PHYSICAL REVIEW 0 VOLUME 40, NUMBER 8 15 OCTOBER 1989

Time and the interpretation of canonical quantum gravity

William G. UnruhCosmology Program, Canadian Institute for Aduanced Research, Department ofPhysics,

Uniuersity ofBritish Columbia, Vancouuer, British Columbia, Canada V6T2A6and Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

Robert M. WaldEnrico Fermi Institute and Department ofPhysics, 5640 S. Elh sAue'nue, Uniuersity of Chicago, Chicago, Illinois 60637

{Received 15 December 1988)

The unsatisfactory status of the interpretation of the wave function of the Universe in canonicalquantum gravity is reviewed. The "naive interpretation" obtained by straightforwardly applyingthe standard interpretive rules to the canonical quantization of general relativity is manifestly unac-ceptable; the "WKB interpretation" has only a limited domain of applicability; and the "conditionalprobability interpretation" requires one to pick out a "preferred time variable" {or preferred class ofsuch variables) from among the dynamical variables. Evidence against the possibility of using adynamical variable to play the role of "time" in the conditional probability interpretation is provid-ed by the fact {proven here) that in ordinary Schrodinger quantum mechanics for a system with aHamiltonian bounded from below, no dynamical variable can correlate monotonically with theSchrodinger time parameter t, and thus the role of t in the interpretation of Schrodinger quantummechanics cannot be replaced by that of a dynamical variable. We also argue that the interpretiveproblems of quantum gravity are not alleviated by the incorporation of observers into the theory.Faced with these difhculties, we seek a formulation of canonical quantum gravity in which an ap-propriate nondynamical time parameter is present. By analogy with a parametrized form of ordi-nary Schrodinger quantum mechanics, we make a proposal for such a formulation. A specific pro-posal considered in detail yields a theory which corresponds at the classical level to general relativi-

ty with an arbitrary, unspecified cosmological constant. In minisuperspace models, this proposalyields a quantum theory with satisfactory interpretive properties, although it is unlikely that thistheory will admit suKciently many observables for general spacetimes. Nevertheless, we feel thatthe approach suggested here is worthy of further investigation.

I. INTRODUCTION

Theories typically are formulated in terms of quantitiestaking values in abstract mathematical spaces. In orderto relate these quantities to physical phenomena, oneneeds an interpretation of the theory. In this paper, byan "interpretation" we mean a description, in ordinarylanguage, of what an observer would see or experiencewhen the mathematical quantities used by the theory todescribe the state of the system take on any of their al-lowed values. Thus, it should be noted that by our usageof the term, the Copenhagen and Everett "interpreta-tions" of ordinary Schrodinger quantum mechanics areequivalent, at least for formulations of the Everett "inter-pretation" which are interpretations in our sense, sincethey give the same rules for what an observer "sees." Aninterpretation is an essential part of any theory and,indeed, interpretations of some classical theories, such asgeneral relativity, are not entirely trivial to state. How-ever, it is in the case of quantum theories that the issue ofinterpretations has attracted the most attention.

In fact, ordinary (nongravitational) quantum-mechanical theories such as standard Schrodinger quan-tum mechanics possess an interpretation which is entirelysatisfactory in that their interpretive rules are well

defined and yield physical predictions which agree withexperiment. (The fact that these predictions are proba-bilistic in nature and, apparently, do not admit a con-sistent picture of "objective reality" are not valid objec-tions to an "interpretation" according to our usage of theterm. ) However, as we shall elucidate further below, thenotion of "time" plays a vital role in these interpretiverules in particular, measurements are made at "instantsof time" and probabilities are assigned only to such mea-surements (not, in particular, to "histories"). Therefore,it should not be surprising that severe difficulties arise inthe formulation of an interpretation of quantum theorycorresponding to general relativity, since the nature of"time" and dynamical evolution in a generally covarianttheory has important differences from the correspondingnotion in theories which are not formulated in adift'eomorphism- (or reparametrization-) invariantmanner. This appears to be the main reason why quan-tum gravity, even, say, in minisuperspace models wheremany technical difficulties such as renormalization can beavoided, does not, at present, possess a satisfactory inter-pretation.

In this paper we shall make a proposal for a version ofquantum gravity in which a notion of "time" is presentthat allows a well-defined formulation of interpretive

40 2598 1989 The American Physical Society

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2599

rules. However, our proposal fails to yield a quantumtheory which corresponds classically to ordinary generalrelativity; rather, the specific proposal we shall investi-gate in Sec. III corresponds to Einstein's equation withan arbitrary, unspecified cosmological constant. Further-rnore, although our proposal yields satisfactory interpre-tive rules and is fully satisfactory in minisuperspace mod-els, it is far from clear that there are a suScient numberof allowed observables in general spacetimes. Neverthe-less, we hope that this proposal, or, at the very least, thenature of the proposal, will be of some interest and thatthe related discussion will help elucidate the role whichtime plays in the interpretation of a quantum theory aswell as some of the difBculties that arise from the notionof time in general relativity.

Classical general relativity admits a Harniltonian for-rnulation but it is a constrained Hamiltonian formulation.The configuration variable is taken to be a Riemannianmetric h, b on a three-dimensional manifold X and aspacetime is viewed as being comprised of the develop-ment of h, b with time. However, h, b and its canonicallyconjugate rnornentum m',

m' =h' (K' —Kh' )

and (1.3)], the canonical transformations generated by

fzPH, and f zg Ho correspond to the spacetimediffeomorphisrns of general relativity.

The standard canonical quantization rules for formu-lating a quantum theory in the "coordinate representa-tion" based upon a classical Harniltonian system suggestthe following structure of the quantum theory in thiscase. States of ths system are taken to be wave functionsof the configuration variable, so %=%(r;h,b) where rdenotes the time variable occurring in the Hamiltonianformulation. On the Hilbert space of such states, h, b isrepresented by the multiplication operator and m' is tobe represented by the functional differentiation

i5/—5h, b Th.e time evolution of 4 is then given by theSchrodinger equation

(1.5)

where in &(h,b, ~' ) we replace h, & and ~'" by theiroperator expressions. If, as in our case, constraints arepresent, then following the Dirac prescription we imposethese as conditions on the state vector. Thus, 4 also isrequired to satisfy

(where K,b denotes the extrinsic curvature of X with thesign convention of Ref. 2), cannot be freely specified asinitial data; rather, they are required to satisfy the so-called Harniltonian and momentum constraints,

I g'H, e=O,

f PH. +=O,

(1.6)

(1.7)

O=H (h vr'")

h 1/2[ (3)g +h —1( ab & 2)]

()=H (h b 77 )= —2h D (h ~27/ )

(1.2)

(1.3)

where P and g are, respectively, an arbitrary vector fieldand function qn X, and in H and H', we again replace h, b

and m' by their operator representatives. Note that if wetake g =N and P=N', Eqs. (1.6) and (1.7) have the iin-portant consequence that

An appropriate Hamiltonian for classical general relativi-ty then is

%= J(NHO+N'H, ), (1.4)

where N and X' are, respectively, an arbitrary functionand vector field on X which have the interpretation oflapse function and shift vector in the spacetime con-structed from the time evolution. Variation of & withrespect to N and N' yields the constraints (1.2) and (1.3),whereas Hamilton's equations of motion for h, b and m'"

yield the Einstein evolution equations; see, e.g. , Ref. 2 forfurther discussion.

Note that the infinitesimal canonical transformationsgenerated by the functions fxPH, on phase space (forarbitrary vector fields P) correspond simply to thechanges of h, b and m' resulting from infinitesimaldiffeomorphisrns of X. The situation is more subtle forthe infinitesimal canonical transforrnations generated byfunctions on phase space of the form f zg Ho (for arbi-

trary functions g ): the changes induced on h, b and m'

correspond to infinitesimal spacetime diffeomorphisms (ofthe spacetime metric constructed from h,b, ~'", X, andN') only when the field equations are satisfied. (More-over, these spacetime diffeomorphisms are "field depen-dent. ") Nevertheless, on the "constraint hypersurface"of phase space [i.e., the h, b and ~' satisfying Eqs. (1.2)

and hence by the Schrodinger equation (1.5), that %' is in-dependent of r, i.e., '0='P(h, b ).

Roughly speaking, the quantum constraints (1.6) and(1.7) can be interpreted as requiring the invarianceof 4' under the infinitesimal canonical transformationsgenerated by f zg Ho and fxPH„which, as discussed

above, correspond to infinitesimal spacetime diffeo-morphisrns on the manifold of solutions. For themomentum constraint (1.7), this holds literally, since Eq.(1.7) implies that %(h,b) is unchanged when aninfinitesimal diffeomorphism on X is applied to h, b. Thismakes 4 a functional on "superspace, " the set ofdiffeomorphism equivalence classes of metrics on X. Itdoes not appear possible to give as literal an interpreta-tion of the quantum Hamiltonian constraint (1.6), knownas the Wheeler-DeWitt equation, as corresponding to theinvariance of 4' under a variation of h, b corresponding toan infinitesimal diffeomorphism in spacetime whichmoves points on X in the direction orthogonal to X. Nev-ertheless, this interpretation of the quantum constraintscan be viewed as accounting for why B%/8~=0 in theformalism.

In the following we shall consider canonical quantumgravity formulated by the above rules. To avoid some ofthe severe technical problems which arise, such as renor-

2600 WILLlAM G. UNRUH AND ROBERT M. WALD

gj=N — m., + m +a'V(P)1 2 1

12a ' 2a(1.10)

In quantum theory, the momentum constraints are au-tomatically satisfied, so the only equation imposed upon4 is the Wheeler-DeWitt equation arising from the classi-cal constraint

=0.With the "Laplacian" factor ordering chosen for the "ki-netic terms" in Eq. (1.10), this equation takes the explicitform

a a+ 1 a'e12a Ba Ba

J

(1.12)

(This form holds even if the lapse function N is allowed todepend upon a and P; i.e., the ordinary wave operator isconformally invariant in two dimensions; for an n-dimensional minisuper space, the appropriate factor-ordering choice would appear to be the conformally in-variant wave operator. ) Although this is a highlysimplified model of quantum gravity, the features leadingto interpretative difficulties remain fully present. Itshould be noted that some technical issues also remain inthe model; in particular, there are some "factor ordering"ambiguities in the definition of the Wheeler-DeWitt equa-tion. (As already mentioned, the factor ordering we havechosen has the advantage of being conformally invariantas well as invariant under redefinition of the "coordi-nates" a, P on superspace. ) In addition, for quantumcosmology based upon the model, the issue arises as towhat boundary conditions should be imposed on solu-tions of the Wheeler-DeWitt equation in order to obtainthe solution corresponding to our Universe; several suchproposals have recently been given ' and widely dis-cussed. However, we shall not address any of these issueshere, since our entire concern in this paper is the inter-pretation of +. If the wave function of the Univer'se isgiven by the mathematical expression %(a, P), what doesan observer of the Universe "see" or experience?

A possible approach toward formulating an interpreta-tion of 4 is to treat 4 in essentially the same way as wetreat wave functions in ordinary quantum mechanics.We will refer to this as the "naive interpretation" ofcanonical quantum gravity. Thus, we could give a

malization, and also for simplicity and definiteness, wewill, for the most part, focus attention below on a simpleminisuperspace model involving a Robertson-Walkercosmology and a (spatially homogeneous) Klein-Gordonscalar field P, self-interacting via a potential V(P). (How-ever, our discussion will not depend upon the details ofthis model. ) Thus, the configuration variables for thissystem are simply P and the Robertson-Walker scale fac-tor a. The "wave function of the Universe" is a functionof these two variables: %=%(a,P). For the case of acosmologically flat (k =0) model,

ds = Ndd+—a (r)(dx +dy +dz ),a suitable classical Hamiltonian (in units where Smg =1)is

Hilbert-space structure to the wave functions 4 bydemanding that they be square integrable with respect toa and P. Indeed, such a Hilbert-space structure is impli-cit in the above canonical quantization rules. We thencould interpret %', as in ordinary quantum mechanics, asgiving, the amplitude that an observer, making a measure-ment at parameter time r, will find the values a and P.However, the fact that 4 is forced to be independent of ~by the Wheeler-DeWitt equation makes this interpreta-tion not viable. Indeed, the Wheeler-DeWitt equationmust hold exactly at all times, including after measure-ments are made. Thus, the situation is analogous todemanding in ordinary quantum mechanics, not only thatthe wave function be in an eigenstate of zero energy (sothat the wave function is time independent), but also thatno measurements can be made which distu. rb this condi-tion. This restricts the allowed observables in the theoryto be those which commute with the Hamiltonian, i.e., tothose which are time independent. Hence, in this inter-pretation of 4, dynamical variables such as a and Pwould not be measurable (since they fail to commute with&), and the Universe would appear to be strictly time in-dependent with respect to those very few observableswhich are measurable.

In this connection, it should be noted that for somedynamical systems, the requirement that an observablecommute with the Hamiltonian H can be extremely re-strictive. In particular, for a classical system which is er-godic in the sense that the dynamical trajectories are"mixing" on each surface of constant energy, any classi-cal observable (i.e, measurable function on phase space)which has vanishing Poisson brackets with the Hamil-tonian must be constant on each dynamical trajectoryand hence constant on each surface of constant energy.Thus, for such systems, the only classical observableswhich have vanishing Poisson brackets with H are func-tions of H. Hence, in the quantum theory, apart fromfunctions of H, there presumably are no quantum observ-ables corresponding to classical observables which com-mute with H. Thus, the suggestion by Page and Wootersthat dynamical evolution can be described in terms of sta-tionary observables as a dependence upon internal clockreadings is manifestly not viable for sufficiently compli-cated (i.e., ergodic) systems: There are no nontrivial sta-tionary observables and "internal clocks" would correlatewith other observables in a random manner.

The difficulty with the naive interpretation can betraced directly to the conflict between the role of time inquantum theory and the nature of time in general rela-tivity. In quantum mechanics, all measurements aremade at "instants of time"; only quantities referring to theinstantaneous state of a system haue physical meaning In.particular, "histories" are unmeasurable in quantumtheory. On the other hand, in general relativity "time" ismerely an arbitrary label assigned to a spacelike hyper-surface. The physically meaningful quantities must be in-dependent of such labels; they must be diffeomorphisminvariant. In other words, only the spacetime geometryis measurable; i.e., only histories haue physical meaning.Thus, it should not be surprising that when one naivelycombines quantum theory and general relativity, the only

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2601

meaningful quantities which survive are those which areboth instantaneously measurable (i.e. , refer to quantitiesdefined on a spacelike hypersurface) and yet depend onlyon the spacetime geometry (i.e., are independent ofchoice of spacelike hypersurface). These are precisely theconserved quantities, i.e., the observables which commutewith the Hamiltonian. Note that the "naive quantiza-tion" of any classical theory which has a timereparametrization invariance will share this feature ofquantum general relativity. Thus, further insight intowhat has gone wrong can be obtained from comparisonwith the parametrized form of ordinary Schrodingerquantum mechanics, which will be discussed below.

A possible modification of the above rules would be topostulate that quantities, such as a and P, which fail tocommute with the Hamiltonian constraint (i.e., Wheeler-DeWitt) operator are, nevertheless, measurable. Howev-er, one then would be faced with the problem of how toincorporate the information obtained from the results ofmeasurements of such quantities into future predictions.If one "reduces" the wave function in the standard way,the Hamiltonian constraint will be violated. If one pro-jects the resulting reduced wave function onto the con-straint subspace (or does not reduce it at all), an immedi-ate repetition of the same measurement could yield a to-tally different result. Thus, there does not appear to beany way of making the naive interpretation viable.

A possible reaction to the failure of the naive interpre-tation would be to object that our simplified model of theuniverse does not include the presence of observerswithin the model, and that it is not fair (and, perhaps,even not necessary) to require an interpretation of thetheory until observers are explicitly incorporated into themodel. However, this same comment could be made withequal force in ordinary (nongravitational) quantumtheory, where human observers also have not been prop-erly incorporated into any model system. It is true thatbecause of the universal nature of gravitation, in quan-tum gravity the influence of the observer on the observedsystem (i.e., gravitational field) cannot be strictly zero.However, it seems clear that the influence of a human ob-server on the global state of the Universe should, in fact,be far more negligible than the influence of such an ob-server on a typical laboratory experiment in ordinaryquantum theory. Thus, in the following discussion, weshall proceed on the assumption that, as in ordinaryquantum theory, it should not be necessary to explicitlyincorporate observers into the model in order to obtainan interpretation. We will return to this issue at the endof this section and argue that, in any case, such an incor-poration is not likely to alleviate any of the interpretiveproblems which occur without their explicit presenceand, indeed, it creates additional difFiculties.

In the literature on quantum cosmology, statementsabout physical phenomena usually are extracted from thewave function of the Universe, V(a, P), by one of twomeans, which we shall refer to, respectively, as the"WKB interpretation" and the "conditional probabilityinterpretation. " We now discuss these two interpreta-tions in turn.

The WKB interpretation is applicable only if %' takes

the WKB form

%=A exp(iS), (1.13)

where 4 is a real function which is rapidly varying com-pared with A. In that case, the Wheeler-DeWitt equa-tion implies, to a good approximation, that 4 satisfies theclassical Hamilton-Jacobi equation. As in ordinary parti-cle mechanics, associated with any solution of theHamilton-Jacobi equation is a family of classical trajec-tories on configuration space. These trajectories aretangent to the "current vector" j"=~A

~V "4 on super-

space, which is conserved (in the WKB approximation) asa consequence of the Wheeler-DeWitt equation. (In thisformula, the superspace index 3 is raised using the natu-ral metric on superspace arising from the "kinetic terms"in the Hamiltonian. ) The WKB interpretation consists ofsaying that, if the wave function of the Universe takes theform (1.13), then an observer in the Universe who makesmeasurements of a and P, which are not so precise as tosignificantly disturb the Universe, would obtain resultsconsistent with one of these classical trajectories. Itwould be natural to supplement this statement by usingthe conserved current j to assign a probability densityfor observing a given classical trajectory. This can bedone if superspace can be foliated by hypersurfaces suchthat each classical trajectory crosses each hypersurfaceonce and only once. If the metric on superspace werepositive definite, or if the "potential term" ' 'R in theWheeler-DeWitt equation had a definite sign, the surfacesof constant 4 would automatically satisfy this property(in the "classically allowed" region) and the conservedprobability density (which is proportional to V "SV„S)would be non-negative. However, since the metric on su-

perspace is not positive definite and ' 'R also is, in gen-eral, of indefinite sign, there is no reason why V SVsSneed be non-negative. Thus, there appear to be seriousdifhculties in obtaining an everywhere non-negative prob-ability density for the classical trajectories valid in all cir-cumstances.

In its range of applicability, the WKB interpretation isa genuine interpretation in our sense, and essentially all"predictions" given in the recent literature in quantumcosmology have been made by means of it. However, itsuffers from the obvious shortcoming of having only avery limited range of applicability; i.e., it applies only tocases where the wave function of the Universe is "verynearly classical. " Furthermore, even in cases where theWKB approximation holds, it typically will be valid onlyin a limited region of a-P space. What is the probabilitythat an observer will measure values of a and P in thisnon-WKB region? [Here, by "non-WKB region" we in-clude any region where the "Euclidean WKB form"%=A.e holds, since no satisfactory interpretation existsin that case either, as there are no classical trajectories,and the conserved current j"=—,'(%*V "4—%V "4")van-

ishes there. ] What would the Universe look like to an ob-server who finds himself in a non-WKB regime?

The WKB interpretation plausibly can be extended tothe case where the wave function of the Universe takesthe WKB form only with respect to some dynamical vari-ables and all the non-WKB variables can be treated as

2602 WILLIAM G. UNRUH AND ROBERT M. WALD

"small perturbations. " ' However, there is no reason, apriori, why + should take this form on all of superspace,nor is there any evidence from the solutions of theWheeler-DeWitt equation in minisuperspace models thatit does. The WKB interpretation also has been applied tocases where 4 takes the form of a sum of WKB solutions.For example, in the solutions of Ref. 4, 4 is real andhence cannot take the form (1.13); at best, it can be a sumof two WKB solutions. The interpretation which hasbeen proposed in that case is that the Universe corre-sponds to a classical trajectory of either of the two WKBsolutions. This is analogous to interpreting the wavefunction resulting from a two-slit experiment in ordinaryquantum mechanics as corresponding to classical trajec-tories in which the particle goes through one (and onlyone) of the two slits. In fact, in ordinary quantummechanics the predictions made from this interpretationof the wave function for the two-slit experiment will becorrect for the case in which accurate enough positionmeasurements are made near the slits so that it can bedetermined through which slit the particle has passed—through not so accurate that the particle motion issignificantly disturbed. (Interactions of the particle withthe environment near the slits would have the same effectas a measurement. ) Furthermore, even if no measure-ments or interactions occur, highly precise position ormomentum measurements may be needed to observe theinterference effects. However, in principle, highly non-classical behavior can be observed. The inability of theWKB interpretation to account for this and thus to prop-erly extend even to the case of a sum of two WKB solu-tions highlights its very limited range of applicability.

One might attempt to rescue the WKB interpretationfrom its shortcoming of limited applicability by postulat-ing that observers can exist only when the Universe isvery nearly classical, so an interpretation is required onlyin that case. However, even if one were to accept thisrather radical postulate, one would still be in the highlyunsatisfactory situation of having only an approximateinterpretation. How close to a WKB solution must theUniverse be before observers can exist? How large canthe departures be from the predictions of the WKB inter-pretation due to the fact that the WKB form does nothold exactly? These equations must be answered if thetheory is to have any predictive power, but it does not ap-pear that such answers are possible unless there exists an"exact" interpretation of the theory which does not relyupon the WKB approximation for its validity.

We turn, now to a discussion of the conditional proba-bility interpretation. The conditional probability inter-pretation asserts that if an observer measures a particularvalue of one of the dynamical variables, then the wavefunction of the Universe (evaluated at that value of thatvariable) yields the amplitude for measuring the variouspossible values of the remaining dynamical variables.Thus, for example, in our minisuperspace model, if an ob-server measures the radius of the Universe to be ao, thenthe probability density for measuring the value of the sca-lar field to be P would be given by

&(y)=l+(&o y)l'/f lq«o y)l'de~ ("4)where d p& is a measure on P space (and the "probability

density" is specified with respect to this measure). Alter-natively, if the observer measures the value Po of the sca-lar field, then the probability for measuring the value of awould be given by

&(a)=I +(~,yo) I'/ f ~

~(a, po) ~dp. ,

where dp, is a measure on a space.It is far from clear how dp& and dp, are to be chosen.

(They, of course, contain as much information about theprobability distribution as does %.) However, an evenmore serious difIiculty with the interpretation is that it isclear that not all quantities are suitable for playing therole of the variable which "sets the conditions" for theother variables. For example, even in the context of ordi-nary Schrodinger quantum mechanics of a particle, al-though at fixed t =to, the wave function P(to, X, Y, Z)gives the amplitude for finding the particle at position(X, Y, Z), at fixed X =Xo, P(t, XO, Y,Z) is not in any senseproportional to the amplitude that the other coordinatesof the particle are ( Y,Z) and the time is t. (Note that forthe purposes of making this point here, we have blurredthe distinction between the time parameter t and thedynamical variables X, Y;Z. We will return to this pointbelow in the context of a parametrized theory, where t isreplaced by a dynamical variable T.) In minisuperspacemodels, this difhculty often manifests itself by the factthat in typical solutions V(a, P) chosen for study,0'(ao, P) fails to be square integrable in P, and/or %(a,Po)fails to be square integrable in a, thus making formulas(1.14) and (1.15) meaningless.

Since not all variables are suitable for "setting the con-ditions" for the other variables, the conditional probabili-ty interpretation must specify which variables are suit-able to "fix,"so that %' yields the conditional probabilitiesfor the remaining variables. This problem has not, as yet,been solved, so the conditional probability interpretationremains seriously deficient.

An interesting variant of the above conditional proba-bility interpretation which makes effective use of theavai1abIe geometricaI structure of superspace is implicitin the work of Misner. ' Consider the case of a spatiallyhomogeneous class of models, with an n-dimensionalminisuperspace. The *'kinetic terms" in the Hamiltonian(1.4) define a metric on minisuperspace (or, really, a con-formal metric, since the lapse function X could be chosento depend arbitrarily on the spatial geometry). Thismetric has Lorentz signature because the square of themomentum ~ conjugate to the "conformal degree of free-dom" enters the Hamiltonian with sign opposite that ofall the other squared momentum terms [see Eq. (1.2)].Consequently, the Wheeler-DeWitt equation (1.6) for thewave function of the Universe + on minisuperspace isformally identical to the equation for a linear scalar fieldN on an n-dimensional curved spacetime [with an addi-tional "external potential" corresponding to the term in-volving the scalar curvature of space in Eq. (1.2)]. Ofcourse, there is a crucial physical difference between thetwo cases: the scalar field W on spacetime is physicallymeasurable and should be represented as an operator onthe Hilbert space of states, whereas 4 is not, in any sense,measurable on minisuperspace; rather the measurable

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2603

quantities now would be mathematical analogs of thespacetime coordinates. Thus, it would not appear tomake any sense to define a quantum theory for + analo-gous to a "second quantized" theory for 4&. (Such atheory for 4 often is referred to as a "third quantized"theory. ) However, it would make sense to define for iP

the analog of a "first quantized" theory for N, i.e., tomake quantum cosmology the mathematical analog ofthe theory of a relativistic particle (as opposed to a rela-tivistic field) on curved spacetime. The basic idea wouldbe to take the Hilbert space of states in quantum cosmol-ogy to be the analog of the Hilbert space of single-particlestates for a scalar field on spacetime. The conditionalprobability interpretation would then be used in the fol-lowing sense: A configuration variable q, on minisuper-space would be taken as appropriate for "setting the con-ditions" if its "level surfaces" (i.e., surfaces of constantvalue) are Cauchy surfaces in minisuperspace. One thenwould hope to define operators (analogous to theNewton-Wigner position and momentum operators for ascalar field in flat spacetime) to represent the remainingconfiguration variables and their conjugate momenta onminisuperspace at "time" q.

From the theory of quantum fields on curved space-time, it is known that if minisuperspace is globally hyper-bolic, then mathematically consistent prescriptions canbe given for constructing a Hilbert space of states fromsolutions to the Wheeler-DeWitt equation. However, inorder to single out a prescription, additional mathemati-cal structure must be specified. This additional structureis most conveniently expressed in terms of a bilinear mapp, on the space of solutions (see proposition 3.1 of Ref.11). Unfortunately for this program, there does not ap-pear to be any natural choice for such a p for minisuper-space. Indeed, if minisuperspace possessed a timelikeconformal Killing field which scaled the "potential term"in the Wheeler-DeWitt equation at the same rate as themetric, and if this potential term were of the correct signon all of minisuperspace, then this structure would pro-vide such a natural choice of p by the procedures used forquantum fields in stationary spacetimes. ' However, al-though full superspace does possess a timelike conformalKilling field, it does not properly scale the potentialterm, ' and, in any case, in general Inodels the potentialterm is not everywhere of the correct sign. Thus, there isa serious problem in this approach with the constructionof the Hilbert space of states. Furthermore, even if thisobstacle could be overcome, it is not clear that sensibleanalogs of the Newton-Wigner operators will exist. Fi-nally, it is far from clear that these ideas can be general-ized to spatially inhomogeneous models, since the metricon superspace will no longer have Lorentz signaturewhen there is more than one conformal degree of free-dom; i.e., the resulting superspace metric signature hasboth multiple plus and minus signs. Thus, at present, thisapproach does not appear to be viable, although it cer-tainly would appear to be worthy of further investigation.

The basic difficulty with the conditional probability in-terpretation can be restated in a language which indicatesits direct connection to the issue of time is quantum grav-ity. The basic property that a variable C should satisfy in

(1.16)

with Lagrangian L of the form

L =L(X,dX jdt ), (1.17)

where it is assumed that L is nondegenerate in the sensethat the relation between momentum and velocity [seeEq. (1.22) below] is one to one. We rewrite this theory inthe following manner. We introduce a new "parameter

order to be appropriate for "setting the conditions" isthat, for each fixed value of C, a measurement of any ofthe other dynamical variables must yield one and onlyone value; it is only under this circumstance that onecould expect to meaningfu11y talk about probability dis-tributions for these other variables. Thus, for example, inordinary Schrodinger quantum mechanics, one should ex-pect the particle position variable X to be inappropriatefor setting the conditions, because when X=XO, thedynamical variable Y could take on many values (sincethe particle could be at X=Xo at many different times) orno value (since the particle might never be at X=Xo). Asdiscussed further by one of us elsewhere, ' this propertyfor a variable needed to properly set the conditions forthe other variables is a key feature of our intuitive notionof "time, " as is well expressed in the aphorism "time isthat which allows contradictory things to occur. " Fol-lowing Ref. 14 we will refer to this as the "Heraclitianproperty" of time, on account of Heraclitus' view of theAow of time as a "war of opposites. " Thus a variablewhich is suitable for setting the conditions for theremaining variables could reasonably be referred to as a"time variable. " The central difticulty with the condi-tional probability interpretation is that in quantum gravi-ty the time variable (or various allowed possible choicesof time variable) has not been specified. (Such aspecification is given in the approach described in theprevious two paragraphs, but as concluded above, thisapproach does not appear to be viable. )

In most theories the notion of "time" that is present atthe classical level can be taken over directly in the formu-lation of the quantum theory. In classical general rela-tivity a spacelike hypersurface in spacetime provides anappropriate realization of the notion of an "instant oftime. " Thus, at the classical level, the specification of afoliation of spacetime by spacelike hypersurfaces defines asatisfactory notion of time. The problem with this notionof time is that it is closely analogous to the notion of timein a so-called "parametrized version" of particle mechan-ics, and this notion of time is unsuitable for use in quan-turn theory in the same manner as the time of a"deparametrized" theory. We now shall brieAy reviewparametrized particle mechanics and the quantum theorybased upon it. In doing so we shall also elucidate some ofthe points raised above regarding the above "naive" andconditional probability interpretations of canonical quan-tum gravity.

Consider the theory of a nonrelativistic particle mov-ing in one dimension, with dynamical variable X and witht denoting the ordinary (Galilean) time parameter. Letthe action S be given by

WILLIAM G. UNRUH AND ROBERT M. WALD

time" ~ given in terms of t by an arbitrary monotonicfunction r=r(t) W. e now view t as a dynamical variable,which may be thought of as the readings of a perfectclock. To emphasize this new viewpoint we shall use thenew symbol T to denote this quantity. We rewrite the ac-tion S as

S= fL(X, T,X, T)d7-, (1.18)

L(X,T,X, T)—:TL(X,X/T) . (1.19)

For the Lagrangian L, the momenta canonically conju-gate to Xand Tare

where the overdot denotes derivatives with respect to ~and where

of T, L depends on X and T only in the combinationX/T. Indeed, if the original Lagrangian L is nondegen-erate as we have assumed, then we can use Eq. (1.20} toeliminate X/T in favor of Px =Px. By Eq. (1.21) we thenhave

PT =L (X,X/T ) (X/—T )P~, (1.26)

&=TPT+XP~ L= T—[PT+H(X, P~)], (1.27)

thus showing that PT can be expressed as a function of Xand Pz and thus cannot be an independent variable.

Nevertheless, we can give a Hamiltonian formulationwherein Eq. (1.26) is treated as a constraint. To do so wedefine the quantity & by the usual formula for a Hamil-tonian

p =aI. yax=p

PT =aL /a T =L (X/T—)P~,

(1.20)

(1.21)

where H(X, P~) is the Hamiltonian associated with theoriginal Lagrangian L:

where P~ denotes the momentum conjugate to X for theoriginal Lagrangian I:

p =aI. /a(dx/dt) .

The Euler-Lagrange equations for I. are

H(X, PX) = Px L=(X—/T)PX L. —

Note that the constraint (1.26) then is equivalent to

(1.28)

(1.29)

aI. . aLax ax ' (1.23)

P = =0.T aT(1.24)

It is not difficult to verify that Eq. (1.24) is redundant;i.e., it follows from Eqs. (1.21) and (1.23). It also is easilyseen, using Pz=P~ [see Eq. (1.20)], that Eq. (1.23) isequivalent to the original Euler-Lagrange equations forI.,

dP~dt

aI.ax

(1.25)

in the sense that the relation X=X(T) between thedynamical variables X and T implied by Eq. (1.23) is thesame as the relation X=X(t) between the dynamicalvariable X and the time parameter t implied by Eq. (1.25).[That this must be the case can be seen from the fact that,with the identification t = T= f T dr, both Euler-Lagrange equations (1.23) and (1.25) arise from the samevariational problem for S.] In this sense, at the classicallevel, the theory described by the original Lagrangian I.(with dynamical variable X and tiine parameter t) isequivalent to the theory described by L (with dynamicalvariables X, T, and time parameter r).

The above parametrized theory can be given a Hamil-tonian formulation which is closely analogous to theHamiltonian formulation of general relativity. Thecanonical momenta Pz and PT were already obtainedabove [see Eqs. (1.20) and (1.21)]. Normally, one wouldproceed by inverting these formulas to eliminate X and Tin favor of Pz and PT. However, this cannot be donehere (for any choice of the original Lagrangian L), sincethe Jacobian matrix of second partial derivatives of Iwith respect to "velocities" X and T is degenerate, as aconsequence of the fact that, apart from an overall factor

The inability to eliminate (T,X) in favor of (PT, P~)comes into play in Eq. (1.27) in that T cannot be relatedto PT and Pz. However, we can proceed by viewing T asa "Lagrangian multiplier" which enforces the constraint(1.29). To emphasize this viewpoint we write N(r)= Tand view N as an arbitrary, unspecified function. Thus,we write

&(N; T,X,PT, P~) =N[PT+H(X, p~)] .

Hamilton's equation of motion then yield

aT—apr

a

(1.30)

(1.31)

(1.32)

aHX— —X (1.33)

a& aHax ax (1.34)

which, when supplemented by the constraint (1.29) ob-tained by variation of & with respect to N, are easilyverified to be equivalent to the previously derived equa-tions of motion. Thus, we have succeeded in obtaining aHamiltonian formulation of the parametrized particletheory. The very close similarity of this formulation tothe Hamiltonian formulation of general relativity is notaccidental, since the presence of the constraint &=0 inboth theories is directly related to the fact that in boththeories "time evolution" is equivalent to a "gauge trans-formation" (i.e., a spacetime diffeomorphism or areparametrization of r).

We turn now to the quantum theory of theparametrized particle. By the standard canonical quanti-zation rules for the "coordinate representation" alreadymentioned above, states of the system are described by

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . .

wave functions 4(r; T,X). The operators T and X arerepresented by multiplication whereas P~ and P~ arerepresented as i—d/dT and i—d/dX T. ime evolution isgoverned by the Schrodinger equation

(1.35)

thus showing that 4 is independent of ~. Thus, this sys-tem possesses many of the features of canonical quantumgravity, but, of course, it has the advantage that thecorrect quantum theory is known, namely, that obtainedfrom the original Lagrangian L and Ham. iltonian H.

Note that the constraint (1.36), which is the analog ofthe Wheeler-DeWitt equation (1.6), is just the ordinarySchrodinger equation for 'P(T, X):

i +—H(X, P~ )4=0 . (1.37)

Thus, the above rules for the canonical quantization ap-plied to our parametrized theory yield the correct equa-tion for %. The above "naive interpretation" applied tothis case would demand that 4( T,X) be square integrablewith respect to T and X, and would then attempt to inter-pret 4 as giving the (time-independent) amplitude formeasuring T and X. This clearly does not make sense;indeed, there presumably do not even exist any solutionsof (1.37) which are square integrable with respect to Tand X. The "WKB interpretation" yields results con-sistent with the standard quantum theory, but, of course,has a severely limited range of applicability. Finally, the"conditional probability interpretation" with T chosen asthe variable which "sets the conditions" would assertthat, given that the time is T, %(T,X) yields the ampli-tude for ending the particle at X. If the measure is takento be dpz=dX, this corresponds precisely to the stan-dard quantum theory of a particle. However, the "condi-tional probability interpretation" with X chosen as thevariable which "sets the conditions" would assert that atgiven particle position X, %(T,X) should give the ampli-tude that the time is T. This does not make sense, andthus illustrates the point already mentioned above thatonly certain variables are suitable for "setting the condi-tions. "

The analogy with the quantum theory of parametrizedparticle mechanics may be viewed as supporting the"conditional probability approach" to the interpretationof the wave function of the Universe in quantum gravity,provided that a suitable "time variable" is identified fromamong the collection of dynamical variables. Inparametrized particle mechanics, the dynamical variableT can be distinguished from the other dynamical vari-ables by the fact that its canonical momentum PT ap-pears only linearly in the Hamiltonian %. However, incanonical quantum gravity, although there has been con-siderable research effort on this issue, no suitable "timevariable" has yet been identi6ed. In particular, theWheeler-DeWitt constraint (1.6) is quadratic in all the

However, the constraint (1.29) is imposed by the condi-tion

(1.36)

momenta. We refer the reader to the reviews ofKuchar"' and references cited therein for extensive fur-ther discussion of this issue.

Thus, the conditional probability interpretationpresently remains in the unsatisfactory state describedabove. Clearly, one approach to the interpretation of thewave function of the Universe would be to attempt tosolve the problem of identifying a suitable time variable.However, we are skeptical that such an approach willsucceed. One reason for our skepticism is that no solu-tion has yet emerged after over 20 years of effort. Amore fundamental reason is that the selection of a pre-ferred time variable (or class of time variables) would ap-pear to be in conAict with the generally covariant natureof general relativity. The ability to "deparametrize" theabove particle theory by choosing T as a preferred timevariable appears to be directly related to the fact that theunderlying classical theory possesses a preferred time pa-rametrization; however, there does not appear to be anyanalogous preferred time slicing of generic spacetimes ingeneral relativity. Finally, perhaps the strongest reasonfor our skepticism is that we do not believe that any real-istic dynamical variable can satisfy the "Herachtianproperty" required for a time variable. The main supportfor this view arises from the fact that in the context of or-dinary Schrodinger quantum mechanics, no dynamicalvariable in a system with Hamiltonian bounded frombelow can act as a perfect clock in the sense that there isalways a nonvanishing amplitude for any realistic dynam-ical variable to "run backwards. " (Note that, by con-struction, the dynamical variable T in our parametrizedparticle model does act as a perfect clock; however, itsHamiltonian HT =PT is unbounded from below. ) In par-ticular, any realistic dynamical variable may take on thesame value at two distinct Schrodinger parameter times.Hence, it would appear that in Schrodinger quantummechanics, all other dynamical variables can be mul-tivalued at a given value of any realistic dynamical vari-able which is selected to be a time variable. We proceed,now, to state and prove this "no perfect clock"theorem. '

Consider any arbitrary system in ordinary Schrodingerquantum mechanics, restricted only by the requirementthat its Hamiltonian H be bounded from below. We seekan observable (i.e., operator) T which can serve as a"monotonically perfect clock" in the sense that, for somechoice of initial state, its observed values increase mono-tonically with Schrodinger time t. Since T may have con-tinuous spectrum we formulate a minimal condition onsuch an operator T as follows. Break up the spectrum ofT into nonoverlapping intervals of 6nite size. We requireof T that there exist an infinite sequence of states

~ To ),~ T, ), ~ T2 ), . . . having the following properties. (i) Each~ T„) is an eigenstate of the projection operator onto thespectral interval centered around the value T„, withTo & T, & T2 &. . . , (ii) for each n there exists an m ) n

and a t )0 such that the amplitude to go from~ T„) to

~T ) in time t is nonvanishing (i.e., the "clock" has anonzero probability of running forward in time); (iii) foreach m and for all t &0, the amplitude to go in time tfrom ~T ) to any ~T„) with n &m vanishes (i.e., the

2606 WILLIAM G. UNRUH AND ROBERT M. WALD

"clock" cannot run backward in time). We then have thefollowing theorem.

Theorem If the Hamiltonian H is bounded from below,then there does not exist an operator T satisfying proper-ties (i)—(iii) above.

Proof. Let n, m be as in condition (ii). Consider thequantity

f ( t }= ( T„ l exp( —iat ) l T ) (1.38)

for tEC. Since H is bounded from below, f is holo-inorphic in the lower half t-plane and hence f cannotvanish on any open interval of real t unless it vanishesidentically for all t with imaginary part 0 (see, e.g., Ref.17). Since by (iii) we have f(t)=0 for t) 0, it followsthat f (t) =0 for all real t. However, for all t )0 we have

I exp( i'Ht ) I 7; ~—= ~ T

1exp( +iHt ) I

&

(1.39)

which contradicts property (ii).The above theorem can be interpreted as saying that

any realistic "clock" in Schrodinger quantum mechanicswhich can run forward in time must have a nonvanishingprobability to run backward in time. Of course, thetheorem allows this probability to be very small. Never-theless, it shows that if we attempt to replace theSchrodinger time parameter t by a dynamical variable T,we cannot expect to obtain more than a crude, approxi-mate interpretation of the theory. Indeed, since clockstypically degrade (so that they are "good" only for afinite period of Schrodinger time) one should expectsevere problems to arise if one attempts to replace thecondition "at Schrodinger time t" with the condition"when the clock is measured to read T'; the latter condi-tion may include occurrences in the distant future (inSchrodinger time t), when the clock is working very poor-ly and has taken a large jump backward, or has fallenapart completely, leaving the hand pointed at some ran-dom digit.

The Hamiltonian, Eq. (1.4), of general relativity is un-bounded from below (or above), so the above theorem isnot directly applicable to it. Nevertheless, we see noreason to expect the presence of a dynamical variablewhich satisfies the "Heraclitian property. "

Our belief that a "Heraclitian time variable" is neededin quantum theory but that dynamical variables are un-suitable for this role leads us to seek a formulation ofcanonical quantum gravity where a suitable parametertime is explicitly present in the theory. We shall proceedto do so in the following manner. In the next section wereformulate Schrodinger quantum mechanics in a mannerentirely equivalent to the usual formulation, but using anarbitrary time parameter ~ rather than the usualSchrodinger time t, in order to help distinguish betweenthe roles of t as a "Heraclitian time variable" and as avariable containing dynamical information. (This pro-cedure of introducing an arbitrary parameter time afterquantization should be distinguished clearly from theabove procedure of parametrization prior to quantization

where the original time t is treated as a dynamical vari-able T.) In the reformulation, r itself is unmeasurable,but the usual predictions can be expressed in terms ofcorrelations between dynamical variables at (ordered) se-quences of ~ times. Furthermore, if one of the dynamicalvariables T is a "good clock" (i.e, at each r its possiblevalues are sharply peaked about a value which variesmonotonically with r), then one can pass (approximately)from the Schrodinger wave function, g to an "effectivewave function" 4 which depends only on the dynamicalvariables and (approximately) satisfies Eq. (1.37). Thiseffective wave function conveniently encodes the correla-tions between clock readings T and the values of the oth-er dynamical variables; however, one must return to theoriginal wave function f in order to obtain a sensible,precise interpretation of the theory.

The above results motivate our proposal for canonicalquantum gravity given in Sec. III. The "efFective wavefunction" 4 of Sec. II, is closely analogous to the wavefunction of the Universe, and it satisfies Eq. (1.37), whichis closely analogous to the Wheeler-DeWitt equation(1.6). Thus, we seek a theory possessing a nondynamical"Heraclitian time variable" ~, which, in the presence of a"good clock" dynamical variable T, gives rise to an"effective wave function" 4' satisfying the Wheeler-DeWitt equation. In Sec. III we present a proposal forsuch a theory in our minisuperspace model based directlyupon analogy with the analysis of Sec. II. This theoryhas the desired qualitative properties, but it turns outthat the "effective wave function" 0 satisfies an equationwhich difFers from the Wheeler-DeWitt equation (unlessT is a "perfect clock, " with Hamiltonian Hr =Pr) We.study a slightly modified version of this proposal which isseen to correspond classically to Einstein's equation withan arbitrary, unspecified cosmological constant. Thismodified proposal yields a mathematically consistent, in-terpretable quantum theory in minisuperspace models,but for general spacetimes it may suffer from a shortageof measurable observables similar to (though possibly notas severe as) the situation for the "naive interpretation"of canonical quantum general relativity discussed above.

We conclude this section with a discussion of the issueof obtaining an interpretation of the wave function of theUniverse by explicitly incorporating observers into thetheory. It often is suggested that the interpretivedifhculties of canonical quantum gravity arise from thefailure to include dynamical variables representing an ob-server in the wave function of the Universe. ' What thesediscussions appear to have in mind is the following. LetY denote the relevant dynamical variables associated withan observer (which we may view, perhaps, as representinghis "state of consciousness" ). Let X denote all of theremaining dynamical variables (e.g., in our minisuper-space model a and P). Then the wave function of theUniverse is a function of Y and X, 4 =+( Y,X). We view'P as describing the correlations between Y and X.Specifically, given that the observer is in state Yo,4( YO, X) gives the amplitude for the various possiblevalues of the remaining dynamical variables.

It is easily seen that this interpretation is just the above"conditional probability interpretation, " with the "pre-

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2607

ferred time variable" chosen to be Y. However, if onetakes seriously the treatment of an observer as an ordi-nary dyanmical quantum system (as is done in thisviewpoint), we see no reason why Y should have this spe-cial property. Indeed, presumably at best, Y could be ex-pected to be a "Heraclitian time variable" only while theobserver is alive. How is this limitation to be implement-ed in this viewpoint? It may well be that the proper in-corporation of observers into the theory will have manyimportant (and, perhaps, radical) implications for quan-tum theory in general and for quantum gravity in partic-ular. However, it seems to us highly unlikely that suchan incorporation can be attained simply by means oftreating observers as ordinary dynamical systems. In anycase, unless some argument, presumably based on a de-tailed dynamical model of observers, can be given show-ing that the "observer variables" Y have the desiredproperty of a "preferred time variable, " the above pro-posed interpretation based upon observers amounts tonothing more than asserting that some (unspecified)"time variable" can be selected so that the conditionalprobability interpretation is valid. Further indicationthat explicit incorporation of observers should not berelevant to the resolution of the interpretive issues underconsideration here comes from the fact that the same in-terpretive difticulties occur for the above quantum theoryof a parametrized particle, but the accepted version ofthe theory which resolves these difficulties (namely, ordi-nary Schrodinger quantum mechanics) does not rely uponthe incorporation of observers into the theory.

Note that when observers are incorporated into thetheory, although the Wheeler-DeWitt equation (1.6) isimposed upon the wave function of the Universe 4'( Y,X),the quantity 4( YO, X) [or, more precisely, the projectionof 'P(Y, X) onto the perceived "eigenstate of observervariables"] will, in general, fail to satisfy the Wheeler-DeWitt equation. ' Thus, it is far from clear that an ob-server would perceive that the Wheeler-DeWitt equationis satisfied and that Einstein s equation holds in the clas-sical limit. Actually, as we shall now discuss, the situa-tion is considerably worse, since an observer presumablyhas access only to Y and cannot directly determine any-thing at all about the "universe variables" X.

The difficulty just alluded to is, of course, nothingmore than the age-old problem of skepticism, but it arisesin a particularly virulent form when one examines thelogical consequences of taking seriously the incorporationof observers into any (classical or quantum) theory sothat they are treated as ordinary dynamical systems. Thekey point is that, even if the dynamics of observers iscompletely specified (which, of course, realistically is farfrom the case), the theory loses all predictive power con-cerning the Universe unless some new principle of phys-ics governing the correlations of observers with theUniverse (presumably formulated in terms of initial con-ditions) can be invoked. To illustrate this point explicit-ly, consider the simple example of the theory of classicalmechanics of point particles. We describe the state ofour dynamical system by dynamical variables (X;,Pr )

and we model our observer as similarly described bydynamical variables (Y;,Pr ). The various possible per-

I

ceptions (including memories) of the observer would thenbe assumed to correspond to particular regions of ob-server phase space. The difficulty is that unless severe re-strictions are imposed upon initial conditions, then allkinematically possible observer states are dynamicallypossible. In particular, the observer may "perceive" or"remember" things about the (X;,Pz ) system which did

I

not occur or even were dynamically impossible. Thus,what may have started out as a theory of the dynamics ofthe (X;,Px ) system would become transformed, by the in-

I

corporation of observers, into a theory dealing primarilywith the kinematical restrictions on observer phase spaceand with the allowed initial states for observers. Indeed,the requirement that these initial states be such that theobservers' perceptions and memory correlate properlywith what actually occurred in the (X;,Px ) system is not

1

experimentally testable, nor is even the existence of an(X;,P~ ) system, and would have to be viewed as an en-

tirely ad hoc hypothesis. Exactly the same difficultiesoccur, of course, in quantum theory. The refusal to in-corporate observers into the theory as ordinary dynami-cal systems does not solve the problem of skepticism, butit does, at least, remove it from one's doorstep.

Thus, in view of all the above discussion, it does notappear to us to be fruitful to attempt to resolve interpre-tive issues of quantum gravity by invoking the explicit in-corporation into the theory of observers, treated as ordi-nary dynamical systems. (We also do not have any seri-ous proposals for treating observers as extraordinary sys-tems. ) Thus, for the remainder of this paper we will con-tinue to treat observers in the same "phenomenological"way that they have been treated in all prior physicaltheories.

II. SCHRODINGER QUANTUM MECHANICS INAN ARBITRARY TIME PARAMETRIZATION

The parameter t which appears in Schrodinger quan-tum mechanics has a rather unusual status. It is not an"observable" in any normal sense; in particular, there isno operator on the Hilbert space of states which corre-sponds to measuring "time." Rather, t primarily playsthe role of a nondynamical, "Heraclitian time variable"(see Sec. I) which "sets the conditions" for measurementsof the dynamica1 variables. It is not true that one "mea-sures" t by examining a clock, since by examining a clockone is simply measuring some ordinary dynamical vari-able (e.g. , the position of the hands of the clock). Indeed,we showed in the previous section that no realisticdynamical variable can even monotonically correlatewith t with certainty. Nevertheless, one can make infer-ences about t by performing measurements of dynamicalvariables. In this section we shall attempt to clarify therole of t in ordinary Schrodinger quantum mechanics byreformulating the theory in terms of an arbitrary time pa-rameter ~. In this way, the roles of t as a "Heraclitiantime variable" and a quantity which is, in some sense,"observable, "can be clearly distinguished.

Qur main purpose in giving this reformulation is that itwill aid us in explaining how information concerning

2608 %WILLIAM G. UNRUH AND ROBERT M. %'ALD

correlations between measurements of a "good clock'"dynamical variable T and another dynamical variable Xcan be expressed in terms of an "e6'ective wave function"%'(T,X) which depends only upon observable dynamicalvariables and which (approximately) satisfies theSchrodinger equation. However, unless the clock is "per-fect," the effective wave function %(T,X) will possess aninterpretation only in a crude, approximate sense; onemust return to the exact Schrodinger wave functiong(r; T,X) in order to formulate a precise interpretation ofthe theory. Our proposal for formulating an interpret-able quantum theory of gravity, to be given in the nextsection, consists, in essence, of simply paralleling for thewave function of the Umverse, %'(a, P), the steps whichlead backward from 4( T,X ) to g(r; T,X ).

Our reformulation of Schrodinger quantum theoryconsists simply of replacing the parameter t in theSchrodinger equation by a parameter w related to t by anarbitrary monotonic function w=r(t). (Note that, as al-ready pointed out in the previous section, this is notequivalent to formally quantizing a classical parametrizedsystem with "time" treated as a dynamical variable. )

Thus, we take the Hilbert space to be as in standardquantum theory; i.e., if the collection of all the dynamical(configuration) variables are denoted as Z, states areagain represented by wave functions g(Z), which varywith time ~. However, the Schrodinger equation now isreplaced by the following condition. There exists a func-tion N(r) not known or specified in advance, such that

(2.1)

where H is the Hamiltonian operator of the usual formu-lation. (We assume that H has no explicit time depen-dence, so that the operator H is independent of r.) Theinterpretation of g is the usual one. At a fixed value ofparameter time ~, P(r;Z) gives the amplitude for thevalues of the dynamical variables to be Z.

The above formulation easily can be seen to beequivalent to the usual formulation via the simple substi-tution t= fN(r)d~ However, . the above formulation

suggest a viewpoint which we feel corresponds muchmore closely to the observable structure actually presentin ordinary quantum theory. In this viewpoint, an ob-server has access to time orderings of his observationsgiven by the label ~ whose numerical values are of nosignificance except for the ordering they provide. (Inkeeping with the "phenomenological" approach to ob-servers taken at the end of the previous section, we donot attempt to explain the mechanism by which the ob-server obtains access to this ordering defined by r.) How-ever„an observer does not have any "innate" knowledgeof the "metrical" properties of time, as given by t; as al-ready mentioned above t can be determined only by infer-ence from measurements of dynamical variables. [As willbe discussed further below, this is easily done if a "goodclock" dynamical variable is present; otherwise muchmore sophisticated methods would have to be employed.The probability distribution for Z can be expressed interms of the unknown parameter t, and techniques of sta-tistical inference (such as the maximum-likelihood

@(r;T,X)=X(r; T)$(r;X)and we have

(2.2)

Hg=(Hr+Hx)Q, (2.3)

where Hz- is independent of X, and Hz is independent ofT; (ii) at each 7 EI, g(~; T,X) is sharply peaked in T aboutthe value f(r), where f is a monotonic function of ~. Al-though we proved in the previous section that if H isbounded from below, then no dynamical variable T canserve as a "perfect monotonic clock," there is no obstacle

method) could be used to infer t ].Thus, the arbitrary (ex-cept for monotonicity) label time r provides the essential"background structure" of quantum mechanics. Toavoid confusion, we emphasize here that we have in mindthe measurements of a single observer (who, of course,may send out probes, or other observers, to spatially dis-tant regions and later measure the readings of theseprobes). The interpretation is formulated for this singleobserver. If a family of observers is present, we do not as-surne that different members of this family have access tothe same ~; i.e., we do not assume that a family of ob-servers has any innate knowledge of simultaneity.

Since N(i) is unknown, the Schrodinger equation (2.1)does not yield unambiguous predictions for the wavefunction P(r;Z) at time r. (This, of course, correspondsto the fact that r is an arbitrary "label time. ") Predic-tions primarily consist of statements concerning the pos-sible values and correlations of the dynamical variablesZ. Thus, the situation is much like that occurring in gen-eral relativity, formulated in terms of arbitrary coordi-nates x". Einstein's equation does not unambiguouslydetermine the value of the metric components g„evalu-ated at the point labeled by x", and typical statementsabout the theory expressed in this manner usually aremeaningless because they refer to the arbitrary, un-measurable labels x".

The meaningful statements in general relativity aretypically formulated in the following manner: "There ex-ists a point (i.e., event) in spacetime such that the mea-surements of. . . made at that point by the specific pro-cedure. . . would have the outcome. . . ." Similarly, inthe above reformulation of quantum mechanics, themeaningful statements typically would take the form,"There exists a time r such that the probability ofmeasuring the dynamical variables to be Z isStatements concerning whether this time ~ came "before"or "after" a time at which certain other measurementswere made also would be meaningful.

In general, these meaningful statements may be rathercumbersome to formulate. However, as we shall now ex-plain, if one of the dynamical variables is a "good clockvariable, " then meaningful statements can be formulatedin a very simple manner (although these statements willhave only approximate validity). We say that a dynami-cal variable, denoted 1, is a good clock Uariabl'e over theinterval I=[a„b] of r time if the state vector f andHamiltonian H satisfy the following two conditions: (i)For all i&I, T (nearly) decouples from all the otherdynamical variables, which we denote as X, in the sensethat g (nearly) takes the product form

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2609

to the existence of a "good clock variable" in the abovesense.

In condition (i) it should be noted that it is not requiredthat the Hamiltonian operator be generally expressible asH=Hr+Hz, i.e., there may be states (other than f) forwhich Eq. (2.3) fails. In condition (ii), since H is indepen-dent of r, it follows from the Schrodinger equation (2.1)that f can depend on ~ only in the combination

fN(r)d~. By redefining T, if necessary, we will assume,without loss of generality, that, in fact, f(r) = fN(~)dr.Thus, Eq. (2.2) and assumption (ii) imply that g takes theform

P(~; T,X)=A. v; T fN—(~)dr P(~;X), (2.4)

where for each r, the function A, (r; ~ ) is sharply peakedaround zero. Furthermore, Eq. (2.3) further implies thatA, and P satisfy

i =N(v )(H~+ C)A. ,-a~ = (2.5), av

i =N(~)(HX —C)P,. 8a7

(2.6)

where C is constant. By redefining A, and P by multiply-ing and dividing, respectively, by the phase factorexp[iC fN(r)dw], we may assume that C=O. We alsomay normalize A, and P so that they each have unit normin the T and X Hilbert spaces, respectively.

Now, since A,(r; ~ ) is peaked sharply about zero, we willmake little error in evaluating f(r; T,X) if, in Eq. (2.2),we replace P(w; X) by the "efFective wave function"%(T;X) defined by

e( T,X)=P(~( T);X),where r( T) is determined by the peak in A, , i.e., by

T= N(~)d~ .

(2.7)

(2.8)

Thus, if T is a "good clock variable" we have

q(~;T, X)=A, r;T fN(r)dr %—(T,X) . (2.9)

The factor A, contains the information concerning theprobability for obtaining "false readings" from measuringthe "clock variable" T. To the extent that the probabilityfor the measured value of T to deviate significantly from

fN(v)d ~ is negligibly small (i.e., to the extent that T is a"good clock" variable), all the remaining informationconcerning the system is usefully encoded in %(T,X).Indeed, to the extent that T is a "good clock variable, "we have the following simple "interpretation" of %. "Ifthe clock variable is measured to have the value T, thenthe amplitude for measuring the remaining dynamicalvariables to have the value X is 4( T,X)." This interpre-tation, of course, is only an approximate one, and if T is a"poor clock" variable, then one must return to the fullwave function P(~; T,X) in order to formulate the predic-tions of the theory, which would have to be stated in themanner discussed above.

It follows immediately from Eqs. (2.6), (2.7), and (2.8),that if T is a good clock variable, then 4( T,X) satisfies

+H 4=0.. B%gT x (2.10)

This equation is, of course, formally the same as the usualSchrodinger equation, except that the Schrodinger pa-rameter time t has been replaced by the dynamical vari-able T. More importantly for our purposes, Eq. (2.10) isformally identical to the constraint equation (1.37) whicharose in the quantization of a parametrized classicaltheory in Sec. I. Thus, we may summarize the results ofthis section as follows. We began with Schrodinger quan-turn mechanics —a well-defined and interpretable quan-tum theory, possessing no constraints. We gave an en-tirely equivalent reformulation of this theory in terms ofan arbitrary label time ~. We then showed that if one ofthe dynamical variables in the theory is a "good clock, "one can pass to an "efFective wave function" %(T,X)which depends only upon the dynamical variables andsatisfies Eq. (2.10), which is formally equivalent to theconstraint equation (1.37). Furthermore, one has an in-terpretation of 4' only in the approximate sense describedabove; the "exact" interpretation of the theory can onlybe formulated in terms of the original wave functiong(r; T,X).

The relevance of the above discussion for the interpre-tation of the wave function of the Universe should nowbe clear. The wave function of the Universe %(a,P) isclosely analogous to the "effective wave function"%(T,X), with the Wheeler-DeWitt equation playing therole of Eq. (2.10). Thus, we propose to view %(a,P) as an"effective wave function" —an approximate concept, val-id only when a "good clock" dynamical variable ispresent. We propose that it arises from an exact theorywhich possesses a "label time" ~ and is we11 defined andinterpretable in all circumstances. We shall attempt toimplement this proposal in the next section by reversingthe steps above which led from g(r; T,X) to 4( T,X).

III. A PROPOSAL FOR CANONICALQUANTUM GRAVITY

In this section we shall explore the possibility that the"exact" quantum theory of gravity is such that a "Hera-clitian time parameter" ~ is explicitly present. In such atheory with time parameter ~, if a "good clock variable"is present, we can pass to an "effective wave function" 4,which depends only on dynamical variables, in themanner described in the previous section. Our goal is toobtain a theory whereby this eff'ective wave functionsatisfies the Wheeler-DeWitt equation, since presumablythis would be necessary (and, presumably, also sufficient)for the theory to reduce to general relativity in the classi-cal limit.

We shall proceed by making a straightforward propo-sal for the desired theory in the simple context of aminisuperspace model (see Sec. I). We shall then explainwhy this proposal fails to yield the Wheeler-DeWitt equa-tion for the effective wave function unless the "goodclock" actually is a "perfect clock" in the sense that itsHamiltonian is Hz-= —iB/BT. Nevertheless, we thenwill proceed to describe a general (i.e., not restricted tominisuperspace models) formulation of a class of propo-sals having the character of our minisuperspace proposal.

2610 WILLIAM G. UNRUH AND ROBERT M. WALD

We will focus attention on one of them and show that itcorresponds classically to general relativity with an arbi-trary, unspecified cosmological constant. Thus, althoughthis proposal fails to do what was originally intended(since it does not correspond classically to ordinary gen-eral relativity) and although (as discussed further below)it may suffer from serious difhculties in the context ofgeneral spacetimes, we hope that both the nature of theattempt and the resulting theory will be of some interest.

Our proposal in the minisuperspace case is baseddirectly upon analogy with the discussion of the previoussection. It consists, in essence, of simply omitting theconstraint (1.12) from the theory. Thus, the wave func-tion of the Universe is taken to be a function of a "Hera-clitian time parameter" ~ and the dynamical variables,which, for simplicity and definiteness, we take to be a and

P as in the model of Sec. I. Thus, g=g(r;a, P) is directlyanalogous to an ordinary Schrodinger wave function. Itsatisfies the Schrodinger equation

(3.1)

which, for our model takes the explicit form

. ay 1 a a@B~ ]2@2 Ba Ba

a21 Bg 3(

2a BP

(3.2)

[see Eq. (1.10) above] but now the constraint equation&/=0 is not imposed. As a consequence, P has a non-trivial time development. Indeed, if we normalize g in aand P and use the natural (metric) volume elementa da dP on minisuperspace, we may consistently give thestraightforward interpretation of f(r;a, P) as yielding theamplitude for an observer to measure the values a and Pat time ~.

Thus, our proposal for the minisuperspace case can bemade to yield a mathematically sensible quantum theorywhich can be interpreted in a straightforward way. How-ever, it remains to be seen whether it corresponds to gen-eral relativity (or, some other physically viable theory) inthe classical limit. To investigate this issue we assumethat one of our dynamical variables, denoted T, is a"good clock variable" in the sense of the previous sec-tion. Then, as discussed in the previous section, the wavefunction g will approximately factor as

g(r; T,X)=X(r; T)V( T,X), (3.3)

. 8%' +H 4=0.X (3.4)

where X denotes the remaining dynamical variable(s). (Inour simple model there would be only one such additionaldynamical variable, but our discussion does not dependupon the details of this model and would apply if manymore dynamical degrees of freedom were present. ) As inthe previous section, the Schrodinger equation (3.2) to-gether with our "good clock" assumptions will imply anequation for our effective wave function 4'( T,X). Indeed,by precisely the same derivation as led to Eq. (2.10) weobtain

However, except in the case Hr= i—BIBT (correspond-ing to the Hamiltonian of a perfect clock), this is not thesame as the %'heeler-DeWitt equation which, in thepresent circumstances, takes the form

H, %+H~% =0 . (3 5)

Thus, the correlations among the dynamical variables im-plied by Eq. (3.4) will not, in general, be the same as thoseimplied by Eq. (3.5). Unless the dynamical variable T is a"perfect clock, " we will not obtain Einstein's equation inthe classical limit.

The reason why Eqs. (3.4) and (3.5) are different can beunderstood as follows. The only property of the "goodclock variable" that enters the derivation of Eq. (3.4) isthe behavior of the clock, i.e., the correlation between Tand ~. On the other hand, the relevant feature of theclock with regard to the Wheeler-DeWitt equation (3.5) isits energy, i.e., its contribution as a source of gravitationin the Hamiltonian constraint equation. But it is easy tobuild two clocks which have the same behavior (i.e.,essentially the same correlation between T and r) butvastly different energies. For example, one could use theposition of a free particle as a clock. ' By using particlesof different masses, one can obtain a simple example oftwo "clock variables" which would make the same con-tribution to Eq. (3.4) but very different contributions toEq. (3.5). Only in the case of a clock whose energy is re-lated to its behavior in the same way as-for a "perfectclock" (i.e., a dynamical system whose true Hamiltonianis Hz. = i r)lr)T) ar—e the contributions of the clock toEqs. (3.4) and (3.5) the same. Note that one can have an"extremely good clock" such that Eqs. (3.4) and (3.5)differ drastically; the diff'erence between Eqs. (3.4) and(3.5) has to do with the "active gravitational mass" of theclock, not how well it runs.

Given that our proposal does not correspond to ordi-nary classical general relativity, one nevertheless may in-quire further as to its physical viability by determiningwhat classical theory (if any) it does correspond to. Inthe minisuperspace context in which it was formulated, itis clear that it corresponds classically to a theory inwhich the Hamiltonian constraint is not imposed, but theusual Einstein evolution equations are retained. Howev-er, to investigate its viability, it is worthwhile to general-ize our proposal and obtain the corresponding classicaltheory for arbitrary spacetimes. [Note that there aremany possible ways to generalize the proposal since, inparticular, there are many inequivalent (on general space-times) theories which reduce to the same theory inminisuperspace models. ] Since our proposals for a quan-tum theory in the minisuperspace case involved droppingthe applicable constraint equation, one way of generaliz-ing our proposal to arbitrary spacetimes would be simplyto drop all of the constraint equations of general relativi-ty (at both the classical and quantum levels). More pre-cisely we could take for the classical theory the sameHamiltonian, Eq. (1.4), as occurs in ordinary general rela-tivity, but now view X and X' as fixed (thoughunspecified) functions, which are not to be varied in ob-taining Hamilton's equations of motion. In quantumtheory, the Schrodinger equation (1.5) would remain, but

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2611

the constraints (1.6) and (1.7) would be absent. Such atheory would possess the desired "Heraclitian time pa-rameter" but would correspond classically to a theorywhich differs sufficiently from ordinary general relativity(in that it allows many more solutions to the field equa-tions) that it would not be physically viable.

However, it is not necessary to take such a drastic stepin generalizing the proposal to arbitrary spacetimes. Inparticular, there is no need to drop the momentum con-straints (1.7). Thus, we could again take the Hamiltonianto be given by Eq. (1.4), now treat X' as a Lagrange mul-tiplier (to be varied in obtaining the equations of motion)as in ordinary general relativity, but treat X as a fixed(though unspecified) function. Again, however, the clas-sical theory obtained in this manner will admit manymore solutions than ordinary general relativity. Indeed,it is not dificult to show that solutions of the vacuumfield equations in this theory can be put in correspon-dence with solutions of the ordinary Einstein equationwith arbitrary irrotational dust matter.

However, it should be noted that the Hamiltonian con-straint equation (1.2) of ordinary general relativity actual-ly represents infinitely many constraints (namely, oneholding at each point of space), but only one constraintneed be dropped in order to obtain a nontrivial Heracli-tian time variable. Thus, we could obtain a classicaltheory which corresponds much more closely to ordinarygeneral relativity by allowing all but "one degree of free-dom" of X also to be varied in the Hamiltonian. For ex-ample, we could fix only the spatial average of X andthereby obtain a classical theory which corresponds toordinary general relativity with irrotational dust of uni-form density on the orthogonal hypersurfaces. Thiswould represent only a "one-parameter generalization" ofgeneral relativity, and thus would not differ as greatlyfrom ordinary general relativity as the previous propo-sals. However, the classical theory has the undesirablefeature of possessing locally preferred Lorentz frames.

Rather than investigate the nature of this theory at theclassical and quantum level, we will consider a slightmodification of it, whereby X is required to be a fixedfunction of the dynamical variables (so that it still cannotbe varied independently) rather than a fixed (or partiallyfixed) function on spacetime. As we shall see, by a partic-ular such choice of X, we can obtain a "one-parametergeneralization" or ordinary classical general relativitywhich preserves "local Lorentz invariance, " and, indeed,corresponds classically to general relativity with an arbi-trary cosmological constant. The canonical quantizationof this theory will possess a Heraclitian time parameter ofthe type we have been seeking. We shall proceed by giv-ing Lorentzian and Hamiltonian formulations of thistheory at the classical level and then investigating the na-ture of the quantum theory in our minisuperspace model.This theory has been proposed by a number of authors,primanly with regard to the cosmological-constant prob-lem. It has been described elsewhere by one of us ' withregard to the problem of time in quantum gravity; forcompleteness we shall repeat some of the discussion ofRef. 21 here. An equivalent proposal has been made re-cently by Sorkin, who was motivated by considerations

5S= fG' 5g,b (3.7)

it is clear that the extrema of S are precisely the solutionsof the trace free Einstein's equat-ion

gab l 6 ab 04 (3.8)

More generally, if matter fields are present, the field equa-tions become

Gab ) Ggah Tab ] Tgah4 (3 9)

where T,b is the usual stress-energy tensor of the matterfields (obtained by functional differentiation of the matteraction with respect to the metric). As in the ordinaryEinstein case, by the equations of motion of the matterfields, T,b will satisfy V'T, b =0. Hence, taking the diver-gence of Eq. (3.9) and using the Bianchi identity we ob-tain

V, (G —T)=0, (3.10)

i.e.,

which were different from (though not entirely unrelatedto) ours.

Our theory involves, as usual, a spacetime metric g, b

on a four-dimensional background manifold M. Howev-er, we now additionally require that M be orientable andwe fix a volume element g,b« =gt, b«~ on M as part of thebackground structure. We require the metric to satisfyg=det(g, b)= —1, where the determinant is calculated

's det~~m~~~d by ~' ' rt, b,d=4!. In othewords we require the natural volume element associatedwith g, b to agree with g,b«. Locally, this requirementplaces no restriction on the spacetime geometry in thesense that, locally (i.e., in any sufficiently small open set),any metric is related to one satisfying g= —1 by adiffeomorphism. Globally this need not always be truesince, in particular, the total volume of M computed us-

ing g,b and g,b«would have to be the same in order for

g,b to be diffeomorphic to a metric with g= —1. Suchglobal restrictions actually could be avoided in the for-malism by requiring g to be any fixed (i.e., not to bevaried as g,b is varied) but unspecified function on space-time (rather than —1). In any case, such global restric-tions will not affect the derivation of the equations ofmotion in the classical and quantum cases, which we ulti-mately shall take as defining theory, so we shall ignorethese possible global restrictions.

(3.6)

where the volume element g,b,d (which is required toagree with the natural volume element of g,b) is under-stood in the integral. It is clear that any solution of theusual Einstein's equation (in the "gauge" g = —1) will bean extremum of S, and, thus, will satisfy the new classicalfield equations. However, more solutions of the new fieldequations are possible because S need only be an ex-tremum with respect to variations 5g,b which preserveg = —1, i.e., which satisfy g' 5g,b =0. Since by the usualcalculation we have

2612 WILLIAM G. UNRUH AND ROBERT M. %'ALD

G —T=const—=4A .

Thus, the equations of motion (3.9) are equivalent to

G ab Ag ab Tab

(3.11)

(3.12)

tegrated momentum constraint J PH, (where P is an ar-bitrary vector field on X) we obtain the additional con-straint

ab S5h, b

again is given by

(3.13)

where A is a constant. This is precisely the form ofEinstein s equation with a cosmological constant. How-ever, here A is not prescribed in advance but rathercomprises part of the initial data; solutions with all possi-ble values of A are allowed. Thus, as claimed above, theclassical theory obtained from the action (3.6) with therestriction g= —1 is the same as general relativity withan arbitrary, unspecified cosmological constant.

In order to proceed with the canonical quantization ofthis theory we must cast the classical theory in Hamil-tonian form. This can be done by the same procedure asin ordinary general relativity. As in that case we identifythe induced metric h, b on a three-dimensional hypersur-face X as the configuration variable. The canonicallyconjugate momentum variable

0= &fPH .=fPD, (h ' H)X X

(3.18)

This implies that

H =h' A0 7 (3.19)

where A is a spatial constant. The equations of motionthen imply that A does not vary with time either. Thus,in essence, in the present theory the Hamiltonian con-straint HO=0, of ordinary general relativity, is replacedby Eq. (3.19). Note that the derivation of Eq. (3.19) cor-responds in the Hamiltonian formulation to the deriva-tion of Eq. (3.12) above.

The canonical quantization of the theory proceeds as inthe case of ordinary general relativity as described in Sec.I. The state vector is taken to be a functional of thedynamical variable h, b on X (as well as the function oftime), P=f(t;h, b). The momentum constraint corre-sponding to (3.17) is imposed as before by requiring g tosatisfy

~~b = i & &2(~ obp, ~b~ ) (3.14) f (D, gl, ) =0 (3.20)

where E,b is the extrinsic curvature of X andh =det(h, b) [Here. the volume element ' 'g, &, =g,b,dton X is used to calculate h, where t' is the "time How vec-tor field" (see, e.g., Ref. 2).] The Hamiltonian again takesthe form

which, again, has the interpretation that g depends onlyon the three-geometry. However, in place of the Hamil-tonian constraint (1.6), we now have the weaker condi-tion arising from the constraint (3.18):

&=f (NHO+N'H, ),X

(3.15) 'D, h ' Ho =0. (3.21)

where X and N' again have the intepretation, respective-ly, of lapse function and shift vector, and Ho and H, aregiven by the same expressions as in ordinary general rela-tivity [see Eqs. (1.2) and (1.3) above]. (Here and in all in-tegrals over X below, the volume element ' 'g, b, is under-stood. ) The major difFerence which occurs here is that,on account of the condition g = —1, the lapse function Nno longer is an independent variable. Rather, it is givenin terms of the dynamical variables h, b by

(3.16)

As already indicated in our discussion above, the mostimportant consequence of this change of status of N isthat the Hamiltonian constraint equation, Ho =0, of ordi-nary general relativity, which is obtained by independentvariation of X, no longer occurs as a constraint equationhere and hence the Hamiltonian & does not vanish iden-tically for solutions. The momentum constraints

(3.17)

do remain present, since they are obtained by variation ofthe shift vector N', which remains an independent vari-able.

Although the Hamiltonian constraint is not present, itis almost recovered by the condition that dynamical evo-lution preserve the momentum constraints (3.17). Takingthe Poisson brackets of the Hamiltonian & with the in-

Again, P will evolve via the Schrodinger equation

(3.22)

However, since the right-hand side of Eq. (3.22) does notvanish, g will have a nontrivial time dependence, and twill play the role of a Heraclitian time parameter. Notethat the theory does not have a "reparametrization in-variance" t~r(t, x ) on account of Eq. (3.16), which fixesthe lapse function in terms of the dynamical variables.Thus t is analogous to the preferred time of Schrodingertheory. However, one could parametrize Eq. (3.22)directly (in the same manner as done in Sec. II) to write itin the form

(3.23)

where JV' is an arbitrary positive function, so that it ap-pears more directly analogous to Eq. (2.1). As in ordi-nary Schrodinger quantum mechanics, 'neither t nor ~ aredirectly measurable. The predictions of the theory mustbe formulated in terms of the correlations of the measur-able dynamical variables. As discussed in Sec. II, suchpredictions are easiest to state when a "good clock"dynamical variable is present, although of course, thepresence of such a variable is not necessary either for theformulation or interpretation of the theory.

TIME AND THE INTERPRETATION OF CANONICAL QUANTUM. . . 2613

Note that the solutions to Eq. (3.22) formally are su-perpositions of "eigenstates of the cosmological con-stant":

&itjA= f h ' H qI =A/ (3.24)

i.e., formally, the general solution of the Schrodingerequation (3.22) is of the form

r

g(t)= f dAa(A)exp i —fAr

(3.25)

where g~ satisfies Eq. (3.24) and the integral in the phasefactor is over the spacetime region between the initialslice and the time slice of interest, using the volume ele-ment g,b,d. (Thus, the phase factor has the interpretationof being simply the cosmological constant times thefour-volume of this region of spacetime. ) From Eq. (3.25)it can been seen that the nontrivial time dependence of Parises from the superposition of states corresponding todifferent values of the cosmological constant.

The new features of the wave function of the Universein this theory are best elucidated by examining in con-crete detail the nature of our minisuperspace model withdynamical variables a and P. We again consider thecosmologically fiat ("k =0") case but now write themetric in the form

ds = —a dt +a (dx +dy +dz ) (3.26)

so that g = —1. The Hamiltonian for this minisuperspacemodel in our new theory is

4m, + sm~+V(P) .12Q 2Q

(3.27)

Since the model is homogeneous, all of the constraints areautomatically satisfied. Thus, any solution %(t;a, P) ofthe Schrodinger equation

r

a'lp + 1 a ag 1 a 1i + v(y12a' aa aa 2a' ay'(3.28)

is a possible wave function of the Universe. Here wehave again chosen the "Laplacian" factor ordering forthe momentum terms. Note that, even when written inthe parametrized form (3.23), this equation is not thesame as Eq. (3.2) above, as a consequence of the fact thathere the lapse function is a function of dynamical vari-ables and hence is an "operator" rather than a "c num-ber. " Nevertheless, Eqs. (3.2) and (3.28) are qualitativelysimilar, and the remarks made at the beginning of thissection concerning the inequivalence of Eq. (3.2) to theWheeler-DeWitt equation apply with equal validity toEq. (3.28).

The Hamiltonian operator appearing on the right-handside of Eq. (3.28) is formally self-adjoint; i.e., more pre-cisely, it is symmetric on the domain of smooth functionsof compact support in L (a, P), where we now use thenatural measure a da d P on minisuperspace [whicharises from the metric on minisuperspace obtained fromthe kinetic terms in Eq. (3.27)]. Since this operator is"real, " it always admits a self-adjoint extension {seeTheorem X.3 of Ref. 23). Thus, the quantum dynamicscan be made rigorously well defined, and both the

mathematical structure and the interpretation of thismodel is exactly as in ordinary Schrodinger quantummechanics.

In the case V=O, corresponding to a free, masslessKlein-Gordon scalar field, Eq. (3.28) becomes simply

3 a aQ + 1 ag4p ap ap

+2p a

(3.29)

where p=a, and the measure in the new variables (p, P)is p dp dP. Remarkably, the Hamiltonian is identical (upto trivial factors) to that of the Laplacian operator in po-lar coordinates in a two-dimensional Aat space, except forthe minus sign occurring in the first term and the factthat here P has the range —~ to oo rather than 0 to 2m.Thus, it can be seen immediately that the "eigenstates ofcosmological constant" have the form

PA ~(p, P) =exp( ipse) J;„((—4A/3)'~ p), (3.30)

where J denotes the Bessel function of order v andi) = ( —,

' )' p. These eigenstates correspond to solutions ofthe Wheeler-DeWitt equation for ordinary general rela-tivity with a fixed value A of the cosmological constant;they fail to be square integrable on minisuperspace.However, there is, of course, no difhculty in obtaining su-perpositions of the form (3.25) which a're square integr-able. Thus, we see how a persistent problem for obtain-ing a probabilistic interpretation in the usual approach,namely, the failure of the wave function to be normaliz-able, is avoided here.

Typical dynamical variables, such as a, will fail to com-mute with the Hamiltonian. In contrast with the usualapproach, this does not pose any di%culty with regard tothe measurability of these quantities. However, it shouldbe noted that such a measurement will inhuence theprobability distribution for the value of the "energy, " i.e.,cosmological constant. Thus, in this theory, observerscan, in effect, change the value of the cosmological con-stant by making measurements. However, one would ex-pect that the uncertainty in the cosmological constant in-duced by a measurement would be of order b,A=6,E/V,where AE is the uncertainty induced in the energy and Vis the spatial volume of the Universe (which would befinite for a three-torus model), in which case b A would benegligible for any physically realistic measurement.

In the context of minisuperspace models, the onlydifFiculty with the above theory is its physical viability. Itmight appear that there is a serious problem in this re-gard, since it corresponds classically to general relativitywith an arbitrary value of A. We know that the observedvalue of A in our Universe is extremely small ( & 10 ' ),but there apparently is nothing in the theory to protect usfrom solutions with much larger values of A. However, itshould be noted that a similar "cosmological-constantproblem" occurs in the usual approach to quantum gravi-ty. There A is a fixed parameter and one could argue thatit is natural to set the "bare" value of A equal to zero.However, there is then apparently nothing to protect usfrom having the effective (renormalized) value of A takeon much larger values. Thus, what we have is a modifiedversion of the usual cosmological-constant problem. The

WILLIAM G. UNRUH AND ROBERT M. WALD

"problem" now is shifted from the issue of why aneffective coupling constant is so small to the issue of whythe initial conditions of the Universe were such that afreely specificable quantity in the theory is so small.

Although in the context of minisuperspace models theabove proposal provides a mathematically (and, perhaps,physically) viable quantum theory of gravitation, it is farfrom clear that it will continue to do so in the context ofgeneral spacetimes. Although an observable in the gen-eral theory no longer need commute with the Hamiltoni-an, it still must commute with the constraints (3.21)(which are trivial in the minisuperspace case on accountof homogeneity). Thus, the theory may well possess thesame type of difhculties that plague the "naive interpreta-tion" of canonical quantum gravity discussed in Sec. I. Itis clear that much more work will be required before onecan determine whether any proposal of the type con-sidered here (i.e., introducing an external parameter time)

can provide a viable solution to the problem of time inquantum gravity. However, we hope that it may providea step along the road to finding such a solution.

ACKNOWLEDGMENTS

We have benefited from numerous discussions withmany colleagues, including Robert Geroch, James Har-tle, Karel Kuchar", and Rafael Sorkin. This research wassupported in part by the Canadian Institute for Ad-vanced Research, by the Natural Sciences and Engineer-ing Research Council of Canada, and by NSF Grant No.PHY 84-16691 to the University of Chicago. One of us(W.G.U.) wishes to thank the Institute for TheoreticalPhysics (supported by NSF Grant No. PHY 82-17853,supplemented by funds from NASA) for hospitality andsupport.

For a discussion of the role of time in canonical quantum grav-ity which differs in some important respects from ours, see J.B. Hartle Phys. Rev. D 37, 2818 (1988); and {to be pub-lished).

R. M. Wald, General Relativity (University of Chicago Press,Chicago, 1984).

In fact, the Ashtekar representation may have significant tech-nical advantages over the coordinate representation in gen-eral relativity. [See A. Ashtekar, New Perspectives in Canonical Gravity (Bibliopolis, Naples, in press). ] In addition, in thecoordinate representation for general relativity, the usualform of the canonical co'mmutation relations imposed on h, b

and m' are not really appropriate on account of the "cone-space" structure of the configuration space [see C. J. Isham in

RelatiUity, Groups, and Topology II, proceedings of the LesHouches Summer School, Les Houches, France, 1983, edited

by R. Stora and B. S. DeWitt (Les Houches Summer SchoolProceedings Vol. 40) (Elsevier, Amsterdam, 1984).] Howev-

er, neither of these points should affect the interpretive issuesunder consideration here.

4J. B.Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983).5A. Vilenkin, Phys. Rev. D 37, 888 (1988).6D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983).7A. Vilenkin, Phys. Rev. D 39, 1116(1989).T. Banks, W. Fischler, and L. Susskind, Nucl. Phys. B262, 159

(1985).J. J. Halliwell and S. W. Hawking, Phys. Rev. D 31, 1777

(1985).~OC. W. Misner, in Magic 8'ithout Magic, edited by J. Klauder

(Freeman, San Francisco, 1972).B.S. Kay and R. M. Wald (unpublished).

~2A. Ashtekar and A. Magnon, Proc. R. Soc. London A346,375 (1975);B. S. Kay, Commun. Math. Phys. 62, 55 (1978).

~3K. Kuchar", J. Math. Phys. 22, 2640 (1981).~4W. G. Unruh, in Proceedings of the Fourth Seminar on Quan

turn GraUity, edited by M. A. Markov, V. A. Berezin, and V.P. Frolov (World Scientific, Singapore, 1988).

'~K. Kuchar", in Quantum Gravity 2, edited by C. J. Isham, R.Penrose, and D. W. Sciarna (Clarendon, Oxford, 1981); K.Kuchar", in the Proceedings of the Osgood Hill Conference,

May, 1988 (unpublished).' This theorem may be viewed as a strengthening of a result

which appears in %. Pauli, Die Allgemeinen Prinzipien derWellenmechanik, edited by S. Flugge (Handbuch d. Physik,V) (Springer, Berlin, 1958), p. 60. Note that our theoremconAicts with a statement in A. Peres, Am. J. Phys. 48, 552(1980).R. F. Streater and A. S. Wightman PCT, Spin and Statistics,and All That (Benjamin, New York, 1964).

~~Note that in such discussions the importance of the Everettinterpretation usually is stressed. However, we are concernedhere solely with the specification of rules for what an observer"sees," not with extraneous language (such as "reduction ofthe wave packet"). Some discussions of the Everett interpre-tation appear to evade this issue of what an observer "sees,"in which case they do not provide an "interpretation" in oursense. Those which do not evade this issue provide ruleswhich are equivalent to those of other interpretations.This point also has been made by D. N. Page, in Quantum

Gravity, proceedings of the Fourth Seminar, Moscow, USSR,1987, edited by M. A. Markov, V. A. Berezin, and V. P. Fro-lov (World Scientific, Singapore, 1988).A. Einstein, Siz. Preuss. Acad. Scis. (1919) translated as "DoGravitational Fields Play an Essential Role in the Structureof Elementary Particles of Matter" in The Principle of RelatiUity, by A. Einstein et al. (Dover, New York, 1952); J. J. van

de Bij, H. van Dam, and Y. J. Ng. Phys. Acta A 116, 307{1982);F. Wilczek, Phys. Rep. 104, 111 (1984); A. Zee, in

Gauge Interactions, proceedings of the 20th Orbis Scientiae:Dedicated to P. A. M. Dirac s 80th year, Miami, Florida,1983, edited by B. Kursunoglu and A. Perlmutter (Plenum,New York, 1984); S. Weinberg, Rev. Mod. Phys. 61, 1 (1989);M. Henneaux and C. Teitelboim, University of Texas report,1988 (unpublished).

2 W. G. Unruh, Phys. Rev. D 40, 1048 (1989).R. Sorkin, in History of Modern Gauge Theories, edited by M.Dresaen and A. Rosenblurn (Plenum, New York, in press).M. Reed and B. Simon, Fourier Analysis, Self Adjointness(Academic, New York, 1975).