From path integrals to the Wheeler-DeWitt equation: Time evolution in spacetimes with a spatial boundary
aa r X i v : . [ g r- q c ] N ov From path integrals to the Wheeler-DeWitt equation: Time evolution in spacetimeswith a spatial boundary
Justin C. Feng and Richard A. Matzner Department of Physics, The University of Texas at Austin, Texas 78712, USA (Dated: November 15, 2017)We reexamine the relationship between the path integral and canonical formulation of quantumgeneral relativity. In particular, we present a formal derivation of the Wheeler-DeWitt equationfrom the path integral for quantum general relativity by way of boundary variations. One featureof this approach is that it does not require an explicit 3+1 splitting of spacetime in the bulk. Forspacetimes with a spatial boundary, we show that the dependence of the transition amplitudes onspatial boundary conditions is determined by a Wheeler-DeWitt equation for the spatial boundarysurface. We find that variations in the induced metric at the spatial boundary can be used to describetime evolution—time evolution in quantum general relativity is therefore governed by boundaryconditions on the gravitational field at the spatial boundary. We then briefly describe a formalismfor computing the dependence of transition amplitudes on spatial boundary conditions. Finally, weargue that for nonsmooth boundaries, meaningful transition amplitudes must depend on boundaryconditions at the joint surfaces.
CONTENTS
I. Introduction 1II. Weiss Variation 2A. Mechanics 2B. Gravitational action without spatial boundary 3C. Superspace: Rewriting the gravitational Hamiltonian 5D. Gravitational action with spatial boundary 7III. From the Path Integral to the Schr¨odinger Equation 8A. Variation of the transition amplitude 8B. Path integral-operator correspondence rule 11IV. Wheeler-DeWitt Equation 13A. Variation of the path integral 13B. Wave functional, operators, and commutators 15C. Hamiltonian operator and the Wheeler-DeWitt equation 16V. Quantum general relativity for spacetimes with spatial boundary 18A. Functional integral and its variation 18B. Spatial boundaries and time evolution 20C. Dependence of transition amplitude on spatial boundary conditions 23VI. Summary and Future Work 23Acknowledgments 24References 24A. The Weiss variation of the gravitational action 25
I. INTRODUCTION
In this paper, we present a formal derivation of the Wheeler-DeWitt equation from the path integral for quantumgeneral relativity, and extend our formalism to describe the dependence of path integral transition amplitudes onspatial boundary conditions. The reader may be aware of existing derivations of the Wheeler-DeWitt equation inthe literature that use the path integral as a starting point [1–4] (there is also an old paper [5] that argues that thepath integral for quantum general relativity satisfies the Wheeler-DeWitt equation). We also note that the matterof spatial boundary conditions has been addressed before [6] in the form of a boundary Schr¨odinger equation. Theformalism we present in this paper, however, has a feature that is not present in the existing derivations; in particular,it does not require a 3+1 splitting of spacetime in the bulk. The absence of this requirement is particularly usefulfor studying spacetimes with spatial boundaries and allows us to establish the dependence of transition amplitudeson spatial boundary conditions. In particular, we find that time evolution for such transition amplitudes correspondsto variations of the induced 3-metric at the timelike spatial boundary. While the general idea that spatial boundaryconditions establish time evolution in quantum gravity may be found in the existing literature, the results we presentformalize this idea and elucidate the relationship between time evolution and spatial boundaries. Furthermore, ourresults demonstrate that the boundary Schr¨odinger equation in Ref. [6] is in fact kinematical; the dependence of thetransition amplitudes on spatial boundary conditions is determined by a boundary Hamiltonian constraint on thepath integral transition amplitudes—the “Wheeler-DeWitt” equation for timelike spatial boundaries.The derivation we present in this paper is based on the Weiss variational principle [12, 13]. In the Weiss variation,the boundaries of the action integral are not held fixed—we include displacements of the boundaries/end points.Though the Weiss variation is rarely referred to as such in the literature, the Weiss variation itself is a well-knownresult and should be recognizable to readers familiar with the derivation of Noether currents from the action. Thispaper is organized as follows. We begin by briefly describing the Weiss variation in mechanics and its applicationto the gravitational action; a more detailed discussion may be found in Ref. [14]. We also review the concept ofsuperspace, which will be used to motivate an operator-path integral correspondence. We then review the derivationof the Schr¨odinger equation from the quantum mechanical path integral and discuss correspondence rules betweenpath integrals and operators. Afterward, we present our derivation of the Wheeler-DeWitt equation for spacetimes without a spatial boundary. Finally, we extend our derivation to include transition amplitudes for spacetimes with aspatial boundary and discuss the matter of time evolution for such transition amplitudes. II. WEISS VARIATION
We summarize the Weiss variational principle in mechanics [12, 13, 15] and briefly discuss its application to thegravitational action. A detailed discussion of the Weiss variation and its application to the gravitational action maybe found in Ref. [14] (also see references contained therein).
A. Mechanics
Consider a mechanical system with N degrees of freedom q i ∈ Q , with Q being a manifold called the configurationspace manifold. The system is described by the action: S [ q ] := Z t t L ( q, ˙ q, t ) dt, (2.1)where the degree of freedom index i is suppressed in the argument of the Lagrangian, which is assumed to benondegenerate in ˙ q i . In the Weiss variation, we include temporal displacements of the end points of the form: t ′ = t + λ τ t ′ = t + λ τ , (2.2)where λ is an infinitesimal parameter. We perform variations for q i ( t ), q ′ i ( t ) = q i ( t ) + λ η i ( t ) , (2.3) Our formalism is conceptually similar to that of the general boundary formulation of quantum field theory[7, 8] in that we constructtransition amplitudes for compact regions of spacetime with a connected boundary. See for instance the following papers on the general boundary formalism: Refs. [8, 9]. This idea is discussed in Ref. [10] and workedout for loop quantum gravity (our approach differs in that we work exclusively in the 3-metric representation, and that our startingpoint is the path integral formalism, rather than the canonical formalism). Also, Smolin Ref. [11] points out that the problem of timeis avoided in asymptotically flat spacetimes. where η i ( t ) are functions of t that define the variations. The first-order (in λ ) variation of (2.1) takes the form: δS := S [ q ′ ] − S [ q ] = ε Z t t (cid:18) ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19)(cid:19) η i ( t ) dt + (cid:18) ∂L∂ ˙ q i λ η i ( t ) + L ∆ t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t t , (2.4)where ∆ t | t := λ τ and ∆ t | t := λ τ . It is convenient to define the total change in the end point values for q i ,∆ q i := q ′ i ( t ′ ) − q i ( t ) = λ ( η i ( t ) + τ ˙ q i ( t )) + O ( λ )∆ q i := q ′ i ( t ′ ) − q i ( t ) = λ ( η i ( t ) + τ ˙ q i ( t )) + O ( λ ) , (2.5)and also the quantities p i := ∂L∂ ˙ q i (2.6) H := p i ˙ q i − L, (2.7)which the reader may recognize as the conjugate momentum and Hamiltonian. From the above definitions, it is notdifficult to rewrite (2.4) in the following form: δS = λ Z t t (cid:18) ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19)(cid:19) η i ( t ) dt + (cid:0) p i ∆ q i − H ∆ t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t t . (2.8)The above expression for δS is the Weiss variation of the action. The Weiss variational principle demands thatphysical paths be described by functions q i ( t ) for which δS consists exclusively of end point/boundary terms, whichis equivalent to the statement that the integral over t in (2.8) vanishes; it follows that the Weiss variational principleimplies the Euler-Lagrange equations. B. Gravitational action without spatial boundary
We assume a four-dimensional spacetime manifold M and use the Misner-Thorne-Wheeler [16] signature ( − , + , + , +)for the metric tensor g µν . Greek indices refer to coordinates on the spacetime manifold M and its subsets, which wedenote as U or W ; U will be used to indicate a spacetime region without a spatial boundary, and later on, W will beused to denote a spacetime region with a spatial boundary. Coordinates on M , U , and W will be denoted x µ with x = t being the time coordinate. Lowercase latin indices refer to either mechanical degrees of freedom or coordinateson hypersurfaces—the distinction should be apparent from the context. Capital latin indices from the beginning ofthe alphabet will refer to two-dimensional surfaces in M .For simplicity, we first consider a globally hyperbolic spacetime without a spatial boundary; in particular, we require M to be a Lorentzian manifold with the topology R × Σ, where t ∈ R is a time coordinate and Σ is a three-dimensionalmanifold without a boundary. We then consider a subset U ⊂ M with boundary surfaces Σ I and Σ F , which need notbe surfaces of constant t . We require the surfaces Σ I and Σ F to be spacelike, with a positive-definite induced metric γ ij . The gravitational action over U may then be written as S GR [ g µν ] = 12 κ Z U R p | g | d x + 1 κ Z ∂ U K ε p | γ | d y, (2.9)where x µ denote coordinates on the spacetime manifold M , R is the Ricci curvature scalar of U , K is the meancurvature of the boundary surface ∂ U = Σ I ∪ Σ F , and we have defined γ := det( γ ij ) (with γ ij ) and ε := n µ n µ = ± n µ is the unit normal vector to ∂ U (here, ε = − κ = 8 πG , where G is Newton’s gravitationalconstant.The Weiss variation includes boundary displacements. To characterize boundary displacements, we begin by placingcoordinates y i on the boundary ∂ U . The boundary may be parametrically defined by functions x µ ( y ), which specifythe position of the boundary ∂ U in M . The induced metric γ ij on the boundary may then be written as: γ ij = ∂x µ ∂y i ∂x ν ∂y j g µν , (2.10)and it is well known that the induced metric takes the form γ µν = g µν − ε n µ n ν in the bulk coordinate basis.Displacements of the boundary in the manifold M may be characterized by adding to x µ ( y ) a function δx µ ( y ) so thatthe displaced boundary may be parametrically defined by the functions: x ′ µ ( y ) = x µ ( y ) + δx µ ( y ) . (2.11)The presence of the boundary term (called the Gibbons-Hawking-York boundary term) in the gravitational action S GR [ g µν ] complicates the derivation of the Weiss variation of S GR [ g µν ]. One may obtain the variation of the boundaryterm by recognizing that the boundary term is the first variation of the area formula; the variation of the boundaryterm is then given by the second variation of the area formula. We shall not reproduce the derivation here; we referthe reader to [14] for the full details of the derivation [a partial justification for (2.12) is given in the Appendix]. TheWeiss variation of the gravitational action takes the form δS GR = 12 κ Z U G µν δg µν p | g | d x + ε κ Z ∂ U (cid:18) p ij ∆ γ ij + (cid:2) n µ (cid:0) R − ε ( K − K ij K ij ) (cid:1) − D α p αβ γ µβ (cid:3) δx µ (cid:19)p | γ | d y, (2.12)where G µν := R µν − R g µν is the Einstein tensor, R and K ij are the respective Ricci scalar and extrinsic curvaturetensor for the boundary ∂ U , and D µ denotes the covariant derivative on a hypersurface—in this case, D µ is thecovariant derivative on ∂ U . We define the following quantity in the basis of the bulk coordinates x µ and the basis ofthe boundary coordinates y i , p µν := K µν − K γ µν p ij := ∂x µ ∂y i ∂x ν ∂y j ( K µν − K γ µν ) , (2.13)where K µν is the extrinsic curvature tensor in the bulk coordinate basis (for K µν and K = g µν K µν , we use the signconvention found in Refs. [17, 18]). These two expressions for the tensor p µν are equivalent because the tensor p µν istangent to the boundary surfaces, or that p µν n µ = p νµ n µ = 0. For later use, we define the quantity: P µν := ε κ p µν p | γ | P ij := ε κ p ij p | γ | . (2.14)The quantity ∆ γ ij is the total change in the induced metric at the boundary surface, in the same way that ∆ q i is thetotal change in the variable q i at the end points [cf. Eq. (2.5)].We stress that despite the appearance of a hypersurface Ricci scalar R and the extrinsic curvature tensor K ij , theWeiss variation of the gravitational action (2.12) makes no reference to a 3+1 split in the bulk manifold U ; in anotherpaper [14], we show that Eq. (2.12) may indeed be derived without performing a 3+1 split in the bulk manifold U .As stated before, the hypersurfaces Σ I and Σ F that form the boundary ∂ U need not be surfaces of constant t . As aresult, the Ricci scalar R and the extrinsic curvature tensor K ij are properties of the boundary surfaces Σ I and Σ F and do not necessarily correspond to the surfaces of constant time coordinate t in the bulk manifold U .We note that if the vacuum Einstein field equations G µν = 0 are satisfied, then the vacuum Hamiltonian andmomentum constraints take the form2 G µν n µ n ν = − ε ( R − ε ( K − K ij K ij )) = 0 (2.15) γ µβ G µν n ν = D α p αβ = 0 ⇒ γ ik D k p ij = 0 . (2.16)The above constraints in turn suggest that the term proportional to δx µ in the variation δS GR (2.12) vanishes. Thisresult suggests that the Hamiltonian and the canonical energy-momentum “tensor” for general relativity vanish onvacuum solutions of the Einstein field equations [14].Without loss of generality, we may choose the boundary displacement δx µ ( y ) to be proportional to the unit normalvector n µ : δx µ ( y ) = n µ ∆ τ ( y ) . (2.17)This is because the portion of δx µ ( y ) tangent to the hypersurface corresponds to infinitesimal diffeomorphisms onthe boundary surface ∂ U . We interpret the quantity ∆ τ ( y ) as the amount (measured in proper time) by which theboundary ∂ U is displaced in the normal direction. The variation (2.12) simplifies to δS GR = 12 κ Z U (cid:18) R µν − R g µν (cid:19) δg µν p | g | d x + Z ∂ U (cid:18) P ij ∆ γ ij − H gf ∆ τ (cid:19) d y. (2.18)where we make use of the expression γ µβ n µ = 0, and we define the “gauge-fixed” Hamiltonian density H gf ( P ij , γ ij ) := − κ (cid:20) R − ε ( K − K ij K ij ) (cid:21)p | γ | , (2.19)where K ij and K depend on P ij via the expressions K ij = 2 κ ε √ γ (cid:18) P ij − γ ij γ kl P kl (cid:19) K = − κ ε √ γ P ij γ ij , (2.20)which can be easily obtained from (2.14). Finally, we integrate (2.19) to obtain the gravitational Hamiltonian: H GR [ P ij , γ ij ] = Z U H gf ( P ij , γ ij ) d y = − κ Z U (cid:20) R − ε ( K − K ij K ij ) (cid:21)p | γ | d y. (2.21) C. Superspace: Rewriting the gravitational Hamiltonian
We now take the opportunity to briefly motivate and review the concept of superspace, the space of Riemannian3-geometries [19–22] (also see [23] and references therein). The concept of superspace will be useful for us becauseit provides a formalism for general relativity that resembles particle mechanics. We begin by using (2.20) to rewrite(2.19) as H gf ( P ij , γ ij ) = − ε κ P ij G ijkl P kl − κ R p | γ | , (2.22)where we define the following: G ijkl := 2 κ p | γ | (cid:0) γ ik γ jl + γ il γ jk − γ ij γ kl (cid:1) G ijkl := p | γ | κ ( γ ik γ jl + γ il γ jk − γ ij γ kl ) . (2.23)The tensor G ijkl is constructed to satisfy the following property: G ijab G abkl = 12 (cid:0) δ ki δ lj + δ li δ kj (cid:1) . (2.24)We may then construct a “supermetric” from the expression G ijkl : G ij : kl ( y, y ′ ) := 12 (cid:0) G ijkl ( y ) δ ( y ′ − y ) + G ijkl ( y ′ ) δ ( y − y ′ ) (cid:1) . (2.25)The supermetric G ij : kl ( y, y ′ ) may be regarded as a metric on the space of Riemannian inverse 3-metric fields γ ij ( y )and γ ij ′ ( y ), which we call iRiem (Σ). In particular, G ij : kl ( y, y ′ ) defines an inner product for tangent vectors ˙ γ ij ( y )of iRiem (Σ). Now, consider two inverse 3-metrics γ ij ( y ) and ˜ γ ij ( y ) that are related by coordinate transformations;we observe γ ij ( y ) and ˜ γ ij ( y ) correspond to distinct points in iRiem (Σ). A more physically relevant concept is thatof superspace, which is the space of coordinate-independent Riemannian 3- geometries (as opposed to coordinate-dependent inverse 3-metrics). Given a manifold Σ, one may define superspace S (Σ) by the formal construction S (Σ) = iRiem (Σ) Dif f (Σ) , (2.26)in which we mod out iRiem (Σ) by the space of diffeomorphisms [denoted Dif f (Σ)] on Σ. Not to be confused with the coordinate space of a supermanifold (usually discussed in the context of supersymmetry), which is alsocalled “superspace.” Recall that supermanifolds possess both commuting (bosonic) and anticommuting (fermionic/Grassmann-valued)coordinates. To avoid confusion, we propose referring to the coordinate space of a supermanifold as
Grassmann superspace and S (Σ)as Riemannian superspace when a distinction is needed (we shall not do this outside this footnote). The supermetric defined above differs from the DeWitt supermetric [19, 22] by a factor of | γ | /
4; this is due to our convention that weused the inverse γ ij , rather than the metric γ ij , as coordinates on superspace [we interpret γ ij = γ ij ( y, γ kl ( y )).]. While superspace S (Σ) is a more physically relevant concept than iRiem (Σ), the difficulty with using S (Σ)rather than iRiem (Σ) lies in the fact that S (Σ) is not a manifold. In particular, it has been pointed out that thedimensionality of S (Σ) can change at points corresponding to geometries that possess a high degree of symmetry[21, 22]. It was (nonrigorously) argued in Ref. [22] that one may nevertheless extend S (Σ) to obtain a manifoldwhich is referred to as extended superspace S ex (Σ). It is beyond the scope of this article to review this matter anyfurther (we refer the reader to Refs. [21], [22], [24], and [23] for further discussion); for our purposes, it suffices toassume that there exists a manifold S ex (Σ) that contains S (Σ) as a subset and admits a surjection (a map that isonto) from S ex (Σ) to S (Σ).Since Σ is assumed to be compact, we may require S ex (Σ) to have countable (though infinite) dimension. Thisrequirement is motivated by the observation that functions defined on certain compact manifolds, the n -torus T n andthe n -sphere S n for instance, admit a complete countable basis for functions defined on them (the respective discreteFourier basis and n -spherical harmonic basis) so that the function spaces on T n and S n have countable dimension. Ifthe manifold Σ has a boundary, we impose the appropriate boundary conditions to ensure that the function spaceshave countable dimension. If S ex (Σ) has countable dimension, then given coordinates ξ a on S ex (Σ), and a map γ ij ( y, ξ ) : S ex (Σ) → iRiem (Σ), we may formally write the supermetric G ij : kl ( y, y ′ ) in the coordinate basis on S ex (Σ)in the following manner: G ab := Z Σ (cid:18)Z Σ ∂γ ij ( y, ξ ) ∂ξ a ∂γ kl ( y ′ , ξ ) ∂ξ b G ij : kl ( y, y ′ ) d y (cid:19) d y ′ . (2.27)Note that, since S ex (Σ) is infinite dimensional (again, we assume that the dimension is countably infinite), there isno upper bound on the values of the indices a , b ∈ N ( N being the set of natural numbers).Given the superspace metric G ab , we may regard S ex (Σ) as an infinite-dimensional pseudo-Riemannian manifold, and since S ex (Σ) is of countable dimension, the standard formulas of Riemannian geometry apply to quantities definedon S ex (Σ). We may, for instance, define an inverse superspace metric G ab that satisfies the condition ∞ X t =1 G at G tb = δ ab (2.28)and the connection coefficients Γ cab := 12 ∞ X t =1 G ct (cid:18) ∂ G tb ∂ξ a + ∂ G at ∂ξ b − ∂ G ab ∂ξ t (cid:19) , (2.29)from which we may construct covariant derivatives on S ex (Σ).These definitions, combined with (2.22), permit a rewriting of the gravitational Hamiltonian H GR in the form: H GR ( P a , ξ a ) = − ε κ ∞ X a =1 ∞ X b =1 G ab P a P b ! + Φ( ξ ) , (2.30)where Φ( ξ ) := − κ Z Σ 3 R p | γ | d y, (2.31)with γ ij = γ ij ( y, ξ ) and R = R ( y, ξ ). The integral over y i in the second term ensures that Φ( ξ ) is strictly a functionof the superspace coordinate ξ a .We note that the Hamiltonian H GR (2.30) resembles the Hamiltonian of particle mechanics on a Riemannianmanifold. A typical Hamiltonian for such a system takes the following form H ( p, q ) = 12 m g ij p i p j + V ( q ) , (2.32)where g ij is a Riemannian metric on the configuration space Q and V ( q ) is the potential. A comparison of theHamiltonian H GR (2.30) with the particle Hamiltonian (2.32) suggests that the first term containing the conjugatemomenta is a kinetic term 12 m g ij p i p j ↔ κ ∞ X a =1 ∞ X b =1 G ab P a P b ! , (2.33) That S ex (Σ) is a pseudo-Riemannian manifold can be seen by noting that for the special case γ ij = δ ij , the independent componentsof G ijkl have five positive roots and one negative root [19]. FIG. 1. An illustration of a cylindrical boundary for a spacetime region W , with boundary ∂ W = Σ I ∪ B ∪ Σ F . The verticaldirection is timelike, so Σ I and Σ F are spacelike surfaces of codimension 1, and B is a timelike surface of codimension 1. Thesurfaces S I and S F are surfaces of codimension 2 (assumed to have exclusively spacelike tangent vectors) that form boundariesbetween Σ I , B , and Σ F . The unit normal vectors (shown in red) are defined to be outward pointing; n I = [ n µI ] is the unitnormal to Σ I , n B = [ n µ B ] is the unit normal to B , and n F = [ n µF ] is the unit normal to Σ F . See Sec. II D. and that the function Φ( ξ ) is a potential. Of course, the idea that general relativity can be recast as a problem ofparticle motion in an infinite-dimensional manifold is not new [22, 25]. However, the formalism developed in thissection will be useful for motivating a correspondence between path integral and operator expressions in quantumgeneral relativity, which we will present in later sections.
D. Gravitational action with spatial boundary
We now consider the case in which we consider spacetime regions W with a spatial boundary; in particular, werequire W to have a boundary ∂ W as described in Fig. 1. The boundary ∂ W consists of three regions, Σ I , B and Σ F ,and is nonsmooth on the two-dimensional surfaces S I and S F . We require the surfaces Σ I and Σ F to be spacelike,meaning that they admit a positive-definite induced metric γ ij with signature (+ , + , +). On the other hand, werequire the surface B to be timelike, meaning that the induced metric q ab on B has a signature ( − , + , +).This time, the gravitational action takes the form S GR, B [ g µν ] = 12 κ Z W R p | g | d x − κ Z Σ F K p | h | d y + 1 κ Z B K p | q | d y − κ Z Σ I K p | h | d y + S C , (2.34)where underlined quantities are defined on the boundary surface B . For nonsmooth boundaries with spacelike junctionsurfaces S I and S F , we must include the term S C , which is often referred to as the corner term [26–29] (also see Refs.[30–34]), S C := 1 κ Z S I η I p | σ | d z + 1 κ Z S F η F p | σ | d z, (2.35)where the rapidity angles η I and η F are formed from the inner product between the unit normal vectors at the junctionsurfaces S I and S F : η I := arcsinh ( h n I , n B i| S I ) η F := arcsinh ( h n F , n B i| S F ) . (2.36) In fact, Ref. [22] goes further; the Hamiltonian constraint was used to show that solutions of Einstein’s field equations can (withappropriate gauge conditions) be interpreted as geodesics in superspace.
We may ignore this term if the inverse metric tensor g µν is held fixed at the junction surfaces S I and S F , and theboundary displacements δx µ ( y ) vanish in a neighborhood of the junction surfaces S I and S F . In other words, werequire h n I , n B i| S I and h n F , n B i| S F to be held fixed and that δx µ | S I = 0, δx µ | S F = 0, δg µν | S I = 0, and δg µν | S F = 0.Without loss of generality, we may choose the boundary displacement δx µ to take the following form: δx µ | Σ I = n µ ∆ τ i ( y ) for y ∈ Σ I δx µ | B = n µ ∆ r ( y ) for y ∈ B δx µ | Σ F = n µ ∆ τ f ( y ) for y ∈ Σ F . (2.37)The Weiss variation of the action is (we stress that I and F are labels—they are not indices to be summed over) δS GR, B = 12 κ Z W G µν δg µν p | g | d x + Z Σ I (cid:18) P Iij ∆ h ijI − H I ∆ τ I (cid:19) d y + Z B (cid:18) P ab ∆ q ab − H B ∆ s (cid:19) d y + Z Σ F (cid:18) P Fij ∆ h ijF − H F ∆ τ F (cid:19) d y (2.38)where we define the momentum densities P Iij := − κ ( K ij − K h
Iij ) p | h I | P ab := 12 κ ( K ab − K q ab ) p | q | P Fij := − κ ( K ij − K h
Fij ) p | h F | (2.39)and the Hamiltonian densities H I := 12 κ P Iij G ijkl P Ikl − κ R p | h I | H B := − κ P ab G abcd P cd − κ R p | q | H F := 12 κ P Iij G ijkl P Ikl − κ R p | h F | , (2.40)where H I is defined over Σ I and H F is defined over Σ F and the quantities G ijkl and G abcd are given by G ijkl := 2 κ √ h (cid:0) h ik h jl + h il h jk − h ij h kl (cid:1) G abcd := 2 κ √ q (cid:0) q ac q bd + q ad q bc − q ab q cd (cid:1) . (2.41)Note that for vacuum solutions of the Einstein field equations, the right-hand sides of all three equations in (2.40)vanish. We also note that the inverse 3-metric q ab defines an extended superspace that is different than S ex (Σ) definedearlier, since q ab defines a pseudo-Riemannian 3-geometry, rather than a Riemannian 3-geometry. We shall denotethe extended superspace for pseudo-Riemannian 3-geometries on a 3-manifold Σ by S ex (Σ). Underlined quantitieswill either refer to quantities defined on S ex (Σ) or quantities defined on a pseudo-Riemannian 3-manifold. III. FROM THE PATH INTEGRAL TO THE SCHR ¨ODINGER EQUATIONA. Variation of the transition amplitude
We now show that wave functions constructed from Feynman path integrals formally satisfy the Schr¨odinger equa-tion [35]. We begin by considering the position-basis transition amplitude h q i , t | q i , t i = K ( q i , t ; q i , t ) , (3.1) Our derivation at first follows closely the one found in the lecture notes [36], but we extend it to include a discussion of the operator-pathintegral correspondence. where | q i , t i is the position-basis state vector and the function K ( q i , t ; q i , t ) is given by the path integral expression[we shall henceforth refer to the function K ( q i , t ; q i , t ) as the transition amplitude] K ( q i , t ; q i , t ) = Z D q e ( i/ ~ ) S [ q ] , (3.2)where the action functional S [ q ] (2.1) is implicitly a function of the end point values q i , t , q i , and t and R D q represents a measure on the space of functions q i ( t ) subject to the Dirichlet boundary conditions q i ( t ) = q i and q i ( t ) = q i ; in particular, we hold the function q i ( t ) fixed to the values q i and q i at the respective times t and t .We do not provide a rigorous definition for the measure; to proceed, we only need the property that the measure isinvariant under shifts in the function q i ( t ) of the form q i ′ ( t ) = q i ( t ) + δq i ( t ). Under this shift, we require the measureto satisfy Z D q ′ F [ q ′ ] = Z D q F [ q + δq ] (3.3)for some functional F [ q ]. The property (3.3) is motivated by the analogous property for the usual single-variableintegral for shifts x ′ = x + δx in the integration variable x : Z ∞−∞ f ( x ′ ) dx ′ = Z ∞−∞ f ( x + δx ) dx. (3.4)We now consider what happens when we vary the function q i ( t ) while holding the end point values q i , t , q i , and t fixed. We may use (3.3) to write the following: Z D q ′ e ( i/ ~ ) S [ q ′ ] = Z D q e ( i/ ~ ) S [ q + δq ] . (3.5)We may perform a relabeling of the integration variable q i ′ → q i to write Z D q ′ e ( i/ ~ ) S [ q ′ ] = Z D q e ( i/ ~ ) S [ q ] (3.6)Equations (3.5) and (3.6) may be used to obtain: Z D q e ( i/ ~ ) S [ q + δq ] = Z D q i e ( i/ ~ ) S [ q ] ⇒ i ~ Z D q δ S e ( i/ ~ ) S [ q ] = 0 (3.7)where the subscript 0 in the variation of the action δ S indicates that the end point values q i , t , q i , and t are heldfixed. Note that, since δq i ( t ) is independent of q i ( t ), we may pull it out of the path integral. The demand that Eq.(3.7) must hold for all δq i ( t ) implies Z D q (cid:18) ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19)(cid:19) e ( i/ ~ ) S [ q ] = 0 . (3.8)We now consider what happens when the end points are displaced; in particular, we change the values of q i , t , q i , and t . Equation (3.6) is no longer valid, since the displacement of the end points prevents us from performing achange in the integration variable. The change in the transition amplitude is given by the following difference in pathintegrals: δ K = Z D q e ( i/ ~ ) S ′ [ q + δq ] − Z D q e ( i/ ~ ) S [ q ] . (3.9)Since the measure itself is invariant under displacements of the end points (the displacements in t and t may beabsorbed into a redefinition of t ), the change in the amplitude is given by the change in the integrand. In particular,the change in the amplitude is given by the change in the action: δ K = i ~ Z D q δS e ( i/ ~ ) S [ q ] . (3.10) We absorb normalization factors into the definition of the measure. For a rigorous treatment of path integrals, we refer the reader to the formalism of Cartier and DeWitt-Morette: Refs. [37, 38]. q i ′ ( t ) = q i ( t ) + δq i ( t ) t ′ = t + ∆ t t ′ = t + ∆ t , (3.11)where δq i ( t ) = ε η i ( t ) and we assume ∆ t ∝ ε and ∆ t ∝ ε . If the action contains no more than first-order timederivatives in q i (and a Legendre transformation can be performed), then to first order in ε , the change in the actionis given by the Weiss variation of the action (2.8), δS = Z t t (cid:18) ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19)(cid:19) δq i dt + (cid:0) p i ∆ q i − H ∆ t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t t , (3.12)where p i = ∂L/∂ ˙ q i , H is the Hamiltonian, and∆ q i := q i ′ ( t ′ ) − q i ( t ) = ε ( η i ( t ) + ˙ q i ( t )) + O ( ε )∆ q i := q i ′ ( t ′ ) − q i ( t ) = ε ( η i ( t ) + ˙ q i ( t )) + O ( ε ) . (3.13)Using (3.8), the change in the amplitude is given by δ K = i ~ Z D q i (cid:0) p i ∆ q i − H ∆ t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t t e ( i/ ~ ) S [ q ] . (3.14)Now, consider what happens when we hold q i and t fixed, so that ∆ q A = 0 and ∆ t = 0. We may write the above(3.14) as δ K| ( q i ,t ) → fixed = i ~ (cid:18)Z D q p i, e ( i/ ~ ) S [ q ] (cid:19) ∆ q i − i ~ (cid:18)Z D q H | t e ( i/ ~ ) S [ q ] (cid:19) ∆ t , (3.15)where we introduce the notation p i, = p i | t and p i, = p i | t . Given an initial state function ψ ( q i ) (equivalent to thestate vector | q i , t i ), where ψ ( · ) is some normalizable complex function, we may define the wave function Ψ( q i , t ) interms of the transition amplitude in the following way:Ψ( q i , t ) := Z K ( q i , t ; q i , t ) ψ ( q i ) dq i . (3.16)To simplify the analysis, we consider the case in which the initial state ψ ( q i ) is sharply peaked around some valuefor q i ; we may recover the general results by integrating the results over all values of q i [in particular, for some pathintegral expression P I = P I ( q i , t ; q i , t ), we perform the integral R P I ψ ( q i ) dq i ].One may infer from Eq. (3.15) the expression for the differential of Ψ( q i , t ), d Ψ = i ~ (cid:18)Z D q p i, e ( i/ ~ ) S [ q ] (cid:19) dq i − i ~ (cid:18)Z D q H | t e ( i/ ~ ) S [ q ] (cid:19) dt , (3.17)which in turn yields the following expressions for the derivatives of the wave function: i ~ Z D q H | t e ( i/ ~ ) S [ q ] = − ∂ Ψ ∂t (3.18) i ~ Z D q p i, e ( i/ ~ ) S [ q ] = ∂ Ψ ∂q A . (3.19)Equation (3.19) motivates the following definition for the momentum operator for wave functions Ψ( q , t ) written inthe position basis: ˆ p i, := − i ~ ∂∂q i . (3.20)1We may motivate the Schr¨odinger equation from (3.18), but to obtain an unambiguous expression for the Schr¨odingerequation, we must first establish an operator-path integral correspondence. In particular, we must construct anoperator corresponding to the path integral: Z D q ( p i ...p i n q j ...q j m ) | t e ( i/ ~ ) S [ q ] (3.21)This is often referred to as the problem of operator ordering. In the next section, we shall explore this further. If theHamiltonian H | t can be expressed as a polynomial function of p i, and q i , the path integral (3.18) consists of termsof the form (3.21). If one can establish a correspondence rule for operators and path integrals of the form (3.21), onemay construct a Hamiltonian operator ˆ H | t = ˆ H (ˆ p i , ˆ q i ) | t ; Eq. (3.18) may be rewritten as i ~ ∂ Ψ ∂t = ˆ H (ˆ p i , ˆ q i ) | t Ψ , (3.22)which is, of course, the time-dependent Schr¨odinger equation. B. Path integral-operator correspondence rule
We now motivate a correspondence rule for path integral expressions and their operator counterparts. We begin byestablishing an operator expression for factors of q i that appear in the path integral. Since q i is evaluated at the endpoint, we may trivially write Z D q q i e ( i/ ~ ) S [ q ] = q i Z D q e ( i/ ~ ) S [ q ] = q i Ψ Z D q q i ...q i n e ( i/ ~ ) S [ q ] = (cid:0) q i ...q i n (cid:1) Z D q e ( i/ ~ ) S [ q ] = (cid:0) q i ...q i n (cid:1) Ψ , (3.23)which motivates the following definition for the operator ˆ q i :ˆ q i Ψ = q i Ψˆ q i ... ˆ q i n Ψ = (cid:0) q i ...q i n (cid:1) Ψ . (3.24)The definition (3.24) for the operator ˆ q i may be used to establish an operator expression for the following pathintegral: Z D q ( p i q j ) | t e ( i/ ~ ) S [ q ] . (3.25)Naively, the above expression may be identified with the operation (ˆ p i ˆ q j ) | t Ψ or (ˆ q j ˆ p i ) | t Ψ. The reader familiarwith quantum mechanics will be fully aware of the fact that these two operations are inequivalent, since the operatorsˆ p i | t and ˆ q i do not commute. Indeed, we obtain the explicit expressions ( δ ji being the Kronecker delta)(ˆ p i ˆ q j ) | t Ψ = − i ~ ∂∂q i ( q j Ψ) = − i ~ (cid:18) δ ji Ψ + q j ∂ Ψ ∂q i (cid:19) (ˆ q j ˆ p i ) | t Ψ = − i ~ q i ∂ Ψ ∂q i , (3.26)which yield the commutation relation when combined:([ˆ p i , ˆ q j ]) | t = (ˆ p i ˆ q j − ˆ q j ˆ p i ) | t = − i ~ δ ji . (3.27)We now establish the operator expression corresponding to (3.25). To do this, we expand the second line of (3.26)in the manner (ˆ q j ˆ p i ) | t Ψ = − i ~ q j ∂ Ψ ∂q i = q j Z D q p i, e ( i/ ~ ) S [ q ] = Z D q q j p i, e ( i/ ~ ) S [ q ] = Z D q ( q j p i ) | t e ( i/ ~ ) S [ q ] = Z D q ( p i q j ) | t e ( i/ ~ ) S [ q ] , (3.28)2where the last equality follows from the expression q j p i = p i q j . The computation (3.28) demonstrates that the pathintegral expression in (3.25) corresponds to the operator ordering (ˆ q B ˆ p i ) | t Ψ, in which the momentum operator ˆ p i, appears to the right of the position operator. The correspondence suggested by (3.28) applies only to wave functionsΨ( q , t ) in the position basis; the correspondence rule for path integrals of the form (3.25) is basis dependent.One may expect that, given the definition (3.20) for the momentum operator ˆ p i , one has the following correspondencerule: ( i/ ~ ) n Z D q ( p i ...p i n ) | t e ( i/ ~ ) S [ q ] ≈ ( i/ ~ ) n (ˆ p i ... ˆ p i n ) | t Ψ = ∂ n Ψ ∂q i ...∂q i n (3.29)The difficulty with this correspondence rule (3.29) is that it is not compatible with coordinate transformations onthe configuration space Q . In particular, we expect Ψ and the Schr¨odinger equation to transform as scalars undercoordinate transformations on Q , but this is not in general true for a Schr¨odinger equation constructed using thecorrespondence rule (3.29). To see this, consider the expression:( i/ ~ ) n Z D q ( T i ...i n p i ...p i n ) | t e ( i/ ~ ) S [ q ] ≈ (cid:0) T i ...i n | t (cid:1) ∂ n Ψ ∂q i ...∂q i n , (3.30)where T i ...i n = T i ...i n ( q ) is a tensor on the configuration space manifold Q . Since the conjugate momenta p i transform as cotangent vectors, the integrand on the left-hand side of (3.30) transforms as a scalar (the action S [ q ]is assumed to be invariant under coordinate transformations). For n >
1, the right-hand side of (3.29) does nottransform as a scalar under coordinate transformations on Q .The problems with establishing a correspondence rule may be attributed to ambiguities in the definition of themeasure. Suppose that, for a given definition of the measure R D q , one obtains the correspondence rule (3.29). Sincethe integrand of (3.29) transforms as a scalar, any nonscalar transformation law must come from the transformationof the measure R D q . Thus, any nontensorial transformation law for the left-hand side of (3.29) must come from themeasure. Though we do not propose a definition for the measure, we require the measure to satisfy the property thatthe left-hand side of (3.30) transforms as a scalar. This suggests that the measure yields the correspondence rule Z D q ( p i ...p i n ) | t e ( i/ ~ ) S [ q ] = ( − i ~ ) n ( ∇ i ... ∇ i n ) | t Ψ Z D q ( p i ...p i n q j ...q j m ) | t e ( i/ ~ ) S [ q ] = (ˆ q j ... ˆ q j m ∇ i ... ∇ i n ) | t Ψ , (3.31)where ∇ i | t is a connection (covariant derivative) on the configuration space manifold Q , constructed from the partialderivatives ∂/∂q i (with q i ∈ Q ) and connection coefficients Γ ijk ; for a second rank tensor T ij , it takes the form ∇ i T jk | t = ∂T ij ∂q i + Γ jil T lk − Γ lik T jl . (3.32)The connection coefficients Γ ijk are assumed to be functions of q i and are defined to satisfy the following transformationrule: Γ ′ ijk = ∂q ′ i ∂q a ∂q b ∂q ′ j ∂q c ∂q ′ k ! Γ abc − ∂q b ∂q ′ j ∂q a ∂q ′ k ∂ q ′ i ∂q b ∂q a ! . (3.33)For a particle on a Riemannian manifold, it is natural to choose Γ ijk to be the Christoffel symbols.We stress that the correspondence rule (3.31) for factors of the momenta is a property that we require of the measure,rather than a derived result. An important question that should be addressed is whether one can explicitly constructa time-sliced path integral that yields the correspondence rule (3.31). For our purposes, the full correspondence rule(3.31) is not necessary, as we only need a correspondence rule for the following expression: Z D q ( g ij p i p j ) | t e i ~ S [ q ] = − ~ g ij ∇ i ∇ j Ψ = − ~ √ g ∂∂q i √ g g ij ∂ Ψ ∂q j ! . (3.34) We assume that q i and p i are c -number valued; the present analysis excludes Grassmann-valued degrees of freedom. By this, we mean that the momentum operator ˆ p i, is applied to Ψ first. Z D q ( · ) := lim n →∞ n − Y k =2 Z Q d N q τ k p | M | exp (cid:18) i ~ Z R dt (cid:19) ( · ) , (3.35)where we use the replacement ˙ q iτ k → ( q iτ k − q iτ k − ) / ∆ t in the argument of the right-hand side. Though we note thatthe Laplace-Beltrami ordering (3.34) follows from our correspondence rule (3.31), it is not yet clear to us that thetime-sliced measure (3.35) yields the correspondence rule (3.31). We will not investigate the compatibility of themeasure (3.35) and the correspondence rule (3.31) and leave it for future work—this question is beyond the scope ofthis article, as the Laplace-Beltrami ordering is sufficient for our purposes. In the next section, we use a Laplace-Beltrami-type ordering for the Wheeler-DeWitt equation by demanding that transition amplitudes be invariant undercoordinate transformations on superspace. IV. WHEELER-DEWITT EQUATION
Again, we consider a spacetime M that has no spatial boundary, so that it has the topology R × Σ , where Σ is athree-dimensional manifold without a boundary. We begin with the gravitational action S GR , (2.9), which we rewritehere, S GR [ g µν ] := S EH [ g µν ] + S GHY = 12 κ Z U R p | g | d x + 1 κ Z ∂ U K ε p | γ | d y, (4.1)where U has no spatial boundary (or where ∂ U = Σ I ∪ Σ F , where Σ I and Σ F are spacelike hypersurfaces without aboundary). The most straightforward approach to the quantization of the gravitational field is to simply write downthe path integral (first written down in Ref. [44]) K J γ ijF ; γ ijI K = Z D [ g αβ ] e i ~ S GR [ g αβ ] , (4.2)where the brackets J K indicate a functional of functions defined over the boundary surface ∂ U and R D [ g αβ ] denotesa functional integral over all functions g αβ ( x ) up to those that differ by a diffeomorphism. We require the integrationmeasure R D [ g αβ ] to be invariant under the shift (field redefinition) g µν → g µν + δg µν , , and we also require themeasure to be defined so that the resulting amplitude K J γ ijF ; γ ijI K is independent of coordinate transformations on Σ I and Σ F . The path integral (4.2) formally defines an unnormalized transition amplitude between a spacetime with aninitial inverse 3-metric γ ijI := γ ij | Σ I for the initial hypersurface Σ I and a final inverse 3-metric γ ijF := γ ij | Σ F for thehypersurface Σ F .In this section we attempt to derive a Schr¨odinger equation from the transition amplitude (4.2), but we insteadfind that the transition amplitude must lie in the kernel of the formal Hamiltonian operator; the transition amplitudeis independent of the time parameters t I and t F . This is known as the problem of time , and though it is primarilydiscussed in the context of the canonical formulation of quantum general relativity, we shall show in the next sectionthat it is also present in the path integral formulation. A. Variation of the path integral
We now perform the variation of the path integral (4.2), assuming that the measure R D [ g αβ ] is invariant underthe shift g µν → g µν + δg µν . The variation of the path integral may then be written in terms of the variation of theaction: δ K = i ~ Z D [ g αβ ] δS GR e i ~ S GR [ g αβ ] . (4.3) It is well known that quantum general relativity [as given by functional integrals of the form (4.2)] is perturbatively nonrenormalizable (seeRefs. [42] and [43] for an overview). The analysis we present is formally nonperturbative, so the question of perturbative renormalizabilitywill not enter into our analysis [one might imagine that in writing down the path integral (4.2) we are studying the low-energy limit ofsome effective field theory for gravity]. For the present analysis, we ignore the details of gauge fixing for the measure R D [ g αβ ] or any other procedure for modding outdiffeomorphisms; we only assume that the shift δg µν is compatible with the procedure for gauge fixing or modding out diffeomorphisms. δS GR = 12 κ Z U (cid:18) R µν − R g µν (cid:19) δg µν p | g | d x + ε κ Z ∂ U (cid:18) p µν ∆ γ µν + (cid:2) D α p αβ γ µβ + n µ (cid:0) R − ε ( K − K ij K ij ) (cid:1)(cid:3) δx µ (cid:19) p | γ | d y, (4.4)where the boundary of U is given by the expression ∂ U = Σ I ∪ Σ F , and [cf. Eq. (2.13)]: p µν := ( K µν − K γ µν ) . (4.5)Note that the indices of p µν are tangent to the boundary ∂ U . From the invariance of the path integral (4.3) under achange in integration variable [recall Eq. (3.5)] and the invariance of the measure R D [ g αβ ] under shifts of the form g µν → g µν + δg µν , one may show the following: Z D [ g αβ ] Z U (cid:18) R µν − R g µν (cid:19) δg µν p | g | d x e i ~ S GR [ g αβ ] = 0 ⇒ Z U (cid:18)Z D [ g αβ ] (cid:18) R µν − R g µν (cid:19) p | g | e i ~ S GR [ g αβ ] (cid:19) δg µν d x = 0 ⇒ Z D [ g αβ ] (cid:18) R µν − R g µν (cid:19) p | g | e i ~ S GR [ g αβ ] = 0 . (4.6)The second line comes from the fact that the shift function δg µν can be chosen independently of the integrationvariable g µν , and the last line can be inferred from the second line by requiring the second line to hold for all choicesof δg µν . The last line is the statement that in the absence of matter the vacuum Einstein field equations are satisfiedwithin the path integral. The variation of the path integral thus becomes δ K = i ~ Z D [ g αβ ] (cid:20)Z ∂ U (cid:18) P µν ∆ γ µν − H ν µ n ν δx µ (cid:19) d y (cid:21) e i ~ S GR [ g αβ ] , (4.7)where we have defined the following: P µν := ε κ p µν p | γ | = ε κ ( K µν − K γ µν ) p | γ | (4.8) H νµ := − ε κ (cid:20) ε D α p αβ γ µβ n ν + δ νµ (cid:0) R − ε ( K − K ij K ij ) (cid:1)(cid:21)p | γ | . (4.9)We choose δx µ to take the form δx µ = n µ ∆ t, (4.10)where ∆ t is a constant. We then make use of the expressions p µν ∆ γ µν = p ij ∆ γ ij and γ µβ n µ = 0 to obtain thevariation of the gravitational path integral, δ K = i ~ Z D [ g αβ ] (cid:20)Z ∂ U (cid:18) P ij ∆ γ ij − H gf ∆ t (cid:19) d y (cid:21) e i ~ S GR [ g αβ ] , (4.11)where H gf is given by (2.19), which we rewrite here: H gf = − κ (cid:20) R + K − K ij K ij (cid:21)p | γ | . (4.12)In the above expression, we have set ε = −
1, appropriate for the case of spacelike boundary surfaces in generalrelativity.5
B. Wave functional, operators, and commutators
We begin defining an initial state ψ J γ ijI K for some value of t I . The wave functional may be formally defined as:Ψ = Ψ J γ ij K ( t ) := Z D [ γ klI ] K J γ ij ; γ klI K ψ J γ klI K , (4.13)where we substitute t in place of t F (the surface Σ F will be denoted Σ t ) and R D [ γ klI ] denotes a functional integral overthe inverse 3-metric γ ijI on the surface Σ I (up to coordinate transformations on Σ I ). As before (in the derivationof the quantum mechanical Schr¨odinger equation), we consider an initial state Ψ J γ ijI K that is sharply peaked aroundsome particular function γ ijI to eliminate the functional integral R D [ γ klI ] (to recover the general result, reintroducethe functional integral R D [ γ klI ]). To first order, infinitesimal changes in the wave functional Ψ will take the form δ Ψ = Z Σ t δ Ψ δ Σ t γ ij ∆ γ ij d y + ∂ Ψ ∂t ∆ t, (4.14)where we use the notation δ/δ Σ ϕ to denote the functional derivative with respect to a function ϕ ( x ) restricted toa hypersurface Σ; for a hypersurface with the parametrization x µ ( y ) and the function ¯ ϕ ( y ) := ϕ | Σ = ϕ ( x ( y )), thismeans δδ Σ ϕ := δδ ¯ ϕ . (4.15)Upon comparing (4.15) with (4.11), we identify the following expressions: δ Ψ δ Σ t γ ij = i ~ Z D [ g αβ ] P ij | Σ t e i ~ S GR [ g αβ ] (4.16) ∂ Ψ ∂t = − i ~ Z D [ g αβ ] (cid:18)Z Σ t H gf d y (cid:19) e i ~ S GR [ g αβ ] . (4.17)Equation (4.16) suggests that the momentum operator ˆ P ij satisfiesˆ P ij Ψ := − i ~ δ Ψ δ Σ t γ ij . (4.18)On the other hand, Eq. (4.17) suggests that we may formally define the quantum theory by constructing a Hamiltonianoperator ˆ H GR such that its action on Ψ is formally equivalent to the following path integral:ˆ H GR Ψ = Z D [ g αβ ] (cid:18)Z Σ t H gf d y (cid:19) e i ~ S GR [ g αβ ] . (4.19)Since the wave functional Ψ is a functional of quantities defined on Σ t only, a valid Hamiltonian operator must bedefined in terms of operators at the surface Σ t . Otherwise, the time evolution may become nonlinear in the timederivatives, and one is no longer doing quantum physics. To construct a Hamiltonian operator, we must first identifythe operators that can be defined for the wave functional Ψ. In general, an operator ˆ O acting on Ψ must have a pathintegral expression of the form ˆ O Ψ = Z D [ g αβ ] O| Σ t e i S Pal / ~ , (4.20)where O| Σ tF is some quantity defined on Σ t F . We have already identified the momentum operator ˆ P ij in Eq. (4.18).We may define the inverse metric operator as follows (we explicitly write the arguments of Ψ for clarity ):ˆ γ ij Ψ J γ ij K ( t ) := γ ij | t Ψ J γ ij K ( t ) = Z D [ g αβ ] γ ij | t e i ~ S GR [ g αβ ] . (4.21) Since it depends on the formally defined measure R D [ γ klI ], the right-hand side of (4.13) is only defined in a formal sense. This isessentially the problem of defining the inner product (see Sec. 5.2.2 of Ref. [43], and also see Ref. [45] for a gauge-fixed definition of theinner product measure). If superspace has countable dimension, it may be possible in the superspace representation to define an innerproduct in the usual manner—this will be left for future investigation. Since we have required K J γ ijF ; γ ijI K to be independent of coordinate transformations on Σ I and Σ F , the wave functional Ψ J γ ij K must beindependent of coordinate transformations on Σ F . This is to indicate that the nonoperator quantities that appear outside of the wave functional depend on the values of the quantitiesthat appear in the arguments of the wave functional. γ ij is held fixed at Σ t in the path integral, we can pull factorsof γ ij | t into the path integral, since γ ij | t is effectively constant with respect to the functional integral R D [ g αβ ].Since the 3-metric γ ij and the volume element √ γ can in principle be obtained algebraically from γ ij , then we candefine a metric operator ˆ γ ij from ˆ γ ij , as well as the operators ˆ √ γ and (1 / ˆ √ γ ) in the following manner (we explicitlywrite the arguments of Ψ for clarity): ˆ γ ij Ψ J γ ij K ( t ) := γ ij | t Ψ J γ ij K ( t )ˆ √ γ Ψ J γ ij K ( t ) := √ γ | t Ψ J γ ij K ( t )(1 / ˆ √ γ ) Ψ J γ ij K ( t ) := (1 / √ γ ) | t Ψ J γ ij K ( t ) . (4.22)Since the three-dimensional Ricci scalar (3) R depends only on the values of the metric at the surface Σ F , we mayconstruct the following operator for the Ricci scalar (we explicitly write the arguments of Ψ for clarity): (3) ˆ R Ψ J γ ij K ( t ) := (3) R | γ ij ( t ) , t Ψ J γ ij K ( t ) . (4.23)To obtain commutation relations for the operators ˆ γ ij and ˆ P ij , we first examine the expressions [cf. (3.26)]:( ˆ P kl ˆ γ ij ) | t Ψ = − i ~ δδ Σ t γ kl ( γ ijF Ψ) = − i ~ (cid:18) δ ik δ jl δ ( y − y ′ ) Ψ + γ ij δ Ψ δ Σ t γ kl (cid:19) (ˆ γ ij ˆ P kl ) | t Ψ = − i ~ γ ij δ Ψ δ Σ t γ kl , (4.24)where we have made use of the expression in the first line, δ γ ij ( y ) δ Σ t γ kl ( y ′ ) = δ ik δ jl δ ( y − y ′ ) , (4.25)with δ ( y − y ′ ) being the Dirac delta function. We now write down the formal commutation relations: h ˆ γ ij ( y ) , ˆ P kl ( y ′ ) i = i ~ δ ik δ jl δ ( y − y ′ ) ˆ I . (4.26) C. Hamiltonian operator and the Wheeler-DeWitt equation
We begin by generalizing the computation (3.28) to quantum general relativity:(ˆ γ ij ˆ P kl ) | t Ψ = − i ~ γ ij δ Ψ δ Σ t γ kl = γ ij | Σ t Z D [ g αβ ] P kl | Σ t e i ~ S GR [ g αβ ] = Z D [ g αβ ] ( γ ij P kl ) | Σ t e i ~ S GR [ g αβ ] . (4.27)The above expression demonstrates that quantities of the form ( γ ij P kl ) | Σ t = ( P kl γ ij ) | Σ t that appear in the integrandof path integrals will yield the operator ordering (ˆ γ ij ˆ P kl ) | t .For path integrals containing higher factors of the momenta P ij , it is instructive to recall our earlier discussion ofsuperspace S (Σ) and its manifold extension S ex (Σ). Recall that in terms of the coordinates ξ a on extended superspace S ex (Σ) (which we assume to be of countable dimension) the gravitational Hamiltonian H GR is the Hamiltonian fora particle on an infinite-dimensional pseudo-Riemannian manifold. In the superspace representation, we demandthat the wave function Ψ( ξ ) is invariant under coordinate transformations on S ex (Σ), which suggests the followingcorrespondence [cf. Eq. (3.34)]: Z D ξ ( G ab P a P b ) | t e ( i/ ~ ) S [ q ] = ( − i ~ ) ( ˆ G ab ∇ a ∇ b ) | t Ψ . (4.28)This is the Laplace-Beltrami operator on the extended superspace S ex (Σ). In the superspace representation, theHamiltonian operator H GR takes the form [cf. Equation (2.30)]:ˆ H GR = ε ~ κ ∞ X a =1 ∞ X b =1 ˆ G ab ∇ a ∇ b ! + ˆΦ , (4.29) We drop the absolute value symbols since we now assume ε = − It should be mentioned that our approach differs from that suggested by DeWitt in Ref. [19], who proposed using a Laplace-Beltramioperator on the six-dimensional manifold coordinatized by the 3-metric components γ ij (as opposed to the inverse 3-metric functions γ ij ( y )). Our approach instead uses the Laplace-Beltrami operator on the infinite-dimensional extended superspace manifold S ex (Σ). ξ ) in (2.31).To obtain the standard representation, we construct a map ξ a J γ ij K : iRiem (Σ) → S ex (Σ) and an inverse map γ ij ( y, ξ ) : S ex (Σ) → iRiem (Σ). We then make use of the chain rule δδγ ij = ∞ X c =1 δξ c δγ ij ∂∂ξ c ∂∂ξ c = Z Σ d y ∂γ ij ( y, ξ ) ∂ξ c δδγ ij (4.30)to obtain the expression for the action of ˆ H GR on the wave functional Ψ = Ψ J γ ij K ,ˆ H GR Ψ = 12 κ Z Σ t (cid:20) ε ~ G ijkl δδγ ij δ Ψ δγ kl + ε ~ C mn ( y, ξ J γ ij K ) δ Ψ δγ mn − ˆ R ˆ p | γ | Ψ (cid:21) d y, (4.31)where any “unhatted” factors are multiplicative operators, and we define C mn ( y, ξ J γ ij K ) := ∞ X a =1 ∞ X b =1 ∞ X c =1 G ab (cid:20)Z Σ t (cid:18) ∂γ pq ( z, ξ ) ∂ξ a δξ c δγ pq ∂ γ mn ( y, ξ ) ∂ξ c ∂ξ b (cid:19) d z − Γ cab ∂γ mn ∂ξ c (cid:21) , (4.32)where Γ cab is given by (2.29). The meaning of the second functional derivative (4.31) will be discussed in an upcomingpaper.We now return to Eq. (4.17), which becomes [using (4.12)]ˆ H GR Ψ = 12 κ Z D [ g αβ ] (cid:18)Z Σ t (cid:20) K ij K ij − K − R (cid:21) √ γ d y (cid:19) e i ~ S GR [ g αβ ] . (4.33)The Hamiltonian constraint forms part of the Einstein field equations, and with the choice α = 1 and β i = 0 (validon the boundary ∂ U ), we may write 2 R − R g = K ij K ij − K − R. (4.34)If we recall the earlier result (4.6), namely, that the vacuum Einstein field equations are satisfied within the pathintegral, Z D [ g αβ ] (cid:18) R µν − R g µν (cid:19) p | g | e i ~ S GR [ g αβ ] = 0 , (4.35)we find that (4.34) and (4.35) imply that the right-hand side of (4.33) must vanish, soˆ H GR Ψ = 0 ⇒ κ Z Σ t (cid:20) ε ~ G ijkl δδγ ij δ Ψ δγ kl + ε ~ C mn ( y, ξ J γ ij K ) δ Ψ δγ mn − ˆ R ˆ p | γ | Ψ (cid:21) d y = 0 . (4.36)Since the Hamiltonian operator determines the time evolution [recall (4.17)], this suggests that the wave functionalΨ must be independent of the time parameter t : ∂ Ψ ∂t = 0 . (4.37)The time independence of the wave functional presents both conceptual and technical difficulties. This is known asthe “problem of time” in the canonical formulation of quantum general relativity. A full discussion of the conceptualand technical aspects of the problem of time in the canonical theory (in particular for spacetimes without spatialboundary) is beyond the scope of this paper—we refer the reader to the review articles [46] and also Ref. [43] for abrief overview. Our results explicitly demonstrate that the problem of time also persists in the path integral approachto quantum general relativity in the case of spacetimes without a spatial boundary. Other derivations of the Wheeler-DeWitt equation, for instance the derivation in Ref. [1], might also lead one to infer that the problem of time persists We will later provide an example of how one might construct such a map. A coordinate condition on γ ij ( y ) is needed in order to map an element of S ex (Σ) uniquely to an element of iRiem (Σ). g ). We will later demonstrate that one can gain some degree of control over this issue in the case ofspacetimes with a spatial boundary.We stress that (4.36) is not what is usually referred to as the Wheeler-DeWitt equation; the Wheeler-DeWittequation [20], [19] is the local counterpart to (4.36), which one may infer from the 00 component of (4.35),2 κ ˆ H gf Ψ = ε ~ G ijkl δδγ ij δ Ψ δγ kl + ε ~ C mn ( y, ξ J γ ij K ) δ Ψ δγ mn − ˆ R ˆ p | γ | Ψ = 0 , (4.38)where C mn ( y, ξ J γ ij K ) is defined in Eq. (4.32). Again, the meaning of the second functional derivative in (4.38) willbe discussed in an upcoming paper. Note that this equation is local in y i , rather than a global one over the whole ofΣ t . This form of the Wheeler-DeWitt equation differs from the form presented by DeWitt in Ref. [19]; in particular,DeWitt argues that one can ignore operator ordering issues by requiring that multiple functional derivatives δ/δγ ij vanish when acting on the same spacetime point. It was later argued by Tsamis and Woodard that one can indeedignore operator ordering issues when employing dimensional regularization—see Ref. [48]. In such cases, we may dropthe term containing C mn ( y, ξ J γ ij K ).Before proceeding, we note that the 0 i component of (4.35) leads to the momentum constraint γ ik D k κ p | γ | δ Ψ δγ ij ! = 0 , (4.39)which follows from (2.16). One may infer from the general Weiss variation (2.12) that this constraint corresponds tothe invariance of the wave functional Ψ J γ ij K under coordinate transformations on Σ t ; in other words, the functionalΨ J γ ij K must be covariant. Earlier, we required the measure D [ g αβ ] to be defined so that the wave function has thisproperty, but if one were to seek functionals Ψ J γ ij K that solve the Wheeler-DeWitt Eq. (4.38) in the absence of apath integral definition, one should check that the solutions Ψ J γ ij K also satisfy the constraint (4.39). V. QUANTUM GENERAL RELATIVITY FOR SPACETIMES WITH SPATIAL BOUNDARYA. Functional integral and its variation
We now consider what happens when we consider path integrals for regions of spacetime W with a spatial boundary,as described by Fig. 1. In particular, we begin with a functional integral of the form [with S GR, B defined in (2.34)], K J h ijI ; h ijF ; q ab K = Z D [ g αβ ] e i ~ S GR, B [ g αβ ] , (5.1)where h ijI is the induced metric on Σ I , h ijF is the induced metric on Σ F , and q ab is the induced metric on B . Here,we view the path integral K J h ijI ; h ijF ; q ab K as a transition amplitude for the gravitational field, subject to the spatialboundary condition that the induced metric on B is given by q ab . This viewpoint is essentially the same as that ofthe general boundary formulation of quantum field theory [7, 49], in which the state of a quantum field is specifiedon the (connected) boundary of a compact region of spacetime—as pointed out in Refs. [8, 9], this approach avoidsthe problem of time, since time evolution is specified by the boundary conditions. In the remainder of this section,we will demonstrate how the boundary conditions specify time evolution.For any functional of the form K J h ijI ; h ijF ; q ab K , one may write δ K = Z Σ I δ K δh ijI ∆ h ijI d y + Z B δ K δq ab ∆ q ab d y + Z Σ F δ K δh ijF ∆ h ijF d y, (5.2)where ∆ h ijI , ∆ q ab , and ∆ h ijF are variations of the respective functions h ijI , q ab , and h ijF . The methods outlined in thepreceding section may be used to obtain path integral expressions for the functional derivatives. To do so, we considerboundary conditions { h ijI ; h ijF ; q ab } that admit solutions of the vacuum Einstein field equations. The variation of thepath integral K J h ijI ; h ijF ; q ab K takes the form δ K = i ~ Z D [ g αβ ] ( δS GR, B ) e i ~ S GR, B [ g αβ ] . (5.3)9As before, we require the variations to satisfy δx µ | S I = 0, δx µ | S F = 0, δg µν | S I = 0, and δg µν | S F = 0, and again, wealso require h n I , n B i| S I and h n F , n B i| S F to be held fixed under the variations so that the variation of the Haywardterm vanishes. From the arguments in the preceding section, we may infer that the Einstein field equations (rescaledby a factor of the volume element p | g | ) are satisfied inside the path integral so that Z D [ g αβ ] (cid:18) R µν − R g µν (cid:19) p | g | e i ~ S GR, B [ g αβ ] = 0 . (5.4)It follows that the Hamiltonian and momentum constraints (which are formed from the Einstein field equations) aresatisfied inside the path integral. Particularly important are the Hamiltonian constraints, which take the followingform (again, recall that an underline denotes quantities defined on B ) Z D [ g αβ ] H I e i ~ S GR, B [ g αβ ] = − κ Z D [ g αβ ] (cid:20)(cid:0) R + ( K − K ij K ij ) (cid:1) √ h (cid:21) Σ I e i ~ S GR, B [ g αβ ] = 0 Z D [ g αβ ] H B e i ~ S GR, B [ g αβ ] = − κ Z D [ g αβ ] (cid:20)(cid:16) R − ( K − K ab K ab ) (cid:17) √ q (cid:21) B e i ~ S GR, B [ g αβ ] = 0 Z D [ g αβ ] H F e i ~ S GR, B [ g αβ ] = − κ Z D [ g αβ ] (cid:20)(cid:0) R + ( K − K ij K ij ) (cid:1) √ h (cid:21) Σ F e i ~ S GR, B [ g αβ ] = 0 . (5.5)where the Hamiltonian densities H I , H B and H F are given in Eq. (2.40). Using (2.38) and the above constraints,we find that the variation of the transition amplitude K takes the form (again, we stress that I and F are labels, notindices to be summed over) δ K = i ~ Z D [ g αβ ] (cid:18)Z Σ I P Iij ∆ h ijI d y + Z B P ab ∆ q ab d y + Z Σ F P Fij ∆ h ijF d y (cid:19) e i ~ S GR, B [ g αβ ] , (5.6)where [cf. Eq. (2.39)] P Iij := − κ ( K ij − K h
Iij ) p h I P ab := 12 κ ( K ab − K q ab ) p | q | P Fij := − κ ( K ij − K h
Fij ) p h F . (5.7)We now compare (5.6) with the general expression (5.2) for the variation δ K to establish a relationship between thefollowing operators and their corresponding path integral expressions:ˆ P Iij K J h ijI ; h ijF ; q ab K = − i ~ δ K δh ijI = Z D [ g αβ ] P Iij e i ~ S GR, B [ g αβ ] ˆ P B ab K J h ijI ; h ijF ; q ab K = − i ~ δ K δq ab = Z D [ g αβ ] P ab e i ~ S GR, B [ g αβ ] ˆ P Fij K J h ijI ; h ijF ; q ab K = − i ~ δ K δh ijF = Z D [ g αβ ] P Fij e i ~ S GR, B [ g αβ ] . (5.8)We may define the following operators:ˆ h ijI K J h ijI ; h ijF ; q ab K = h ijI K J h ijI ; h ijF ; q ab K ˆ q ij K J h ijI ; h ijF ; q ab K = q ij K J h ijI ; h ijF ; q ab K ˆ h ijF K J h ijI ; h ijF ; q ab K = h ijF K J h ijI ; h ijF ; q ab K . (5.9)Equations (5.5) then demand that the functional K J h ijI ; h ijF ; q ab K must satisfy the following functional differentialequations, which we collectively call the extended Wheeler-DeWitt equations , since we extend the formalism to spatial0boundaries (for simplicity, we suppress the argument of K J h ijI ; h ijF ; q ab K ),2 κ ˆ H I K = − ~ G ijklI δδh ijI δ K δh klI − ~ C mn ( y, ξ J h ijI K ) δ K δh mnI − ˆ R I ˆ p h I K = 02 κ ˆ H B K = ~ G abcd δδq ab δ K δq cd + ~ C ab ( y, ξ J q cd K ) δ K δq ab − ˆ R B ˆ p | q | K = 02 κ ˆ H F K = − ~ G ijklF δδh ijF δ K δh klF − ~ C mn ( y, ξ J h ijF K ) δ K δh mnF − ˆ R F ˆ p h F K = 0 , (5.10)where C ab ( y, ξ J q cd K ) is given by the formula (4.32) with all quantities replaced by underlined quantities and γ ij replaced with q ab . Equations (5.10) form the extended Wheeler-DeWitt equations for spacetimes with a spatialboundary. To ensure that K = K J h ijI ; h ijF ; q ab K is invariant under coordinate transformations on the surfaces Σ I , Σ F ,and B , the extended Wheeler-DeWitt equations must be supplemented by the momentum constraints: h ikI D k κ √ h I δ K δh ijI ! = 0 q ac D c κ p | q | δ K δq ab ! = 0 h ikF D k κ √ h F δ K δh ijF ! = 0 . (5.11)The problem of finding the dependence of the transition amplitude K on h ijI , h ijF , and q ab amounts to solving thesefunctional differential equations.It should be cautioned that Eqs. (5.10) and (5.11) may not be sufficient to fully determine the transition amplitude K . So far, we have ignored the dependence of the transition amplitude on the 2-surfaces S I and S F , in particular,the role of the rapidity angles η I and η F [defined in (2.36)]. A simple example suggests that meaningful transitionamplitudes must also depend on the rapidity angles η I and η F . Consider the Schwarzschild solution in Painlev´e-Gullstrand coordinates [17, 50, 51]: ds = − (cid:18) − GMr (cid:19) dT + 2 r GMr dT dr + dr + r ( dθ + sin θ dφ ) . (5.12)One may construct a boundary in the manner of Fig. 1 from cylindrical hypersurfaces of constant r and (flat)hypersurfaces of constant T that is isometrically embeddable in Minkowski spacetime; the induced metrics h ijI , h ijF ,and q ab will be identical for Schwarzschild spacetime and Minkowski spacetime. Meaningful transition amplitudestherefore cannot be solely dependent on h ijI , h ijF , and q ab . On the other hand, note that due to the cross-terms in(5.12), the rapidity angles η I and η F (2.36) at the corners of the cylinder (the 2-surfaces S I and S F ) differ betweenSchwarzschild spacetime and Minkowski spacetime. This suggests that meaningful transition amplitudes must dependon the rapidity angles η I and η F at the 2-surfaces S I and S F (in addition to the induced metric on S I and S F ). To obtain the dependence of the transition amplitude K on the rapidity angles, one must also consider variations ofthe corner terms S C in the gravitational action and the variations due to displacements of the 2-surfaces S I and S F .This will be left for future work. B. Spatial boundaries and time evolution
One obstacle to addressing the question of time in the case of spacetimes without spatial boundaries is the factthat the path integral involves summing over all functional forms (up to diffeomorphisms) for the spacetime metric g µν . Consider a timelike geodesic in U , defined by initial conditions (the initial position and 4-velocity) at a point Upon closer analysis, one may note that, while the surfaces of constant T are flat, they each contain the point r = 0, which correspondsto the location of the Schwarzschild singularity. The questions of whether a cylindrical boundary constructed from a surface of constant T is admissible, and whether one must include an additional boundary surface formed from excising curvature singularities from themanifold (which may require prior assumptions about the structure of singularities and topology of the manifold), are left for futureanalysis. I . The proper time along the geodesic segment between the hypersurfaces Σ I and Σ F depends on the spacetimemetric g µν . The path integral over g µν therefore prevents one from unambiguously establishing a notion for the timeelapsed between Σ I and Σ F .In spacetimes with spatial boundary, the aforementioned obstacle may be used to provide a resolution for theproblem of time. Recall that the spatial boundary conditions for the metric tensor g µν are provided by the inducedmetric q ab , which in turn specifies the geometry of the boundary surface B . Now, consider a timelike geodesic on B (a geodesic with respect to the boundary metric q ab ), defined by an initial 3-velocity and an initial starting point on S I . The induced metric q ab determines the elapsed proper time along the geodesic segment between the surfaces S I to S F . The boundary metric q ab therefore provides a measure of time elapsed between the hypersurfaces Σ I and Σ F .This suggests the following view: time should not be treated as a local parameter but as a property of the geometryof the spatial boundary B . More explicitly, one may imagine describing time evolution as a displacement of the surface Σ F in the future timedirection, with a corresponding “stretch” of the boundary B [53]. One may note, however, that the displacementof the surface Σ F by way of δx µ is a diffeomorphism and is mathematically indistinguishable from an infinitesimalcoordinate transformation. On the other hand, a stretch of the boundary B in the timelike direction that correspondsto an increase in the proper time of timelike geodesics cannot be represented as a change in coordinates. In fact,we may ignore boundary displacements altogether and characterize the stretch in the boundary by changing thecomponents of the boundary metric q ab .To illustrate how the components of the boundary metric q ab may be used to stretch the boundary in the timelikedirection, we consider a boundary B with coordinates ( t, θ, φ ), with the domain t < t < t < θ < π < φ < π. (5.13)We place a metric q ab on B , which admits the following line element: ds = q ab dy a dy b = − α dt + r (cid:0) dθ + sin θ dφ (cid:1) . (5.14)The boundary metric q ab corresponds to a particular set of boundary conditions on B . If α and r are constants, thenthe proper time along a timelike geodesic on B defined by θ = constant and φ = constant is given by the expression T = α ( t − t ). The physical stretching of the boundary B in the timelike direction corresponds to an increase inthe value of α . Now, one might note that changes in the value of α may also be interpreted as a rescaling of thetime coordinate t , which in turn may be interpreted as a coordinate transformation. However, what distinguishes ourconstruction from a coordinate transformation is that the domain (5.13) of the coordinates on the manifold B is heldfixed—in particular, the coordinate values t < t < t that define the boundary B are held fixed. The only thing wechange is the component of the metric q = − α .To see how such a stretch in the boundary affects the action (and by extension the path integral K J h ijI ; h ijF ; q ab K ),consider the 3-volume for B , which may be written as V ( B ) = 4 π Z t t α r dt = 4 π r T, (5.15)where, again, T = α ( t − t ) is the proper time of a geodesic defined by defined by θ = constant and φ = constant.For the boundary metric in (5.14), the formula above establishes a relationship between the proper time of certainobservers on the boundary and the 3-volume V ( B ). Note that the presence of the Gibbons-Hawking-York boundaryterm in the gravitational action S GR, B ensures that, even for vacuum solutions of the Einstein field equations, thegravitational action S GR, B has a nonvanishing value. The Gibbons-Hawking-York boundary term for a spacetimewith the spatial boundary B with line element (5.14) will ultimately depend on α , since the 3-volume (in particular,the volume element for B ) depends on α . The path integral K J h ijI ; h ijF ; q ab K for the same spatial boundary B will in One might recognize the relationship between this viewpoint and Mach’s principle (see Ref. [52] for a modern discussion of Mach’sprinciple in GR), noting that the geometry of the spatial boundary B is determined by the spacetime geometry outside of the region W , which in turn depends on the matter configuration outside of W . The definition of time as a property of the boundary geometry for B suggests that time is fundamentally nonlocal quantity that depends on the spacetime geometry and matter configurations in distantregions. The results and arguments presented in this section are equivalent to those of Brown and York in Ref. [53]. To better see that displacements of the boundary are not sufficient to describe physical time evolution, consider the action S GR, B [ g µν ]evaluated on a vacuum solution (in which the Ricci scalar takes the value R = 0) with a cylindrical boundary as described in Fig. 1.Displace a portion of the boundary Σ F in the normal direction by a function δx µ ( y ) that vanishes on S F . At the same time, perform aninfinitesimal variation of the bulk metric tensor g µν under the condition that the boundary metric γ ij is held fixed when the boundaryis displaced; in other words, we perform a variation such that δx µ = 0 and ∆ γ ij = 0. Equation (2.38) suggests that the variation δS GR, B [ g µν ] vanishes by virtue of the Hamiltonian constraint H I = 0 [where H I is defined in (2.40)] on the boundary. Displacementsof the boundary surface (in particular the t F surface Σ F ) have no effect on the value of the action S GR, B [ g µν ] (but stretching of theboundary surface B does). α as well; time evolution for spacetimes with spatial boundaries admitting a line element of the form(5.14) corresponds to an increase in α .For more general boundary geometries, we may consider a boundary metric q ab written in a form adapted to thefoliation induced by the coordinate t . In particular, we construct the line element on B ds = q ab dy a dy b = − ( α + σ AB β A β B ) dt + σ AB β A dz B dt + σ AB dz A dz B , (5.16)where z = θ , z = φ , σ AB is the induced metric on (two-dimensional) surfaces of constant t , and the quantities α and β A are in general functions of t and z A . A reader familiar with the Arnowitt-Deser-Misner(ADM) formalism [16, 54]will recognize the above as the ADM decomposition for the boundary line element ds = q ab dy a dy b . As before, astretch in the boundary corresponds to an increase in the value of boundary “lapse” function α . That an increase in α corresponds to a stretch in the boundary can be seen in the expression for the 3-volume of B V ( B ) = Z B p | q | d y = Z t t Z π Z π α p | det( σ AB ) | dθ dφ dt, (5.17)where we have made use of the expression p | q | = α p | det( σ AB ) | . From the above expression for the 3-volume V ( B ),it is clear that an overall increase in the value of α will increase the 3-volume of the boundary B .The notion that the lapse function α governs time evolution has been explored before in Ref. [53]; variations ofthe action with respect to α were used to obtain the Brown-York quasilocal energy. We may obtain the Brown-Yorkquasilocal energy (up to a reference term) from the expression (2.38) for δS GR, B by first noting that the components q ab depend on α in the following way: q = − α − q A = α − β A q AB = − α − β A β B + σ AB . (5.18)The Brown-York quasilocal energy /κ R S ¯ n a ¯ n b ( K ab − K q ab ) p | σ AB | d z is given by the integral (over a spacelike2-surface S ⊂ B ) of the functional derivative α p | q | δS GR, B δα = 2 1 α p | q | (cid:18) δS GR, B δq − δS GR, B δq A β A + δS GR, B δq AB β A β B (cid:19) = 1 κ ¯ n a ¯ n b ( K ab − K q ab ) , (5.19)where [¯ n a ] := (1 /α, − β A /α ) are vectors tangent to B that have unit norm and are normal to surfaces of constant t . Following Ref. [53], one may obtain similar expressions for momentumlike and stresslike quantities by performingvariations with respect to β A and σ AB .Finally, the above expression for the variation may be used to obtain the change in the transition amplitude withrespect to changes in α , ˆ α δ K δα = − (cid:16) ˆ P B − ˆ β A ˆ P B A + ˆ β A ˆ β B ˆ P B ab (cid:17) K J h ijI ; h ijF ; q ab K , (5.20)where the operators ˆ α and ˆ β A pick out the value of the quantities α = −| q | − / and ˆ β A = − q A /q from theinverse boundary metric q ab : ˆ α K J h ijI ; h ijF ; q ab K = −| q | − / K J h ijI ; h ijF ; q ab K ˆ β A K J h ijI ; h ijF ; q ab K = − q A /q K J h ijI ; h ijF ; q ab K . (5.21)Equation (5.20) is a form of the boundary “Schr¨odinger equation” in Ref. [6]. Though it is tempting to regard (5.20)as the Schr¨odinger equation for the transition amplitude K J h ijI ; h ijF ; q ab K , it is in fact a kinematical expression as itmerely expresses one functional derivative of K J h ijI ; h ijF ; q ab K in terms of other functional derivatives [more pointedly,(5.20) is simply a statement of the chain rule for functional derivatives]. Equation (5.20) therefore does not determine We use the definition given in Eq. 4.3 of Ref. [53]. Our result (5.21) uses a slightly more general form of the Brown-York quasilocal energy than that of the boundary Schr¨odinger equationin Ref. [6]. K J h ijI ; h ijF ; q ab K satisfies the extended Wheeler-DeWitt equations(5.10) and the momentum constraints (5.11). The Wheeler-DeWitt equations (5.10) and the momentum constraints(5.11) determine the functional dependence of the transition amplitudes K J h ijI ; h ijF ; q ab K on h ijI , h ijF , and q ab . Oncethe solutions to (5.10) and (5.11) are found, Eq. (5.20) may then be used to extract the explicit time dependence forthe transition amplitude K J h ijI ; h ijF ; q ab K . C. Dependence of transition amplitude on spatial boundary conditions
We briefly describe how the formalism presented in this paper might be used to compute the dependence of transitionamplitudes on spatial boundary conditions. If the spatial boundary B has the cylindrical topology ¯ R × S (recall that¯ R is a compact subset of the real line R ) and the appropriate boundary conditions are imposed on S I and S F , onemay decompose the induced metric in the manner q ab ( t, θ, φ ) = q ab ( t, θ, φ ) + X lmn ( A ml n sin( π n t/T ) + B ml n sin( π n t/T )) Y ml ( θφ ) , (5.22)where A ml n and B ml n are constant coefficients, T is the length of the real line segment ¯ R , and q ab ( t, θ, φ ) are thecomponents of inverse metric for the cylindrical line element: ds = − dt + r (cid:0) dθ + sin θ dφ (cid:1) . (5.23)Since (5.22) is formed from a complete, orthogonal basis for functions on ¯ R × S , the coefficients A ml n and B ml n maybe used to coordinatize the manifold S ex ( B ). This may be used to convert the spatial boundary Wheeler-DeWittEq. (5.10) from a functional differential equation to an infinite number of PDEs on an infinite-dimensional manifold(note that the spatial boundary Wheeler-DeWitt equation is a local equation in the metric basis—it is a function of y ∈ B ). One may truncate the series (5.22) to obtain a finite number of PDEs on a finite-dimensional manifold; ifthese can be solved, one can obtain the dependence of the transition amplitudes on spatial boundary conditions upto truncation errors. The remaining Wheeler-DeWitt equations (5.10) may be solved in a similar manner to obtainthe dependence of the transition amplitude on h ijI and h ijF .A potentially tricky aspect of this procedure is to obtain the functional ξ c J q ab K , which is a map from the space of3-metrics q ab ( y ) to the coordinates ξ c on S ex ( B ). The orthogonality of the basis functions, which we write as e c ( y )may be exploited to obtain the functionals ξ c J q ab K from the functions q ab ( y ). Using the shorthand ¯ q ( y ) to represent q ab ( y ) (we suppress indices for simplicity), we may write ¯ q ( y ) = ξ c e c ( y ). The functional ξ c J ¯ q K may then be writtenas (no sum over c ) ξ c J ¯ q K = R B ¯ q ( y ) e c ( y ) d y R B e c ( y ) e c ( y ) d y . (5.24)This may be varied in order to obtain an expression for δξ c /δq ab in the chain rule formula (4.30). In turn, one mayuse the chain rule (4.30) to rewrite the Wheeler-DeWitt equation on B (5.10) in terms of coordinates on superspace S ex ( B ). VI. SUMMARY AND FUTURE WORK
We claim that when spatial boundaries are included transition amplitudes in quantum general relativity satisfy theextended Wheeler-DeWitt Eq. (5.10) and the momentum constraint (5.11), which in turn determine the dependenceof the transition amplitude on the components of the boundary metric q ab , which constitute boundary conditionsfor the metric tensor g µν at the spatial boundary B . We have argued that time evolution for transition amplitudescorresponds to a stretching of the spatial boundary B in the timelike direction, which in turn may be described by One should be aware that two sets of values for the coefficients may correspond to the same point in S ex ( B ), as they may describeequivalent 3-geometries that differ by a coordinate transformation. Since the transition amplitude satisfies more than one equation, the system is overdetermined, and there is a danger that truncationmay lead to an inconsistent set of equations. It may be necessary to multiply the Wheeler-DeWitt equation by a factor of the inverse metric q ij ( y ); the dependence on the coordinate y acts as part of a “free index”, and the factor of the inverse metric q ij ( y ) allows one to convert the Wheeler-DeWitt equation to a setof PDEs indexed by the superspace coordinate index c . q ab . In short, we find that spatial boundary conditions determinetime evolution in quantum general relativity; our results formalize, validate, and sharpen the general idea[8, 9] thattime evolution is determined by boundary conditions for a connected boundary of a compact spacetime.As argued at the end of Sec. V A, the formalism presented in this paper is not sufficient, even at a formal level,to fully determine the transition amplitude K ; further development of the formalism is needed. Critically importantis the dependence of K on the rapidity angles η I and η F and the induced metric at the 2-surfaces S I and S F . Todo this, one must obtain the variation of S GR, B when variations in the rapidity angles η I and η F , variations in theinduced metric on S I and S F , and the displacement of the surfaces S I and S F are included.We have briefly described a way to convert the functional Wheeler-DeWitt equation to a set of PDEs on an infinite-dimensional manifold; one may truncate the function space in order to obtain PDEs on a finite -dimensional manifold.The next step is to write down the explicit expressions for the resulting PDEs and to study their general properties.This may be attempted for both the case of spatially compact 3-geometries without a boundary and compact regionsof spacetime with a spatial boundary. ACKNOWLEDGMENTS
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As stated earlier, the full justification for Eq. (2.12) is given in Ref. [14]. However, for the benefit of the reader,we present a partial justification for (2.12) in this Appendix, valid for spacelike boundary surfaces ( ε = − infer Eq. (2.12) from results in the literature (excluding Ref. [14]) and theWeiss variation formula (2.8) for mechanical systems, which we rewrite here [setting δq i ( t ) = λ η i ( t )]: δS = Z t t (cid:18) ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19)(cid:19) δq i ( t ) dt + (cid:0) p i ∆ q i − H ∆ t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t t . (A1)We begin by recalling that for the gravitational action in (2.9) the quantity to be held fixed on the boundary ∂ U in Hamilton’s principle is the induced metric γ ij [55]. Equivalently, we may instead require the inverse metric γ ij to be held fixed on the boundary in Hamilton’s principle. This suggests that a natural choice for the configurationvariables in gravity is the inverse induced metric γ ij ( x ). From Ref. [55], the variation of the gravitational action (2.9)is (excluding boundary displacements) δS GR = 12 κ Z U G µν δg µν p | g | d x + ε κ Z ∂ U ( K ij − K γ ij ) δγ ij p | γ | d y. (A2)The reader should be aware that the definitions for K ij and K in Ref. [55] differ from the ones here by a sign. Also,the boundary variation in Ref. [55] is expressed in terms of δγ ij ; we use the first-order expression δγ ij = − γ ia γ jb δγ ij to obtain (A2). Equation (A2) suggests the definition for the conjugate field momentum tensor p ij := K ij − K γ ij , (A3)which is equivalent to the definition in (2.13).6It is well known (see Refs. [17, 51]) that for spacetimes without a spatial boundary the Hamiltonian density for thegravitational field may be written in the form H = α H gf + β i C i , where α and β i are the respective lapse functionand shift vector of the ADM formalism [54], H gf is gauge-fixed Hamiltonian density of (2.19), and C i is C i := − κ D j p ij p | γ | (A4)Explicitly, the gravitational Hamiltonian on a boundaryless spacelike hypersurface Σ t is [17, 51] H t = 12 κ Z Σ t (cid:2) α ( K ij K ij − K − R ) − β i D j ( K ij − K γ ij ) (cid:3) p | γ | d y. (A5)The basis vector ∂/∂t has components t µ = δ µ and may be decomposed in the manner t µ = ε α n µ + β µ , (A6)where α satisfies α = t µ n µ and β µ = γ µν t ν are the components of the shift vector in the bulk coordinate basis. Since β i = γ ij β j , its bulk-basis counterpart is γ µν β ν = γ µν t ν . We may therefore rewrite the Hamiltonian (A5) in thebulk basis H t = 12 κ Z Σ t (cid:2) n µ t µ ( K ij K ij − K − R ) − γ µν t µ D ρ ( K νρ − K γ νρ ) (cid:3) p | γ | d y, (A7)which we rewrite as H t = 12 κ Z Σ t (cid:2) n µ ( K ij K ij − K − R ) − γ µβ D α p αβ (cid:3) t µ p | γ | d y. (A8)Given formulas (A1), (A2) and (A8), one may infer that for ∂ U = Σ t ∪ Σ t (with Σ t and Σ t spacelike andboundaryless) the Weiss variation takes the form δS GR = 12 κ Z U G µν δg µν p | g | d x − κ Z ∂ U p ij ∆ γ ij p | γ | d y + ( H t ∆ t ) | t t (A9)where ∆ t (assumed to be infinitesimal and constant over the boundary) is the amount by which the boundary surfaceis displaced in the coordinate t . We identify the displacement vector δx µ = t µ ∆ t and write δS GR = 12 κ Z U G µν δg µν p | g | d x − κ Z ∂ U (cid:0) p ij ∆ γ ij + (cid:2) n µ ( K ij K ij − K − R ) − γ µβ D α p αβ (cid:3) δx µ (cid:1) p | γ | d y. (A10)Since (A10) applies for any choice of time coordinate t with spacelike hypersurfaces, it can be used to describe generalboundary displacements, provided that the boundary surfaces are spacelike. Equation (A10) is therefore equivalentto (2.12) for spacelike boundary surfaces ( ε = − A word of caution: while β = 0, the lowered component β = 0. In fact, it is not too difficult to show that β is the norm of β ii