Quantum Cosmology and Baby Universes~tqw~_darksiderg

QUANTUM COSMOLOGYAND

BABY UNIVERSES

bySteve Weinberg

Published by

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QUANTUM COSMOLOGY AND BABY UNIVERSES

Copyright 1991 by World Scientific Publishing Co. Pte. Ltd.

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vPREFACE

Quantum gravity is notoriously a subject where problems vastly outnumberresults. This is no surprise at short distances, on the order of the Planck length,where most of us expect quantum fluctuations in the metric to cause space-timeitself to lose its meaning and gravity to be subsumed (along with the other so-calledfundamental interactions) into some more comprehensive, consistent, and radicaltheory. However, even at larger scales, where one would naively expect quantumcorrections to be small and controllable, reasonable people disagree not only on whatare the right answers but also on what are the right questions. Recently there hasbeen a flurry of activity in this area, inspired both by the importation of ideas fromstring theory and by speculations about connections between topology-changingquantum fluctuations and the vanishing of the cosmological constant. This last wasthe focus of the Seventh Jerusalem Winter School, on "Quantum Cosmology andBaby Universes", held from December 27 1989 to January 5 1990.

Lectures were given by S. Coleman (Harvard), J. Halliwell (M.LT.), J. Har-tle (University of California, Santa Barbara), S. Hawking (Cambridge), A. Stro-minger (University of California, Santa Barbara), L. Susskind (Stanford), and C.Teitelboim (Centro de Estudios Cientificos de Santiago). Halliwell's "Introduc-tion to Quantum Cosmology", Hartle's "The Quantum Mechanics of Cosmology" ,and Teitleboim's "Hamiltonian Formulation of General Relativity" dealt with thefoundations of the subject, while my "Wormhole Dynamics", Hawking's "Worm-holes and Non-simply Connected Manifolds", Strominger's "Baby Universes" , andSusskind's "Critique of Coleman's Theory of the Vanishing Cosmological Constant"dealt with more recent developments. The more responsible lecturers submittedlecture notes to be published in this book.

Everyone involved in the school, both lecturers and students, displayed a veryhigh level of energy and enthusiasm; I had a wonderful time and learned a lot, andI suspect others did also.

The success of the school owed much to the efforts of my co-directors, JimHartle and Steve Weinberg, and to those of the local organizer, Tsvi Piran. We aregrateful to the Israeli Ministry of Science and to the Alfred P. Sloan Foundationfor their support.

Sidney ColemanCo-Director

CONTENTS

PREFACE

HAMILTONIAN FORMULATION OF GENERAL RELATIVITYClaudio Teitelboim

vii

v

l. INTRODUCTION 3

2. HAMILTONIAN FORMULATION OF GAUGE THEORIES (PRE-ERST) 3

3. ERST HAMILTONIAN FORMULATION OF GAUGE THEORIES 4

4. DYNAMICS OF GRAVITATIONAL FIELD 5

5. DOES THE HAMILTONIAN VANISH? GENERAL COVARIANCEAS AN 'ORDINARY' GAUGE INVARIANCE 75.l. Generally covariant systems 85.2. Time as a canonical variable. Zero Hamiltonian. 95.3. Time reparametrization invariance 125.4. 'True dynamics' versus gauge transformations 15

6. MUST TIME FLOW? 176.l. Gauge independence of path integral for a

parametrized system illustrated. Equivalence ofthe gauges t=7 and t=O. 18

7. ERST CHARGE OF GRAVITATIONAL FIELD 22

8. ELEMENTS OF ERST THEORY 248.l. The ghost, you've come a long way, baby 258.2. BRST symmetry in classical mechanics 388.3. Quantum BRST theory 52

viii

9. ACKNOWLEDGEMENTS

CONTENTS

63

THE QUANTUM MECHANICS OF COSMOLOGYJames B. Hartle

1. INTRODUCTION

2. POST-EVERETT QUANTUM MECHANICS2.1. Probability2.2. Decoherent histories2.3. Prediction, retrodiction, and history2.4. Branches (illustrated by a pure p)2.5. Sets of histories with the same probabilities2.6. The origins of decoherence in our Universe2.7. Towards a classical domain2.8. The branch dependence of decoherence2.9. Measurement2.10. The ideal measurement model and the Copenhagen

approximation to quantum mechanics2.11. Approximate probabilities again2.12. Complex adaptive systems2.13. Open questions

3. GENERALIZED QUANTUM MECHANICS3.1. General features3.2. Hamiltonian quantum mechanics3.3. Sum-over-histories quantum mechanics for

theories with a time3.4. Differences and equivalences between Hamiltonian

and sum-over-histories quantum mechanics fortheories with a time

3.5. Classical physics and the classical limit of quantummechanics

3.6. Generalizations of Hamiltonian quantum mechanics

4. TIME IN QUANTUM MECHANICS4.1. Observables on spacetime regions

67

70707378818283899293

9498

100101

102102103

104

107

109111

111112

CONTENTS ix

4.2. The arrow of time in quantum mechanics 1144.3. Topology in time 1164.4. The generality of sum-over-histories quantum mechanics 120

5. THE QUANTUM MECHANICS OF SPACETIME5.1. The problem of time5.2. A quantum mechanics for spacetime5.3. The construction of sums over spacetime histories5.4. Some open questions

6. PRACTICAL QUANTUM COSMOLOGY6.1. The semiclassical regime6.2. The semiclassical approximation to the quantum

mechanics of a non-relativistic particle6.3. Semiclassical prediction in quantum cosmology

ACKNOWLEDGEMENTS

REFERENCES

APPENDIX: BUZZWORDS

INTRODUCTORY LECTURES ON QUANTUM COSMOLOGYJonathan J. Halliwell

120120123132136

137137

137140

145

146

152

1. INTRODUCTION 159

2. A SIMPLE EXAMPLE 161

3. THE HAMILTONIAN FORMULATION OF GENERAL RELATIVITY 167

4. QUANTIZATION 1694.1. Interpretation 173

5. MINISUPERSPACE - GENERAL THEORY 1735.1. Canonical quantization 1765.2. Path integral qUlUltization 177

x ( ONTENTS

5.3. The probability measure 179

6. CLASSICAL SPACETIME 1806.1. The general behaviour of the solutions 182

7. THE WKB APPROXIMATION 1847.1. The measure on the set of classical trajectories 186

8. BOUNDARY CONDITION PROPOSALS 1908.1. The no-boundary proposal 1918.2. The tunneling boundary condition 199

9. NO-BOUNDARY VS. TUNNELING 204

10. BEYOND MINISUPERSPACE 20810.1 Quantum field theory in curved spacetime 20910.2. Inhomogeneous perturbations about minisuperspace 211

11. VACUUM STATES FROM QUANTUM COSMOLOGY 21411.1. De Sitter-invariant vacua 21511.2. The no-boundary vacuum state 216

12. SUMMARY 220

ACKNOWLEDGEMENTS 221

13. A GUIDE TO THE LITERATURE 221

REFERENCES 227

WORMHOLES AND NON-SIMPLY CONNECTED MANIFOLDS 245s. W. Hawking

REFERENCES 267

CONTENTS

BABY UNIVERSESAndrew Strominger

1. INTRODUCTION

2. TOPOLOGY CHANGE AND THIRD QUANTIZATIONIN 0+1 DIMENSIONS2.1. Third quantization of free one-dimensional universes2.2. Third quantization of interacting one-dimensional

unIverses2.3. The single-universe approximation and dynamical

determination of coupling constants2.4. The third quantized uncertainty principle

xi

272

274275

279

285291

3. THIRD QUANTIZATION IN 3+1 DIMENSIONS 2953.1. The gauge invariant action 2963.2. Relation to other formalisms 298

4. PARENT AND BABY UNNERSES 3004.1. The hybrid action 3024.2. Baby universe field operators and spacetime couplings 304

5. INSTANTONS - FROM QUANTUM MECHANICS TOQUANTUM GRAVITY 3085.1. Quantum mechanics 3085.2. Quantum field theory 3165.3. Quantum gravity 3195.4. Axionic instantons 3265.5. The small expansion parameter 329

6. THE AXION MODEL AND THE INSTANTON APPROXIMATION 332

7. THE COSMOLOGICAL CONSTANT 3367.1. The Hawking-Baum argument 3367.2. Baby universes and Coleman's argument 338

ACKNOWLEDGEMENTS

REFERENCES

340

341

CONTENTS

CIUTlqllE OF COLEMAN'S Til EORY OF TilE VANISIIINGCOSMOLOGICAL CONSTANT 347

Leonard Susskind

REFERENCES 351

QUANTUM COSMOLOGYAND

BABY UNIVERSES

HAMILTONIAN FORMULATION OFGENERAL RELATIVITY

CLAUDIO TEITELBOIM 1

Centro de Estudios Cientificos de Santiago,Casilla 16443, Santiago 9, Chile

and

Departamento de Matematicas, Facultad de Ciencias,Universidad de Chile Casilla 653, Santiago, Chile.

lLbng Term Member, Institute for Advanced Study, Princeton.

1

21. INTRODUCTION

CLAlilllO TI':ITI':LIlOIM

2. HAMILTONIAN FORMULATION OF GAUGE THEORIES (PRE-BRST)

3. BRST HAMILTONIAN FORMULATION OF GAUGE THEORIES

4. DYNAMICS OF GRAVITATIONAL FIELD

5. DOES THE HAMILTONIAN VANISH? GENERAL COVARIANCEAS AN "ORDINARY" GAUGE INVARIANCE

6. MUST TIME FLOW?

7. BRST CHARGE OF GRAVITATIONAL FIELD

8. ELEMENTS OF BRST THEORY

9. ACKNOWLEDGMENTS

HAMILTONIAN FORMULATION OF GENERAL RELATIVITY

1 INTRODUCTION

3

The lectures were a review of aspects of gravitation theory that are mostly

well understood. There appeared to be no point in writing lectures notes assuch, since almost all the material was well covered in the existing literature.What is given below is a bibliography, with some comments when deemednecessary. The bibliography is, in turn by no means complete. Rather, it isthe opposite, consisting of the small number of references which were heavily

relied upon in the presentation given in the school.

There are two exceptions to the above. It was felt by the editors thatBRST theory was not familiar to many people interested in quantum cos-mology and that the reference used for that lecture was not easily accessible.For that reason that paper is reprinted here in full in Sec VIII. The sameapplies to the comments on generally covariant systems given in Secs. Vand VI which, in the form given below, are taken from a book that has not

appeared at the time of this writing.

2 HAMILTONIAN FORMULATION OF GAUGETHEORIES (PRE-BRST).

1. Dirac, P.A.M. 1950. "Generalized Hamiltonian Dynamics", Can. J.Math., 2, 129.

2. Dirac, P.A.M. 1950. "Generalized Hamiltonian Dynamics", Proc. Roy.Soc., (London), A 246, 326.

These papers develop the general hamiltonian formalism for gauge sys-tems. They give the general definition of a gauge system as one for whichthe general solution of the equations of motion contains arbitrary functions

4 CLAlJlHO TEITELIIOIM

of time. It is shown thaI. this property leads to the presence of constraintsamong the canonical variables and the general properties of constrainedhamiltonian systems are then developed.

3. Dirac, P.A.M. 1967. Lectures on Quantum Mechanics, Yeshiva Univer-sity, New York: Academic Press. Despite its title, this is a beautifulsummary of the classical theory of constrained hamiltonian systems.

4. Hanson, A., Regge, T. and Teitelboim, C. 1976. Constrained Hamilto-nian Systems, Rome: Accad. Naz. dei Lincei. This book contains anaccount of the general theory of constrained hamiltonian systems andapplications, including the gravitational field. The role of surface inte-grals as asymptotic symmetry generators for gauge fields is also treated.

3 BRST HAMILTONIAN FORMULATIONOF GAUGE THEORIES.

5. Henneaux, M. 1985. "Hamiltonian Form of the Path Integral for The-ories with a Gauge Freedom", Phys Rep. 126, 1. A lucid account,including some of Henneaux' s important own contributions.

6. Teitelboim, C. 1987,1988. "Elements ofBRST Theory", in V th BrazilianSchool of Cosmology and Gravitation, M. Novello, ed. Singapore,1987:World Scientific. Also in Connections Among Particle Physics, NuclearPhysics, Statistical Physics and Condensed Matter, J.J. Giambiagi et aleds. Singapore,1988: World Scientific. (Proceedings of the 1987 LatinAmerican School of Physics held in La Plata, Argentina.) Reprinted

.

here in Sec VIII.

7. Henneaux, M. and Teitelboim, C. 1991. Quantization of Gauge Systems.Princeton: Princeton University Press. A complete and self contained

HAMILTONIAN FORMULATION OF GENERAL RELATIVITY 5

presentation of the general theory of gauge systems, classical and quan-tum, which deals at length with BRST theory. The application to thegravitational field is not discussed. Some sections of this book are in-cluded below in Sees. V and VI.

4 DYNAMICS OF GRAVITATIONAL FIELD

The classical references that give the complete form of the hamiltonian are,

8. Dirac, P.A.M. 1958. "The Theory of Gravitation in Hamiltonian Form",Proc. Roy. Soc. (London) A246, 333.

9. Arnowitt, R., Deser, S. and Misner, C.W. 1962. "The Dynamics ofGeneral Relativity" in Gravitation, an Introduction to Current Research,1. Witten, ed, New York: Wiley.

A summary, which includes some later developments, may be found IIIref. 4 of this list.

In higher dimensions the most general action that gives first order field equa-tions is not the Hilbert action. It is given, rather, by a sum of terms, each ofwhich is continuation of the Euler characteristic of an even dimension belowthe one in consideration. That theory has the same degrees of freedom (spa-tial geometry and its conjugate) as the Einstein theory but its hamiltonianis different, see

10. Teitelboim, C. and Zanelli, J. 1987. "Gravitation Theory Generated byDimensional Continuation of the Euler Characteristic as a ConstrainedHamiltonian System" in Proceedings of the Workshop on ConstraintsTheory and Relativistic Dynamics, G. Longhi and L. Lusanna eds, Sin-gapore: World Scientific.

Ii ('I.AlJllIO TEll LI.lIOIM

When the space is not compact a gauge transformation that does not becomethe identity at infinity is "improper". Its generator is not a constraint but,rather, it is given by a non-vanishing surface integral that must be added tothe hamiltonian and whose value depends on the field configuration. Thosesurface integrals generate the global (non-gauge) symmetries of the system.In the case of gravity they include the energy, momentum and angular mo-mentum of the field. See

11. Regge, T. and Teitelboim. C. 1974. "Role of Surface Integrals in theHamiltonian Formulation of General Relativity", Ann. Phys., N.Y. 88,286.

A brief account may be found in ref. 4.

The commutation rules of the constraint-generators of gravity capture thegeometrical content of the theory. See

12. Teitelboim, C. 1973. "How Commutators of Constraints Reflect theSpace-time Structure", Ann. Phys., N.Y. 79, 542.

13. Teitelboim, C. 1980. "The Hamiltonian Structure of Space-time" IIIGeneral Relativity and Gravitation, Vol 1, 195, A. Held, ed. Plenum,N.Y.

There is an analog of gravitation theory in two space-time d.imensions. Thisanalog is remarkably rich and has received much attention lately in connec-tion with string theory. See

14. Teitelboim, C. 1983. "Gravitation and Hamiltonian Structure in TwoSpace-time Dimensions", Phys. Lett., 126B, 41.


15. Teitelboim, C. 1984. "The Hamiltonian Structure of Two- DimensionalSpace-time and its Relation with the Conformal Anomaly" , in QuantumTheory of Gravity, S. Christensen, ed. Bristol: Adam Hilger.

16. Polchinski, J. 1989. "A Two-Dimensional Model for Quantum Gravity",Nucl. Phys. B324, 123.

17. Cooper, A., Susskind, 1., Thorlacius, 1. 1991. "The Classical Limit ofQuantum Gravity Isn't". SLAC-PUB-5413-T.

5 DOES THE HAMILTONIAN VANISH?GENERAL COVARIANCE AS AN "ORDINARY"GAUGE INVARIANCE.

The hamiltonian for the gravitational field is a linear combination of theconstraints. Its numerical value for a solution of the equations of motion iszero. This fact caused some perplexity in the early days, but it was then re-alized that zero hamiltonian was closely tied with general covariance. Indeedwhat one normally calls "time" for "ordinary systems" is included among thedynamical variables of the gravitational field. See in this context

18. Baierlein, R. F., Sharp, D. H. and Wheeler, J.A. 1962. "Three- Dimen-sional Geometry as a Carrier of Information About Time". Phys. Rev.126, 1684.

and also

19. Wheeler, J.A. 1963, in Relativity, Groups and Topology, C. DeWitt andB. DeWitt, Eds., Gordon and Breach, N.Y.

8 CLAUDIO T1';I'I'ELIlOIM

There was then a line of work devoted to "disentangle the time" from the"true dynamical variables". It is important however to understand thatalthough such a disentanglement might be useful technically it is by no meansnecessary. Indeed one may treat general covariance exactly along the linesof an "ordinary" gauge theory, even within the hamiltonian framework.

The following remarks on that problem are taken from Chapter 4 of ref 7.They are included here because that reference is not yet in print at the timeof this writing.

5.1 GENERALLY COVARIANT SYSTEMS

One normally describes the motion of the system by giving the canonicalvariables as a function of time. The time is assumed to have a direct physicalsignificance but is not itself a dynamical variable.

There exists a different formulation of the dynamics in which the physicaltime and the dynamical variables of the system are treated more symmetri-cally. This formulation includes the time among the dynamical variables anddescribes the relations ("correlations") between the original dynamical vari-ables and the physical time by giving the enlarged set of canonical variablesin terms of an arbitrary parameter. The arbitrary parameter does not pos-sess any physical significance, and the formalism is therefore invariant underreparametrizations of it, or, as one says, it is "generally covariant". In fieldtheory, one also introduces arbitrary labels for the spatial coordinates, andthe theory becomes then invariant under arbitrary changes of the space-timecoordinates.

In practice, generally covariant systems arise in two different ways. Onemay have a system in which originally the physical time was not includedas a canonical variable and proceed to "parametrize" the theory to achieve


general covariance. This can always be done. Or the system may be "alreadygenerally covariant". The already generally covariant system "par excellence"is the gravitational field in general relativity.

Attempts at "deparametrizing" already generally covariant systems havenot been quite successful. It seems thus preferable to aim at both formulatingand answering questions while treating all variables on the same footing. Aswe shall see below this amounts to treating the motion as the unfolding of agauge transformation.

5.2 TIME AS A CANONICAL VARIABLE. ZERO HAMILTO-NIAN

5.2.1 Parametrized Systems

Consider a system with canonical variables qi, Pi, Hamiltonian Ho(q,p) and,for simplicity, no constraint. The action reads

[ i ] l t2 (dqi )S q (t),pi(t) = h Pia; - Ho dt. (5.1 )Introduce now the time t == qO and its conjugate Po as canonical variables byreplacing (5.1) by

where the dot denotes derivative with respect to the parameter T.

The motion obtained by demanding that (5.2) be stationary is equivalentto that following from extremizing (5.1). This is seen by extremizing first(5.2) with respect to U O and Po, which yields

10

and

')'0 == Po + Ho = 0 (5.3)

(5.4)

Equations (5.3) and (5.4) may be solved to express those variables that werevaried in terms of the others. It is then legitimate to replace in (5.2) Poby -Ho (and uo by i which is not seen) to obtain a reduced action for theremaining variables. That reduced action depends only on qlJ.(T) (p, = 0, i)and Pi (T) and reads

(5.5)

For Eq. (5.5) to hold t must be a monotonous function of T so that itsinverse exists. However, the generally covariant version does not need thatassumption and is, therefore, more flexible. For example, in the path integral,one admits trajectories that go back in the time t, even in nonrelativistic par-ticle mechanics. However, in that case, due to the linearity ofthe constraints(5.3) on po, the net contribution of the histories that go back in t cancelsout, and the resulting path integral coincides with that of the reduced ac-tion (5.1). This accidental feature may not hold for systems that are alreadygenerally covariant. For instance, in the case of the relativistic particle, thehistories going back in time do contribute a net amount corresponding toantiparticle propagation.

5.2.2 Zero Hamiltonian

The action (5.2) contains one extra canonical pair over (5.1) but also containsthe constraint ')'0 ~ O. This constraint -being the only one- is first class.


Thus, the number of independent degrees of freedom is the same for (5.1)and (5.2), in agreement with the discussion leading to (5.5).

An important property of (5.2) is that there is no non- vanishing first-class Hamiltonian in it. Thus, the extended Hamiltonian contains only theconstraint 10. Therefore, the motion is the unfolding of a gauge transforma-

tion.

If the original theory has other gauge generators la' (a' = l, ... ,m) andsecond-class constraints X'" = 0 to start with, the action reads

s = 17'l (p,Jl' - HE) drTl

(5.6)

where the extended Hamiltonian is now a combination of all the constraints

HE = ua'Ya +u"'Xc> (a = D, ... ,m)

5.2.3 Parametrization and Explicit Time Dependence

(5.7)

Incorporating time as a canonical variable in a theory originally not writtenin generally covariant form does not only enable one to reach a more symmet-rical formulation of the motion. It is also useful in practice to deal with prob-lems in which there is an explicit time dependence in the constraints. Such adependence complicates the formalism because the equations expressing thepreservation in time of the constraints involve "explicit time derivatives" andare not formulated just in terms of brackets.

This difficulty disappears when time is introduced as a canonical variable,because after that step is taken no explicit time derivative is left. The anal-ysis proceeds then as in the ordinary case but in the enlarged phase space

12 CLAUDIO TEITELBOIM

containing t and Po. The notion of first class is, in particular, understood toinclude also Po + H (p, q, t) among the constraints.

5.3 TIME REPARAMETRIZATION INVARIANCE

5.3.1 Form of Gauge Transformations

The action (5.7) is invariant under the transformation

8q = qc (5.8a)8p = pc (5.8b)

8uOl = (uOlc)' (5.8c)8ua = (uac)' (5.8d)

with

(5.9)

The transformation (5.8) is an infinitesimal reparametrization of amountc(7), i.e., is obtained from 7 ---t f = 7 - C(7). Equations (5.8a)-(5.8b)state that q and p transform as scalars, whereas (5.8c)-(5.8d) state that themultipliers ua , u Oi transform as scalar densities. The condition (5.9) statesthat the endpoints 71 and 72 are left invariant.

The reparametrization (5.8) differs from the gauge symmetries generatedby the fa by and "equation-of-motion symmetry" and, therefore, is not' anindependent symmetry. Indeed, (5.8) is equal on-shell to a gauge invarianceof the action wi th parameters

(5.10)


5.3.2 Must the Hamiltonian be Zero for a Generally Covariant

System?

We have just shown that the extended action BE with zero Hamiltonian isinvariant under arbitrary reparametrizations of the r-variable. One may

then ask the converse question, namely, whether general covariance implies

a zero Hamiltonian.

If the q's and p's transform as scalars under reparametrizations, the pqterm in the action transforms as a scalar density, and its time integral is

therefore invariant by itself. Furthermore, since the 'Ya and the x'" are func-tions of p and q, they also transform as scalars. One therefore adjusts thetransformation law for the multipliers u a and u'" to be that of a scalar density

so as to make the ua'Ya and u"'x", terms also scalar densities. Thus, once it is

assumed that the p's and q's are scalars, it is crucial to have an independent

variable in front of each constraint function to be able to achieve invariance.

If one had in the action a nonzero first-class Hamiltonian H' (p, q) besidesthe ua'Ya +u"'x", term, one could not achieve reparametrization invariance,since H' (p, q) would transform as a scalar and not as a density.

Thus, if q and p transform as scalars under time reparametrizations, theHamiltonian is (weakly) zero for a generally covariant system. This is thecase found so far in practice because one normally formulates theories in

terms of variables with a simple geometrical meaning. For that reason, one

usually says that a zero Hamiltonian is the distinguishing mark of a generally

covariant system.

However, it is important to realize that one may also have general covari-

ance with a nonzero Hamiltonian. Indeed, one knows that a time- dependent

canonical transformation changes the value of the Hamiltonian. Thus, by per-

forming a canonical transformation that depends explicitly on the parameterr, one may bring in a nonzero Hamiltonian while keeping time reparametriza-


tion invariance of the action. Since r itself does not transform as a scalar, thenew canonical variables will not be scalars in agreement with the discussionabove.

5.3.3 Simple Example of a Generally Covariant System with aNonzero Hamiltonian

A simple and interesting example of this possibility is to consider the action

(5.2) and define the variable

T(r) = t(r) - rwhile keeping all the other variables unchanged. The action becomes

(5.11)

(5.12)

with ,\ = U O - 1. This action is generally covariant and possesses a nonzeroHamiltonian Ho. The canonical coordinate T does not transform as a scalarunder reparametrizations but rather inhomogeneously as a "connection"

8T = TIS +IS (5.13)

The example (5.12) shows clearly that there is really no distinguishingtechnical feature of general covariance over other gauge transformations. In-deed, once one has a gauge generator, the action can always be made time

reparametrization invariant by appropriately defining the transformation of.

the variables: One first makes a time-dependent canonical transformationto go to a zero Hamiltonian and one defines the new canonical variables totransform as scalars. This can be achieved without introducing an explicittime dependence in the constraints.


5.4 "TRUE DYNAMICS" VERSUS GAUGE TRANSFORMA-TIONS

5.4.1 Interpretation of the Formalism

Once there is an arbitrary function of time in the general solution of theequations of motion, one cannot tell from the equations themselves whetherthis is because some of the canonical variables are not observable or whetherthe time (i.e., the variable over which the Lagrangian is integrated to formthe action) is not observable.

Normally, this choice is made based on information provided from outside.For example, in Maxwell's equations in Minkowski space one assumes that the

time variable has an objective meaning and interprets the gauge invarianceas indicating that not all components of the vector potential are observable.Ultimately, this interpretation is based on the fact that the Maxwell field isjust a subsystem of the universe and that there are other systems interactingwith it that are used to provide a time standard.

Conversely, when one deals, for example, with a parametrized nonrelativis-

tic particle, one normally assumes that the variables t and qi have a physicalmeaning and that T is unobservable. Again, this is based on information notcontained in the action itself.

However, when one tries to apply the formalism to the universe as a whole

-as one does in general relativity- one has no "outside" available, and it isdesirable to have a way to handle the formalism that does not need foreigningredients. For that purpose, the most symmetric from of the theory isthat in which the Hamiltonian is weakly zero, i.e., in which it is a linear

combination of the constraints. One then says that all physical questions

should be formulated in terms of functions that have zero brackets with theconstraints. By doing so, one treats on the same footing "ordinary gauge-


invariant quantities" (snch as the magnetic field in electromagnetism) andconstants of the motion (such as the initial position or the energy for a freeparticle). This is because, when the Hamiltonian weakly vanishes, the firstclass functions with no explicit r-dependence are constants of the motion.

At first sight this approach would seem to exclude asking questions such

as "what is the position q at time to" for the particle. However, this is not so.Indeed, there is a constant of the motion qto (r) that is equal to the positionat time to. For a free nonrelativistic particle with mass m and momentum P,it is simply

p(r)qto(r) = q(r) - - (t(r) - to)

m(5.14)

and the above question is equivalent to "what is the value of the first-class

function qto (r )?"

5.4.2 Reduced Phase Space

The functions having vanishing brackets with the constraints are the "classi-

cal observables" and are defined over the reduced phase space. So when the

extended Hamiltonian is a linear combination of the constraints, an observ-

able takes the same value on an entire classical history (and on the gauge-related ones). It may thus be thought of as a function in the space of classicalsolutions of the equations of motion ("covariant phase space").

The set of observables is easily characterized in the case of a parametrized

system. The constraint Po +Ho = 0 can be solved for Po. Accordingly, func-tions on the constraint surface can be viewed as functions of qi, Pi and t. The

condition [A(l,pi, t),po +Hol ::::: 0 can then be integrated, at least in prin-ciple. This is equivalent to solving the equations of motion and completely

determines the time dependence of A.


The set of the 0 bservables is thus isomorphic to the set of functions A(q, p)of the initial data. The space of the initial data q,p is itself isomorphic tothe space of the q's and p's at any fixed time. Therefore, the observablesare in bijective correspondence with the dynamical variables of the original,unparametrized theory.

Going to the reduced phase space does not amount to freezing the dy-namics at a given instant of time. Rather, it is equivalent to describing thesystem in terms of constants of the motion. In these canonical coordinates theHamiltonian vanishes and the equations of motion reduce to dAjdT = aAjaT.

One may reach the reduced phase space for a parametrized system byimposing the canonical gauge condition t = to. This condition defines a slicethat cuts all the histories once and only once. In the gauge t = to, q and pbecome equal to the constants of the motion qto and Pto; i.e., these constantsof the motion are the gauge-invariant extensions of q and p off the canonicalgauge t = to.

In practice, the explicit construction of the reduced phase space in termsof the given canonical variables may not be possible, since it requires solvingthe equations of motion for arbitrary initial data. This is a hopeless task for

systems with a complicated Hamiltonian, not to mention general relativity.Therefore, it appears necessary to develop the theory without reducing thesystem.

6 MUST TIME FLOW?

The considerations of Sec V are of practical importance for the path integral.Indeed, within the hamiltonian framework there is nothing that calls, forexample, for a positive lapse function. Thus one may use gauges that arewidely regarded as not permissible. For example N = 0, or tn = 0 for a


compact space are quite alright for gravity, although they seem not to allowthe time to flow. [M. lIenneaux and C. Teitelboim, to be published].

For the technically simpler case of the free particle the issue is illustratedin the following analysis taken from Chapter 16 of ref 7.

6.1 GAUGE INDEPENDENCE OF PATH INTEGRAL FOR APARAMETRIZED SYSTEM ILLUSTRATED. EQUIVALENCOF THE GAUGES t = rAND t = 0

We illustrate in this subsection the general analysis of the reduced phasespace path integral and its expressions in terms of gauge fixings, the keypoint being that different choices of gauge conditions are mere rewritings ofone and the same reduced phase space amplitude.

We will deal with the parametrized free nonrelativistic particle with con-straint

p2H = Pt +- = 02m

and action

S[q,p, t,Pt, N] = J(pq +Pt i - NH)dr.(H was denoted 'Yo in (5.3) and N was called U O in (5.2).)

(6.1 )

(6.2)

6.1.1 Reduced Phase Space Transition Amplitude as a ReducedPhase Space Path Integral

For any value of the constant c, a complete set of observables is given by


q; = q - E..(t - C), p* = p. (6.3)m

These are constants of the motion, which coincide with q and p in the canon-

ical gauge t = c. The Heisenberg operators q~(r) are independent of r. Fordifferent values of c, they are related by a unitary transformation

* p2 * p2qc = exp z-2 (Cl- C2) qc exp-z-(Cl - C2),1 m 2 2mand the corresponding eigenstates are related by

The mixed transition amplitude

(6.4)

(6.5)

(6.6)(m )1/2 [m(q* _ q* )2](q;2,r2Iq;1,rl) = 2 O( ) exp - 2 0t Cl)

7I"Z C2 - Cl Z C2 - Cl

can be written as a sum over reduced phase paths q~=o(r) == q*(r),p;=o(r) ==p*(r) of the exponential of the reduced phase space action SR[q*(r),p*(r)].These paths obey the boundary conditions

k-~Cl] (rl)[q* - ~ C2] (r2)

(6.7)

(6.8)

obtained by using the relation between q;l' q~2 and q*, p*. Since the reducedHamiltonian H(q*, p*) vanishes, the reduced action differs from Jp*q*dr bya boundary term adapted to the boundary conditions (6.7)-(6.8),

(6.9)


6.1.2 Canonical Gauge Conditions

The reduced phase space path integral

(6.10)

can be written in terms of q,p, t,Pt and gauge conditions. To that end, oneobserves that the action (6.2) weakly differs from the reduced action (6.9) bya surface term,

[2 ] 7'lP 7 - 71SR::::: S + - (C1 + --(C2 - C1) - t)2m 72 - 71

7"1

We shall consider gauge conditions of the form

t-f(7)=0

(6.11)

(6.12)

for which the determinant [t - f(7),Pt + HoJ is unity. The reduced phasespace path integral (6.10) is equal to

The paths are subject to the boundary conditions


[q - ~ (f(7) - C1)] (71) =[q - ~ (f(7) - C2)] (72) =

6.1.3 Gauge t = 0

q~,

(6.15)

(6.16)

In the gauge t = 0 the Hamiltonian vanishes. One finds from ecs. (6.13)-(6.16) that the path integral can correctly be described in that gauge as

with the boundary conditions

( ) p(71) * () p(72) *q 71 +--C1 = qCl' q 72 +--C2 = qC2.m m

One sees that there is nothing wrong with "the time not flowing".

6.1.4 Gauge t ex 7

(6.17)

(6.18)

One may also let the time flow as it is ordinarily done. Although t = 7 ispermissible, it is a bit simpler here to adjust the endpoints so that t(7d = C1

22

with

CLAUDIO TEITELBOIM

(6.20)

(6.21)

If C2 - Cl # 0, one can further transform the right hand side of (6.20) byrewriting the integral as integral over t. This yields the usual form of thetransition amplitude

J[Dp][Dq] exp i 1:2 (p ~~ - ::) dt. (6.22)It is thus quite possible to derive the standard transition amplitudes of thetheory by treating the transformations generated by Pt + Ho ~ 0 as ordi-nary gauge symmetries. This point of view, which treats all the first classconstraints on the same footing, possesses the further advantage that it isnot necessary to undig a physical time variable in order to compute physicalamplitudes.

7 BRST CHARGE OF GRAVITATIONAL FIELD

In the "minimal" BRST formulation of gravitation theory one introducesbesides the spatial metric 9ik and its conjugate momentum 7["ik, a set offour anticommuting ghost fields .,,1., ."i and their conjugates Pl., "Pi. Thedynamics of the field is then summarized in a single global function, theBRST generator n given by


J 3 [.1 . . .1 .1 .1. . k]n = d X TJ 1{.l +TJ'1{i - 'P'TJ,i TJ - (P.lTJ ),iTJ' - 'PjTJ:kTJwith

and

23

(7.1)

(7.2)

(7.3)The meaning of the ghosts and of n is explained in the following section.


8 ELEMENTS OF BRST THEORY

This is a reprint of ref. 6, whith minor corrections. Its content is the follow-ing.

1. THE GHOST, YOU'VE COME A LONG WAY BABY1.1 Introduction1.2 Quantum mechanics, the art of finding and combining simple elementary pro-

cesses

1.3 Ghosts necessary to keep elementary process simple1.4 BRST symmetry: ghosts and matter become different component of a simple

geometrical object

1. BRST SYMMETRY IN CLASSICAL MECHANICS2.1 Ghosts have role in classical mechanics2.2 Gauge invariance and constraints2.3 Classical mechanics over Grassmann algebra necessary2.4 Higher order structure functions2.5 Rank defined. Open algebras2.6 Ghosts. Ghost number. BRST generator as generating function for structure

functions.2.7 Ab..liallizat ion of constraints. Existence of!12.8 Ulliqueness of !12.9 Classical BRST cohomology

Ill. QllANTUM BRST THEORY3.1 States and operators3.2 Ghost number3.3 BRST invariant states3.4 Quantum BRST cohomology3.5 Equivalence of BRST physical subspace with conventional gauge invariant sub-

space3.6 Action principle3.7 Path integral. BFV theorem3.8 Contact with lagrangian notation3.9 BRST invariant boundary conditions


I. THE GHOST, YOU'VE COME A LONG WAY BABY

1.1 Introduction

25

The BRST symmetry shows that ghosts are real. I have chosen to startlike this, because I think that, at least for a theoretical physicist, the statementis correct.

Furthermore, in order to have a least a chance of fully exploiting the richnessinherent in the BRST symmetry, one must try to go all the way. That cannot bedone if one does not feel at home with ghosts, or if one thinks that they are somekind of embarrassing mathematical artefact that must be included to fix up somedetails. Therefore, it is important to cross that threshold immediately, so that onecan relax and enjoy what follows.

In order to make that step easier, I would first like to make some general com-ments on the historical evolution of the idea of ghost in physics and its relation withthe basic principles of quantum mechanics. These comments will not be logicallyneeded for the subsequent presentation, which does not follow the historical route.

1.2 Quantum mechanics, the art of finding and combining simple ele-mentary processes

In particle physics language, a ghost is a particle that obeys the wrong relationbetween spin and statistics. The first kind of ghosts were introduced by Feynman [1]in 1963, when studying quantum theory of gravitation. They were vector particles,that is, particles of integer spin (1 and 0) obeying Fermi statistics. Probably thefirst mention of the ghost in this sense III the physics literature are the followingphrases in Feynman's article,

.. J found it by trial and error you must substract from the answer... theresult you get imagining that an artificial dopey particle is coupled to it(the gmviton). It's a vector particle.

Since 1963 the ghost has come a long way. We do not, or rather we shouldnot think of it anymore as an "artificial" or "dopey" thing, although many of usstill have that tendency. Rather, the lesson that we have been taught, sometimestbe hard way, is that the distinction between ghosts and "real matter" is not oneto be taken too seriously.

To see this, it is best to look at quantum mechanics from the point of viewof the sum over histories. In that formulation, the amplitude K (2,1) to propagate

26 CLAUDIO TEITELBalM

from configuration 1 to configuration 2 is given by,

K(2,1) =hi"toriea

joining 1 and 2

The idea is illustrated in Fig. 1.

2

1

e~ S[hi.tory] (1.1)

Figure 1 The quantum amplitude is obtained by summing over all possible historiesjoining the configurations 1 and 2.

In Dyson's words [2] on occasion of the Einstein centennial in 1979, this formulais describ..d as follows,

... Thirty-one ymr.< ago, Dick Feynman told me about his 'sum over histo-ries' l'ersion of quantum mechanics. 'The electron does anything it likes,'he said. 'It just goes in any direction at any speed, forward or backwardin time. however it likes, and then you add up the amplitudes and it givesyou the wave-function.' I said to him, 'You're cmzy.' But he wasn't.In the classical limit of actions which are large compared to Planck's constant,

due to destructive interference, only histories close to the one which makes theaction stationary contribute to the sum. This is why action principles arise mclassical physics.

Now, the point here is that alt.hough the resulting amplitude K(2, 1) may bequite complicat.ed, the amplitude for an elementary process, or history, is simple.It takes the form,

lexp lis (1.2)


But this is not all, another important ingredient must be added. In field theory,(and by field theory here I mean something very general, including things likequantum gravity and strings), the action may be written as,

S = Skin (free) + Sint.raction (1.3)

The splitting (1,3) is not just a technicality or a calculational tool. It deter-mines our whole physical picture of objects (particles, say) that propagate freelyin between interactions. If we did not have this splitting, there would not be thatmuch information contained in (1.2). We would only know that the elementaryamplitude must have an absolute value equal to unity, and that is just too flexible.

The sum over histories, together with the splitting of the action into free partand an interaction, lead directly to Feynman diagrams, like the familiar one ofquantum electrodynamics shown in Fig. 2; which describes the interaction betweentwo electrons due to the exchange of one photon. To obtain the amplitude for thatprocess one must sum, according to the general rule, over all alternative ways ofexchanging a photon.

Figure 2 Interaction between two electrons due to exchange of a photon.

There are different ways of doing this. The most convenient and physicallytransparent one is that which exhibits manifest Lorentz invariance. In that case,one includes in the sum over all possible exchanged photons, not only those which arepolarized transverse to the direction of motion, but also those whose polarizationvector lies along the momentum (longitudinal photons) and along the time axis


(time-like photons). There is also another way of calculating the same amplitude inwhich only transverse photons are exchanged, but a supplementary instantaneousCoulomb interaction must then be added.

Here we see in a familiar context the first example of a general occurrence whichappears to be of importance; namely, if we insist in formulating a theory in termsof as few variables as possible, the elementary machinery becomes less transparentand we lose understanding rather than gaining it.

In fact, look at what happens with the exchanged photons. We may view theinstantaneous Coulomb interaction as resulting from performing first the sum overlongitudinal and time-like modes, while leaving the sum over transverse polarizationyet undone. However, the result of that partial sum is no longer of the form exp(*5) with 5 of the form (1.3).

Thus, this seemingly innocent (and technically correct) step carries with itabandoning the picture that the interaction is completely accounted for by an ex-change of particles. This is not just a philosophical loss, since the Lorentz invariancebecomes also obscured. Conversely, if one insists in a formulation based on a simple,uniform, elementary process, one gains as a bonus Lorentz invariance.

What I am trying to say is that it appears to be a good principle to insist onan el"'lJlentar~' amplitnde of the form E.Tp( *5). If we start giving np and allowingthings like .4 fJ'p(iB) we might as well give up completely and try to write the fullamplitude !\' (2, I) right away, which is not likely to work.

This import ant example of quantum electrodynamics also teaches us anotherlesson. We must be willing to pay a certain price in order to have a simple, uniform,elementary process. We must not panic too easily. The time-like photons havenegative kinetic energy and their contribution to the probability enters with a minussign. Yet we must have them; they are a good thing.

It is often said that no catastrophe takes place due to these minus signs becausethe time-like and (Iongtitudinal) photons are not "real" but they are just "virtual."By this it is meant that they do not take part in processess like the one illustratedin Fig. 3b but they only appear in diagrams such as the one of Fig. 3a.

TillS is correct, but the terminology is somewhat unfortunate and misleading.Indeed, the virtual photons are not less real that the other ones in the sense ofhaving observable physical effects. They contribute to energy levels as reflected, forexample, in a small shift (the Lamb shift) they produce in the spectrum of lightfrom excited hydrogen atoms. It is just that they cannot be observed directly with


something like a photo-cell.

29

a b

Figure 3 The process shown in (3a) is possible for both real and virtual photons whilein (3b) only a real photon can appear.

I would like to indulge here in a slight digression: The more familiar one getswit h Cjualltlull mechanics, the more blurred in one's mind the distinction between"rear' and "virtual" processes becomes. Indeed, relativistic quantum mechanics issomehow engineered so that it is consistent to have a distorted view of the world.For example, a relativistic particle has a non-zero amplitude to cross the light-coneand can even turn back in time. Yet we can, because of that very fact, reinterpretthings so that our usual notions of causality are not violated. In the same way,the consistency of a world in which only traverse photons can be observed by aparticle detector may be thought of as being possible because of the existence of"virtual" photons of all four polarizations, otherwise the sum of all probabilities forall mutually exclusive "real" processes would not be equal to unity.

It would seem that the more we progress toward systems that are less familiarand reach into smaller scales, the more useful it should be to take quantum me-chanics as it naturally comes, without manacling it early in the game with thingslike the observer or the measuring process.

As an example of how respectable people are willing to think along this line,I would like to quote from a paper by Hawking {3J on the path integral approachto quantum gravity. To put the quotation in context I should mention that for the


gravitational field, the "initial configuration" is one three-dimensional space and the"final configuration" is another three-dimensional space. The elementary processor history which interpolates between them is a four-dimensional space-time whichhas the initial and final three-spaces boundaries.

final

----------~".------ initial

Figure 4 Initial and final configurations for the gravitational field are space-like hyper-surfaces.

The quotation is the following:

l '[timatdy I suspect that one should do away with all boundary .-tilllt>s. Tht> wholt> tht>ory would rt>ft>r only to an t>normous maze of"virtual" space-times.

At the present state of development this is not established. I have mentioned itonly to illustrate the way of thinking.

1.3 Ghosts necessary to keep elementary processes simple

We have so far been using quantum electrodynamics as an example. There,one may view the need for the longitudinal and time-like photons as arising fromthe demand that the sum of aU the probabilities for processes involving only "real"pcrrticles in the initial and final states should be equal to unity. However this turnsout not to be sufficient in more complicated theories. One needs in general, another,

IIAMILTONIAN FORMULATION OF GENERAL RELATIVITY 31

more exotic kind of "virtual" particle, and the price to pay in mental flexibility inorder to have a simple and uniform elementary process is stiffer. One must acceptthat the new modes obey a relation between spin and statistics which is oppositeto that of usual matter. These new modes are called ghosts. They seem to be moreshocking or, I suppose I should say, "ghostlier" than, say, the time-like photons. Thereason is that they are either fermions with integer spin or bosons with half-integerspin. However, in my opinion this is mostly psychological since we have had alreadyforty years or so to learn how to live with those other ghosts- the longitudinal andtime-like photons.

One may view this "wrong" connection between spin and statistics in the sameway as we viewed above the fact that in quantum mechanics particles can propagatefaster than light and backward in time. That is, at a basic level there is reallyno connection between spin and statistics. Anything can happen. One can havebosons with integer spin, fermions with integer spin, bosons with half-integer spinand fermions with half-integer spin. Yet, things are somehow engineered so that inthe world that is directly accessible to us, we may consistently imagine that particlesdo not turn around in time, do not travel faster than light and they do obey thespin- statistics theorem.

To go further, I have to explain what I meant by "more complicated theories"above. I meant a nonabelian gauge theory.

A gauge theory is one which is invariant under a symmetry that acts indepen-dently at different points. Since one may define a geometrical object as somethingwhich is invariant under a set of transformations, one may say that gauge theoriesare field theories with geometrical content. This is what makes them theoreticallyattractive.

The simplest of gauge theories is electrodynamics and the gauge transformationfor the photon is the familiar one,

( 1.4)

Here, A = A( x) is an arbitrary function over space-time. The fact that there is onlyone function involved, means that the synunetry upon which the theory is built, hasonly one parameter and hence it is abelian. One then says that electrodynamicsis an abelian gauge theory. The extension of this idea to a nonabelian group isdue to Yang and Mills. In that case one has a set of functions Aa = Aa(x) wherethe index A runs over the generators of a Lie group. One also has in that kind


of theory, not just one "photon" field but a whole collection of them A~(x). Thegauge transformation (1.4) is modified to read,

(1.5)

where the real numbers Cbc are the structure constants of the (nonabelian) gaugegroup.

The symmetries present in electrodynamics and in its extension, the Yang-Millstheory, are what we call internal gauge symmetries. Technically this appears in thefact that the gauge transformation does not contain derivatives of the fields andhence does not connect different space-time points. There is another kind of gaugesymmetry, which is perhaps even more interesting, and which does act on space-time and hence it is not internal. In that case, the gauge parameter A is labeledby a space-time index instead of an internal gauge index. For example, in generalrelativity one has the gauge transformation for the graviton (reparametrization),

(1.6)

and the gange parameter carries a vector index. Another important instance of anoninternal symmetry is the gange snpersynunetry ("the square root" of a repara-metrization) present in snpergravity, where the change in the "gravitino" field isgiven by

(1. 7)In this case, the gauge parameter A carries a spinor index A and it is anticommu-tative. For this reason the transformation mixes fermions with bosons and is calleda supersymmetry transformation.

The general practical rule for the appearance of ghosts in a gauge theory cannow be stated. In each case there will be a pair of new fields, usually called "ghost"and "anti-ghost," for each gauge function present in the gauge transformation. Theghost field will be anticommuting (fermionic) if the corresponding gauge parameterwas commuting and vice versa.

Therefore, in electrodynamics there is one pair (CG) of fermionic scalar (spin0) ghosts, whereas in the Yang-Mills case we have a whole collection (ca and Ga)of fields with the same characteristics.

It is important to emphasize that there are indeed ghosts in quantum electro-dynamics. The reason that they were not seen to be necessary before is that in the


path integral for a gauge theory one has the freedom of choosing a gauge condition.In electrodynamics there is a simple gauge condition, the Lorentz gauge, which islinear in the fields and does not destroy the simple nature of the elementary processof photon exchange. If that gauge condition is used, the ghosts decouple completelyfrom the photon and can therefore be ignored. However, if one would use other,nonlinear, gauge conditions it would not be possible to ignore the ghosts.

The rise of the ghost has an interesting history with its ups and downs whichhave reflected the influence of other advances in field theory. As I said at thebeginning, the first ghost that was seen to be necessary was the fermionic vectorghost of quantum gravity, coming from the AI'- in (1.7). This ghost was foundby Feynman as he says, "by trial and error." He was trying to make the sum ofall the probabilities for graviton-graviton scattering to be equal to unity and herealized that even including the time-like and longitudinal modes of the graviton,as in electrodynamics, would not do. But of course he, for one, was very fond ofFeynman diagrams and tried to overcome the difficulty by bringing in new diagramswhich would involve new particles.

There is good reason to be fond of the diagrams because they are not just acalculational tool. They carry with them the key message that the whole compli-cated theory is Imilt by putting together a few simple elementary processes whichare very similar to each other. Thus, with this idea in mind, it was natural toattempt to c\lre the disease by simply extending what had already been done forelectrodynamics, and "introducing" yet another "virtual" degrees of freedom be-sides the longit udinal and time-like gravitons. I have used here quotation marksfor the words "introducing" and "virtual," to emphasize that one is not puttingsomething artificial in by hand, but rather one is discovering something quite realthat was there all the time.

This approach, let to a satisfactory formulation which was already written inthe desired form in which all fields present, including the time-like and longitudinalmodes and the ghost, appeared on the same footing describing particles that prop-agate freely in between interactions. That is, the total action was written in theform (1.3)

Next, there took place another development which has been hailed with goodreason, but from the point of view taken here could be thought of as "reactionary."Faddeev and Popov [4] observed in 1967 that, since the gauge transformations thatwe wrote before do not change the physical fields (by definition of a physical field),


one should include in the sum over histories only classes of equivalence of gauge-related histories, rather than histories themselves. From this nice geometrical pointof view, they concluded that the amplitude for an elementary process was not ofthe form exp( kS) but instead of the form

ldet M (exp tiS) (1.8)

where the Faddeev-Popov determinant det M, is a certain functional of the fields.They then went on to say [5],

the expression...for the S-matrix contains the nonlocal functional det M andtherefore does not look like the familiar integral of the Feynman functionalexp (i action) ... We may, however, use for det M the integral representa-tion...

Now, this "integral representation" was given, of course, in terms of the ghostfields and brought us back to the expression previously obtained directly from thediagrams. One gained with this a nice and useful connection between the ghosts andthe geometry of the gauge field. There was, however, at the same time, a negativeeffect that came from the psychological impact of the word "representation." \Vebecame used to thinking that the understandable thing was the Faddeev-Popovdeterminant and that the ghosts were merely a technicality to represent it. Perhapswe would have been more flexible if we had kept in mind that this way of thinkingwas analogous to taking the instantaneous Coulomb interaction in electrodynamicsmore seriously than the time-like and longitudinal photons.

1.4 BRST symmetry: ghosts and matter become different componentsof single geometrical object

Perhaps this psychological blockade is one of the reasons that made it necessaryfor seven more years to pass before the important discovery [6] which put the ghostonce and for all at the same level with "real" matter, the BRST symmetry. Theother, and more significant reason for this delay would seem to be that the concept ofsupersymmetry [7] was only formulated in 1973. Indeed, otherwise the developmentin question could have naturally taken place already in 1967.

The BRST symmetry, where the initials stand for Becchi-Rouet-Stora andTyutin, may be taken to be the basic invariance of the quant urn mechanics of ageometrical system. It contains and extends the concept of gauge invariance. In it,


the "original fields" get mixed with the ghosts. For example, the BRST transfor-mation of the Yang-Mills field is given by

(1.9)

where is an anticommuting parameter. It is necessary that should be anticom-muting, because the field A~ is bosonic, whereas the ghost c a is fermionic. In thissense, the BRST transformation is a supersymmetry transformation.

The fact that the ghosts and the other fields take part in an invariance whichmixes them with each other, means that they are all to be thought of as componentsof a single geometrical object. This is much in the same way as we consider theelectric and magnetic fields as components of one single field, the electromagneticfield, because they get mixed under the Lorentz transformation. Thus, from now on,we have to get used to thinking in terms of "BRST multiplets" instead of "originalfields" and ghosts. The task of finding geometrical quantum theories is then thetask of developing a BRST "tensor calculus."

This would be already attractive by itself but it is made even more so by aspectacular property of the BRST transformation: If we apply it twice we get zero.That is, if we can 0 the BRST generator, we have the equation

(1.10)

and we say that 0 is nilpotent.The construction of a nilpotent BRST charge for any gauge system, including non-internal symmetries, was completed with the work of Fradkin, Batalin, Vilkoviskyand Henneaux [8].

Equation (1.10), may be taken to express that "the boundary of the boundaryis zero" as John Wheeler likes to put it. Thus, one expects that there should be adeep connection between BRST invariance and topology. This relation has not yetbeen fully unraveled.

The BRST invariance has already had a number of useful practical applications,both in Yang-Mills and string theory. I win just mention some of them here.

The first one is connected with the business of the Faddeev-Popov determinant.If one takes the original Faddeev-Popov expression for the amplitude and "repre-sents" the determinant as an integral over "additional variables" caned ghosts, ther~ulting ghost action is always quadratic in the ghost field. This means that theghosts interact with the other fields of the theory but they do no interact directly


among themselves. However, if one looks in detail, in terms of diagrams, at thebalance of probability in a complicated theory like supergravity, one finds that it isnecessary to include terms which are quartic in the ghosts, that is, self-interactionsof the ghosts. Now, there is no way in which this can come out of a direct applica-tion of the Faddeev-Popov presectjption which only gives a quadratic ghost action.This came as a bit of a shock because it clearly invalidates the view that the ghostsare simply a way to represent a determinant. However, if one looks at the situationfrom the point of view of BRST invariance this is not an exceptional case at all;the interactions of the ghosts among themselves are necessary in such a case just toensure BRST invariance.

Another important application of the BRST invariance is the one which actuallydirectly motivated its discovery. It is the fact that it is responsible for the Ward-Slavnov identities which are cmcial in proving the renormalizability of the Yang-Mills theory. As a result of this understanding, a greatly simplified proof of therenormalizability became available.

The importance of the BRST invariance becomes even greater for generally co-variant theories like gravity, supergravity and strings. The reason is that in that kindof theory, gauge invariance is everything. There is no distinction between dynamicsand a gauge transformation. because the latter is just a localized displacement inspace-time.

The condition for the BRST invariance of permissible states,

(1.11 )

is then the equation of the theory.In string theory, the nilpotency condition provides a particularly transparent

way to understand the need for the critical dimension. Indeed, if one computes 0 2

for, say, the free bosonic string immersed in a D-dimensional spacetime, one findsthat it has two contributions. One of the form 0 times a factor coming from the"original fields," and another equal to -26 times the same factor, coming from theghosts. The condition D = 26 results thus from precisely balancing the "matter"and the "ghosts." What else could one ask to be convinced that the ghosts are onthe same footing with matter!

The applications of the BRST symmetry, and hence, of the concept of ghost.which I have just described are quite significant. But one has the feeling of havingseen only the tip of the iceberg. Probably the deepest developments along theselines should lie in developing theories where elementary extended objects (strings in


string theory, three dimensional spaces in quantum gravity) can undergo quantummechanical decay. A field theory of strings along these lines has been proposed byWitten [9], not long ago. Another interesting line of development, perhaps morereachable, lies along the understanding of anomalies in gauge theories. Of course allthese developments, being based on BRST invariance, could not be even thoughtof without allowing ghosts to playa prominent role.

The ghost has in..leed come a long way since it was called in 1963 an "artificialdopey particle" by its own discoverer.


II. BRST SYMMETRY IN CLASSICAL MECHANICS

2.1 Ghosts have role in classical mechanics

As it was reviewed in the first lecture, the need for ghosts and the symmetrythat reveals their profound importance were first established in quantum mechanics.It was only afterwards realized that they have a natural a,nd necessary place withinclassical mechanics as well. Indeed, the BRST symmetry could have been discoveredin the last century within a strictly classical context by mathematicians dealing withthe geometry of phase space had they only been willing to extend their analysis toGrassmann variables.

Having said this, it should be immediately clarified that I am not advocatinga direct physical meaning for ghosts within classical mechanics. Their physicalimplications can only be infered by explicit use of quantum mechanical laws. It isnevertheless extremely useful to be able to discuss them and the BRST symmetryclassically. One can then bring in these concepts as a powerful tool in the actualconstruction of the quantum theory.

TIlt' role of ghosts in classical mechanics emerges most clearly through theHamiltollian formnlation of the classical dynamics of gauge systems, which providesthe IIIl"t !2:t'lwral ground for sY$tematically discussing the BRST symmetry. Thiswillllt' tilt' st artill/!. point for onr whole discussion of BRST invariance. It will appearthat throll!2:h Hanliltonian methods one obtains a formulation of great generalityalld power. In particular one frees olleself from the assumption that the gaugetransforlllat iUIl$ obey a group conlposition law. The results cover therefore, thegeneral case of an "open algebra." Furthermore, they are valid "off-shell."

2.2 Gauge invariance and constraints

One says that a dynamical system is a gauge system if the general solution ofthe equations of motion contains arbitrary functions of time which are not fixed bythe initial conditions.

In practice, a gauge system is most often given by specifying the action integralin lagrangian form. The procedure for passing from the lagrangian to the hamil-tonian was worked out by Dirac long ago [10]. It will be assumed here that th:system is already given in hamiltonian form.

By definition, one says that all the histories which spring from the sanle initialcondition are physically indistinguishable and are related to each other by a gauge


transformation. The gauge transformation turns out to be a canonical transforma-tion whose generators will be denoted by Ga(q,p).

2.3 Classical mechanics over Grassmann algebra necessary

The notation (q, p) is being used to represent a set of canonically conjugate co-ordinates in phase space. Some of these coordinates might be commuting and otheranticommuting. More precisely, they will be assumed to be elements of a Grass-mann algebra with definite Grassmann parity. One must allow for anticommuting(q, p) in order to have a classical description of fermions, but even in a theory whichhas no fermions to start with, they will be brought in when the ghosts are included.Therefore, we have to be prepared and have a classical mechanics capable of deal-ing with anticornmuting numbers, so we allow for that possibility from the verybeginning. Also, it is not necessary for what follows to have canonically conjugatepairs. Indeed, there are cases of interest such as the classical description of spin, inwhich the dimension of phase space is odd and conjugate pairs are not available.Those cases are included in the present discussion which just needs the existence ofa Poisson bracket. I have chosen to use the notation (q, p) anyway, because it ha.sa phase space ring.

2.4 Higher order structure functions

In order to avoid unnecessary cluttering of the equations with sign factors inthis illtrooud ory account, I will only deal with the case in which there are nofennionic coordinates among the (q,p). Actually it suffices to assume that the Gaare commuting. All the important results carry over unchanged to the generalBose-Fermi case.

It will also be assumed for simplicity that the constraints G a are independentor "irreducible," as one says. That is, the Jacobian nlatrix (8G a /8q, 8Ga /8p) isof maximal rank everywhere on the constraint surface. This means that one canlocally take the Ga's as the first m (non-canonical) coordinates in phase space. Thereducible case will not be dealt with. Again, the main results still hold (providedone adds even more ghosts!- usually called "ghosts of ghosts").

To begin with, the constraints G a will be also called "zeroth order structure(0)

functions" and will be indistinctly denoted by U ' From the point of view of the

action principle these constraints are not uniquely determined. They can always be

40

replaced by an equivalent set,

CLAUDIO TEITELBOIM

(0) (0)U .(q,p) == M;(q,p) U b(q,P), det M i- 0 (2.1)

~) ~because the equations (J = 0 are equivalent to U = O. This change in the

generators implies of a change in the description of the gauge transformations. If,for example, the structure functions C~b associated to the first set are independentof (q,p) (structure constants of a group), that property will not hold for the newset. One expects, of course, that the two descriptions will be equivalent, but thatequivalence is not transparent. It will be one of the virtures of bringing in the ghoststo make the equivalence manifest.

A new notation and a new name will be also introduced for the structurefunctions C~b (q, p) appearing in

(2.2)

11) 11)They will be deno! cd by -2 U :. and the U 's will he called "first ord ..r stmet ure

functions." The first class property of the constraints reads then,

[(0) (0)]U .' U b

11) (0)-2 V d V

ab d(2.3)

The first order structure functions carry with them an ambiguity over and above(0) (0)

that implied by the anlbiguity (2.1) in U . Indeed, once U is fixed, equation (2.3)

(1)determines U only up to

(1)[TC

ab

with

(1) (I) (0)[T ~ b = [T ~b + M~t U d (2.4a)

(2.4b)


(0) (1)Having introduced an appropriate notation for U and U, the analysis pro-

(0)ceeds as follows. One takes the Poisson bracket of equation (2.3) with U d and

fully antisymmetrizes in a, band c. The left-hand side then vanishes by virtue ofthe Jacobi identity, while the right side becomes, with the help of (2.3), a linear

(0)combination of the U 'so This yields,

(0) ([(1) (0)]U d U tab' U c]

(1) (1) )+ 2 U lab U ~]. =0 (2.5)

(0)Now, (2.5) does not imply that the coefficient of U vanishes, because that equation

is clearly identically satisfied if one sets

[II) (0)]U [ab l U c]

(1) (l) (2) (0)?U U d -?U d U+ - lab cJ. - - abc e (2.6)

(2)",liter" the thereby.defined second order structure functions U ~bc(q,P) are antisym-

(0) 11) (2)metric in (d.e) and (a,b.c). Again, once U and U are fixed, the U's bring in

their own ambiguity. They are determined up to

(2) (2) (0)U d. U d. Md.! Uabc --+ abc + abc f (2.7)

where M::! is completely antisymmetric in both the upper and lower indices. Itmay be shown [8] that, under the assumption that the constraints are irreducible,equation (2.6) with the ambiguity (2.7) is the most general solution of (2.5) and italways exists.

The construction leading to the appearance of the second order structure func-tions may be systematically continued. Thus, third order structure functions will

(0)appear by taking the Poisson bracket of (2.6) with U ! and fully antisymmetrizing

('LAlJDIO TEITELIIOIM

in a, b, c and f. lJsill~ lhr- Jacobi identity and the defining equations (2.3) and (2.6)of the structure functions of lower order, yields the identity

(0)U a, = 0

where,(2-q) ]U aq+l.a,

bq + 2 ..b..

(2.9)

From (2.9), it follows that

(2.10)

where the "third order structure functions" are completely antisymmetric in boththe a and b indices.

For the higher order structure flllldions, one finds, in an analogolls manner,the identit ies

(2.11)

which imply in turn the existence of the structure functions of order n + 1

obeying

(n+1)U a, ...a n +,

b1 b n +2

(n+1)U [a, an+tl

[b, bn +2 ]

(n)In (2.11) and (2.12), D :::::~~bn+' stands for

(2.12)

(n)D ala" -

b1 bn + 2 -


(2.13)n-I (q+l) (n-q)

_,,( + l)(n - + 1) U a, a q C U aq+, ...a n (- t(q+t)L q q b, bq +, bq +3 bn+,cq=O

and only involves structure functions of order::,:: n.Finally, one easily checks that (2.12) determines the structure functions of order

n + 1 only up to

(n+t)U al.an.+l --+

bt bn + 2 (2.14)

(for given structure functions of order::':: n), where A!:,'.::b::~2 possesses the appro-priate antisymmetry properties.

The proof of the existence of the structure functions of order two and highermay be found in [8]. It will not be dealt with here. I will turn instead to therelation between the structure functions and to the BRST generator n. Afterwards,an independent proof of the existence of n will be given. That proof will, therefore,provide in turn a proof of the existence of the structure functions. However, thisindirect method does not make the direct construction superfluous for two reasons.The first is that the direct construction provides an explicit way to write down theBRST generator for a given set of G n 's. This is done by systematically followingthe steps indicatt:>d in lilt:> prt:>vious paragraph. Second, the indirect prouf as givenbelow hulds only locally in phase space, while no such restriction applies to theother prucedure. (The indirect proof call also be extended to hold globally. See[I1J.)

2.5 Rank defined. Open algebras

One knows that for a Lie group, all the (local) geometric structure is contained(2)

in the structure constants C~b' In that case one may take the U and all the higher

order structure functions equal to zero. However, in the general case, this will notbe possible and structure functions up to some order n will appear. One then saysthat the theory is of rank n. It should be emphasized here that the notion of rankis not intrinsic to a given theory, but it depends on the choice of the constraints Ga(structure functions of order zero) and also on how the ambiguities in the choiceof the structure functions of order one and higher are resolved. Indeed, as we shallsee below, one may always choose, at least locally in phase space, a set of Ga suchthat any theory is of rank zero ("abelianization"). However, in general, that choice

(2.14)

CLAUIlIO TI':l'n:LIIOIM

is cumbersome. For a tidd t 11t"ury it typically leads to generators which are nonlocal in space. In practice, there is always a choice, or perhaps a few choices, ofthe Ga which are privileged because of locality properties, covariance, etc. Thus,when I indulge, from now on, in speaking of "rank of a theory" I will have in mindthe lowest possible rank associated with those natural choices. In this sense, theMaxwell field ha~ rank zero, the Yang-Mills field, the relativistic string and Einsteintheory of gravitation have rank 1, N = 1 supergravity in four space-time dimensionshas rank 2, and the n-dimensional relativistic membrane has rank n. For a theoryof rank n, the idea is that the local geometrical structure is contained not only inthe first order structure functions, but also in those up to order n.

When the first order structure functions are not constant, one often says that"the gauge algebra only closes on-shell". This means that the commutator of twogauge transformations is a new gauge transformation only on the constraint surface.Note that for this to happen it is not necessary that the rank be higher than one.The gravitational field is an example of a theory of rank one whose Hamiltoniangauge algebra only closes on-shell.

The only on-shell closure property arises as follows. The commutator of twogauge transformations with parameters f~ and f 2, is a new transformation obtainedby acting; with C;.f';f~Ge' This is a consequence of the Jacobi identity. However,the action l>F of that transformation on a generic function F is

fJF = [F,C~bf~f~Ge] = C~bf~f~[F,GeJ+ f~f~[F,C~b]Ge

and this is generated by Ge only on the surface in which G e is equal to zero.Otherwise the second term of the right side of (2.14) makes this no longer true forevery F unless C~b is independent of q and p.

[Actually, this last assertion is a bit too strong. Indeed, if the C~b dependedon the q's and p's only through the generators Ga themselves the second term onthe right of (2.14) would also have an overall factor [F,Ge]. However, in such a casethere is no quarantee that the rank would be equal to one].

2.6 Ghosts. Ghost number. BRST generator as generating function forstructure functions

The original phase space of the q's and the p's will e enlarged by introducingan additional canonical pair (Tt, Pa ) for each first class constraint G a present. Thecanonical pair will be taken to be of Grassmann parity opposite to that of the


corresponding Ga. Thus, if the G's are all commuting, as we have been assumingfor simplicity, theFT's and the P's will be anticornmuting. These extra variables willbe called ghosts. More specifically, Tf will be called ghost and P anti-ghost.

The ghosts may initially be thought of as a useful bookkeeping device forconcisely taking into account the properties of the structure functions. However, thisnarrow view lends itself to rapid change once one realizes that those properties implythe existence of a symmetry which mixes the ghosts with the original variables. Thismeans that the appropriate space for describing the dynamics of the gauge systemis not the original space of the q's and the p's, but rather, the extended one whichincludes the ghosts.

The basic properties of the ghosts are, therefore,

and

[Pa'Tfb] = [Tfb, Pal = -6:(Tfa)* = Tfa (Pa)" = -P"

(2.15)(2.16)

(2.17)

!\lote that '1 is taken to be real which implies that the conjugate Pis imagiuary. Thisproperty. allel also the sy11l111etry of t he bracket (2.1.5) are due to the ilnticollllllutillgcharacter of tilt' ghosts for bosonic Ga.

lt is also convenient to define an additional stmclure on the extended phasespace, that of ghost number. This is done by attributing the following ghost numberto the canollical variables: the qi, p;'s have ghost number zero, the ghosts Tfa haveghost number one, and the antighosts Pa have ghost number minus one. Moreover,one requires that the ghost number of a product of variables is equal to the sum oftheir ghost numbers.

Consider now the following function on the extended phase space

(n)n = 2: r/ n + ....r/ U ::..:.....~:+l Pan ...Pa

n;?:Q

(2.18)

Note that due to the antisymmetry properties of [T, there is no loss of informationwhen contracting it with the anticommuting TJ'S and P's. Indeed, one may recoverthe U's by repeatedly differentiating n with respect to the ghosts and setting themequal to zero afterwards.


The function n will bt' callt'd the BRST generator. It has the following funda-mental properties

o is real, 0 = 0*

o has ghost number + 1, 9(0) = 1o is anticommuting , (O) = 1

here denotes Grassmann parity. More importantly, it is nilpotent

[0,0] =0

(2.19)(2.20)(2.21 )

(2.22)

Properties (2.19) through (2.21) follow because the n-th term in 0 contains n prod-ucts r(P and one "loose" 71. Each product TJP is real, has ghost number zero andeven Grassmann parity while the loose 71 is also real, has ghost number +1 and isanticommuting.

The crucial property of 0, its nilpotency, involves the detailed properties ofthe structure functions. Indeed, one may check directly that (2.22) is equivalent tothe identities (2.12), (2.13) which define the U's. This shows that the generator 0captures in a nut shell the complete gauge structure of the system. For this reason,it is nat nral to consider 0 as the central geometrical object in a gauge theory.

A remarkable feature of 0, in the present Hanliltonian formulation, is that thenilpotency holds "off-shell," namely at all points in the extended phase space. Thisis so even for systems in which the gauge algebra in the original phase space of theq's and the p's only closes on the constraint surface, as discussed in Sec. 2.5.

2.7 Abelianization of constraints. Existence of fI

As was mentioned in Sec. 2.5, the structure functions can be changed byreplacing the constraints G a by linear combinations of themselves. Since one admitsin the linear combination coefficients which depend on q and p, the flexibility in thestructure functions is enormously greater than the one available when changing thebasis of generators of a Lie algebra. Indeed, one can even achieve in principle thatthe constraints become locally abelian, that is that all the structure functions vanishin a region of phase space. This possibility is useful for proving general propertiessuch as the one we are interested in now, the existence of 0, but it is not onewhich is of much use in practice. The reason is that simplicity requirements on thefunctional form of the Ga , which are specially important in the passage to quantum


mechanics, such as polinomial structure or locality in field theory, are absent in theabelianized constraints.

The problem one faces is t.he following. Given a set of constraints G obeying

(2.23)

one wants to find an invertible matrix M:(p, q) such that

(2.24)

obeys(2.25)

One may find a general proof of the existence of M in [8]. I would just liketo mention here that the passage from G to F amounts to solving the constraintequation Ga = 0 t.o express some of the momenta in t.erms of the remaining variables.For example in the Yang-Mills case, one solves Gauss's law, for t.he longitudinalcomponent of t.he "electric field." This may only be done in a perturbation seriesin t.he coupling strength, which illustrates that F exit.s only locally in that. case. It.follows from the work of Gribov [12] that the solut.ion does not exist. globally.

For the constraint F, t.he existence of 0 is innnediate. One simply writes

(2.26)

which obeys all the propert.ies list.ed in (2.19-2.22).The question now is how to infer from OF the BRST generator OG correspond-

ing to the constraint Ga. Clearly, if one could show that 0G is related to OF bya canonical transformation, the problem would be solved. The reason is that t.hekey nilpotency property [0,0] = 0, being written in terms of Poisson brackets, isinvariant under canonical t.ransformations. The reality, ghost number 1 and Grass-mann parity 1, properties of 0G would be assured, if the generat.or of t.he canonicaltransformation is real, has ghost number 0 and Grassmann parity O.

The canonical transformation that. I anl talking about here, should be a canon-ical transformation in the extended phase space and should unavoidably mix theghost with the original p's and q's. This is so, because there is no way t.hat F andG could be related by a canonical transformation in the p's and the q's only sincethe Poisson brackets are different.

(I.AlJmO TEITEI.1I0lM

The solutioll may he ell.sily found in the case when the constraints G a and Fadiffer infinitesimally,

(2.27)

with ~ (p, q) small. The general case can then be obtained by exponentiation.The question reduces then to that of finding the generator C of the canonical

transformation such that

where

o(G) _ O(F) := [O(F), C]

(1)o(G) - O(F) = r//,Fa + r/r,u U ~bPc + 0(2)

(2.28)

(2.29)

(the higher structure functions of the set Ga may be taken to be of order 2 orhigher).

This generator C is obviously given by

(2.30)

and has all the desired properties.The above argument only covers the case in which the invertible linear trans-

formation ]\.f(p, q) is in the connected component of the identity, i.e., has positivedeterminant. The general case with hoth positive and negative detemlinant is eas-ily included by observing that the particular matrix .M: = diag(-l,l, ... l), whichpossesses negative determinant, can be generated by the ghost canonical transfor-mation

2.8 Uniqueness oro

The central idea of the discussion that I have been giving is to take the BRSTapproach as the basic description of the idea of gauge invariance. The reasoning ofthe preceding paragraph shows that this view has an important pay-off. It makesevident the equivalence between the descriptions based on different choices of theconstraints Ga , an equivalence which is by no means transparent in the originalphase space of the q's and the p's. Indeed, it emerges that the BRST generator is


unique. For a given system, the 0 obtained from one choice of the whole tower ofstructure functions (including, in particular, the choice of the G a ) is related to thatobtained from another choice by a canonical transformation in the extended phasespace.

One may say that this important result shows that the "canonical covariance"of the theory becomes manifest only when one enlarges the original phase space toinclude the ghosts. Once more, simplicity and understanding are gained by addingvariables and not by eliminating them.

When going over to quantum mechanics, it is impossible to realize all canoni-cal transformations as unita;:y transformations in Hilbert space. Therefore, in thequantum case, different choices of the constraints may lead to BRST generatorswhich are not unitary equivalent. This is not a problem of BRST theory, but rathera general problem of the passage from classical mechanics to quantum mechanics.In practice, one is happy if one can find a choice of the constraints which will lead toan 0 simultaneously satisfying the key requirements of nilpotency and hermiticity.

2.9 Classical BRST cohomology

The off-shell nilpotency of the BRST transformation enables Olle to introducethe Ilotion of das~il'al (noll quantum) BRST cohomology for any gauge theory. Thisconcept permits o\]e to imple1111'nt the ideas of gange transformation alld gaugeinvarialH'" in BRST terms.

Indeed, it follows from the nilpotency (2.22) of 0 and the Jacobi identity forthe graded Poisson bracket that

[[A, 0], 0] = 0 (2.31)

for any A(q,P,l1, P) on the extended phase space. Hence, one can define BRST-closed functions as functions which are BRST invariant,

[A, 0] =0while BRST-exact functions are given by

A=[K,O]

(2.32)

(2.33)

for some K, and are clearly BRST-closed.The following idea proposes itself for exploration. To what extent can one

identify two BRST-closed functions wich differ by a BRST-exact function? Or,


what is the same, to what extent is the addition of a function of the fonn (2.33) theBRST analog of a gauge transfonnation? This is the question which is addressedby BRST cohomology.

As a result of the uniqueness of n, the classical BRST cohomology only dependson the first-class constraint surface defining the dynamical system under consider-ation, and not on how one represents this surface by the equations Ga(q,p) = 0,or on how one removes the ambiguity in the structure functions entering in theconstruction of n.

Because the BRST charge possesses definite ghost number one can study co-homology classes with given ghost number. Two equivalent functions will thendiffer by a su

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