l i e s t s t r i n g
t h e o r y 5) h a d a c r i t i c a l d i m e n s i o n D - - 2 6
while t h e
theory
(the bosonic
spinning6 string
theory
w h i c h also i n c o r p o r a t e d f e r m i o n s had D - 10.
I t was n o t i c e d t h a t t h e s p e c t r u m of
t h e s p i n n i n g s t r i n g t h e o r y c o u l d be t r u n c a t e d
to g i v e a s u p e r s y m m e t r i c s p e c t r u m 7
i.e. a t This
every
mass level t h e r e
gave
rise
supersymmetry
to
over
the
superstring
theories.
contrast
the
ground
to
states
the
supersymmetry
last
five
equal
string
states
multiplets
I
of
theories
shall
refer
s t a t e s of s u p e r s t r i n g
theories
with
n u m b e r of b o s o n a n d f e r m i o n s t a t e s .
construction y e a r s 8.
The g r o u n d
earlier i.e.
are an
explicit
which
negative
were
corresponding
to
to
these
by
These
the
space-time theories
theories are
plagued
(mass)2).
with
having
massless
familiar
as
m a s s l e s s (in tachyonic
states
massless
form
states
in
tan-dimensional super-Yang-Mills and supergravity. The m a s s s c a l e s e t b y t h e s t r i n g t e n s i o n is s u p p o s e d t o b e t h e P l a n c k s c a l e (in ten dimensions). scales
much
T h i s m e a n s t h a t , f o r m a n y p u r p o s e s , w h e n c o n s i d e r i n g momentum
less
than
the
Planck
scale
the
higher
mass
states
are
effectively
infinitely massive and t h e y decouple leaving an effective " l o w - e n e r g y " t h e o r y of the massless ground
states.
This
supergravity
and
(the quarks,
leptons, gauge
is
just
super-Yang-mills.
a conventional point field
theory
The f u n d a m e n t a l p a r t i c l e s o b s e r v e d
that
the
low e n e r g y
theory suggests
as
in n a t u r e
particles,....) should occur among the massless g r o u n d
s t a t e s s i n c e t h e i r m a s s e s a r e n e g l i g i b l e c o m p a r e d to t h e P l a n c k mass. fact
such
theory
has
arisen
from
an
almost
However, t h e
unique
superstring
t h a t t h e p a r a m e t e r s m e a s u r e d in e x p e r i m e n t s ( s u c h a s t h e m a s s e s
and coupling strengths)
s h o u l d b e d e t e r m i n e d w i t h little a m b i g u i t y f r o m t h e t h e o r y .
At m o m e n t u m s c a l e s a r o u n d t h e P l a n c k s c a l e t h e m a s s i v e s t a t e s of t h e s t r i n g c a n b e e x c i t e d so t h a t
superstring
theory.
T h i s s c a l e is j u s t
because
they
theory
differ from E i n s t e i n ' s t h e o r y
these scales that certain superstring space-time picture the strings appear as
then
d i f f e r s r a d i c a l l y from a n y
p o i n t field
where the problems with quantum gravity arise. (or a n y
supergravity
I t is
field theory)
theories avoid quantum inconsistencies.
at
In a
h a v e a n a v e r a g e s i z e of t h e P l a n c k l e n g t h so t h e y
points when looked at c o a r s e l y b u t
their
n o n - z e r o e x t e n s i o n is c r u c i a l
w h e n c a l c u l a t i n g q u a n t u m f l u c t u a t i o n s a t small s c a l e s .
SUPERSTRING DYNAMICS
I will g i v e a v e r y s k e t c h y o u t l i n e o f t h e w a y in w h i c h t h e d y n a m i c s of a f r e e superstring As
is f o r m u l a t e d . a
world-sheet
string just
as
moves a
through
point
space-time
particle
traces
it out
sweeps a
out
a
world-line.
(two-dimensional) The
space-time
138 c o o r d i n a t e o f a n y p o i n t on t h e s t r i n g a t a g i v e n time, X~(O,r), is a f u n c t i o n o f t h e two p a r a m e t e r s o f t h e w o r l d - s h e e t , a a n d T, a n d /J ( = 0,1,...9) is a s p a c e - t i m e v e c t o r index.
In
superstring
coordinates
oa(a,T)
theories
which
are
there Weyl
are
additionally one
spinets
(which
d i m e n s i o n s ) l a b e l l e d b y t h e i n d e x a = 1,2,...16.
have
or
two
16
anticommuting
components
in
ten
These spinor coordinates embody the
s u p e r s y m m e t r y of t h e t h e o r y (X~ a n d e a a r e s u p e r s p a c e c o o r d i n a t e s ) . The
classical
dynamics of
a
relativistic
string
is
obtained
principle t h a t g e n e r a l i z e s t h a t of a relativistic point particle.
from
an
action
J u s t as the action for
a r e l a t i v i s t i c p o i n t p a r t i c l e is t h e l e n g t h of i t s w o r l d - l i n e , t h e a c t i o n f o r a s t r i n g is taken
to b e p r o p o r t i o n a l to t h e a r e a
quantity
which
parametrized.
does In t h e
not
depend
of t h e on
w o r l d - s h e e t 9.
the
way
case of the s u p e r s t r i n g
in
This
which
theories the
is a g e o m e t r i c a l
the
world-sheet
is
notion of the a r e a
is
g e n e r a l i z e d so t h a t , r o u g h l y s p e a k i n g , t h e a c t i o n i s p r o p o r t i o n a l t o t h e a r e a o f t h e world-sheet
in
superspace.
The
fact
that
the
action,
S, is i n d e p e n d e n t
of
the
p a r a m e t r i z a t i o n o f t h e t w o - d i m e n s i o n a l w o r l d - s h e e t m a k e s i t like a t h e o r y o f g r a v i t y in t h e t w o - d i m e n s i o n a l a - r s p a c e 10
S = I 4m dT ¢~ n ~ g¢~ 8~X~ gBXu + e terms w h e r e gcq~ is a t w o - d i m e n s i o n a l m e t r i c
(5)
(c%~--O,T) a n d
g is i t s d e t e r m i n a n t .
This
m e t r i c is a n o n - d y n a m i c a l a u x i l i a r y f i e l d in two d i m e n s i o n s w h i c h c a n b e e l i m i n a t e d by substituting
t h e s o l u t i o n o f i t s e q u a t i o n o f motion b a c k i n t o t h e a c t i o n .
t e r m s in (5) a r e
d e s i g n e d to e n s u r e
the s u p e r s y m m e t r y of the action.
The e
The a b o v e
a c t i o n d e s c r i b e s a s t r i n g m o v i n g in flat t e n - d i m e n s i o n a l Minkowski s p a c e w h e r e r ~ is
the
flat
space-time
metric.
g e n e r a l i z a t i o n s to b a c k g r o u n d the
compactified
string
has
concerned
s p a c e s w i t h six c o m p a c t i f i e d d i m e n s i o n s .
Requiring
theory
Much
to
be
work
of
consistent
recent
puts
months
severe
restrictions
on
the
p o s s i b l e b a c k g r o u n d s p a c e - t i m e s a s I will d e s c r i b e l a t e r . An i m p o r t a n t f e a t u r e of t h e a c t i o n , in a d d i t i o n to t h e m a n i f e s t r e p a r a m e t r i z a t i o n i n v a r i a n c e , is i n v a r i a n c e u n d e r function. (the
rescalings g~
-~ Ag¢xB w h e r e A(a,r) is a n a r b i t r a r y
T h e s e s y m m e t r i e s allow t h e c h o i c e of a c l a s s of g a u g e s in w h i c h g0~ = 1
conformal
gauges}
and
eonformal transformations.
in This
which
the
theory
is
conformal invarianee
invariant
plays
under
a crucial
(pseudo}
role in
the
c o n s i s t e n c y o f t h e q u a n t u m m e c h a n i c s o f a s i n g l e f r e e s t r i n g in e n s u r i n g t h a t t h e s t a t e s c r e a t e d b y t h e t i m e - l i k e o c i l l a t i o n s of t h e s t r i n g d e c o u p l e f r o m t h e p h y s i c a l space of s t a t e s . are
This is i m p o r t a n t s i n c e t h e t i m e - l i k e m o d e s h a v e n e g a t i v e n o r m a n d
therefore ghost
states.
The c h o i c e of s u c h
a gauge
i s o n l y p o s s i b l e in t h e
q u a n t u m t h e o r y in t h e c r i t i c a l d i m e n s i o n w h i c h is t e n f o r s u p e r s t r i n g is
conceivable
that
superstring
theories
could
be
obtained
in
theories.
lower
It
dimensions
(D = 3, Lt o r 6) b y t e c h n i q u e s a d v o c a t e d b y P o l y a k o v 11 b u t t h a t i s s u e is s o m e w h a t murky at p r e s e n t . purely
transverse
Only in t e n just
as
gauge
dimensions are fields
are
the physical modes of the
transversely
polarized
in
string
Yang-Mills
139 theories. The
solutions
e x p a n d e d in a n
of
the
classical
equations,
derived
from
the
action,
can
i n f i n i t e s e t of normal m o d e s w h i c h c a n
then
be q u a n t i z e d .
be The
s p e c t r u m d e p e n d s on the b o u n d a r y conditions. A string
with free e n d p o i n t s can c a r r y
internal quantum
w i t h a c l a s s i c a l g r o u p 12 (SO(n), U(n) o r U S p ( n ) ) .
numbers associated
The c h a r g e s a r e a t t a c h e d t o t h e
e n d s o f t h e s t r i n g ( r a t h e r like t h e old p i c t u r e of a m e s o n a s a s t r i n g w i t h a q u a r k at one end and an a n t i - q u a r k
at the other).
It turns
out that a string with free
e n d p o i n t s has a massless v e c t o r particle among its massless states.
This a p p a r e n t l y
a c c i d e n t a l f e a t u r e is t h e r e a s o n w h y s t r i n g t h e o r i e s r e d u c e to Yang-Mills t h e o r i e s in t h e l o w - e n e r g y limit 13 ( w h e n all t h e m a s s i v e s t a t e s e f f e c t i v e l y d e c o u p l e ) .
A closed associated
with
string the
contains
fact
that
a massless the
low
spin-2
energy
particle
effective
which theory
is
the
graviton
contains
general
r e l a t i v i t y 14. When t h e
interactions
between
strings
u n i f i c a t i o n b e t w e e n g r a v i t y a n d Yang-Mills. ( k n o w n a s t y p e I t h e o r i e s ) two o p e n s t r i n g s
are
included
there
is
a remarkable
In the t h e o r i e s c o n t a i n i n g o p e n s t r i n g s interact by joining at their endpoints
to f o r m a s i n g l e o p e n s t r i n g o r a n o p e n s t r i n g s p l i t s i n t o two s t r i n g s .
Fig.(1) T h i s is a local i n t e r a c t i o n a n d
consistency requires
t h e same i n t e r a c t i o n to c o u p l e
t h e two e n d s o f a s i n g l e o p e n s t r i n g to f o r m a c l o s e d s t r i n g a s i l l u s t r a t e d b y
Fig.(2) so t h a t t h e e x i s t e n c e o f o p e n s t r i n g s existence
of
closed
strings
(and
( a n d h e n c e t h e Yang-Mills s e c t o r ) r e q u i r e s t h e
hence
the
gravity
sector).
The
gravitational
140
c o n s t a n t j K, a n d t h e Yang-Mills c o u p l i n g , g, a r e r e l a t e d b y K ~ g2T. T h e r e a r e also t h e o r i e s w i t h o n l y c l o s e d s t r i n g s . describe
closed
strings
which
have
an
For example, t y p e II t h e o r i e s
orientation
i.e.
they
have
excitations
c o r r e s p o n d i n g to waves r u n n i n g a r o u n d the s t r i n g i n d e p e n d e n t l y in e i t h e r direction ( t h e II r e f e r s
to t h e f a c t t h a t t h e s e t h e o r i e s h a v e twice a s much s u p e r s y m m e t r y ) .
T h e s e t h e o r i e s may h a v e
no n e t c h i r a l i t y
( t y p e IIa) o r may b e c h i r a l
( t y p e IIb).
The l a t t e r t h e o r y is s t r i k i n g s i n c e i t s low e n e r g y limit y i e l d s a p o i n t f i e l d t h e o r y 15 w h i c h is f r e e from all g r a v i t a t i o n a l a n o m a l i e s 16. have an internal symmetry group and
H o w e v e r , t y p e II t h e o r i e s do n o t
so do n o t r e d u c e in a n y o b v i o u s way to a
chiral four-dimensional theory. The m o s t i n t e r e s t i n g k i n d of s u p e r s t r i n g
t h e o r y is t h e h e t e r o t i c s u p e r s t r i n g 17,
This d e s c r i b e s c l o s e d s t r i n g s w h i c h c a r r y i n t e r n a l s y m m e t r y ( u n l i k e t h e o t h e r c l o s e d superstring
theories} with
string.
These
running
around
charges
theories are the string
built
which are from
smeared out
m o d e s of t h e
densities along
the
ten-dimensional superstring
in o n e s e n s e w i t h m o d e s o f t h e 2 6 - d i m e n s i o n a l b o s o n i c
s t r i n g t h e o r y r u n n i n g a r o u n d in t h e o t h e r s e n s e : o f d i m e n s i o n a l i t i e s is
as
reconciled by
the
This a p p a r e n t l y
bizarre mixture
i d e n t i t y 18 26 = 10+16 w h e r e t h e f i r s t
ten
d i m e n s i o n s o f t h e r i g h t p o l a r i z e d m o d e s a r e t a k e n to be t h e s p a c e - t i m e d i m e n s i o n s . The
other
sixteen
dimensions
become i n t e r n a l
associated with a sixteen dimensional lattice. this
lattice
to
be
even
and
self-dual.
coordinates
forming
a
hypertorus
The c o n s i s t e n c y of t h e t h e o r y r e q u i r e s
There
are
known
to
l a t t i c e s 19 w h i c h a r e r e l a t e d to t h e r o o t l a t t i c e s o f t h e g r o u p s
be
only
two
such
E 8 x E 8 a n d SO(32)
(or, more accurately, the group (Spin 32)/Z 2 which has the same algebra as SO(32)). Therefore
the heterotic string
were already k n o w n
theory is only consistent
for the two groups
to be selected by requiring the absence of anomalies.
that
In the
heterotic string theory K ¢¢ g/CT.
SUPERSTRING INTERACTIONS
Superstring
s c a t t e r i n g a m p l i t u d e s c a n b e c a l c u l a t e d in p e r t u r b a t i o n
theory by
c o n s t r u c t i n g a s e r i e s o f d i a g r a m s t h a t g e n e r a l i z e t h e F e y n m a n d i a g r a m s o f familiar point field theories.
Tree Diagrams The t r e e a p p r o x i m a t i o n to t h e s c a t t e r i n g a m p l i t u d e of, f o r example, f o u r c l o s e d s t r i n g s is r e p r e s e n t e d two o u t g o i n g s t r i n g s .
b y a c o n t i n u o u s w o r l d - s h e e t t h a t j o i n s t h e two i n c o m i n g a n d
141
Fig.(3) This diagram describes
two i n c o m i n g c l o s e d s t r i n g s
which join together
at a point to form one intermediate closed string which subsequently two f i n a l s t r i n g s
(time is t a k e n to b e i n c r e a s i n g
by touching
splits into the
f r o m l e f t to r i g h t ) .
I t is p o s s i b l e
t o d e r i v e t h e a m p l i t u d e f o r d i a g r a m s l i k e fig.(3) e i t h e r b y a s t r i n g g e n e r a l i z a t i o n of Feynman's path integral approach formalism expressed strings. theory
Unfortunately, for
understanding calculations.
configuration.
the
geometric
A string
create
and
destroy
light-cone
structure
of t h e
field, ~[X(a),e(a)],
important
aspect
gauge)
takes
complete
which
theory
but
is a f u n c t i o n a l
of c l o s e d
string
the world-sheet
place) corresponding
is
not
suffices of t h e
theories
only involve terms which are cubic in closed-string
can be seen by slicing through the interactions
g a u g e 21 ( t h e
A particularly
that the interactions
fields which
f o r t h e m o m e n t t h e o n l y c o m p l e t e f o r m u l a t i o n of t h e f i e l d
perturbative
string
to q u a n t u m m e c h a n i c s o r f r o m a s e c o n d - q u a n t i z e d
of s t r i n g
of s t r i n g s 20 is i n a s p e c i a l
satisfactory for
in terms
is
f i e l d s (as
of fig.(3) a t t h e p l a c e w h e r e o n e of
to t h e local j o i n i n g o r s p l i t t i n g of t h e
strings
Fig.(4) ( w h i c h is t h e c l o s e d - s t r i n g are
no higher
gravity
based
order on
contact the
interaction
terms
gray/tons.
All t h e s e
arising vertices.
from
the
a n a l o g u e of t h e o p e n - s t r i n g interactions
Einstein-Hilbert
involving
contact
contact
exchange
of
whereas action
there
interactions
i n t e r a c t i o n of fig.(1).
in the perturbative are
between
terms
emerge
as
the
massive
string
low e n e r g y states
an
infinite
arbitrary effective between
T h i s is a n a l o g o u s t o t h e w a y i n w h i c h t h e f o u r - F e r m i
There
treatment
of
number
of
numbers
of
interactions cubic
string
model of t h e w e a k
i n t e r a c t i o n s i s now k n o w n to e m e r g e a s a n e f f e c t i v e t h e o r y f r o m t h e W e i n b e r g - S a l a m t h e o r y a t e n e r g i e s m u c h l e s s t h a n t h e W o r Z b o s o n mass.
142 Fig.(3) g e n e r a l i z e s t h e f o u r - g r a v i t o n amplitude
of E i n s t e i n ' s
states).
(since
the
graviton
is o n e of t h e
=rP°intlresultfield t h e o r y }
s,t,u
are
the Mandelstam
x
string
r'(1-8~T)r'(1-8~T)F(1-8~ T)
invariants
defined
by
(6)
s = ( p l + P 2 ) 2, t = ( p l + P 4 ) 2,
and u = (pl+P3) 2 where Pl,P2,P3,P4, are the external momenta). all t h e
massless
T a k i n g t h e e x t e r n a l s t a t e s to b e g r a v i t o n s t h i s a m p l i t u d e is g i v e n b y
T(s,t,u)
(where
theory
t r e e d i a g r a m c o n t r i b u t i o n to t h e s c a t t e r i n g
string
features
are contained in the F functions.
s,t,u ~ T the expression
manifestly reduces
In this expression
I n t h e low e n e r g y
to t h e f a m i l i a r r e s u l t
limit
based on general
relativity.
Loop D i a g r a m s
Higher order loop
diagrams
world-surface represented
diagrams in the perturbation
analogous
to
those
of
point
e x p a n s i o n a r e g i v e n b y a s e r i e s of field
theories.
For
example,
the
of t h e o n e - l o o p c o n t r i b u t i o n to t h e s c a t t e r i n g of f o u r c l o s e d s t r i n g s i s b y e i t h e r of t h e d i a g r a m s .
Ca)
(b)
Fig.(~) T h e r u l e s of s t r i n g t h e o r y m a k e t h e s e two d i a g r a m s e q u i v a l e n t b e c a u s e t h e y c a n b e distorted
into
world-sheet
each other.
is
a
striking
This
property
a n a l o g u e in p o i n t field t h e o r y . 5(b)
looks
theories
like
there
interpretation propagator set
a tadpole
equivalence of
between
string
different
theories
{"duality")
Fig. 5(a) l o o k s like a s t r i n g
has
no
spinning
string simple
in
configuration
the
the original
which
box d i a g r a m while fig.
This infinity has a very
of
In
of t h e
is a n i n f i n i t y 22 i n t h i s a m p l i t u d e . terms
diagram.
distortions
tadpole
diagram
bosonie
and
(fig.
5(b)).
The
i n t h e leg of t h e t a d p o l e is s i n g u l a r s i n c e i t i n c l u d e s ( a m o n g a n i n f i n i t e
of s t a t e s )
the
contribution
which has the form 1/k 2 where conservation.
from
the
massless
to s t r i n g
partner
of
the
graviton
t h e m o m e n t u m i n t h e leg, k, i s z e r o b y m o m e n t u m
T h i s d e s c r i p t i o n of t h e d i v e r g e n c e
e f f e c t is u n i q u e
scalar
theories.
The
of a loop d i a g r a m a s a n i n f r a - r e d
discovery
that
the
t h e o r i e s a r e f i n i t e a t o n e loop 23 was t h e f i r s t i n d i c a t i o n t h a t
type
II s u p e r s t r i n g
superstring
theories
143
might
be
consistent
evaluated
in
ultra-violet
10
quantum
dimensions
divergences).
heterotic superstring
field where
This
theories
(remember
ordinary
result
point
that
field
h a s also r e c e n t l y
loop
integral
have
is
terribly
been established for
the
t h e o r y 24.
I t h a s a l s o now b e e n e s t a b l i s h e d t h a t t h e o p e n - s t r i n g gauge group
the
theories
one-loop amplitudes with
SO(32) a r e f i n i t e w i t h f o u r 25 (or m o r e 26) e x t e r n a l s t a t e s a n d i n f i n i t e
for any other gauge group. It
is
associated
probable with
the
that
any
possible
emission of
generalized tadpoles.
divergences
massless
scalar
at
higher
particles
at
loops zero
can
also
momentum
be via
F o r example, a t t w o l o o p s t h e d i v e r g e n t t a d p o l e c o n t r i b u t i o n
to a c l o s e d - s t r i n g a m p l i t u d e is r e p r e s e n t e d
b y t h e ("E.T.") d i a g r a m .
Fig.(6) From t h i s it follows t h a t t h e c o n d i t i o n f o r a n a m p l i t u d e to be f i n i t e a t a n y n u m b e r of loops is t h a t ~
= 0 where
~
is t h e o n - s h e l l c o u p l i n g o f t h e
m a s s l e s s s c a l a r p a r t i c l e to t h e g e n e r a l t a d p o l e . requirement that there
be an u n b r o k e n
o n l y b e b r o k e n in p e r t u r b a t i o n
But t h i s c o n d i t i o n is p r e c i s e l y t h e
supersymmetry.
Since s u p e r s y m m e t r y can
t h e o r y in t e n d i m e n s i o n s if t h e r e a r e a n o m a l i e s it
follows t h a t f r e e d o m f r o m a n o m a l i e s *-~ f i n i t e n e s s 27. I t is i m p o r t a n t to e s t a b l i s h b y e x p l i c i t c a l c u l a t i o n w h e t h e r t h e t h e o r i e s a r e f i n i t e to all o r d e r s . complete
This
is
being
p r o o f of f i n i t e n e s s
intensively (at
least
for
studied the
at
type
the II
present and
time28, 29 a n d
heterotic
a
superstring
t h e o r i e s ) s h o u l d b e f o r t h c o m i n g in t h e n e a r f u t u r e .
A N O M A L I E S A N D THEIR C A N C E L L A T I O N
The s i g n a l f o r a n a n o m a l y is t h e p r e s e n c e of a n o n - z e r o c o u p l i n g b e t w e e n a n u n p h y s i c a l l o n g i t u d i n a l mode of a g a u g e p a r t i c l e a n d a n y p h y s i c a l t r a n s v e r s e Just
a s in f o u r
dimensions an
dimensions an anomaly can arise
anomaly
can
arise
from a
fermions and external gauge particles.
hexagon
from a triangle diagram
modes.
diagram, in ten
with c i r c u l a t i n g
ehiral
F o r example, t h e e v a l u a t i o n o f t h e a n o m a l y in
t h e Yang-Mills c u r r e n t in a t y p e I t h e o r y r e q u i r e s t h e c a l c u l a t i o n of
144
Fig.(7) where the i n t e r n a l lines are o p e n s t r i n g p r o p a g a t o r s and one of the external s t a t e s is a
longitudinal
modes.
mode o f
a
Yang-Mills p a r t i c l e
while
the
others
are
transverse
The r e s u l t of t h i s c a l c u l a t i o n is t h a t t h e anomaly v a n i s h e s 3 w h e n t h e g a u g e
g r o u p is SO(32). T h i s may a p p e a r p u z z l i n g b e c a u s e t h e h e x a g o n d i a g r a m s o f t h e c o r r e s p o n d i n g low e n e r g y
massless
point
field
theory
do
not
give
a vanishing
anomaly.
The
explanation is b as ed on the fact t h a t c e r t a i n o p e n - s t r i n g hexagon diagrams contain closed-string
bound
states
in v a r i o u s
massless s t a t e s of the s u p e r g r a v i t y (in w h i c h all t h e m a s s i v e s t a t e s
channels.
These bound
states
include
the
sector which means that the low-energy theory
d e c o u p l e ) g e t s c o n t r i b u t i o n s f r o m t h e s e s t a t e s in
addition to the expected anomalous c o n t r i b u t i o n from the massless hexagon diagrams. These
extra
terms
particles are
have
the
exchanged and
form
of
tree
diagrams
in
which
the
supergravity
w h i c h h a v e anomalies w h i c h e x a c t l y c a n c e l t h e
a n o m a l y of t h e m a s s l e s s h e x a g o n d i a g r a m s .
usual
This explains the a b s e n c e of an anomaly
in t h e l a n g u a g e o f t h e l o w - e n e r g y p o i n t f i e l d t h e o r y a s b e i n g d u e to a c a n c e l l a t i o n b e t w e e n t h e e x p e c t e d q u a n t u m a n o m a l y ( d u e to t h e u s u a l h e x a g o n d i a g r a m s ) a n d a new, a n o m a l o u s , t e r m in t h e c l a s s i c a l t h e o r y
(associated with these tree diagrams).
The new t e r m c a n b e t h o u g h t of a s a n a d d i t i o n a l (anomalous) t e r m in t h e e f f e c t i v e p o i n t f i e l d t h e o r y a c t i o n w h i c h is a local polynomial in t h e f i e l d s .
This m e c h a n i s m
d e p e n d s o n a d e l i c a t e i n t e r p l a y b e t w e e n g r a v i t a t i o n a l a n d Yang-Mills e f f e c t s . The a n o m a l i e s a s s o c i a t e d w i t h g r a v i t a t i o n a l c u r r e n t s with
external
gravitons)
and
the
mixed a n o m a l i e s
( d u e to h e x a g o n d i a g r a m s
(due
to h e x a g o n d i a g r a m s
with
e x t e r n a l Yang-MiUs p a r t i c l e s t o g e t h e r w i t h g r a v i t o n s ) h a v e n o t y e t b e e n c a l c u l a t e d in t h e
superstring
theory
was
theories.
carried
anomalies can
out
However, t h e a n a l y s i s of t h e low e n e r g y
for
be cancelled
these
a n o m a l i e s also 3.
by adding
local a n o m a l o u s
It
turned
terms
out
p o i n t field
that
to t h e a c t i o n
all
the
if t h e
Yang-Mills g r o u p , G, is s u c h t h a t : (a)
The d i m e n s i o n of t h e a d j o i n t r e p r e s e n t a t i o n of G - 496 (which g u a r a n t e e s
the a b s e n c e of the gravitational anomalies). (b)
An a r b i t r a r y
T r F 6 _- 1
48
matrix, F, in t h e a d j o i n t r e p r e s e n t a t i o n of G
TrF 2 (TrF 4 _
1
300
(TrF2)2)
satisfies (7)
145
(which
ensures
the
absence
of
the
Yang-Mills
and
mixed
anomalies).
The
only
g r o u p s f o r w h i c h t h e s e c o n d i t i o n s a r e s a t i s f i e d a r e S0(32) a n d E8 x E8 ( a p a r t f r o m t h e p r e s u m a b l y u n i n t e r e s t i n g c a s e s (U(1) 496 a n d E 8 x U(1)248).
The t y p e I t h e o r i e s
do n o t a d m i t e x c e p t i o n a l g r o u p s b u t t h e h e t e r o t i c s t r i n g i n c o r p o r a t e s b o t h of t h e m . F o r c o m p l e t e n e s s i t w o u l d b e d e s i r a b l e f o r t h e a n a l y s i s of p o s s i b l e s u p e r s y m m e t r y a n o m a l i e s to b e c a r r i e d o u t 30. The p r e c e d i n g gauge
discussion referred
transformations.
There
transformations which are
to a n o m a l i e s a s s o c i a t e d w i t h i n f i n i t e s s i m a l
is still t h e
p o s s i b i l i t y of a n o m a l i e s in
the
n o t c o n t i n u o u s l y c o n n e c t e d to t h e i d e n t i t y .
"large"
There are,
f o r example, k n o w n to b e 991 t y p e s of l a r g e g e n e r a l c o o r d i n a t e t r a n s f o r m a t i o n s in ten dimensional spherical space-time. for
the
cases
anomalies31.
in
which
the
gauge
T h e s e h a v e b e e n s h o w n n o t to b e a n o m a l o u s group
is
one
of
those
free
of
infinitessimal
T h i s r e s u l t h a s a l s o b e e n g e n e r a l i z e d 32 with some a s s u m p t i o n s to more
general spaces than the ten-sphere
in t h e c a s e of t h e E 8 x E 8.
COMPACTIFICATION OF EXTRA DIMENSIONS
The
structure
of s p a c e - t i m e a n d
hence
the
p o s s i b i l i t y o f c o m p a c t i f i c a t i o n to f o u r
d i m e n s i o n s s h o u l d b e d e t e r m i n e d b y t h e s o l u t i o n of t h e e q u a t i o n s of t h e s u p e r s t r i n g field t h e o r y (which h a s so f a r o n l y b e e n f o r m u l a t e d in t h e l i g h t - c o n e g a u g e ) . approach has not yet been productive.
t h a t s u g g e s t t h a t r e a l i s t i c f o u r - d i m e n s i o n a l p h y s i c s may well e m e r g e . are
based on topological features
features
This
However, t h e r e are many o t h e r o b s e r v a t i o n s
of t h e t h e o r y .
of t h e a n o m a l y c a n c e l l a t i o n a r g u m e n t
Many of t h e s e
For example, o n e of t h e c r u c i a l
d i s c u s s e d a b o v e is t h e p r e s e n c e
of
t h e m a s s l e s s s e c o n d - r a n k a n t i s y m m e t r i c t e n s o r f i e l d B~rv w h i c h h a s a f i e l d s t r e n g t h H~lj9 d e f i n e d b y
H~9 = a l i v e ]
___ly + L 30 ~ e w~
(8)
where [ ] d e n o t e s o n t i s y m m e t r i z a t i o n o f t h e i n d i c e s and t h e Chern-Simons t e r m s a r e Y d e f i n e d by ~[o~jjvp]=TrF[o~Fljg] (where t h e Y a n g - M i l l s f i e l d s t r e n g t h F ~ i s a m a t r i x L i n t h e a d j o i n t r e p r e s e n t a t i o n o f t h e gauge group) and a[o~/2v@]=trR[o~vp] (where t h e ]nil Riemann c u r v a t u r e R/2v i s a m a t r i x in t h e t a n g e n t s p a c e w i t h m , n = 0 , 1 , . . . 9 ) . The c o n d i t i o n t h a t H/2v~ s h o u l d b e s i n g l e - v a l u e d is t h a t t h e i n t e g r a l of i t s c u r l o v e r a n y f o u r - d i m e n s i o n a l s u b s p a c e s h o u l d v a n i s h 33, i.e.
(trR[o~Rv@]
~0 TrF[°~Fv@]) = 0
(9)
T h i s c o h o m o l o g y c o n s t r a i n t h a s two immediate a n d i m p o r t a n t c o n s e q u e n c e s : (a)
It is t h e
s h o w n to v a n i s h
condition that
ensures
that
the
in f l a t t e n - d i m e n s i o n a l s p a c e
anomalies which were
continue
to
be a b s e n t
previously when
some
146
d i m e n s i o n s are curved. (b) It indicates that in general w h e n there is non-zero curvature, so that R~aj*0, the Yang-Mills
field strength
is also non-zero,
i.e. F~#0.
This
is just what
is
needed, since a non-zero field strength will lead to a breaking of the (very large) gauge
group
of the ten-dimensional theory to a smaller group in the compactified
theory (which will hopefully be of more direct interest for physics).
TOWARDS
FOUR-DIMENSIONAL
A
particularly
proposed
in
PHYSICS
interesting
ref. 34
in
class of possible
which
the
compactified
ten-dimensional
space
is
the
product
with vanishing Ricci curvature.
The holonomy group of this compact six-dimensional
that
leads
supersymmetry. spaces Although
(they
to
a
space
is an SU(3) matrix in the tangent space - a
four-dimensional
theory
possessing
an
unbroken
Such compact Ricci-flat spaces have come to be called "Calabi-Yau" were
conjectured
the original motivation
interest was
compact
of
space and
space is SU(3) i.e. the curvature, R ~ ,
six-dimensional
been
four-dimensional flat Minkowski
condition
a curved
solutions has
to exist by
Calabi and
for suggesting
shown
to exist by
that Ricci-flat spaces
may
Yau). be of
based on analysing the effective point field theory that approximates
the superstring at low energy, it is possible to argue convincingly that they are also solutions of the full string theory. The equations of the low energy effective point field theory also suggest that the curvature and
the Yang-Mills field strengths
should be proportional (which is
yet another example of the unification of the gravitational and Yang-Mills aspects of the theory), i.e.
R~, ~ F ~ (suppressing
(i0) the matrix indices) so that the field strength is also non-zero in an
SU(3) subgroup of E 8 x E 8 (the SO(32) case does not appear to hold m u c h prospect of describing physics).
As a result the symmetry
is broken
down
to a subgroup
that commutes with that SU(3), namely
E6 × E8 The
E6
factor
(11) plays
four-dimensional theory
the
of
a
Grand
Unified
group
for
(familiar from t h e P h e n o m e n o l o g y o f t h e
with certain novel features. has been dubbed
r61e
the
effectively
m i d - 1 9 7 0 ' s 35)
but
The E 8 f a c t o r d e s c r i b e s a n o t h e r s e c t o r o f m a t t e r ( t h a t
' s h a d o w m a t t e r ' ) c o n s i s t i n g of p a r t i c l e s w h i c h a r e n e u t r a l u n d e r
t h e E6 f o r c e s a n d t h e r e f o r e u n d e t e c t a b l e e x c e p t via t h e i r g r a v i t a t i o n a l i n t e r a c t i o n s w i t h t h e m a t t e r t h a t we o b s e r v e .
I t i s h i g h l y n o n - t r i v i a l t h a t i t is p o s s i b l e f o r t h e
147
symmetry
to
break
in
this
manner
and
yet
be
consistent
with
the
topological
c o n d i t i o n s implied b y eq.(9).
Consequences There isn't yet any systematic procedure spaces
but
it
has
been
conjectured
by
f o r c l a s s i f y i n g all p o s s i b l e C a l a b i - Y a u
Yau
that
there
are
a
discrete
but
large
n u m b e r of s u c h s p a c e s
( a p p r o x i m a t e l y 10000!) a n d s o it i s to b e h o p e d t h a t f u r t h e r
theoretical
will
constraints
Calabi-Yau space used
restrict
provide
encompasses known
such
all
spaces
restrictions
features.
close
to
that
for
making
contact
no
the
single
Nevertheless,
explaining
m a k e s u s e of s u c h r e m a r k a b l e f e a t u r e s potential
For
moment
the
particular
parameter
However, the phenomenological considerations
severe
desirable come
choice.
for t h e six compact d i m e n s i o n s is a d i s c r e t e
can be c h o s e n a r b i t r a r i l y . below
the
with
various
space
the
way
aspects
so
in
of
far
which
"low
that
described
constructed s o m e of
energy"
the
physics
t h a t it s e e m s l i k e l y t h a t t h e s c h e m e h a s t h e
physics.
Some
of
the
major
features
of
this
s c h e m e follow. (a)
U n l i k e w i t h c o n v e n t i o n a l k i n d s of u n i f i e d t h e o r i e s , t h e r e a r e f e w a d j u s t a b l e
parameters
once the Calabi-Yau space has been selected.
of s p e c i e s of m a s s l e s s p a r t i c l e s
(and
determined by a topological property characteristic).
An
aspect
phenomenology
is
four-dimensional
theory
consistent
that
with eq.(9)).
of
in
of t h e
this
the lie
hence
that
of fermion g e n e r a t i o n s )
is
chiral
complex
For example, t h e n u m b e r
number
is
s p a c e , n a m e l y , it i s e q u a l to ~ x ( E u l e r
scheme
massless the
the
E6
crucial
fermions
for in
representation
T h e r e is no f r e e d o m to a d j u s t
describing the
27
E6
effectively
(which
is
again
the particle c o n t e n t so t h i s
is a l r e a d y a notable s u c c e s s for t h e scheme. (b)
T h o s e s p a c e s w h i c h g i v e r i s e to a s m a l l n u m b e r o f f e r m i o n f a m i l i e s ( t h e E6
phenomenology restricts
t h e n u m b e r to b e t h r e e o r f o u r ) t u r n
them (they are not simply connected). E6 s y m m e t r y
to b r e a k
unified
point
field
adding
Higgs fields.
no s u c h the
adjustable
This in turn
symmetry
This cannot parameters.
holes in t h e compact space,
of t h e e f f e c t i v e Higgs fields a r e u p to a d i s c r e t e
symmetry group.
breaking
can
symmetry. be
theories
However, loops of E 6 flux can which has
the same effect as
In by
conventional arbitrarily
s i n c e t h e y allow
become trapped having
in
an effective
With t h i s t o p o l o g i c a l m e c h a n i s m t h e v a l u e s
d e t e r m i n e d , u p to a d i s c r e t e c h o i c e , w h i c h in t u r n
choice, t h e way in which E6 b r e a k s
A m o n g p o s s i b l e low e n e r g y
SU(3) x SU(2) x U(1) x U(1) x U(1),
symmetry groups
to a low e n e r g y
a r e 36
SU(3) x SU(2) x S U ( 2 ) , ... A g e n e r i c f e a t u r e of
t h i s m e c h a n i s m is t h a t t h e r e i s a l w a y s e x t r a low e n e r g y standard
introduced
be d o n e i n s u p e r s t r i n g
Higgs field in the adjoint representation.
determines,
p l a y s a k e y rble in allowing t h e
d o w n to a r e a l i s t i c low e n e r g y
theories
o u t to h a v e h o l e s i n
s y m m e t r y in a d d i t i o n to t h e
m o d e l ( t h e r e s i d u a l g r o u p h a s to h a v e a t l e a s t r a n k 5).
The m e c h a n i s m of b r e a k i n g s w i t c h e d off.
t h e s y m m e t r y by flux loops c a n n o t be c o n t i n u o u s l y
T h i s m e a n s t h a t t h e E6 s y m m e t r y i s n e v e r a n e x a c t s y m m e t r y o f t h e
148
four-dimensional theory, even at so-called
Grand
Unified
high energy
schemes}.
(in c o n t r a s t
Although
there
is
to i t s p r e v i o u s r61e in
unification
of
the
gauge
c o u p l i n g s t h e Yukawa c o u p l i n g s do n o t s a t i s f y t h e E 6 r e l a t i o n s w h i c h a v o i d s some of t h e bad p r e d i c t i o n s of E6 m a s s r e l a t i o n s . (c)
All t h e c o u p l i n g s of t h e m a s s l e s s p a r t i c l e s a r e d e t e r m i n e d b y t o p o l o g i c a l
c o n s i d e r a t i o n s 37 a n d Calabi-Yau s p a c e
do n o t
depend
on
detailed
knowledge of the
metric of
the
{which is j u s t a s welt s i n c e n o n e of t h e m e t r i c s of t h e s e s p a c e s
has ever been constructed!). t h e r e is no a p p a r e n t
A possible problem arises with proton stability since
r e a s o n f o r t h e Yukawa c o u p l i n g s r e s p o n s i b l e f o r p r o t o n d e c a y
a t t h e t r e e l e v e l to v a n i s h .
N e v e r t h e l e s s , e x p l i c i t c a l c u l a t i o n of t h e s e c o u p l i n g s in
many o f t h e k n o w n s p a c e s s h o w s t h a t f o r most o f t h e m t h e p r o t o n is s t a b l e 38 (up to the usual considerations about decay caused by radiative corrections}. {d)
T h e r e a r e p o s s i b l e s t a t e s a s s o c i a t e d with s t r i n g s w i n d i n g t h r o u g h h o l e s in
t h e c o m p a c t s p a c e w h i c h would lead to s t a b l e , u n c o n f i n e d p a r t i c l e s w i t h f r a c t i o n a l e l e c t r i c c h a r g e s w i t h m a s s e s a r o u n d t h e P l a n c k m a s s 39 ( t h e v a l u e o f t h e f u n d a m e n t a l c h a r g e d e p e n d i n g on w h i c h Calabi-Yau s p a c e is used}.
Correspondingly, magnetic
m o n o p o l e s a r e p r e d i c t e d b y t h e t h e o r y w h i c h h a v e c h a r g e s w h i c h a r e a multiple of t h e u s u a l Dirac v a l u e . (e)
The p o i n t o f v i e w a d o p t e d in t h i s whole s c e n a r i o is t h a t m u c h p h y s i c s c a n
b e o b t a i n e d f r o m t r e a t i n g t h e low e n e r g y e f f e c t i v e p o i n t field t h e o r y in l o w e s t o r d e r in p e r t u r b a t i o n understanding theory
has
theory. of the
the
understand
However, c e r t a i n
theory.
form of a
how
The
aspects
no-scale supergravity
supersymmetry
clearly require
a much
deeper
effective four-dimensional low-energy classical
gets
broken.
t h e o r y 40 a n d One
it is i m p o r t a n t
s u g g e s t i o n 41
c o n d e n s a t e of t h e g l u i n o s in t h e " s h a d o w " E8 s e c t o r w h i c h t r i g g e r s
invokes
to a
a b r e a k i n g of
t h e s u p e r s y m m e t r y w i t h o u t l e a d i n g to t h e u s u a l p r o b l e m o f g e n e r a t i n g a cosmological constant.
Despite this virtue, the suggested mechanism requires a non-perturbative
e f f e c t t h a t g o e s o u t s i d e of t h e a p p r o x i m a t i o n s o n w h i c h t h e s c h e m e is b a s e d . (f)
(Mildly) p e s s i m i s t i c n o t e .
well-motivated
provided
the
radius
The c o n s i d e r a t i o n of Calabi-Yau s p a c e s c a n b e of
the
compact
dimension
(which
is
a
free
p a r a m e t e r t h a t s h o u l d b e a p p r o x i m a t e l y t h e i n v e r s e of t h e u n i f i c a t i o n mass) i s m u c h more t h a n t h e P l a n c k scale.
The c a l c u l a t i o n s also a s s u m e t h a t t h e s t r i n g c o u p l i n g
c o n s t a n t ( w h i c h s h o u l d u l t i m a t e l y b e d e t e r m i n e d b y t h e t h e o r y ) is weak.
There are
c o n v i n c i n g a r g u m e n t s 42 t h a t n e i t h e r of t h e s e a p p r o x i m a t i o n s c a n b e v a l i d a n d t h a t superstring
theory
is i n t r i n s i c a l l y a s t r o n g l y
coupled theory,
It is i m p o r t a n t to
e s t a b l i s h , in t h a t c a s e w h e t h e r t h e r e s u l t s b a s e d o n t o p o l o g i c a l c o n s i d e r a t i o n s m i g h t still h a v e some v a l i d i t y . Since s u p e r s t r i n g
theories are
so v e r y
d i f f e r e n t f r o m p o i n t f i e l d t h e o r i e s it
would be more s a t i s f y i n g to f i n d a q u a l i t a t i v e l y n e w k i n d o f e x p e r i m e n t a l p r e d i c t i o n rather
than
trying
to
predict
details
of
presently
measured
accelerator
data.
[Examples of s u c h p r e d i c t i o n s a r e t h e e x i s t e n c e of e x t r a low e n e r g y s y m m e t r i e s a n d t h e i r associated gauge particles, the o c c u r r e n c e of u n c o n f i n e d fractionally c h a r g e d
149
p a r t i c l e s a n d t h e e x i s t e n c e of s h a d o w m a t t e r . ]
THEORETICAL DEVELOPMENTS (a) S t r i n g T h e o r i e s i n C u r v e d S p a c e - T i m e . The s t r i n g a c t i o n s o f t h e f o r m o f eq. (5), w h i c h d e s c r i b e t h e motion of a s t r i n g in
a flat
Minkowski
space
background
(with
t w o - d i m e n s i o n a l field t h e o r i e s of g r a v i t y coordinates}.
metric
~)
can
thought
of
as
(in w h i c h t h e " f i e l d s " a r e t h e s u p e r s p a c e
The t r e e d i a g r a m s of t h e c l o s e d - s t r i n g t h e o r y a r e a s s o c i a t e d w i t h a
two-dimensional
world-sheet
(as
in
t o p o l o g i c a l l y e q u i v a l e n t to a s p h e r e .
fig.
3)
which
is
a
perturbation
theory
a
closed
surface
that
is
The n - l o o p c o r r e c t i o n s ( i l l u s t r s t e d i n fig. 5)
c o r r e s p o n d to t w o - d i m e n s i o n a l s u r f a c e s w i t h n h a n d l e s . string
be
string
theory
is
T h e r e f o r e to a n y o r d e r in
equivalent
to
a
g r a v i t a t i o n a l t h e o r y e v a l u a t e d o n a manifold of p a r t i c u l a r g e n u s .
two-dimensional This v i e w p o i n t is
a t h e m e in m a n y i n t e r e s t i n g d e v e l o p m e n t s . (i)
The
heterotic
(world-sheet) coordinate
invariance and
with the other superstring (due
to t h e
noteworthy
s u p e r s t r i n g 17
all
these
ten-dimensional sense.
not
only
two-dimensional
two-dimensional supersymmetry
in common
t h e o r i e s b u t is also c h i r a l in t h e t w o - d i m e n s i o n a l s e n s e
asymmetric t r e a t m e n t of that
possesses
the
properties
right
are
C o n s i s t e n c y of
and
also
string
left
polarized
properties theory
of
the
requires
modes).
It
theory
it
to
be
in
is the
free
of
a n o m a l i e s in t w o - d i m e n s i o n a l c o o r d i n a t e t r a n s f o r m a t i o n s t h a t c a n n o t be C o n t i n u o u s l y connected
to
the
identity
transformations).
This
(these
is t h e
"large"
coordinate
transformations
are
i n g r e d i e n t in t h e h e t e r o t i c s u p e r s t r i n g
modular
theory
that
r e s t r i c t e d the possible g a u g e g r o u p s to j u s t t h o s e p r e v i o u s l y o b t a i n e d b y r e q u i r i n g t h e a b s e n c e o f t h e i n f i n i t e s s i m a l t e n - d i m e n s i o n a l c h i r a l anomalies.
Furthermore, this
g e n e r a l i z e s to t h e s i t u a t i o n in w h i c h t h e t e n - d i m e n s i o n a l s p a c e is c u r v e d t h e i d e n t i f i c a t i o n in eq.(10) is made 43.
provided
T h i s i d e n t i f i c a t i o n is also r e q u i r e d in o r d e r
to e n s u r e t h a t t w o - d i m e n s i o n a l c h i r a l a n o m a l i e s v a n i s h in t h e c o m p a c t i f i e d t h e o r y 44. (ii)
In o r d e r
to d e s c r i b e a s t r i n g
p r o p a g a t i n g in a c u r v e d
background
it is
n e c e s s a r y to r e p l a c e t h e f l a t m e t r i c in eq.(5) b y a c u r v e d m e t r i c , G ~ ( X ) , w h i c h is a function of
X.
This
gives
world-sheet into the curved formulating
consistent
the
action
space-time.
string
theories
of a
non-linear
sigma
model m a p p i n g
the
T h e r e a r e , h o w e v e r , s t r o n g c o n s t r a i n t s on in
a
curved
background
due
to
the
r e q u i r e m e n t t h a t t h e t h e o r y c a n be f o r m u l a t e d i n a p a r a m e t r i z a t i o n i n w h i c h it is c o n f o r m a l l y i n v a r i a n t ( r e c a l l t h a t t h i s w a s n e c e s s a r y to p r o v i d e t h e g a u g e c o n d i t i o n s r e q u i r e d to decouple t h e n e g a t i v e - n o r m e d states). presence
of o t h e r
terms
in
the
I n g e n e r a l t h i s will r e q u i r e t h e
two-dimensional action
involving,
in
addition
to
G~(X)9 a n a n t i s y m m e t r i c t e n s o r b a c k g r o u n d , B ~ ( X ) , a s c a l a r b a c k g r o u n d , ¢(X), a n d fermionic content
t e r m s 45. of
the
[These
string
field
background theory.]
fields In
correspond
addition,
in
to
the
order
for
massless one
of
field these
150
g e n e r a l i z e d sigma models to c o r r e s p o n d to a compac~ification o f t h e t e n - d i m e n s i o n a l superstring
the
transformations
algebra (the
satisfied
"Virasoro"
particular coefficient.
by
the
generators
algebra)
must
of t w o - d i m e n s i o n a l c o n f o r m a l
have
a
central
extension
with
a
The c o n d i t i o n t h a t s u c h a t h e o r y b e c o n f o r m a l l y i n v a r i a n t is
t h a t t h e r e n o r m a l i z a t i o n g r o u p ~ f u n c t i o n s s h o u l d all v a n i s h ( t h e r e is o n e ~ f u n c t i o n f o r e a c h k i n d of b a c k g r o u n d field). group
manifold
can
be
made
It is k n o w n t h a t a n o n - l i n e a r sigma model on a
conformally invarlant
by
the
addition
of
the
term
involving B~
if t h i s is n o r m a l i z e d to a s p e c i a l v a l u e 46 ( t h i s c o r r e s p o n d s to a d d i n g a
torsion term
that
parallelizes the curvature
of the
group
manifold).
H o w e v e r , it
s e e m s u n l i k e l y t h a t s u c h s t r i n g t h e o r i e s 47 c a n b e s u p e r s y m m e t r i c in s p a c e - t i m e a n d t h e y do n o t c o r r e s p o n d to a c o m p a c t i f i c a t i o n of a t e n - d i m e n s i o n a l t h e o r y ( t h e y h a v e t h e w r o n g v a l u e f o r t h e c e n t r a l e x t e n s i o n t e r m in t h e V i r a s o r o a l g e b r a ) . (iii)
The e q u a t i o n s implied b y t h e v a n i s h i n g of t h e #B f u n c t i o n s , a n d
hence by
t h e r e q u i r e m e n t of c o n f o r m a l i n v a r i a n c e , h a v e b e e n s t u d i e d f o r a wide c l a s s of s i g m a models in p e r t u r b a t i o n t h e o r y ( w h e r e t h e e x p a n s i o n p a r a m e t e r is t h e i n v e r s e s t r i n g tension).
As e x p e c t e d
b y g e n e r a l a r g u m e n t s 48 t h e s e e q u a t i o n s a r e
equations for the massless components of the s u p e r s t r i n g
just
t h e field
f i e l d s e x p a n d e d in a p o w e r
s e r i e s in t h e i n v e r s e s t r i n g t e n s i o n i.e. in a low e n e r g y e x p a n s i o n .
This a n a l y s i s
p r o v i d e s more e v i d e n c e , a t l e a s t in low o r d e r s in t h i s e x p a n s i o n , f o r t h e f a c t t h a t R i c c i - f l a t s p a c e s o f SU(3) h o l o n o m y (Calabi-Yau s p a c e s ) a r e s o l u t i o n s of t h e s t r i n g theory together strength (iv) which
with
t h e i d e n t i f i c a t i o n of t h e c u r v a t u r e
with the
Yang-Mills field
(eq.(lO)). T h e r e is now a p r o o f 49 t h a t
Gi/ij is
expansion
the
and
m e t r i c on
so
have
a
s u p e r s y m m e t r i c n o n - l i n e a r s i g m a models in
Calabi-Yau
vanishing
space
are
/3 f u n c t i o n
(and
finite are
to all o r d e r s therefore
invariant) and are suitable candidates for compactified superstring is also e v i d e n c e t h a t crucial
for
the
the
space
in
theories.
There
R i c c i - f l a t c o n d i t i o n is n o t b y i t s e l f s u f f i c i e n t b u t
to
be
Kahler 50
(which
is
equivalent
to
this
conformally
demanding
it is SU(3)
holonomy), t h u s l e n d i n g f u r t h e r s u p p o r t to t h e s c h e m e of ref.(34). iv)
S i n c e no m e t r i c o n a Calabi-Yau s p a c e h a s e v e r b e e n e x p l i c i t l y c o n s t r u c t e d
it is n o t p o s s i b l e to g i v e a n e x p l i c i t s o l u t i o n f o r a s u p e r s t r i n g d i m e n s i o n s a r e c o m p a c t i f i e d on s u c h a s p a c e .
t h e o r y in w h i c h six
However, c e r t a i n Calabi-Yau s p a c e s
r e d u c e , in a s i n g u l a r
limit, to s i x - d i m e n s i o n a l t o r i w i t h d i s c r e t e i s o m e t r i e s d i v i d e d
out.
These
spaces are
with
orbifold
singular
backgrounds
can
called be
"orbifolds".
analyzed
Superstring
explicitly
and
theories
behave
defined
as
if
the
b a c k g r o u n d w e r e a Calabi-Yau s p a c e ( t h e s i n g u l a r i t i e s of t h e o r b i f o l d a r e i r r e l e v a n t in t h e s t r i n g t h e o r y ) 51. (vi)
The c o v a r i a n t f o r m u l a t i o n of s u p e r s t r i n g
s t a r t i n g from t h e s p i n n i n g s t r i n g t h e o r y a n d
t h e o r i e s can e i t h e r be d e d u c e d b y
then truncating
to a s u p e r s y m m e t r i c
s u b s e t of s t a t e s (as m e n t i o n e d e a r l i e r ) o r f r o m a m a n i f e s t l y s u p e r s y m m e t r i c a c t i o n in superspace
as
implied
by
g e o m e t r i c a l i n t e r p r e t a t i o n 52.
eq.(5) 8. It
The
latter
formulation
has
a
h a s also b e e n g e n e r a l i z e d to c u r v e d
much
more
gravitational
151
b a c k g r o u n d s f o r t h e t y p e I t h e o r i e s 53 ( a n d , r e c e n t l y , f o r t h e t y p e II t h e o r i e s 5 4 ) .
It s e e m s u n l i k e l y
that
the
p h y s i c s o f two d i m e n s i o n s will d e t e r m i n e all t h e
c o n s t r a i n t s o n t h e t h e o r i e s a l t h o u g h it is r e m a r k a b l e how r e s t r i c t e d t h e p o s s i b i l i t i e s are
for
constructing
a
suitable
conformally
t w o - d i m e n s i o n a l manifold o f a r b i t r a r y string
effects must
play an
genus.
important
invarlant
sigma
A suggestion
rSle is t h a t
that
model
on
a
non-perturbative
f l a t t e n - d i m e n s i o n a l Minkowski
s p a c e s a t i s f i e s all t h e r e s t r i c t i o n s we k n o w of t h a t follow f r o m t h e t w o - d i m e n s i o n a l v i e w p o i n t a n d y e t we h o p e to p r o v e it is n o t a p o s s i b l e s o l u t i o n of t h e t h e o r y .
(b) T o w a r d s
a g a u g e - i n v a r i a n t field t h e o r y of s u p e r s t r i n g s .
I n o r d e r to a r r i v e a t a more g e o m e t r i c a l u n d e r s t a n d i n g is p r o b a b l y n e c e s s a r y step
was
the
understanding
Lorentz-covarlant
of s t r i n g f i e l d t h e o r y it
to f o r m u l a t e it i n a g a u g e - i n v a r i a n t
gauge
of
using
the
the
free
BRS
bosonic
manner.
string
t e c h n i q u e 55.
This
A preliminary
field has
theory
led
to
a
in
a
gauge
i n v a r i a n t f o r m u l a t i o n o f t h e f r e e s t r i n g field t h e o r y 56 {as well a s some a s p e c t s o f the interactions57).
(c) O t h e r T o p i c s (i)
Throughout
the d e v e l o p m e n t of s t r i n g t h e o r i e s t h e r e has b e e n a parallel
d e v e l o p m e n t o f Kac-Moody a l g e b r a s Kac-Moody a l g e b r a s These
has
infinite-dimensional algebras
string world-sheet.
in m a t h e m a t i c s .
The
deep c o n n e c t i o n s with the express
the
representation
d y n a m i c s of s t r i n g
algebra
o f local
theory
of
t h e o r i e s 58,
currents
in
the
The c o n n e c t i o n b e t w e e n Kac-Moody a l g e b r a s a n d s t r i n g t h e o r i e s
h a s b e e n a c t i v e l y s t u d i e d a n d w a s c r u c i a l in d e v e l o p i n g t h e h e t e r o t i c s u p e r s t r i n g 59, (ii) superstring
A l t h o u g h a p p e a l i n g f r o m a g e o m e t r i c a l p o i n t of v i e w , t h e f o r m u l a t i o n of theories
in
terms
of
a
manifestly
supersymmetric
a c t i o n like eq.(5) h a s n o t b e e n q u a n t i z e d in a c o v a r i a n t m a n n e r . to b e s o l v e d b y e x t e n d i n g t h e s y m m e t r i e s o f t h e a c t i o n 60. been
progress
towards
formulating
theory by directly constructing
a
manifestly
Lorentz-covariant This s e e m s l i k e l y
Furthermore, there has
supersymmetric
first-quantized
t h e q u a n t u m o p e r a t o r s o f t h e t h e o r y 61.
This may
l e a d to a p r o o f of t h e a b s e n c e of s u p e r s y m m e t r y a n o m a l i e s a t n l o o p s f o r t h e t y p e II a n d h e t e r o t i c s u p e r s t r i n g (iii)
t h e o r i e s , a n d h e n c e to t h e i r f i n i t e n e s s .
I t is now p l a u s i b l e t h a t t h e r e q u i r e m e n t s
that a chiral ten-dimensional
t h e o r y w i t h a Yang-Mills g a u g e g r o u p b e s u p e r s y m m e t r i c a n d also f r e e of a n o m a l i e s l e a d s i n e x o r a b l y to s u p e r s t r i n g
theory.
The "minimal" t e n - d i m e n s i o n a l f i e l d t h e o r y
of s u p e r - Y a n g - M i l l s c o u p l e d to s u p e r g r a v i t y by
adding
superstring
extra
terms
theory)
supersymmetry
to
the
theory3
h a s anomalies. (motivated
which spoil its s u p e r s y m m e t r y .
by
These can be cancelled the
low
energy
b y a d d i n g y e t more t e r m s s h o u l d e v e n t u a l l y r e c o n s t r u c t
n u m b e r o f t e r m s t h a t c o n s t i t u t e t h e e x a c t e x p a n s i o n of t h e terms of the massless fields.
limit
The p r o c e s s o f r e s t o r i n g superstring
of the
the infinite theory
in
At t h e l o w e s t n o n - t r i v i a l o r d e r in t h i s e x p a n s i o n t h i s
152
h a s b e e n s h o w n 62 to imply t h e e x i s t e n c e of t e r m s i n t h e low e n e r g y a c t i o n w h i c h are
quadratic
in
the
Riemann
curvature
and
which
have
the
structure
initially
c o n j e c t u r e d in r e f . 63.
CONCLUSION
Superstring mechanics
that
theories have cause
Einstein's theory.
p a s s e d all t h e t e s t s
problems
with
conventional
of c o n s i s t e n c y w i t h q u a n t u m
theories
of
Furthermore, this consistency restricts
the
gravity,
based
on
p o s s i b l e Yang-Mills
g r o u p s almost u n i q u e l y a n d t h e r e f o r e h o l d s t h e e x c i t i n g p r o s p e c t o f a u n i f i e d a n d c o n s i s t e n t q u a n t u m t h e o r y of all t h e i n t e r a c t i o n s . T h e s e a r e e a r l y d a y s , h o w e v e r , a n d t h e r e a r e many q u e s t i o n s to be a n s w e r e d a b o u t how s u p e r s t r i n g now, q u i t e a p a r t
t h e o r i e s may make c o n t a c t w i t h o b s e r v e d p h y s i c s .
Up u n t i l
from the phenomenologieal issues d e s c r i b e d earlier, t h e s e t h e o r i e s
h a v e n o t p r o v i d e d a n a t u r a l e x p l a n a t i o n f o r s e v e r a l of t h e most a c c u r a t e l y k n o w n n u m b e r s in p h y s i c s : - At t h e v e r y l e a s t we m u s t u n d e r s t a n d
how it is t h a t t h e s e t h e o r i e s , f o r m u l a t e d
i n i t i a l l y in D-10 d i m e n s i o n s p r e d i c t t h a t , to a v e r y g o o d a p p r o x i m a t i o n , D=4 in
the
world
scheme
that
outlined
we s e e above,
at
accessible energies
lit
would
at
least
which be
is a n
satisfying
assumption to
in
discover
the that
t e n - d i m e n s i o n a l Minkowski s p a c e is n o t a s o l u t i o n of t h e t h e o r y . ] The f a c t t h a t t h e cosmological c o n s t a n t is z e r o to a n a m a z i n g a c c u r a c y h a s n o t
-
y e t b e e n e x p l a i n e d in a n a t u r a l way in s u p e r s t r i n g
theory.
[Although the scheme of
ref.(34) o u t l i n e d e a r l i e r d o e s n o t g e n e r a t e a cosmological c o n s t a n t i n t h e p r o c e s s o f compactification
there
is
no
obvious
mechanism
that
would
prevent
one
being
g e n e r a t e d in t h e s u b s e q u e n t s y m m e t r y b r e a k i n g t r a n s i t i o n s . ] -
Another outstanding
q u e s t i o n is w h e r e t h e m a s s s c a l e a s s o c i a t e d w i t h weak
s y m m e t r y b r e a k i n g c o m e s from.
The primitive.
present
theoretical
understanding
the two-dimensional world-sheet. and
of
superstring
theories
is
somewhat
The t h e o r i e s a r e f o r m u l a t e d in t e r m s o f i n v a r i a n c e p r i n c i p l e s r e l a t e d to T h e y c o n t a i n b o t h t h e m a s s l e s s Yang-Mills p a r t i c l e
t h e m a s s l e s s g r a v i t o n w h i c h is w h y t h e y r e d u c e , a t low e n e r g i e s , to { s u p e r )
Yang-Mills c o u p l e d
to
(super)
gravity.
However, t h i s a p p e a r s
to b e a f o r t u i t o u s
a c c i d e n t s i n c e t h e t h e o r i e s w e r e n o t e x p l i c i t l y b a s e d o n a n y g e o m e t r i c a l p r i n c i p l e in space-time. the
The d i s c o v e r y of s u c h a p r i c i p l e , w h i c h would b e a g e n e r a l i z a t i o n of
principle
of
general
relativity,
would
lead
to
a
much
more
u n d e r s t a n d i n g of t h e b a s i s o f t h e t h e o r y a n d t h e r e f o r e of i t s p r e d i c t i o n s .
profound
153
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CONFORMALLY INVARIANT FIELD THEORIES IN TWO DIMENSIONS CRITICAL SYSTEMS AND STRINGS J.-L. GERVAIS Physique Theorlque, Ecole Normale Superieure 24 rue Lhomond 75231 Paris cedex 05
At the present time it is hardly necessary to emphasize the fundamental importance of string models since super string theories are the most promising candidates for a completely unified theory of all interactions. An other key point is that string concepts have plaid an important role in the recent developments of theoretical physics and mathematics by suggesting man~,~new important ideas,such as in particular supersymmetry--~J%~ ,and have led to very interesting progress in the related critical models in two dimensions. In these notes I shall mostly concentrate on this latter aspect which is not, presently, so directly aimed at a unified theory of all interactions but is quite interesting in its own right . The unifying feature of string theories and critical systems is that they are both associated with conformally invariant field theories. We shall first review the essential features of this connection considering only, for simplicity, bosonic strings. At the level of the present discussion, supersymmetric strings are not basically different. One essentially replaces the oonformal group by its superconformal generalization. The position of a string at time ~ is specified by a field X ~ ( ~ , ~ ) where ~ distinguishes the various points along the line. Hence one has a two dimensionnal field theory in parameter space. When ~ varies the string sweeps out a world sheet and one sets up the dynamics in such a way that it be invariant under reparametrization of the corresponding geometrical surface. One can rigourously show that it is always possible to choose the parametrization in such a way that the curves ~ =cste and the curves =care intersect at right angle. This choice is not unique since this orthogonality condition is left invariant by all conformal transformations of ~ and ~ .In string theories the conformal group is thus the residual symmetry of the system with an orthogonal choice of O~) parameters. Basically it is the group of all transformations of the form where
f and
g are
two arbitrary
real
functions
of one
157
variable. In the present dicussion we stick to the Lorentz covariant string quantization where conformal invariance is not explicitely broken. Since a physical string has a finite length, ~ ' v a r i e s over a finite range. It is always possible te redefine the parameters in such a way that the dynamics is periodic in with period 2 ~ .It is often very convenient to go to Euclidean time by letting
For real ~ the conformal group can best be the group of analytic transformations of
described as
~ : e~÷~
(3)
Indeed with this variable, the strip o~ ~ ~ ~ represented by the whole complex z plane. In pioture, a conformal transformation in given by
is this
(4)
=
where F is an arbitrary complex function of one variable. In genera)~, a quantity ~ ( % . ~ ] is called eonformally oovarlant ) if it transforms accordlng to •
~
•
•
~
l
~
4
.
where ~ and ~ are parameters depending on the quantity considered which are called conformal weights. This notion was rediscovered recently (~) and the corresponding fields were called primary. If we separate the real and imaginary parts according to
the differential transforms as where ~ is the rotation matrix with angle 0 and where is a dilatation factor. ~ and 8 are given by the differential equations --
Formula (5) becomes
Q9~ z
9~C~
(8)
(s) Since ~ and ~ are the local dilatation and rotation parameters respectively, d is the dimension and J is the spin of the quantity considered. For critical systems in two dimensions, OC~and DC$ a r e the two coordinates. It is well known that a stastistieal system at a point of transition of second order becomes
158
scale invariant. The corresponding rescaling of 9(:aandgC is a particular case of (4).Polyakov has proposed @~h~ critical systems are invariant under the full conformal group (4). One thus sees thas that both string theories and critical systems are based on conformally invariant field theories,with however different descriptions. ~ and are string parameters while x~ ,x~ are the coordinates of the critical system. In a conformally invariant field theory the improved energy momentum tensor is symmetric traceless and conserved. For two dimensional field theories in real ~ ) ~ s p a c e this leads to
)( T:
(in)
1 --o
Due to the periodicity in
@
one can write I
~ (Too , T ~ ) =
Z L~.~ -~
;
3 ; ~ C~-~)
(li)
The operators L m a n d L ~ are the infinitesimal generators of conformal transformations. For a conformally covariant field which satisfies equation (5) one has
(i2)
The operators algebra
~ Lm
+(~)i] and
Lm
0 each
satisfy
the
Virasoro
where the central charge C depends on the model considered. Its actual value is a key point. We shall have more to say about this below, ln general the field theories we are discussing are characterized by C and by the set of conformally oovaria~t fields ~ together with the set of conformal weights h ~ .Since the two V i r a s o r o algebras (13) have the same properties we only consider explicitely the algebra of the L m ' s most of the time. As a first simple example, let us recall the essential
159
features of the standard bosonic (Veneziano) model. In this case one only considers massless two dimensional free fields X ~ . W e shall denote by A ~ the associated Virasoro generator. As it is well known they satisfy eq.(13) with C = ~)
(14)
the vertex for the emission is simply given by •.
where k -~ particle. ~t (12) with
of the lightest
•
e
string state
(15)
is the energy momentum of the emitted is well known that V ~ satisfies condition
The present ~ i s c u s s i o n of string is in the covariant formalim where one has to make sure that the time like components of X ~ deoouple from the physical S matrix. Such a ghost killing mechanism requires first of all that ]:,he vertex V 4 have dimension I. From eq. ( 18 ) this leads to ~ % - ~ . ~ .the emitted particle is a tachyon with ,,ass m 2
=
--2.
For critical models the ~ and ~ are critical exponents. Indeed it is easy to see t h a t the global dilatations, rotations , and translations of x ,x are % f the genrated by L~ ~= L t and "Ll .The vacuum state 1 system must therefore annihilated by these operators. As a result the two point function of any covariant operator can be computed u p _ to a constant factor by means of equation (12).If ~ = ~ for instance, one finds
<01 Hence ~ gives the power behaviour of the two point function at the critical point. It is quite obvious that must be positive for physical operators since the correlation f u n c t i o n s must decrease when the separation increases. As it is well known (~)(9] ,the derivative of a covariant operator is not covariant in general. An important exception is the case of an operator of vanishing weight, lts derivative with respect to has ~ =i , ~ = Q Conversely, assume there exists an operator I(z) with =i Then it is obvious that =o
In stastistical marginal. If they
(18)
mechanics such operators are called exist one has critical lines instead of
160 critical points since they can be added to the action with arbitrary coefficients without destroying conformal invariance. For the Veneziano model we have just recalled that V ~ (z) has weight l.lndeed equation (18) for V~ is the basic ingredient for the decoupling of ghosts. The notion of conformally covariant operator was introduced in string models (~) in order to dicuss posible generalizations of the Veneziano model. We now recall the essential points of this approach. Generalized string models involve other fields besides the free fields X ~ a n d the corresponding two dimensional field theory may have a non trivial interaction. Such is the case, for instance, if we have additional space components which are oompaotified. Quite generally we can consider that X remains a free field which does not mix with the additional two dimensional fields. These will be characterized by the set of covariant operators ~ together with the set of weights ~ .The ghost killing condition now requires the emission vertex to have conformal weight one under the action of the total Virasoro generator where L m is the Virasoro generator of the additional dimensional dynamics. This is realized by c h o o s i n g
two
We therefore see t h a t the spectrum of lightest particles
w i l ~ lnvozve no tachyon provided t h a t ~ ~%
(22)
and this condition selects the conformally invarian~ field theories for which all eovariant fields have weights larger than one. From the view point of critical systems this condition is unusual since it means that the corresponding two point function has a Fourier transform which has at most a logarithmic singularity at zero momentum. We shall come back to this later. The last important general point about string theories is that they make sens only if the central charge of the total Virasoro algebra is equal to the critical value 28.This can be seen in many ways. The simplest one is to notice (~) that the central charge of the Faddev Fopov ghosts is precisely -26 so that, with the above value, the central charge vanishes when all the fields are included and there is actually no breaking of conformal invariance. For the Veneziano model where only the X field enters this means that ~ =26. In the generalized models the total central charge is C + ~ where C is the central charge of the additional dynamics. Hence one must satisfy
161
~5-_ ~ - C
(23)
If C is an integer larger than one, this will effectively allow to lower the space time dimension. At this point it is useful to recall some general properties of representations of the Virasoro algebra( 13).From the group theory viewpoint it can be regarded as being in a Weyl Caftan basis,Lo being the only operator of the commuting subalgebra, and L,t with n O being step operators. Hence a highest weight vector will be such that o
(24)
An irreducible representation is characterized by the values of E and C .The corresponding vector space which is called a Verma module, is spanned by all vectors of the form
where J ~ m { are arbitrary eigenstates of Le"
positive
integers,They
are
All eigenvectors with the same eigenvalues are said to belong to the same- level N. Kae (E) has considered the matrix of all inner products in a given module which is entirely determined :from the V i r a s o r o algebra together with the hermiticity condition
H e n c e it is purely alebraic and values of ~ and C • It obviously of finite matrices at each level. closed formula for each finite quantity
only depends upon the factorizes into products Kac (s) has obtained a determinant. Define the
(28)
where p and q are arbitrary integers, lf we c o n s i d e r a highest weight representation with ~ = ~(p,q) for some given p,q both larger than zero, the Kae determinant vanishes at the level N=pq. T h i s v a n i s h i n g shows that the metric of the Verma module need not be positive definite. The unitarity of the representation is thus in
162 question. It is easy to show that the negative values of the highest weights are all excluded. For p o s i t i v e values we note that, for C >I, ~ (p,q) is always negative for p>l ,q>l , i.e. when it corresponds to a zero of a Kae determinant. Hence, in this region the Kac d e t e r m i n a n t s never change sign f o r . . p o s i t i v e 6 and one can show by explicit construction that there exist a unitary representation for all ~ >O,C>I .For C
C=
(29)
i
where r,p,q are integers. Hence t h e allowed values of g p r e c i s e l y coincide with zeroes of Kac determinants. Going b a c k to c o n f o r m a l l y invariant field thories we recall that, given a c o v a r i a n t operator with w e i g h t ~ ,it is easy to see that the state
0(3)
o>
(30)
is a h i g h e s t weight v e c t o r with ~ = ~ .The spectrum of highest weights coincides with the set of conformal weights which, in general, involves more than one values. The r e p r e s e n t a t i o n of the V i r a s o r o algebra is thus reducible since the H i l b e r t space is the sum of the c o r r e s p o n d i n g V e r m a modules. The c o v a r i a n t operators are intertwening operators between the different irreducible representations. For a r b i t r a r y ~ and C one can c o n s t r u c t an infinite family of c o v a r i a n t operators with weights given by f o r m u l a (28) for all p and q p o s i t i v e or n e g a t i v e as a natural b y p r o d u c t o;_~he exact quantum solution of quantum L i o u v i l l e theory ~) .These operators are not all physical since formula ( 2 ~ is not always positive. ]'his shows n e v e r t h e l e s s that the set of critical dimensions must coincide with Kac formula in general. In v i e w of formula (28), it is clear that one has to d i s t i n g u i s h three regions for the p o s s i b l e values of C. I-The region C
especially
zero modes
because
the q u a s i c l a s s i c a l
in mind we have model
and
satisfactory,
spectrum
with hope to get some kind of a n o n e q u i v a l e n t
of the
The q u a s i c l a s s i c a l
of phase
so the tachyon
of the
of p e r i o d i c
of the
is c o n t i n u o u s
give d i s c r e t e
and
a considerable
treatment
system in cases
Indeed,
excitations
(i)
has a t t r a c t e d
is not c o m p l e t e l y
negative
0
role in the q u a n t i z a t i o n
thereby
of the c o r r e s p o n d i n g
is however
~.
the h a m i l t o n i a n
dynamical
was p e r f o r m e d
obtained
in the p e r i o d i c masses
In particular,
classical
conditions
quantization
0 ( X ~ 2~
of its p o s s i b l e
ilL.
corresponding boundary
The q u a n t u m v e r s i o n
of th~ Hamiltonian.
e.
attention
integrable
and a lattice
integrable
step we o b t a i n e d
Because
of t e c h n i c a l
version
to the end.
among our colleagues in such a p r o g r a m
models
difficu~ies We
and r e a l i z e d ]61.
in
the exactly
In this
that
167 circumstances
we d e c i d e d
version
before
lattice
deformation
in our t r e a t m e n t
The paper L2] -
of the V i r a s o r o
2. F i n a l l y
algebra w h i c h
In sec.
continuous
generalization.
in sec.
on the classical
one.
We think that the naturally
appears
by itself.
as follows.
131 on the c l a s s i c a l lattice
the results
of the q u a n t u m
is i n t e r e s t i n g
is o r g a n i z e d
the natural sec.
to p u b l i s h
the c o m p l e t i o n
model
The
3 we shall
1 we remind
in the form w h i c h
latter will
indicate
the results
allows
be d e s c r i b e d
a possible
quantum
of
in
genera-
lization.
One of the authors ~ de Vega
(L.D.F.)
for their kind h o s p i t a l i t y
O. Babelon,
J.L.
interesting
discussions.
1. C l a s s i c a l
In w h a t
is g r a t e f u l
Gervais,
continuous
follows
M. Jimbo,
to p r o f e s s o r s
at the LPTHE.
R. M a r n e l i u s
B. Diu and
We thank P r o f e s s o r s
and A. Neveu
for
model.
only p e r i o d i c
boundary
conditions
(2)
will
be used.
The main
idea of
t2] -
131 is to use the change
of
variables
(3)
from the initial is supressed)
data
~(x)
)
TrtxJ ---- ~
to new fields w i t h trival
~(a)
equations
(time v a r i a b l e of m o t i o n
(4)
and
some r e a s o n a b l y
described
in detail
(note the absence
simple in
171,
boundary
condition.
One v a r i a n t
can be based on the a u x i l l a r y
of the spectral
of this m a ~ linear
parameter)
(5)
problem
188
where
~(~r)
is a
Q
2x2
m a t r i x p a r a m e t r i z e d by the initial data
..
e~
Let
T(~)
T(x,~) : I
_~
1
be a fundamental m a t r i x solution n o r m a l i z e d by . Then
~ : T(z~, o)
is called a m o n o d r o m y matrix.
particular form of ~ @ ) g u a r a n t i e s that d e t ~ " that ~
(6)
= 1 and tr ~"
The
> 2, so
is hyperbolic.
Using the n o t a t i o n
gc~))
!
Tc~,o)
= To,,)
= I
I
We introduce functions
2A ¢~)
(7)
Co-; and
Aj-(~)
(8)
cx)
C c-J
with the f o l l o w i n g properties
(9)
and
(i0)
where for any f
(il)
A, B, C, D being the m a t r i x elements of the m o n o d r o m y matrix. n o n t r i v i a l observables
so that the form (i0)
They are
of the b o u n d a r y conditions
169
seems not very transparent. The Poisson structure (12)
where
~
plays the role of the coupling constant,
leads to the fun-
d a m e n t a l Poisson bracket relations
(13)
(we use already standard n o t a t i o n from
151) where the
~x
~
m a t r i x r is given by
0
0
0 r"
It follows
o
0
-~' :~"
o
(14)
-.-
0
o
-~
o
0
o
o
0
from an evident r e l a t i o n
and c o m m u t a t i v i t y of the m a t r i x elements of locality)
T(y)
and
T(~,~)
(ultra-
that the f o l l o w i n g relations are true
÷
Y
[~,.~
-
15
~e~) ~]
+ 16
170
where
~(~)
is a sign function and we confine ourselves to the fixed
fundamental domain O < X, y < 2 ~
.
The f o l l o w i n g chain of Ansatze
(18)
~ -
,~,
4 ¢'~,'
J
4 2..
X~" ,~..~.'
(19)
I2o)
and a n a l o g o u s l y for
~4-
is now introduced.
The final object
(the
Schwartz d e r i v a t i v e of ~~- ) is known tO be invariant under the transformation
(ii) and so the simple b o u n d a r y conditions hold
It is i n t e r e s t i n g that the variables i n t r o d u c e d in (18)
-
beautiful Poisson brackets of their own. With notations
~(a)--~[~(¢xj],
(w} ~. y ~ ( K } ]
(20) have
and so on we have the following list of formulae
(22)
so that for
~
the Poisson brackets are field independent.
Conti-
ruing the d i f f e r e n t i a t i o n we get
(23)
171
so that the p and ~ fields decouple.
F i n a l l y for S we acquire the
brackets
¢x-~) (24) and a n a l o g o u s r e l a t i o n for V i r a s o r o algebra.
S (~)
, w h i c h are c h a r a c t e r i s t i c of the
Thus the phase space of the L i o u v i l l e model is essen-
tially the product of two such algebras.
It can be shown that the h a m i l t o n i a n of our model
(25) @
has a simple expression
in terms of S[~.]
and S~,v-]
Z--
the e q u a t i o n of m o t i o n being linear
This allows to call S (
9 z,
~,~
) the angle action variables.
The inverse map is given by the famous L i o u v i l l e formula
(27)
with periodic
~{X} .
We finish this section w i t h a comment on b o u n d a r y conditions
{i0). One
can s i m p l i f y them by d i a g o n a l i z i n g the m o n o d r o m y m a t r i x w h i c h is a c h i e v e d by the t r a n s f o r m a t i o n
172
where
~ , ,
~
are
the
real
2"
The
new v a r i a b l e s
=
~ ---4, Z
points
)
of
~, ~
~z
(29
~"(~] have t h e f o l l o w i n g p r o p e r t i e s
~ (~) and
~(o)
fixed
4 (30
and
satisfy
where
e-P
It is c l e a r function
p
simple
boundary
conditions
is'the
smallest
eigenvalue
that
p is the
zero F ~ u r i e r
of T
, that
coefficient
means
that
p >/ O.
of the p e r i o d i c
[ ~. ~') ]
o
We h a v e the
calculated
following
Ia~,
~}
the P o i s s o n
brackets
of t h e s e
new v a r i a b l e s
with
answer
= r Ec.-~)[,~c~)-~,~;]
~
+
(33)
173
~v" ( x ) z
_
,~_
(.))
(34)
and
(35)
Note the invariance of the first relation w i t h respect to t r a n s f o r m a t i o n (ii) w i t h c o n s t a n t coefficients.
F o r t u n a t e l y the rather c o m p l i c a t e d t r a n s f o r m a t i o n n e c e s s a r y and the o r i g i n a l v a r i a b l e s
~)
(28) is not really
, ~(~)
can be used to
c o n s t r u c t the g e n e r a t o r s of V i r a s o r o algebra as well as H a m i l t o n i a n of L i o u v i l l e model.
It is for them that we are able to find generali-
zation on the lattice both in c l a s s i c a l and q u a n t u m Case.
2. C l a s s i c a l model on the lattice.
W o r k i n g in a c c o r d a n c e w i t h the general spirit of
151 we g e n e r a l i z e
L i o u v i l l e model on the lattice b e g i n n i n g w i t h the a u x i l l a r y linear problem
: instead of
(5) we have now the e q u a t i o n
and L ~
must be c o n s t r u c t e d in terms of v a r i a b l e s
fying the discret form of Poisson brackets
(12)
qT~ , ~
satis-
174
L~
The matrix
is essentially uniquely defined by the requirements :
i. In the continuous where
~
q]'~
d,c~rxj ; ~ , , , =
=
is lattice with length
L. 2.
limit
L,,,.
=
|
~ L~
+. ~ e t ~
"J,,.,.
behaves as follows
÷ 0(~)
(38)
exactly satisfies the fundamental Poisson relation of the
form (13)
{a;a}
=
[,., L. , L . ]
with the r-matrix (14). The explicit
¢4 + ~#'
L4~.
=
(39)
formula is given by
e
Ae
~
;
--~r
(40)
Introducing the transport matrix
A~
L
g~ (41)
we let
A~
C~
The essential role of local relation it leads the same relation for
~'~
(42)
(39) consists in the fact that
17,5
The latter in its turn gives the f o l l o w i n g Poisson bracket relations for
AA~ ~ ~
(44)
(45)
Here the numbers n, m vary in the "fundamental domain" w h e r e N is a length of the lattice and
I
i
n > m
0
n = m
-i
n < m
~
The f o r t u n a t e p r o p e r t y of
~
1 ~ m,n ~ N
is defined as follows
(47)
our lattice f o r m u l a t i o n is that the rela-
tions
(44) -
(46) look as the most naive g e n e r a l i z a t i o n of the rela-
tions
(15) -
(17).
This luck continues in the c o n s t r u c t i o n of analogous
of A n s a t z e
(2O).
(48)
We let
(49)
~..~ 4- ~.~.~
~÷i
_ ,~_,
(50)
(18)
176
5~, = 0-~..,)0.
~',,-.) =
(51) (~
and a n a l o g u o u s l y
Observe
The most
that
for
5~
property
of their
give the f o l l o w i n g
C c~,,,. _
~_~
)
.
is invariant
striking
simplicity
~
÷, - ~ - , )
Poisson
under
the t r a n s f o r m a t i o n s
of the new v a r i a b l e s
consists
brackets; s t r a i g h t f o r w a r d
(ll),
in the r e l a t ~ e
calculations
formulae
t F'., f,..t = i ( r""," - ~ ' . , . . . . ) O - r 2 ) ( ~ - e - )
(53
and
2-
(54
The
last formula
interesting
gives
in its own.
~,,, =
a lattice
generalization
In the continuous
S(~.)
'~
t.t so that the h a m i l t o n i a n
of the V i r a s o r o
algebr~
limit we have
(551
177
4 (56)
is a natural g e n e r a l i z a t i o n of
(26).
The e q u a t i o n s of m o t i o n
(57)
g e n e r a t e d by the h a m i l t o n i a n are known to be c o m p l e t e l y integrable, as was shown by S. M a n a k o v and M. Kac - P. van Moerbeke
. In fact
they a p p a r e n t l y a p p e a r e d first in the e c o l o g i c a l papers of Volterra. So the v a r i a b l e s
S~
(and S ~
c o r r e s p o n d i n g to
~/~
first step in c o n s t r u c t i n g the a n g l e - a c t i o n v a r i a b l e s
) constitute the for the L i o u v i l l e
model on the lattice. At this point we stop the d i s c u s s i o n of the classical
lattice model.
3. Partial q u a n t u m results.
C o n t i n u i n g to w o r k in the spirit of
151 we get the q u a n t u m v e r s i o n of
the lattice model via the c o n s t r u c t i o n of the of the o p e r a t o r s
This
L~
~
must turn to
and
L~-operator
in terms
" ~ ' ~ w i t h the usual c o m m u t a t i o n relations
(40) in the c l a s s i c a l
limit and satisfy the fun-
damental commutation relation
& (L.®
L. > : ( L . ~ L. II<.
with a p a r t i c u l a r C - n u m b e r
The formula
~ x ~
matrix
~
•
{sg)
178
~/~ I/~
"~/2 60)
/~e ~
gives such an object,
e
the R - m a t r i x being
I1
0
0
0
0
ei~"
0
0
0
e i~"
0
01
l_e2i~
0
0
(61)
1
Now we literally repeat what was done before,
namely introduce the
transport matrix
=
(62)
and the operators
~,
=
A~
Bf I
j
v,
chosing a p a r t i c u l a r order of the factors.
: c., D I '
(631
It is g r a t i f y i n g to check
that these operators satisfy rather simple relations
(65)
179
~
~d'~
w here
~
through
~
= ~ 6~
~
, p~
we were
of the lattice This was
learned
encouraging.
version.
new results
of r e p r e s e n t a t i o n s
algebras
his visit
c a s ~ so that
of the
the quan-
in a n o n s a t i s f a c our r e l a t i v e l y
some new d e v e l o p m e n t s
181 on the tensor
191 w h i c h
to the LPTHE
this o p t i m i s t i c
only
generalizations
is still
However
of M. Jimbo
of S k l y a n i n
during
With
model
enters
case.
in the q u a n t u m
Liouville
(66)
the reason w h y we did not p u b l i s h
on the c l a s s i c a l
and in p a r t i c u l a r
,%1-
constant
the S i n e - G o r d o n
variables
tory
old results
(4 -- ~ z )
not able to find natural
and S~,
state.
~-
. Note that the c o u p l i n g
tum v e r s i o n
(L.D.F.)
2d-~
in a n a l o g y w i t h
Unfortunately ~
-~-
products
one of the authors
in april
1985
seem
note we finish this paper.
References.
11
A.M.
Polyakov,
12
J.L.
Gervais,
Phys.
Lett.
A. Neveu,
IO3B
Nucl.
(1981)
Phys.
207.
B199
(1982)
59
; B209
(1982)
125. 3
A. Kihlberg,
4
C.P.
Dzhozdrhadze,
Phys.
4__00(1979)
51 L.D.
Faddeev
61 J.L.
Gervais,
171A.K.
Phys.)
Jimbo,
191 E.K.
Pogrebkov,
Les Houches
preprint M.K.
ed.
preprint
Sklyanin,
A.K.
ITP - G 6 t e b o r g M.K.
preprint
Polivanov,
N82-2 Theor.
(1982). Math.
221.
in
Pogrebkov,
(Math. 181M.
R. Marnelius,
ENS,
Polivanov
S.P.
Novikov,
RIMS
Funct.
Lectures,vol.XXXIX
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