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NORDITA-94/71 P gr-qc/9502023

Square-Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity

arXiv:gr-qc/9502023v2 18 Jul 1995

A. Carlini

?

?

and J. Greensite

?

NORDITA, Blegdamsvej 17 DK-2100 Copenhagen ?, Denmark Email: carlini@nbivax.nbi.dk

?

Physics and Astronomy Dept. San Francisco State University San Francisco, CA 94117, USA Email: greensit@stars.sfsu.edu

Abstract

We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the √ gravitational action should be based on a path integrand exp[ iS ] rather than the familiar Feynman expression exp[iS ], and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.

1

Introduction

Square-root Lagrangians are a feature of many ?eld theories which are invariant under a time-reparametrization. The action of a relativistic particle SP = ?m and the action of the Nambu string SN = T dσdτ (?τ x)2 (?σ x)2 ? (?τ x · ?σ x)2 (2) dτ ?g?ν ?τ x? ?τ xν (1)

are familiar examples. Somewhat less familiar is the Baierlein-Sharp-Wheeler (BSW) form of the gravitational action 1 √ 3 g RGijnm (?t gij ? 2N(i;j ) )(?t gnm ? 2N(n;m) ) d4 x (3) 2 κ which is obtained from the standard ADM action, as reviewed below, by solving the Hamiltonian constraint. It is well known that for a relativistic particle moving in an arbitrary curved background, and for gravity in general, the corresponding quantum theory lacks a well-de?ned probability measure and time-evolution parameter [1], [2], [3]. In this article we will propose a path-integral formulation of these ”square-root” theories which is something of a departure from the standard Feynman expression. For one thing, the integrand of our path-integral will involve an unconventional phase: √ (4) exp[ iS ] rather than exp[iS ] SBSW = ? Secondly, we will regularize the integration measure so as to uncover what we believe to be the true time-evolution parameter of the quantum theory. Third, we will ?nd it necessary to sum over path-segments of both real and imaginary proper-time, i.e. over time-like and space-like trajectories in the case of the relativistic particle; Lorentzian and Euclidean signature manifolds in the case of gravity. It will be shown that the combination of the regularization, the unconventional phase, and the inclusion of imaginary proper-time segments, leads to a unitary evolution of states which corresponds, via the Ehrenfest principle, to the standard classical dynamics. In two previous articles [4], [5] we have advocated a transfer-matrix approach to quantizing time reparametrization-invariant theories. The present article essentially presents the ”real-time” version of our former ”Euclidean” approach. Our previous work did not include fermion ?elds, which involve certain complications in our formulation. In this paper, we will show how the fermionic ?elds are also incorporated into our approach.

2

Minisuperspace Actions

We begin by considering simple quantum-mechanical theories with a time-reparametrization invariance, i.e. the ”minisuperspace” models of the form S = H = dt {pa ?t q a ? NH [p, q ]} (5)

1 ab G (q )pa pb + mV (q ) 2m

where it is assumed that the supermetric Gab has Lorentzian signature (?+++ ...+). The ”square-root” form of the action is obtained by solving for pa in terms of the time-derivatives of the {q a }, i.e. ?t q a = N =? pa = NGab pb ?H = ?pa m (6)

m Gab ?t q b N

and then solving the Hamiltonian constraint for the lapse function 1 ab G (q )pa pb + mV (q ) 2m m = Gab ?t q a ?t q b + mV (q ) 2N 2 Gab a b ? =? N = ?t q ?t q 2V 0 =

(7)

Substituting (6) and (7) into the minisuperspace action then gives the square-root form (8) S = ?m dt ?2V Gab ?t q a ?t q b For V = 1 , this is simply the action for a relativistic particle of mass m, moving in 2 a background manifold with metric Gab . In non-relativistic quantum mechanics, a path-integral is constructed out of elementary integrals which evolve the wavefunction by a small time-interval ?, i.e. ψ (x′ , t + ?) = dD x ?? exp[iS [(x′ , t + ?); (x, t)]/h ? ]ψ (x, t) (9)

= U? ψ (x′ , t)

where S [(x′ , t + ?); (x, t)] is the action of a classical trajectory between the points x at time t and x′ at time t + ?. The measure ?? is chosen so that ψ (x, t + ?) → ψ (x, t) as ? → 0. With this rule, one ?nds U? = exp[?iH?/h ? ] + O (?2 ) (10)

where H (the Hamiltonian) is an ?-independent Hermitian operator. Taking the ? → 0 limit, the evolution operator for ?nite times U?t ≡ lim(U? )?t/?

?→0

= exp[?iH ?t/h ?]

(11)

is a unitary operator. Straightforward imitation of this construction doesn’t work in the case of the square-root theories, due to the time-reparametrization invariance. Because of this invariance, the action of a classical trajectory between an initial point q and an end point q ′ is independent of the time parameters t and t + ? which label those con?gurations, i.e. S [(q ′ , t + ?); (q, t)] = S [q ′ , q ] (12)

The resulting operator U? de?ned from (9) would therefore be ?-independent, and also in general non-unitary. Let us see if it is possible to recover an evolution operator of the form (10) for the square-root actions, by making a slight change to the construction shown in eq. (9). The modi?cation is to multiply the action S [q ′ , q ] by an ?-dependent complex constant c? ψ (q ′ , τ + ?) = dD q ?? exp[c? S (q ′ , q )]ψ (q, τ ) (13)

= U? ψ (q ′ , τ )

which is to be chosen such that U? is a unitary operator (up to order ?) of the form U? = exp[?i??/h ? ] + O (?2 ) (14)

where ? is an ?-independent operator, hermitian in the measure ?? . Begin, for and Gab = ηab ; i.e. the simplicity, with a ”minisuperspace” action having V = 1 2 action of a relativistic particle in ?at D-dimensional Minkowski space. Let x′? = x? ? ?x? so that ψ (x′ , τ + ?) = dD x ?? exp[?c? m ?ηab ?xa ?xb ]ψ (x, τ ) (16) Comparing this expression to the corresponding expression for a free non-relativistic particle δij ?xi ?xj ψ (x, t) (17) ψ (x′ , τ + ?) = dD x ?? exp m (?i?h ?) (15)

1 (18) ?i?h ? with the understanding that the ”time”-step ? now has units of action, and that the branch of the square-root is chosen so that the exponent in (16) has a negative real part. This choice does not quite complete the de?nition of U? , as there is still a question of the range of the integral over x. Should this integral range over all possible x, or should it be restricted so that the path-segment ?xa is timelike? To resolve this issue, we will compute separately the contributions from timelike and spacelike intervals. Following the usual steps leading from the path integral to the Schr¨ odinger ′ equation, expand ψ (x, τ ) in a Taylor series around x √ ?] ψ (x′ , τ + ?) = dD x ?? exp[?m ?η?ν ?xν ?x? / ?i?h c? = √ × ψ (x′ , τ ) + = U? ψ (x′ , t) In order that lim U? = 1 the measure must be chosen to be

1 ?? ? = ?→0

motivates us to try

?ψ 1 ?2ψ ν ? x + ?x? ?xν + ... ?x′? 2 ?x′? ?x′ν (19)

(20)

√ ?] dD x exp[?m ?η?ν ?x? ?xν / ?i?h

(21)

Changing variables ?x → x, we then have U? = 1 + 1 2

√ dD x ?x? xν exp[?m ?η?ν x? xν / ?i?h ? ] ?? ?ν + ... √ r 2 η ?ν exp[?m ?η?ν x? xν / ?i?h ? ] ?? ?ν + ... (D ? 1)

(22)

Denote x? = {t, x} and r 2 = x · x. Then, on grounds of relativistic covariance, U? = 1 + = 1+ 1 2 dD x ?

IB 2 1 ? + ... 2(D ? 1) IA IA ≡ IB ≡

(23)

where ? 2 = η ?ν ?? ?ν , and √ dD x exp[?m ?η?ν x? xν / ?i?h ?] √ dD x r 2 exp[?m ?η?ν x? xν / ?i?h ?]

(24)

We ?rst evalute IA ; the second integral IB will follow easily. Starting with √ √ IA = σ dtdr r D?2 exp[?m t2 ? r 2 / ?i?h ?] (25)

where σ=

2π (D?1)/2 ?1 ) Γ( D2

(26)

divide the integral over t into two contributions, one from timelike and one from spacelike paths: ∞ ∞ √ √ dt exp[?m t2 ? r 2 / ?i?h drr D?2 IA = σ ?] r 0 r √ √ ?] (27) dt exp[?m r 2 ? t2 / i?h +

0

√ √ where the branches of the square roots ?i?h ? and i?h ? are taken with positive real parts, to ensure convergence of the integrals. Next ∞ ∞ ∞ √ √ ?] dy 2yδ [y 2 ? (t2 ? r 2 )] IA = σ drr D?2 dt exp[?m t2 ? r 2 / ?i?h 0 0 r r ∞ √ √ dt exp[?m r 2 ? t2 / i?h + dy 2yδ [y 2 ? (r 2 ? t2 )] ?] = σ = σ where F1 (r ) ≡ F2 (r ) ≡

√ y h ?my/ ?i?? e dy √ 2 y + r2 0 √ r y h ?my/ i?? dy √ 2 e r ? y2 0 ∞ 0 ∞ 0 0 ∞ 0

drr D?2

∞ 0

dy √

drr D?2{F1 (r ) + F2 (r )}

√ √ y y ?my/ ?i?? ?my/ i?? h h √ e e + Θ(r ? y ) 2 2 2 2 y +r r ?y

(28)

(29)

It is easy to see that asymptotically, as r → ∞, both F1 (r ) and F2 (r ) go like 1/r . But that implies timelike paths contribution = σ spacelike paths contribution = σ

∞ 0 0 ∞

drr D?2F1 (r ) drr D?2F2 (r )

is divergent is divergent (30)

This means that if we were to restrict the paths to only timelike, or only spacelike paths, then IA (and IB ) would be hopelessly divergent, and the evolution operator U? would be ill-de?ned. The remarkable thing, which we now show, is that the sum of the two contributions is actually ?nite. Let us deform the contour of y -integration for the integral de?ning F1 (r ) in eq. (29). As it stands, it runs along the real axis from 0 to ∞. Deform it to run along the imaginary axis from 0 to ?ir , and then parallel to the real axis from ?ir to ∞. There are no poles or branch cuts in the way, so the deformation is permissible. Then

F1 (r ) = Change variables, y → ?iy F1 (r ) = ?

r 0

?ir 0

dy +

∞ ?ir

dy √

√ y h ?my/ ?i?? e r2 + y 2

(31)

√ y h ?my/ i?? dy √ 2 ? e 2 r ?y

i∞ r

√ y h ?my/ i?? dy √ 2 e 2 r ?y

(32)

Its not hard to see that it is the ?rst integral which causes the divergence of the r -integration. Adding together F1 and F2 , observe that the ?rst integral in (32) exactly cancels F2 , leaving an expression which decays exponentially as r increases F1 + F2 = ?

i∞ r √ y h ?my/ i?? e dy √ 2 2 r ?y

(33)

The contour of this integral runs parallel to the (positive) imaginary axis. Now rotate the contour by 90 degrees, so that it runs along the positive real axis. Again, there are no poles or branch cuts in the way, and the integral is convergent along any contour intermediate between the initial contour, and the 90 degree rotated contour. This gives F1 + F2 = i

√ y ?my/ i?? h e dy √ 2 y ? r2 r mr = irK1 ( √ ) i?h ? ∞

(34)

Inserting this result into (28), one ?nds IA = iσ mr drr D?1K1 ( √ ) 0 i?h ? √ i?h ? D D+1 D?1 = iσ 2D?2 ( ) Γ( )Γ( ) m 2 2

∞ ∞

(35)

It is trivial to repeat all the above steps for the IB integral, and the result is IB = iσ mr drr D+1K1 ( √ ) 0 i?h ? √ i?h ? D+2 D + 1 D?1 = iσ 2D ( ) Γ( + 1)Γ( + 1) m 2 2

(36)

Finally, inserting (35) and (36) into the expression for the evolution operator, eq. (23), we obtain U? = 1 + i?h ? (D + 1) ?ν η ?? ?ν + O (?2 ) 2 2m = exp[?i??/h ? ] + O (?2 ) h ?2 ? = ? 2 (D + 1)η ?ν ?? ?ν 2m

(37)

where

(38)

The operator ? is ?-independent, and clearly Hermitian for inner products < ψ1 |Q|ψ2 >= The evolution operator U?τ ≡ lim(U? )?τ /?

?→0 ? dD x ?? ψ1 (x, τ )Qψ2 (x, τ )

(39)

= exp[?i??τ /h ?]

(40)

is therefore unitary, as in the usual path-integral approach for theories without a time-reparametrization invariance. It should be emphasized that the unitarity of our proposed evolution operator √ ? , and also on summadepends both on the choice of complex constant c? = 1/ ?i?h tion over both timelike and spacelike path segments. A glance at equations (35) and √ (36) shows that the crucial factor of i? in (37) could only be obtained if a 1/ ?i? factor multiplies the action in (13). Furthermore, the ?niteness of the result depends on keeping contributions to the integrand from both timelike and spacelike path-segments; the integral over either contribution separately is divergent. It is easy to generalize from the relativistic particle action to any minisuperspace square-root action of the form (8). First de?ne the modi?ed supermetric Gab ≡ 2V Gab For ?q a small S [q ′ , q ] = ?m

?t 0

(41)

dt

?Gab ?t q a ?t q b (42)

= ?m ?Gab ?q a ?q b The measure is √ D 1 ′ ? ) lim ?? ? (q ) = ( ?h

?→0

√ dD q √ exp[?m ?Gab ?q a ?q b / ?i?h ?] ( ?h ? )D

(43)

Now introduce Riemann normal coordinates ξ a around the point q ′a , which transforms the modi?ed supermetric into the Minkowski metric Gab = ηab at the point q ′ (ξ = 0). Then

1 ′ ?? ? (q ) =

dD ξ det[ 1 |G (q ′ )| IA

√ ? ?q a ?] ] exp[?m ?ηαβ ξ a ξ b / ?i?h b ?ξ (44)

=

√ Inserting this measure, and c? = 1/ ?i?h ? , into eq. (13), we have ψ (q ′ , τ + ?) =

√ 1 dD ?q |G (q ′ + ?q )| exp[?m ?Gab ?q a ?q b / ?i?h ?] IA ×ψ (q ′ + ?q, τ ) √ 1 1 dD ξ (1 ? Rab ξ a ξ b + ...) exp[?m ?ηab ξ a ξ b / ?i?h = ?] IA 6 1 ?2ψ c d ?ψ ξ ξ + O (ξ 3 ) × ψ (q ′ , τ ) + c ξ c + ?ξ 2 ?ξ c ?ξ d 1 IB 1 2 1 = 1+ { ? ? R + O (?2 )} ψ (q ′ (ξ ), τ ) (D ? 1) IA 2 6 (D + 1) (D + 1) ab ? 2 η ? i?h ? R ψ (ξ, τ ) (45) = 1 + i?h ? 2 a b 2m ?ξ ?ξ 6m2

where R is the curvature scalar formed from the metric Gab . Transforming back from Riemann normal coordinates, we have (D + 1) 1 ? ? (D + 1) ? ψ (q, τ + ?) = ?1 + i?h ? |G|G ab b ? i?h ? R ψ (q, τ ) 2 a 2m ?q 6m2 |G| ?q = = U? ψ (q, τ ) exp[?i??/h ? ] + O (?2 ) ψ (q, τ ) (46)

? ?

As in the relativistic particle case, exp[?i??/h ? ] is a unitary operator, where ? = ?h ?2 ? (D + 1) 1 ? (D + 1) |G|G ab b + h ?2 R 2 a 2m ?q 6m2 |G| ?q (47)

is obviously Hermitian in the measure ?? of eq. (44). Taking the ? → 0 limit, the wavefunction ψ (q, τ ) satis?es a Schr¨ odinger equation ih ? ?τ ψ (q, τ ) = ?ψ (q, τ ) (48)

The Schr¨ odinger evolution equations (47) and (48) for time reparametrization invariant theories have been obtained by us previously, in refs. [4] and [5], from a transfer matrix approach. The transfer matrix involves integration over a purely real integrand, but in our case the cost was not simply a Wick rotation of the evolution parameter τ , but also a rather unnatural rotation of signature of the modi?ed supermetric Gab from Lorentzian to Euclidean. This rotation then had to be undone in postulating the Schr¨ odinger equation (48). We have now seen that this supermetric signature rotation can be avoided, and unitary Schr¨ odinger evolution is derived directly. In refs. [4] and [5] the correspondence of this evolution to the usual classical dynamics was also discussed. In the interest of completeness we will

brie?y review this correspondence here, and refer the reader to the cited references for further details. The classical quantity ?cl corresponding to the operator ? is obtained by replacing derivatives with c-number momenta ?cl [q a , pa ] = lim ?[q a , ?ih ?

h→0 ?

? → pa ] ?q a

(49)

which gives ?cl = (D + 1) The Poisson bracket evolution equation

1 Gab pa pb 2m

mV

(50)

?τ Q = {Q, ?cl } , ?cl = ?E

(51)

is easily checked to be equivalent, up to a time-reparametrization, to the standard brackets ?t Q = {Q, N HE } , HE = 0 where HE = (52)

√ 1 √ Gab pa pb + E mV (53) 2m E The parameter E is classically irrelevant, in the sense that it drops out of the EulerLagrange equations; the square-root action corresponding to HE is √ SE = ? E m dt ?2V Gab ?t q a ?t q b (54)

Since E only appears as a parameter multiplying the action, the fact that it drops out of the Euler-Lagrange equations is obvious. The same can be said for the mass of a relativistic particle in free-fall, the tension of the Nambu string, or Newton’s constant in pure gravity. None of these parameters appears in the equations of motion at the classical level. Because of the classical equivalence of the Poisson bracket equations (51) and (52), it is clear that the Schr¨ odinger evolution (48) will obey an appropriate Ehrenfest principle, with certain quantum corrections due to the measure. The general solution ψ (q, τ ) can be expanded in terms of stationary states ψ (q, τ ) =

E h φE (q )eiE τ /?

(55)

where ?φE = ?E φE (56)

This τ -independent equation can be rewritten in the form

?

h ? 2 (D + 1) h ? 2 (D + 1) V ? ?1 ab ? ?? V + V R + V ? φ E (q ) = 0 |G| G a E 4m2 ?q b E 6m2 |G| ?q

?

(57)

which is a Wheeler-DeWitt equation with a particular choice of operator-ordering and a (classically irrelevant) parameter E , which can be absorbed into a rede?nition of either m or h ?. In the standard Dirac canonical quantization of actions of the form (5), the physical states must satisy a Wheeler-DeWitt equation of the form H φ (q ) = = 0 ?h ? 2 ab ? 2 ”G ” + mV φ(q ) 2m ?q a ?q b (58)

where the quotation marks indicate an operator-ordering ambiguity. However, mul√ tiplying the action by an arbitrary constant E converts this constraint to H E φ (q ) = = 0 √ ?2 ?h ?2 √ ”Gab a b ” + E mV φ(q ) ?q ?q 2m E

(59)

Because E is irrelevant at the classical level, there is no overriding reason that it should be a ?xed parameter at the quantum level. In essence, our approach enlarges the space of physical states as the Hilbert space spanned by all states satisfying H E φ E (q ) = 0 (60)

and it is this enlargement of the space of states which enables us to obtain nonstationary states ψ (q, τ ). Moreover, our approach ?xes the operator-ordering, as seen in eq. (57), at least for the minisuperspace theories. Further discussion of these points may be found in ref. [5]. Returning to eq. (13), the path integral for square-root actions is now de?ned as the limit ψ (qf , τ0 + ?τ ) = Dq (τ0 ≤ τ < τ0 + ?τ ) ec0 S [q(τ )] ψ (q0 , τ0 )

?τ /??1 n=0 ?τ /? ?→0

= lim

= lim(U? )

?→0

ψ (qf , τ0 )

1 dD qn ?? (qn ) exp ? √ ?i?h ?

?

?τ /??1 m=0

S [qm+1 , qm ]? ψ (q0 , τ0 )

?

(61)

where ?τ = τf ? τ0 . We see that the time parameter emerges from a regularization of the path-integral measure: at ?xed ?, a regularized path between the initial point

q0 and the ?nal point qf consists of np = ?τ /? path segments, each segment being a classical trajectory between intermediate points qn and qn+1 . The evolution parameter is therefore a measure of the number of independent con?gurations (points) np in the path joining q0 to qf , multiplied by the regularization interval, i.e. ?τ = np ? (62)

This is a quantum-mechanical time variable with no direct connection to, e.g., the proper time lapse. Nor is it an ”intrinsic” time variable; all dynamical degrees of freedom are treated on the same footing and none is singled out as an evolution variable. Our evolution parameter is here identi?ed as proportional to the number of ”quantum steps” taken by the system in evolving from the initial to the ?nal con?guration. In this formulation the Green’s functions are transitive, and the evolution of states is unitary.

3

The BSW Action

As in the minisuperspace case, the square-root form of the full gravitational action is derived from the ?rst-order ADM action by solving for the lapse function. The ADM action for pure gravity is S = d4 x [pij ?t gij ? N H ? Ni Hi ] 1√ 3 g R κ2

H = κ2 Gijklpij pkl ?

Hi = ?2pik ;k 1 Gijkl = √ (gik gjl + gil gjk ? gij gkl ) 2 g

(63)

where gij is the metric of a 3-manifold and 3 R is the corresponding scalar curvature. The momentum is related to the time-derivative of the metric by ?t gij = N ?H ?pij = 2κ2 NGijkl pkl + Ni;j + Nj ;i 1 = Gijkl (?t gkl ? 2N(k;l) ) 2κ2 N

=? pij

(64)

Solving the Hamiltonian constraint for the lapse function 1√ 3 1 ijkl g R G ( ? g ? 2 N )( ? g ? 2 N ) ? t ij ( i ; j ) t kl ( k ; l ) 4κ2 N 2 κ2 1/2 1 ijkl =? N = G (?t gij ? 2N(i;j ) )(?t gkl ? 2N(k;l) ) √ 4 g 3R 0 =

(65)

and replacing the momenta in (63) by the expression (64), with lapse (65), gives the Baierlein-Sharp-Wheeler action [6] 1 √ 3 g RGijnm (?t gij ? 2N(i;j ) )(?t gnm ? 2N(n;m) ) d4 x (66) 2 κ Before quantizing, it is convenient to ?x the coordinate system by choosing shift functions Ni = 0. Then the corresponding supermomentum contraints δS/δNi = Hi = 0 are to be imposed as operator constraints on the space of physical states. It is straightforward to extend the BSW action to include non-gravitational bosonic ?elds. To compress indices somewhat, we introduce the notation SBSW = ? {a = 1 ? 6} ? {(i, j ), i ≤ j } q a (x) ? gij (x) pij (x) pa (x) ? 2pij (x) (i = j ) (i < j ) (67)

Gab (x) ? Gijnm (x)

and the non-gravitational ?elds are represented by q a (x) with indices a > 6. It is convenient to rescale all non-gravitational ?elds by an appropriate power of κ so that all ?elds, and all components of the supermetric, are dimensionless. The action is d4 x[pa ?t q a ? N H ? Ni Hi ] √ H = κ2 Gab pa pb + gU S = where √ gU = ?

(68)

1√ 3 g R + non-gravitational contributions (69) κ2 Setting the shift functions to zero and repeating the above steps of solving for the lapse, gives again a square-root action 1 √ d4 x ? gUGab ?t q a ?t q b (70) κ The next step is to construct the evolution operator U? for the BSW action in the path-integral approach, following the procedure of the last section. The evolution operator is de?ned by S=? Ψ[q ′ (x), τ + ?] = + Dq (x) ?(q )e?S/

√ ?i?? h

Ψ[q ′ (x)] +

d3 x

δΨ ?q a (x) a δq (x)

1 δ2Ψ d3 xd3 y ?q a (x)?q b (y ) + ... 2 δq a (x)δq b (y ) = Ψ(q ′ , τ ) + [T0 + T1 + T2 ] + O (?2 ) = U? Ψ(q ′ , τ ) (71)

where the Tn represent terms with n functional derivatives of Ψ and one power of ?, and ?S = ? 1 d3 x κ = q a ? q ′a √ ? gUGab ?q a ?q b (72)

?q a

In order to obtain U? , we need to evaluate < ?q a (x1 )?q b (x2 ) > = D (?q ) (?)0 ?q a (x1 )?q b (x2 ) × exp ? where Gab is the modi?ed supermetric 1 Gab ≡ √ UGab g (74) 1 κ √ √ d3 x( g )0 ?(Gab )0 ?q a ?q b / ?i?h ? (73)

and ()0 indicates that the quantity in parenthesis is evaluated at q = q ′ . Clearly, < ?q ?q > is a highly singular quantity, and is only well-de?ned in the context of a regularization procedure. In the absence of a non-perturbative regulator which preserves the exact di?eomorphism invariance, we work with a naive lattice regulator in which the continuous degrees of freedom labeled by x are replaced by a discrete set, labeled by n, associated with regions of volume vn . We have in mind, e.g., a Regge-style discretization of a continuous 4-manifold into a ?xed number Np of simplices of varying volume. As Np → ∞, the choice of vn is of course required to be irrelevant in computing the evolution operator, as long as the regions’ volumes vn → 0 in this limit. As we will see, this requirement is not satis?ed trivially or automatically. We take the naive lattice-continuum correspondences to be ?q a (x) ? ?q a (n) √ d x g ?

3 Np

vn

n=1

δ ? a δq (x) Dq ?

n

δ a δq (n) d q (n)

D

R

≡

g (n) vn

? ?q a (n) (75)

With such a discretization, we have < ?q a (n)?q b (m) > =

j

dD q (j ) ??q a (n)?q b (m) 1 κ √ ? vk ?Gab ?q a (k )?q b (k )/ ?i?h (76)

× exp ?

k

The supermetric Gab for the discretized degrees of freedom {q a (n)} still has Lorentzian signature, and we can follow the steps of the last section in integrating over the q a at each n. The result is < ?q a (n)?q b (m) >= i?h ? (D + 1)κ2 G ab and we ?nd for the T2 term T2 = i?h ? (D + 1) 2 κ 2 (D + 1) 2 κ = i?h ? 2 (D + 1) 2 κ = i?h ? 2

n

1 δ 2 mn vn

(77)

n

n

?2 1 Ψ 2 ?q a (n)?q b (n) vn ?2 √ 1 ab 1 g G 2 a Ψ U vn ?q (n)?q b (n) δ 1 δ √ Gab a b gU δq (n) R δq (n) G ab

Ψ

R

(78)

The term in braces has a simple continuum limit and, if this term were weighted by a volume factor vn , then the continuum limit would be simply T2 = i?h ? (D + 1) 2 κ 2 d3 x U ?1 Gab δ2 Ψ δq a δq b (wrong) (79)

There is, however, no such vn weighting factor in the sum, which means that the contribution of each term at each position n δ 1 √ Gab a gU δq (n) δ

R

δq b (n)

Ψ

R

(80)

is weighted equally, regardless of the cell or simplex volume vn . As a consequence, even in the vn → 0 limit, the ?nal answer for the state evolution would seem to depend on the distribution of volumes {vn }. Such regularization dependence never arises in ordinary quantum ?eld theory. There may be other regularization issues, such as renormalization and anomalies, but certainly one doesn’t encounter this kind of dependence on the distribution of cell volumes in computing the naive continuum limit of the Hamiltonian operator. Since the problem doesn’t arise in ordinary quantum ?eld theory, why does it come up here? The reason, of course, can be traced back to the square-root form of the 2 action, which gives a factor of 1/vn , rather than a factor of 1/vn , in the correlator (77). The additional power of 1/vn is the source of the (apparent) di?culty. There is only one way out, if the evolution operator is not to depend on the {vn } distribution: we must impose a constraint on the physical states Ψ such that the term (80) is independent of the discretized position label n, at least in the vn → 0 limit. In that case, we have T2 = i?h ? δ (D + 1)Np 1 2 ab √ κG a 2 gU δq (n) δ

R

δq a (n)

Ψ

R

(any n)

(81)

The other terms T0 and T1 are operator-ordering contributions which, in the absence of an exact di?eomorphism-invariant regulator, will not be considered further (D + 1)Np into a rede?nition of ?, and taking the here. Now absorbing the factor 1 2 continuum limit, we arrive at 1 δ2 ih ? √ κ2 Gab a b Ψ = ?τ Ψ gU δq δq Expanding Ψ(q, τ ) =

E h aE eiE τ /? ΦE (q )

(all x)

(82)

(83)

eq. (82) requires that for each stationary state δ2 √ κ2 1 √ HE ΦE = ?h ? 2 Gab a b + gU ΦE = 0 E δq δq E (84)

which is simply the Wheeler-DeWitt equation (up to operator-ordering contributions), with an e?ective value of Planck’s constant rescaled by h ? h ? ef f = √ E (85)

Moreover, the Wheeler-DeWitt equation is consistent with, and in fact implies (via the Moncrief-Teitelboim interconnection theorem [7]), the supermomentum constraints Hi ΦE = 0 (86) which are needed to compensate the gauge choice Ni = 0. Thus, each stationary state ΦE satis?es the usual constraint algebra of general relativity, with a rescaled value of Planck’s constant. The Hilbert space of all physical states is spanned by the stationary states, with all possible values of E . Finally, multiplying both sides √ of (82) by N gU , where N is an arbitrary function, integrating over space, and applying the supermomentum constraint (86), we obtain the equation of motion ih ? ?τ Ψ = = 1 κ2 √ d3 x′ gNU d3 x NGab (?h ?2 δ2 ) Ψ δq a δq b

1 mP = ?Ψ where ?= 1 mP

? 2 Gab d3 x ?h ? 2 Nκ

δ2 i + Ni Hx Ψ δq a δq b (87)

? κ2 Gab d3 x ?h ? 2N

δ2 i + Ni Hx δq a δq b

(88)

and ? (x) ≡ mP N

d3 x′

N (x) √ gNU (q )

(89)

with mP an arbitrary parameter of dimension of mass. The evolution equation (87) was obtained in ref. [5] by a transfer matrix approach, and shown to correspond to the usual classical evolution via the Ehrenfest principle. Here we have instead used the path integral to obtain a unitary evolution operator, as in the real-time Feynman approach, and avoided the signature rotation of the supermetric which was required in deriving the transfer matrix. The main point of this section is that, in performing the ”real-time” path-integral of the BSW action Ψ[qf (x), τ0 + ?τ ] = Dq (x, τ ) ec0 S [q(τ )] Ψ[q0 (x), τ0 ]

?τ /??1 n=0 a Dqn (x)?? [qn ] exp ? √

= lim

?→0

?

= lim(U? )?τ /? Ψ(qf (x), τ0 )

h = e?i??τ /? Ψ[qf (x), τ0 ] ?→0

×Ψ[q0 (x), τ0 ]

1 ?i?h ?

?τ /??1 m=1

S [qm+1 (x), qm (x)]?

?

(90)

it is necessary, as in the minisuperspace case, to integrate over all possible paths, including those for which the lapse function 1 N (x) = ? 2 √ Gab ?t q a ?t q b 4κ gU

1/2

(91)

is imaginary. Real-valued lapse functions correspond to Lorentzian 4-manifolds, imaginary values correspond to Euclidean signature. If the paths are restricted to real N (x) only, then we ?nd that due to the Lorentzian signature of the supermetric, the integrals in eq. (76) are singular despite the regularization. The conclusion is that in order to obtain a unitary evolution of states, we are required to sum over 4-manifolds of both Lorentzian and Euclidean signature, and in general over manifolds which may be Lorentzian in some regions, and Euclidean in others. This raises the obvious question of why spacetime seems to have Lorentzian signature, rather than Euclidean or mixed signature. The question ”why is spacetime Lorentzian?” can be raised already at the level of classical general relativity. Einstein’s equations themselves do not specify a choice of metric signature; there are Lorentzian solutions to these equations, and there are Riemannian solutions. Recently, solutions to the Einstein equations in

which part of the manifold is Riemannian (Euclidean signature) and the rest is Lorentzian have been studied [13]; it is conceivable that solutions of this kind are relevant to the very early Universe. In any case, the signature of a manifold solving the Einstein equations is determined in general from initial conditions {gij , pij } satisfying the appropriate constraints. A given initial 3-manifold may trace out either a Lorentzian or Riemannian 4-manifold, depending on the initial choice of conjugate momenta. The dependence of lapse on initial conditions applies also to the quantum theory. The general solution of the evolution equation (48) for the ”relativistic particle” example, with ? given in eq. (38), is ψ (x? , τ ) = It is easy to see that < x? >=< x? >0 + (D + 1) < p? > τ 2 m (93) d4 p f (p) exp ? i h ? (D + 1) 2 p τ + p? x? 2m2 (92)

(recall that τ has units of action). So long as f (p) = 0 for p2 > 0, the expectation value of position follows a timelike path. We would expect the same situation in quantum gravity, for the same reason, namely, the Ehrenfest principle. If the initial ”wavefunction of the Universe” Ψ[q a (x), τ0 ] has expectation values which are peaked around some (equivalence class of) con?gurations and momenta {q, p}0 , then the wavefunction tends to remain peaked in the neighborhood of a classical manifold which solves the Einstein equations for this initial data. Thus, despite the fact that the path integral sums over Lorentzian and Euclidean manifolds, the probability density can still be sharply peaked at one or the other signature. Obviously these remarks do not answer the question ”why is spacetime Lorentzian?”, but only replace it with another question about initial conditions. For an attempt to explain the preference for Lorentzian signature (in the context of non-timeparametrized theories) from an analysis of an e?ective ”signature potential”, see ref. [8].

4

Quantum Theory in Curved Spacetime

Associated with the problem of time in quantum gravity is a ”problem of state.” Let us return, for a moment, to the standard formulation of canonical quantum gravity, which in our language is a restriction to a single value of E , and let H be the Wheeler-DeWitt Hamiltonian. Suppose a physical state Ψ is an eigenstate of

an observable Q; this means that QΨ must also be a physical state. But then H (QΨ) = [H, Q]Ψ = 0 (94)

which is not true, in general, unless [H, Q] vanishes weakly. It is then problematic to construct physical states which are approximate eigenstates of, e.g., 3-geometry, or the position of the hands of a clock. In this section we show how to construct physical states which are sharply peaked around a given 3-geometry and extrinsic curvature. Treating the metric degrees of freedom semiclassically, the dynamics of the other degrees of freedom approximates the standard quantum theory on a curved background. Of course, the WKB treatment can be extended to any other degrees of freedom (such as the hands of a clock) which behave more or less classically. We recall that our path integral leads, in the end, to the following solution for the evolution of physical states: Ψ[q, τ ] =

E ,α h c[E , α]ΦE ,α [q ]eiE τ /?

(95)

where ?ΦE ,α = ?E ΦE ,α (96) and where the subscript α is meant to distinguish between di?erent solutions of (96). As discussed above, eq. (96) is a one-parameter (E ) class of Wheeler-DeWitt equations h ? 2 2 ab δ 2 √ + gU ΦE ,α = 0 ? κG (97) a b E δq δq

each of which can be treated by WKB methods. To get the quantum-theory-incurved-spacetime limit, we follow the approach of Banks [9], treating the metric semiclassically, and expanding in powers of κ2 (back-reaction of matter on metric will be ignored; it can presumably be dealt with following the approach of ref. [10]). Thus, write (96) in the form ? δ2 1√ h ?2 2 E κ Gijkl + 2 gR + Hm ΦE ,α = 0 E δgij δgkl κ (98)

E where Hm is the Hamiltonian density for the non-gravitational ?elds, denoted φ. We then make the WKB ansatz √ E ? ρE (99) ΦE ,g = exp i E S [g, g ]/κ2h V V [g ]ψm [φ]

where S [g, g] is Hamilton’s principal function (the action of a 4-manifold solving the Einstein equations, bounded by the three manifolds with metric g ij and gij ); it

satis?es the Einstein-Hamilton-Jacobi equation in both arguments δS δS √ ? gR = 0 δgij δgkl √ δS δS ? gR[g ] = 0 Gijkl [g ] δg ij δg kl Gijkl

(100)

E The functional ρE V V [g ] is the Van-Vleck determinant, while ψm is a solution of the Tomonaga-Schwinger equation for quantum theory on a curved spacetime background E δψm E E = Hm ψm (101) ih ? ef f δT (x; g, g)

h ? (102) h ? ef f = √ E is the e?ective value of Planck’s constant, and T (x; g, g) is a functional of the background spacetime de?ned by δ δS δ = 2Gijkl δT δgij δgkl Up to this point, we have simply repeated the analysis of ref. [9] However, the semiclassical approach to recovering ordinary quantum ?eld theory, as outlined above, is subject to the following objection: Although the part of the wavefunction involving the non-gravitational ?elds obeys a Tomonaga-Schwinger equation, the metric gij , on which the ”many-?ngered” time parameter T (x; g, g) depends, is still a dynamical degree of freedom, and there is no physical state satisfying the Wheeler-DeWitt equation which has a probability distribution peaked at a particular 3-geometry gij , i.e the wavefunction is not peaked on any particular time-slice of a 4-manifold. In fact, the squared-modulus of the leading term in the WKB approach, i.e. √ ? exp i E S [g, g ]/κ2h

2

where

(103)

=1

(104)

has no dependence on gij at all. The best one can do in the standard formulation (that is, using only a single value of E ) is to superimpose WKB solutions ΦE ,F [g ] = = √ E Dg ij f [g ij ] exp i{ E S [g, g]/κ2 + θ[g ]}/h ? } ρE V V ψm Dg ij F [g ij ]ΦE ,g

(105)

where f [g ] is a real functional peaked (modulo di?eomorphisms) at a particular 3-geometry g0ij , and we de?ne pij 0 ≡ δθ δg ij (106)

| g =g 0

where θ[g ] is the phase of the smearing functional F [g ]. As shown many years ago by Gerlach [11], this superposition is still not peaked at any one 3-geometry, but rather on all three-geometries which are spacelike slices of a certain 4-manifold, satisfying Einstein’s equations with initial data {g0ij , pij 0 }. Thus there is no physical state, and no subspace of physical states, which would correspond to an eigenstate of a non-stationary observable (such as the three-geometry, or the ?elds on a given three-geometry). It is at this point that we make use of the freedom, inherent in our formulation, to superimpose states of di?erent E , and write Ψ[gij , φ, τ ] = √ i E {E τ + E S [g, g]/κ2 } ρE V V ψm [φ, T (x; g, g)] h ? √ i E0 ≈ ψm [φ, T (x; g, g0)] dE Dg F [g, E ] exp {E τ + E S [g, g]/κ2 } ρE VV h ? (g ,p ) E0 = ψm [φ, T (x; g, g0)]ΦF 0 0 [gij , τ ] (107) dE Dg F [g, E ] exp

where it is assumed that F [g, E ] is sharply peaked around E = E0 , g ij = g0ij , and pij 0 is de?ned as in eq. (106) above. The ψm factor can be pulled outside the integral, on the grounds that its variation with E and g is much less than that of the smearing √ function F [g, E ], and the exp[i E S/κ2 h ? ] factor. Now consider the leading WKB term ΦF 0

(g ,p0 )

[gij , τ ] =

dE Dgij f [E , g ] exp

√ i E τ + E S [g, g ]/κ2 + θ[g ] h ?

ρE VV

(108)

This wavefunction will be peaked at con?gurations gij where the phase in the integrand is stationary, with respect to small variations in g ij and E around g0ij and E0 , respectively. In other words, the wavefunction is peaked at metrics gij , at time τ , such that τ = ? pij 0 S [g, g0] E0 √ E 0 δS [g, g] = ? 2 κ δg ij 2κ2 1 √

(109)

| g =g 0

The second of these two equations is satis?ed by the metric gij of any time-slice of a 4-manifold, satisfying Einstein’s equations with initial data {g0ij , pij 0 }. The

?rst equation requires that the action of the 4-manifold between the initial slice g0ij and the given slice gij is proportional to the time-parameter τ . Now consider a foliation of the given 4-manifold parametrized by some variable x0 , with gij = g0ij at x0 = 0. Hamilton’s principal function S [g, g0] is monotonic in x0 , which means that S [g, g0] = 0 only for gij = g0ij . It follows that, at τ = 0, eq. (109) gives us 0 = S [g, g0] =? gij = g0ij (modulo di?eomorphisms)

(g ,p )

(110)

As a consequence, at τ = 0, the wavefunction ΦF 0 0 [g, τ = 0] is peaked at gij = g0ij (modulo di?eomorphisms). Thus, from the de?nition of the many-?ngered time variable, where T (x; g0 , g0 ) = 0, Ψ[gij , φ, τ = 0] = ΦF 0 ≈ ΦF 0

(g ,p0 ) E0 [g, τ = 0] × ψm [φ, T (x; g, g0)]

(g ,p0 )

E0 [g, τ = 0] × ψm [φ, T = 0]

(111)

The importance of eq. (111) is that there exists, in our formulation, a class of states where the metric (and extrinsic curvature) is sharply peaked around a given geometry g0ij (and pij 0 ), and where the state factorizes into a wavefunction (ΦF ) suppressing ?uctuations away from the given 3-geometry, and a wavefunction (ψm ) describing the state of the non-gravitational ?elds on that 3-geometry. Such states can be fairly described as eigenstates of non-stationary observables; these eigenstates are impossible to construct, as physical states, in the standard formulation of canonical quantum gravity. Finally, we consider transition probabilities. Take an initial state of the form Ψin [gij , φ] = ΦF 0 and a ?nal state of similar form Ψf [gij , φ] = ΦF ′0

(g ′ ,p′ 0)

0 [g, 0] × ψ ′ m [φ, 0]

(g ,p0 )

E0 [g, 0] × ψm [φ, 0]

(112)

E

(113)

′ where the smearing function F ′ is peaked around some time slice (g0 , p′0 ) of the classical 4-geometry speci?ed by the initial data (g0 , p0 ). The transition probability for Ψin → Ψf after a time τ is given by the factorized expression h Pin→f (τ ) = | < Ψf |e?i?τ /? |Ψin > |2

= | < Ψf |Ψin (τ ) > |2 = | < ΦF ′0

E (g ′ ,p′ 0)

0 E0 ′ [φ, 0]|ψm [φ, T (x; g0 , g0 )] > |2 × | < ψ′m

[g, 0]|ΦF 0

(g ,p0 )

[g, τ ] > |2

(114)

The ?rst of these factors | < ΦF ′0

(g ′ ,p′ 0)

[g, 0]|ΦF 0

(g ,p0 )

[g, τ ] > |2

(115)

′ , p′0 ) (up gives the probability, after a time τ , to be on the time-slice described by (g0 to a certain uncertainty, speci?ed by the smearing function F ′ ). The second factor

0 E0 ′ | < ψ′m [φ, 0]|ψm [φ, T (x; g0 , g0 )] > |2

E

(116)

is the quantum-?eld-theory-in-curved-spacetime result; it gives the probability for a transition from an initial state ψm of quantum ?elds on the time-slice g0 , to the ′ ′ state ψm on the later time-slice g0 . Both the initial and ?nal 3-manifolds are timeslices of the same 4-manifold speci?ed by the initial data {g0 , p0 }, and the state ψm [φ, T ] evolves according to the Tomonaga-Schwinger equation (101). In this way, we see how approximate eigenstates of geometry and extrinsic curvature may be constructed, and how the standard formalism of quantum ?eld theory in curved spacetime emerges. We will not attempt to go further and discuss the problem of measurement in this context, apart from noting that any of the standard ”realistic” approaches that have been applied to non-relativistic quantum mechanics, e.g. many-universes, decoherence, or Bohm’s theory, can be applied in our formulation as well.

5

Inclusion of Fermions

We have so far assumed that the canonical momenta pa appear quadratically in the Hamiltonian, with indices contracted by the supermetric. The Hamiltonian of a set of Dirac ?elds, on the other hand, is linear in the fermionic momenta, and it is not immediately obvious how such ?elds are incorporated into our approach. In our previous work [5] we found two independent methods for determining the ? operator. The ”undetermined constant” method was based on the trivial observation that the actions S and S ′ = const. × S are equivalent at the classical level; this leads to the fact that the ratio of the kinetic and potential terms of the Hamiltonian (which is the ? functional), is indeterminate at the classical level. The second method, leading to the same quantum theory, is the transfer matrix method, whose ”real-time” or Feynman version was presented in the preceding sections. We will now apply both methods to obtain the ? operator for gravity coupled to a Dirac ?eld. The action for the Einstein-Dirac system is expressed in terms of the fermion ?eld ψ and tetrad ?eld ea? as SED = d4 x det(e) 4 R + iψ (ea? γ a D? ? m)ψ (117)

where D? is the usual covariant spinor derivative. The extension of the canonical ADM formalism to this system was worked out in ref. [12], for the ”time-gauge” e0i = 0 (i = 1, 2, 3) (118)

In this gauge, the Einstein-Dirac action expressed in terms of canonical momenta has the form S= ˙ ? (N H + Ni Hi + ?ij Mij ) d 4 x pa q ˙a + πψ ψ √ gU + Hψ (119)

where the q a are the triad ?elds ec k (x), and H = κ2 Gab pa pb +

a Hψ = πψ Kψ = πψ γ 0 [ei a γ Di ? m]ψ

(120)

The ?rst-class constraints are H = Hi = Mij = 0 (121)

where the supermomenta Hi and the generators of local frame rotations Mij are linear in the momenta. In addition there are 2nd-class constraints, some of which are associated with the time-gauge condition (118), and also which relate πψ to ψ :1 √ πψ = i gψγ 0 (122)

The 2nd-class constraints are handled, according to the Dirac procedure, by replacing Poisson brackets by Dirac brackets. The explicit form of all constraints in terms of the canonical variables, and other details, may be found in ref. [12]. ′ Now consider an alternative action SED which di?ers from SED only by a multiplicative constant √ ′ = E SED SED (123)

′ Obviously, the equations of motion derived from SED are identical to those derived from SED . The constant E is therefore irrelevant at the classical level. In going to the canonical formulation, however, we ?nd that

S′ = HE

˙ ? (N HE + Ni Hi + ?ij Mij d 4 x pa q ˙a + πψ ψ √ √ 1 = √ κ2 Gab pa pb + Hψ + E gU E √ √ πψ = i E gψγ 0 (124)

with 2nd class constraints enforcing

1

We take right derivatives with respect to ψ .

De?ne HE ≡ d3 x (N HE + Ni Hi + ?ij Mij ) (125) and consider a ?eld con?guration {ea i (x, t), ψ (x, t), ψ (x, t)} which solves the Hamiltonian equations of motion derived from H E , for some given value of E . Then it is clear that this con?guration is a solution for any other value of E , since the classical orbits in con?guration space (i.e. solutions of the Euler-Lagrange equations) are independent of E . In general then, the Dirac bracket equation of motion 2 ?t F = {F, H E }D supplemented by the ?rst class constraints HE = 0 for any N, Ni , ?ij (127) (126)

=? HE = Hi = Mij = 0

generates a set of orbits in con?guration space which is independent of E . In this sense E is ”classically irrelevant.” Now observe that the constraint H E = 0 can be written ? = ?E where ? is de?ned implicitly by ?= √ 2 ab ??Hψ ) 1 N ( κ G p p + a b + (Ni Hi + ?ij Mij ) d3 x √ 3 ′ d x gNU mp (129) for any N, Ni , ?ij (128)

and where mp is an arbitrary parameter with dimensions of mass. From this de?nition, it is straightforward to show that, for any functional F = F [q, p, ψ, πψ ], the Poisson bracket with ? is related to the corresponding Poisson bracket with H E via √ ?1 3 E mp N d x N H 1 ψ mp {F, ?} = d3 x 1 + √ √ {F, HE } √ 3 3 d x3 N gU 2 E d x2 N gU +Ni {F, Hi } + ?ij {F, Mij } = {F, H E }N →N where

3 ψ ? (N ) ≡ 1 + √ d x1 N H√ N 3 2 E d x2 N gU ?1 ?

(130) √

E mp N √ 3 d x3 N gU

(131)

2

Of course, {F, H E }D ≈ {F, H E }, since HE = 0 is a ?rst-class constraint.

Eq. (130) is derived by simply carrying out the functional derivatives contained in the Poisson brackets shown, and applying the constraint (128). Then, since the Dirac bracket {F, ?}D is linear in Poisson brackets {.., ?}, eq. (130) implies

N →N mp {F, ?}D = {F, H E }D ?

(132)

De?ning τ = mp t, this demonstrates the equivalence of ?t O = {O, H E }D ? ?τ O = {O, ?}D (133)

? . Note that N (x) and const. × up to a time-reparametrization, expressed by N → N ?. N (x) have the same N We now quantize by replacing Dirac brackets with commutators (in the case of bosonic ?elds), and anticommutators (in the case of fermionic ?elds). Time evolution of states is given by the Schr¨ odinger equation ? τ ] = ?Ψ[q, ψ, ? τ] ih ? ?τ Ψ[q, ψ, with the general solution ? τ] = Ψ[q, ψ,

E h ?]eiE τ /? aE ΦE [q, ψ

(134)

(135)

where

1 ? ≡ g4 ψ ψ

(136)

and ΦE satis?es a Wheeler-DeWitt equation 1 h ?2 h ? δ δ2 ? + √gU ΦE = 0 √ HE ΦE = ? κ2 ”Gab a b ” + i √ Kψ ? E δq δq E E δψ (137)

where the operator-ordering remains to be speci?ed. Note that, as in the purely bosonic case, the classically irrelevant constant E can be absorbed into a rede?nition of h ? h ? (138) h ? ef f = √ E This concludes the ?rst, ”undetermined constant” method for ?nding the ? operator. Next we apply the path-integral approach, following as closely as possible the procedure of the previous section for the purely bosonic case. Since the generalized BSW action for gravity + fermions will contain a factor of Hψ inside the square-root, our strategy will be to expand the path-integrand in powers of Hψ , and evaluate the relevant expressions to some ?nite order (?rst order, in this article). These expressions can then be compared, order by order in Hψ , with results of the ”undetermined constant” method above.

We again set Ni = 0, and also ?ij = 0, which is to be compensated by imposing the corresponding physical state constraints Hi Ψ = 0 Mij Ψ = 0 (139)

Solving for the bosonic momenta in terms of the time-derivatives pa = 1 Gab ?t q b 2κ2 N (140)

inserting into the Hamiltonian constraint 1 4κ2 N 2 Gab ?t q a ?t q b + √ gU + Hψ = 0 (141)

and solving for the lapse function 1 1 N= ? √ Gab ?q a ?q b 2κ ( gU + Hψ ) we arrive at a square-root action S= 1 √ √ d4 x i gψγ 0 ?t ψ ? ?( gU + Hψ )Gab ?t q a ?t q b κ √ d4 x i gψγ 0 ?t ψ ? τ] Ψ[q, ψ, (144) (143)

1/2

(142)

The corresponding path-integral is ?′ , τ + ?τ ] = Ψ[q ′ , ψ ? where DqDψDψ ?(q, ψ, ψ) exp c0 1 κ √ ?( gU + Hψ )Gab ?t q a ?t q b √ Hψ = i gψγ 0 Kψ

(145)

and c0 represents the ? → 0 continuum limit of the regularization-dependent con√ stant c? ∝ 1/ ?i?h ? . Now expand the exponential to ?rst order in Hψ ?′ , τ + ?τ ] Ψ[q ′ , ψ = DqDψDψ ?(q, ψ, ψ) × 1? c0 κ Hψ d3 xdτ √ 2 gU √ ? gUGab ?τ q a ?τ q b ? τ] Ψ[q, ψ, (146)

× exp c0 S0 + where S0 is the bosonic action S0 = ? 1 κ

√ d3 xdτ i gψγ 0 ?τ ψ

√ d3 xdτ ? gUGab ?τ q a ?τ q b

(147)

We now regularize the path-integral according to the lattice prescription (75). The bosonic part of this path-integral, based on the action S0 , leads to the operator evolution ih ? ?τ Ψ = AΨ (148) where (D + 1) A = 2 (D + 1) = 2 and we have de?ned pL ? a (n) ≡ ?ih √

n

n

g 2 ab 1 ?2 ? 2) a κ G 2 (?h U vn ?q (n)?q b (n) √ g 2 1 ab L κ 2 G pa (n)pL b (n) U vn ? ?q a (n)

(149)

(150)

Then to zeroth-order in Hψ , we can identify the derivatives ?τ q in the term proportional to Hψ as proportional to the bosonic momentum operators, according to ?τ q a (n) = ?A ?pL a (n)

2

= (D + 1)κ = where we have introduced pa (n) ≡ so that, as operators, using eq. (150), pa (n) → ?ih ?

g ab L G pb (n) 2 Uvn (151)

√

(D + 1)κ2 ab G pb (n) Uvn √ g pL a (n)

vn

(152)

δ δq a (n)

R

(153)

The regularized path-integral, to ?rst-order in Hψ is now ?′ , τ + ?] = Ψ[q ′ , ψ DqDψDψ ?? (q )(1 ? ?W ) exp ?c?

n

vn {iψγ 0 ?ψ (154)

1 ?′ + ?ψ, ? τ] + ?Gab ?q a ?q b } Ψ[q ′ + ?q, ψ κ where W = c? (D + 1) 2

n

1 √ gU

?κ2 Gab pa pb √ [i gψγ 0 K (ψ ′ + ?ψ )] √ gU

(155)

Carrying out the integrals over ?q , ?ψ , ψ , we ?nd Ψτ + ? = 1+? (D + 1) 2 1 1 2 ab √ κ G pa pb ih ? gU δ ? δψ

?? ? ? ? Ψτ Kψ ?

n

1 +√ gU

?κ2 Gab pa pb √ gU

(156)

R

where the square-root operator is de?ned via spectral analysis. Once again, the requirement that the state evolution is independent, in the continuum limit, of the choice of {vn }, implies that the term in braces is the same in each cell (simplex) n. Therefore, in the continuum limit, 1 ? 2 ab κ G pa pb + ih ? ?τ Ψ = √ gU

?

where the divergent factor Np (D + 1)/2 has been absorbed into a rescaling of τ . For stationary states ΦE we have

? ?κ2 Gab pa pb

?κ2 Gab pa pb δ ?? Ψ ih ? Kψ √ ? gU δψ

?

(all x)

(157)

+

Then, since

?κ2 Gab pa pb δ ? + E √gU ? ΦE = 0 ih ? Kψ √ ? gU δψ

?

(158)

√ ?κ2 Gab pa pb ΦE = E ΦE + order Hψ corrections √ gU

(159)

it follows that, up to ?rst order in Hψ , we recover the same stationary state equation ? h ? δ h ? 2 2 ab δ 2 ? + √gU ΦE = 0 κ ”G ” + i√ Kψ a b ? E δq δq E δψ (160)

that was obtained (eq. (137)) from the ”undetermined constant” approach. As in the purely bosonic case, the imposition of HE ΦE = 0 for all E at every point x is consistent with, and in fact implies (c.f. ref. [7]), the other required ?rst-class constraints Hi Ψ = Mij Ψ = 0, up to the usual operator-ordering issues.

6

Conclusions

For ordinary quantum theories without time-parametrization, the regularized path integral is expressed as a product of integrals, each of which evolves the state function unitarily over a very small time interval. In this article we have examined whether such a construction can be applied to theories with square-root, timereparametrization invariant actions. Our result is that unitarity requires: i) an

unconventional phase in the path-integrand; and ii) summation over con?gurations of both real and imaginary proper-time lapse. In the case of quantum gravity, the second requirement means that path-integration must run over manifolds of Lorentzian, Euclidean and, in general, mixed signature. We have also shown how the formalism extends to fermionic actions. Unitarity, of course, refers to evolution in a certain time parameter. In our formulation, the time parameter is simply a measure of the number of integrations in the (regulated) path-integral, evolving an initial state to a later state. This ”quantum time” parameter is neither a geometrical quantity (such as a proper time lapse), nor a dynamical variable (such as the extrinsic curvature). It is, instead, a parameter which is intrinsic to the path-integral measure. The connection to classical dynamics is established via an Ehrenfest principle. In the standard canonical formulation of quantum gravity, the physical states are solutions of a Wheeler DeWitt equation HΨ = 0. In contrast, an outcome of our formulation is that physical states belong to a Hilbert space which is spanned by the solutions of a family of Wheeler-DeWitt equations HE ΦE = 0, which are distin√ guished by having di?erent e?ective values of Planck’s constant h ? ef f = h ? / E . As discussed in section 4, a superposition of states with varying E (or h ? ef f ) allows us to construct, at the semiclassical level, physical states whose amplitudes are peaked at particular 3-geometries and extrinsic curvatures. The width of the peak, in superspace, is inversely proportional to the dispersion ?E . Projection operators formed from such states, and linear combinations of those projection operators, belong to the physical observables of the theory. It is worth noting that the stationary states (i.e. solutions of a Wheeler-DeWitt equation with a ?xed value of E ) can never be peaked around any one 3-geometry. At best, in the WKB limit, a stationary state is peaked at every possible spacelike slice of some 4-manifold satisfying the Einstein equations. If our view is correct, then the phenomenological value of Planck’s constant is the mean value of a dynamical quantity, having a ?nite uncertainty of quantum origin. How large this uncertainty might be, and whether there could conceivably be testable consequences, are interesting issues for further investigation.

Acknowledgements

We would like to thank Maurizio Martellini for some very stimulating discussions. J.G. is also happy to acknowledge the hospitality of the Lawrence Berkeley Laboratory, and the Niels Bohr Institute. J.G.’s research is supported in part by

the U.S. Dept. of Energy, under Grant No. DE-FG03-92ER40711. A.C.’s research is supported by an EEC fellowship in the ‘Human Capital and Mobility’ program, under contract No. ERBCHBICT930313.

References

[1] K. Kuchar, in Quantum Gravity 2: A second Oxford symposium, ed. C. Isham, R. Penrose, and D. Sciama (Oxford University Press, Oxford, 1981). [2] K. Kuchar, Time and interpretations of quantum gravity, Proceedings of the 4th Canadian Conference on General Relativity and Astrophysics, ed. G. Kunstatter et. al. (World Scienti?c, Singapore, 1992). [3] C. J. Isham, Imperial College preprint IMPERIAL-TP-91-92-25, archive: gr-qc/9210011. [4] J. Greensite, Phys. Rev. D49 (1994) 930. [5] A. Carlini and J. Greensite, Niels Bohr Institute preprint NBI-HE-94-31, archive: gr-qc/9406044. [6] R.F. Baierlein, D.H. Sharp, and J.A. Wheeler, Phys. Rev. 126 (1962) 1864; J. A. Wheeler, in Relativity, Groups, and Topology, ed. C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1964). [7] V. Moncrief and C. Teitelboim, Phys. Rev. D6 (1972) 966. [8] A. Carlini and J. Greensite, Phys. Rev. D49 (1993) 866; J. Greensite, Phys. Lett. B300 (1993) 34. [9] T. Banks, Nucl. Phys. B249 (1985) 332. [10] R. Brout and G. Venturi, Phys. Rev. D39 (1989) 2436. [11] U. Gerlach, Phys. Rev. 177 (1969) 1929. [12] J.E. Nelson and C. Teitelboim, Ann. Phys. 116 (1978) 86. [13] G.F.R. Ellis, A. Sumeruk, D. Coule and C. Hellaby, Class. Quantum Grav. 9 (1992) 1535; T. Dereli and R. Tucker, Class. Quantum Grav. 10 (1993) 365; T. Dray, C. Manogue and R. Tucker, Phys. Rev. D48, (1993) 2587; S.A. Hayward, Class. Quantum Grav. 9 (1992) 1851.

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