The Functional Schrodinger Equation in the Semiclassical Limit of Quantum Gravity with a Gaussian Clock Field

We derive the functional Schrodinger equation for quantum fields in curved spacetime in the semiclassical limit of canonical quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit by a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrodinger equation that we obtain exhibits a time derivative that completes the usual definition of so-called"many-fingered time,"so that the usual Schrodinger-type evolution is recovered for quantum fields in Minkowski spacetime.


I. INTRODUCTION
The Hamiltonian dynamics of quantum fields on a classical curved spacetime of the kind envisioned by Tomonaga and Schwinger [1,2] can be recovered as a WKB-like approximation of canonical quantum gravity [3,4] in the form of a functional Schrödinger equation (we work with units where c = = 1) for the wave functional χ of matter fields governed by the Hamiltonian H φ on the spatial hypersurface Σ [5,6] (see [7] for a recent comparison of this approach to the Born-Oppenheimer method). Here γ ab are the components of the spatial metric, G abcd is the DeWitt super-metric, and S 0 is a solution to the Hamilton-Jacobi-Einstein equation that describes the background geometry. However, the locally defined "mani-fingered time" τ [γ ab (x)] (a functional of the spatial metric) thus emerging at the first level of approximation of the quantum matter field dynamics, and for the sake of convenience identified ad hoc with that appearing in the Tomonaga-Schwinger equation, most remarkably fails for the simplest case, the flat Minkowski spacetime, in which case S 0 is a constant. This important issue of the semiclassical limit of quantum gravity seems to have drawn little to no attention in the literature.
In this paper, we would like to draw attention to this point and provide one viable solution. We do so by considering the semiclassical approximation of the Schrödinger-type evolution for quantum geometrodynamics that is obtained by explicitly introducing a physical reference clock field in the model, in the form of a Gaussian reference dust [8]. The Gaussian time condition is imposed before variation of the action and time reparametrization invariance is recovered by parametrizing the new action and promoting Gaussian time to a scalar field labeled by new free spacetime coordinates. The semiclassical limit of the quantized theory results in a modified functional Schrödinger equation for the quantum state of matter fields as observed on the hypersurfaces determined by the clock field. The time derivative operator in (1) is, in this case, naturally completed into the analogous of a local "material derivative" acting on the quantum state of the matter fields that evolves on the "medium" of a background super-spacetime.

II. CANONICAL REFERENCE DUST
In this section we will briefly review the elements of the canonical description of the Gaussian reference fluid in the context of quantum geometrodynamics relevant to our discussion. As we are only interested in the Gaussian time condition and keep the spatial frame unaffected by reparametrization, we will closely follow a simplified version of [8], which we recommend to the reader for its in-depth analysis of the various issues in the classical and quantum regime related to employing a phenomenological fluid to implement coordinate conditions and address the definition of geometrodynamical observables. For a recent study of the perturbation theory of Gaussian dust in a FLRW cosmological scenario, see [9].
In the Lagrangian formalism of general relativity, the Einstein field equations can be derived from the Einstein-Hilbert action by variation of the spacetime metric g µν (x) of the unbounded manifold M. We take a vanishing cosmological constant for simplicity and do not yet introduce explicitly the gravitational scale before the action in order to simplify notation.
The diffeomorphism invariance of the theory allows for any choice of spacetime coordinates. Consider the Gaussian time condition which fixes as constant everywhere the normal proper time separation between hypersurfaces of constant times. This choice can be imposed before the variation of the action, by introducing the coordinate time condition (3) through a Lagrange multiplier L = L(x) at the level of the action (2). This results in an extra term The broken time reparametrization invariance can be restored by parametrization of the action: we promote the gaussian time to a variable t → T (x) labeled by new arbitrary coordinates x. The new action must be invariant under the transformation of the new coordinates and must reduce to the old action when Gaussian time is adopted, which leads to Variation of (5) with respect to g µν provides the vacuum Einstein field equations obtained from (2) with the stress-energy-momentum tensor where is the four-velocity of the source. Variation with respect to T gives the dynamical equation which describes the source as an incoherent dust. Assuming L > 0, the weak, strong and dominant energy conditions are satisfied and T (x) constitutes a good candidate as a physical clock field.
In the ADM formalism [10] one performs a 3 + 1 decomposition of the metric where N is the lapse function, N i is the shift vector and γ ab with a, b ∈ (1, 2, 3) is the induced spatial metric. Using these new variables, spacetime is described by the time propagation of the three-dimensional hypersurfaces of constant time corresponding to the chosen arbitrary foliation. The parametrized action takes the form where P is the canonical momentum conjugate to T . Variation with respect to N and N i gives the constrained Hamiltonian and momenta where the constrained Hamiltonian and momentum for gravity are and for the clock where n = (1 + γ ab T ,a T ,b ) 1 2 . The equations of motion for γ ab and π ab can be obtained from the super-Hamiltonian of the canonical action (10) We quantize the theory by promoting the canonical variables to operators and requiring the physical state to be annihilated by the constraint operators In the next section, we will focus on the Hamiltonian constraint (15), which gives to a Schroödinger-type evolution in the {T (x)}-representation. Such evolution is consistent and unambiguous with respect to the foliation choice only if the commutators of the constraints vanish, a condition which originates from the quantization of the classical equations of motion. This condition depends, in turn, on the factor ordering of the geometrodynamical operators. As the focus of the present work concerns only the emergence of time in the semiclassical limit, we will not deal with the unresolved technical and interpretational issues of quantum geometrodynamics as long as they do not affect the general validity of the semiclassical approach. In the specific case, the choice of operator ordering results in non-derivative terms in the functional Schrödinger equation for matter fields, thus not affecting the definition of semiclassical time, and we will adopt the trivial ordering.
However, it is worth noticing that, when one quantizes the incoherent dust model, one is not prevented a consistent definition of probability density function for γ ij on the embedding T (x), as one is instead for the more general Gaussian reference fluid where the full Gaussian coordinate conditions are imposed and all four spacetime coordinates promoted to variables. Consequently, the clock field model may help define a conserved positive inner product in superspace and, although the quantized theory may still present other problems, it may provide a valid starting point to consider them.

III. FUNCTIONAL SCHRÖDINGER EQUATION
The description of matter fields with non-derivative coupling to gravity can be included straightforwardly in the canonical formalism. We will use a scalar field φ as their representative. In the {γ ij (x), T (x), φ(x)}-representation the Hamiltonian constraint (15) becomes In this section we will imply spatial integration throughout this equation and its consequences in order to simplify notation.
To the purpose of discussing the semiclassical limit, we will introduce in (2) and in the Hamiltonian formalism derived from it the gravitational scale M := 1/32πG = (M P /2) 2 , M P being the reduced Planck mass. The geometrodynamical Hamiltonian density will read where G ijkl is the index-lowering DeWitt metric of superspace, the configuration space of general relativity. We perform the semiclassical limit of (17) following for the most part [5] and consider an expansion in powers of M of the wave functional where S n ∈ C in general. Substituting in (17) and equating terms of equal power, the highest order (M 2 ) contribution comes only from the kinetic term of the matter Hamiltonian, and gives us that the leading term S 0 = S 0 [γ ab , T ] is independent on the matter fields. Although not necessary, we will require that S 0 is also independent on T , as we want to treat gravity as classical and the clock field as quantum.
At the next order (M 1 ) we retrieve the Hamilton-Jacobi-Einstein equation [11] with Hamilton's principal function which provides a classical description of the vacuum background space equivalent to the G 00 component of Einstein's field equations [12,13]. Equation (20), which is integrated over space alone, holds for any value of coordinate time. Rather than considering the evolution between spatial hypersurfaces that takes place in spacetime [14], we consider the evolution as taking place in the "super-spacetime" S ⊗R that is obtained by extension of the superspace S spanned by the functions γ ab (x, t) defined over all physical space for each fixed value of coordinate time t ∈ R, which we take as an external dimension.
In this augmented superspace, the geometrodynamical "velocities" for γ ab with respect to coordinate time t are obtained from Hamilton's equations of motion with the classical Hamiltonian H G that governs (20) These shall not be mistaken for the physical spacetime velocities that one obtains from the super-Hamiltonian (14) which are identical to (21) only for Gaussian coordinates. Proceeding with the semiclassical expansion, at order M 0 we have We decompose S 1 into a pure complex part only dependent on geometry and a complex part S 1 dependent also on the field configuration and define the matter wave functional where we use the bar in the argument just to distinguish the quantum degrees of freedom from the classical parameters. At this order of approximation, the total wave functional can be written as where we have defined Let us then impose the conservation law G abcd δj cd δγ ab = 0 (28) for the current density Equation (23) then gives Given a choice of spacetime coordinates (t, x), let us restrict the state χ[φ, T |γ ij ] to the hypersurfaces of Gaussian time T (x) = t. This does not break the diffeomorphism invariance, but simply defines the surface on which we observe or register the matter state.
When we interpret T (x) only as the function that gives Gaussian time from any general spacetime coordinate that we choose to adopt, it enters the equation and the matter wave functional as a simple parameter rather than a physical variable. If, instead, we interprete it as the latter, the time evolution of the matter field φ is then implicitly described by its correlations with the value of the clock field, which in principle we can physically measure. We have then an application at the semiclassical level of the conditional probability interpretation proposed by Page and Wootters [15], where the field wave function evolving in time, that we write as χ[φ|γ ab , t], is obtained by conditioning the total "timeless" wave function χ[φ, T |γ ab ], which solves the Hamiltonian constraint, by the state of the clock T = t.
In either case, under such restriction (or conditioning) and using (21) we can finally write the following functional Schrödinger equation where we have reinserted the spatial integration for the sake of clarity and we have defined In analogy with classical continuous mechanics, the "material derivative" (32) expresses the fact that we take into account the intrinsic time evolution of the state (i.e. its dependence on clock time) as well as the evolution of the background "medium" -in this case, the coordinate time-augmented superspace. When one performs the semiclassical limit starting from the usual Wheeler-DeWitt equation without clock field, only the second ("convective") part of (32) is present in the derivative, and it cannot thus account for the time evolution when one uses static space coordinates.
The general solution to (31) at time t can be expressed in terms of any solution at time t 0 as where the line integral is taken along the local classical paths (the "streamlines") of the spatial metric γ ∈ S ⊗ R, with elementary arc length s and endpoints γ(t 0 ) and γ(t).

IV. DISCUSSION AND CONCLUSIONS
In this work, we have shown how the Schrödinger-type equation for quantum gravity that results from the introduction of a physical clock field can provide a semiclassical limit for matter fields with a notion of time that remains valid also when the geometrodynamical momentum vanishes, such as in Minkowski spacetime. The semiclassical limit consists of a classical gravitational background that is still consistent with the "timeless" equations of motion of general relativity in the form of the Hamilton-Jacobi-Einstein equation, and quantum matter fields that evolve according to a functional Schrödinger equation where the time derivative is substituted with the analogous of a material derivative that takes into account both the explicit time dependence introduced by the clock field as well as the dependence of the classical gravitational background on coordinate time. Clock time and coordinate time are identified by the choice of the hypersurface on which we observe the quantum state of the matter fields.
The time derivative operator of the usual semiclassical approximation is completed, in our case, by explicitly introducing a Gaussian time function that can be interpreted as a physical clock field. While this tool has been our choice to introduce explicit dynamics back into gravity in order to recover a viable evolution for quantum fields in curved (and flat) spacetime, various alternative derivations of Schrödinger-type evolutions for quantum gravity have been proposed since the first formulation of the canonical quantization of the theory, especially in relation to the so-called problem of time (for an extensive review, see [16]). Other models may equally provide in a semiclassical approach viable solutions to the problem addressed here. Our primary focus has been the recovery of an appropriate semiclassical limit for the evolution of quantum matter fields, and we do not claim that it provides a viable definition of time in the quantum gravity regime, nor that such a notion is necessary to begin with.
The functional Schrödinger equation that we have obtained differs from the one originally introduced in Tomonaga and Schwinger's works in that the integration is performed along the classical path of spatial geometry in the superspace augmented with coordinate time. It has been questioned whether the Tomonaga-Schwinger equation is indeed able to consistently describe the unitary time evolution of quantum fields between spacelike hypersurfaces in general curved spacetime (see e.g. [17,18]). Our treatment may help to address this issue. Incidentally, we would like to draw attention to the fact that, unlike in our case, in the Tomonaga-Schwinger equation one is working in the interaction picture. This detail does not seem to have received much attention either, although it might have some relevance to the interpretation and regularization of the "wave function of the universe" that solves the Wheeler-DeWitt equation.