Field Theory, Quantum Gravity, and Strings

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

New York Tokyo

Editors H. J. de Vega Universite Pierre et Marie Curie, L.P.T.H.E. Tour 16, ler Stage, 4, place Jussieu, F-75230

Paris Cedex, France

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

Springer-Verlag Springer-Verlag

Berlin Heidelberg NewYork Tokyo NewYork Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 5 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to “Verwertungsgesellschafi Wart”, Munich. 0 by Springer-Verlag Printed in Germany Printing and binding: 2153/3140-543210

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1986 HemsbachIBergstr.

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 ~ 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>

=

(~!)~

(~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 '~

~

:

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~. 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~

' , the new density matrix

:

twice the standard one. But the hermitici-

for the density matrix restrict

re, we must restrict ourselves

invariant.

I~>

k> 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(~/ (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> 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 > 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


>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 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 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 by 3()2 in the equation for 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)

:,



(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-

O presents no more advantages than the case M2M~

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~

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

REFERENCES

(i)

(2)

(3) (4)

(5)

(6)

(7) (8)

(9) (i0) (ii) (12)

(13) (14)

(15)

(16) (17) (18) (19)

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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

: ( L . ~ L. II