Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

Leonardo Chataignier

doi:10.1515/zna-2019-0223

Article Publicly Available

Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

Leonardo Chataignier

Published/Copyright: August 7, 2019

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From the journal Zeitschrift für Naturforschung A Volume 74 Issue 12

Abstract

We make a critical review of the semiclassical interpretation of quantum cosmology and emphasise that it is not necessary to consider that a concept of time emerges only when the gravitational field is (semi)classical. We show that the usual results of the semiclassical interpretation and its generalisation known as the Born–Oppenheimer approach to quantum cosmology can be obtained by gauge fixing, both at the classical and quantum levels. By “gauge fixing,” we mean a particular choice of the time coordinate, which determines the arbitrary Lagrange multiplier that appears in Hamilton’s equations. In the quantum theory, we adopt a tentative definition of the (Klein–Gordon) inner product, which is positive definite for solutions of the quantum constraint equation found via an iterative procedure that corresponds to a weak coupling expansion in powers of the inverse Planck mass. We conclude that the wave function should be interpreted as a state vector for both gravitational and matter degrees of freedom, the dynamics of which is unitary with respect to the chosen inner product and time variable.

Keywords: Born–Oppenheimer Approximation; Gauge Fixing; Quantum Cosmology; Semiclassical Interpretation

1 Introduction

In canonical general relativity, the bulk Hamiltonian is constrained to vanish [1]. This constraint is related to the symmetry of the theory (“general covariance”), which is enforced by the Bergmann–Komar group in phase space [2], [3]. Upon quantisation, one may promote the constraints to operators that annihilate the wave functional, which is equivalent to requiring that physical states are invariant under the action of the symmetry group. In the absence of boundary terms, this implies that physical states are annihilated by the Hamiltonian. Such states are therefore independent of the choice of spacetime coordinates and, in particular, independent of coordinate time. This time independence seems to imply that the wave functional is static and there is no dynamics. This is the so-called “problem of time” in canonical quantum gravity. There are many approaches to understanding and solving this problem (see, e.g. [4], [5], [6] and references therein).

In this article, we will examine and reinterpret one such approach, known as the semiclassical interpretation of quantum gravity (see, e.g. [4], [5] for a review), which proposes that the notion of time emerges if the gravitational wave functional is semiclassical, i.e. if it can be approximated by its Wentzel–Kramers–Brillouin (WKB) counterpart. In this case, the first approximation to the phase of the WKB wave functional is a solution to the Einstein–Hamilton–Jacobi equations [7]. This solution defines a congruence of classical gravitational trajectories and a standard of time with respect to which quantum matter evolves according to the (functional) time-dependent Schrödinger equation (TDSE) [8]. Thus, one is able to derive quantum field theory on a classical gravitational background from the quantum constraint equations for the composite system of gravitational and matter degrees of freedom. If one proceeds to higher orders in the semiclassical expansion, usually performed as a formal expansion in powers of the inverse Planck mass [9], [10], [11], [12], it is possible to compute corrections to the TDSE [11], [12], [13], [14].

In this approach, the concept of time is taken to be inherently semiclassical, and it cannot be defined when the gravitational field is fully quantum. This was argued by Banks [9], who followed an earlier argument of DeWitt [15] that time should be a phenomenological concept in a covariant theory. The view expressed by Banks was shared by many authors in subsequent works in quantum gravity and cosmology [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], as well as in articles regarding the nonrelativistic quantum mechanics of closed, isolated systems [27], [28], [29].

In the present article, we take a different view, motivated by the fact that the choice of time coordinate in canonical general relativity is analogous to a choice of gauge in canonical Yang–Mills theories. More precisely, both general relativity and Yang–Mills theories are constrained systems, and thus, the canonical field equations contain arbitrary Lagrange multipliers. In the Yang–Mills case, the multipliers are fixed by a gauge condition on the vector potential, whereas in the case of general relativity, they are fixed by a choice of spacetime coordinates, i.e. a “coordinate condition.” In analogy to the Yang–Mills case, we refer to the choice of spacetime coordinates as “gauge fixing.” This terminology is unrelated to gauge theories of gravitation, such as Poincaré gauge theory [30].

As it is possible to fix the gauge both in the classical and quantum versions of Yang–Mills theories, we assume that the same is true for (quantum) general relativity. Although it might be indeed meaningless to discuss the interpretation of clock readings in a fully quantum regime, we assume that there is in principle no inconsistency in parametrising the dynamics with respect to a given choice of time coordinate also in the quantum theory. We will thus argue that it is unnecessary to relegate the concept of time to the (semi)classical level and that the usual results obtained in the semiclassical interpretation of quantum gravity coincide with a particular gauge fixing of the theory, both at the classical and quantum levels.

It is not unexpected that the results of the semiclassical approach should coincide with a particular choice of gauge. Indeed, the emergent semiclassical time is a time coordinate associated with the background geometry defined from the phase of the semiclassical gravitational wave functional. Once this coordinate is defined, it may be used to parametrise the dynamics not only of the background geometry, but also of the composite system of gravitational and matter degrees of freedom. Classically, time evolution is only defined once a choice of coordinates has been made, as this corresponds to fixing the arbitrary multipliers that appear in Hamilton’s equations. Therefore, any notion of time (even if it is “emergent”) must correspond to a particular fixation of the multipliers. We will provide closed-form expressions for the Lagrange multiplier (the “lapse”), as well as the classical reduced gauge-fixed Hamiltonian associated with the choice of time given by the phase of the semiclassical gravitational wave functional. To the best of our knowledge, such closed-form expressions for this choice of time have not been derived before.

In [31], Parentani showed that a time-dependent Hamilton–Jacobi equation (TDHJE) for matter fields could be derived from the Einstein–Hamilton–Jacobi equations in a way analogous to the derivation of the TDSE from the quantum constraints in the standard semiclassical approach. He emphasised that such a derivation amounts to a background field approximation, as the notion of time in both classical and quantum cases is defined from the (classical) gravitational background. The higher-order corrections to the TDHJE and TDSE thus depend on the choice of this background. More recently, Briggs [32] has independently derived the TDHJE from the time-independent Hamilton–Jacobi equation. It remained unclear, however, whether the derivation of the approximate TDHJE corresponds to a choice of gauge by fixing the arbitrary multipliers in the equations of motion. As already mentioned, we will show that this is indeed the case.

More generally, one can fix the coordinates to be given by functions of the canonical variables. This “canonical gauge” choice is in line with DeWitt’s view [15] that, in a covariant theory, the contents of the universe itself should be used to define the coordinates and, in particular, time. In this way, the evolution of physical quantities is described through the correlation of their configuration with the trajectory of a quantity chosen to be the “clock” [33], [34]. Moreover, it is important to note that gauge fixing is not merely a mathematical convenience, as different choices of coordinates may also be associated with different reference frames and observers. This marks a difference between the external symmetry of general relativity and the internal symmetries of Yang–Mills theories, for which different gauge choices are unobservable.

Regarding the quantum theory, we take the position that the independence of the wave functional on the choice of coordinates does not preclude its dynamical and probabilistic interpretation. Rather, the invariance of physical states implies that the quantum dynamics is the same for any choice of spacetime coordinates. To fix the gauge in the canonical quantum theory, we proceed in analogy to the classical Hamilton–Jacobi theory. At the classical level, the Einstein–Hamilton–Jacobi equations are gauge-independent. However, if we choose suitable functions of the canonical variables (e.g. the Weyl scalars or matter scalar fields) to define the spacetime coordinates, the solutions to the Einstein–Hamilton–Jacobi equations may be interpreted dynamically, as their variation in the chosen time coordinate will be encoded in their dependence on the canonical variables. If we take the view that the same is true for the quantum constraint equations, the time dependence of the wave functional will be encoded in its dependence on the configuration or momentum variables in the appropriate representation.

The difficulty in establishing such a quantum theory resides in defining the inner product on the space of physical states and assessing whether the theory is unitary with respect to different choices of the time parameter. There have been various proposals for selecting such a physical inner product and constructing the physical Hilbert space (see, for instance, [35], [36] and references therein). This is a subtle issue that is outside the scope of this article. We will adopt a tentative definition of the inner product, with respect to which we will interpret the usual results of the semiclassical approach as a particular gauge fixing. Thus, the quantum theory here presented is provisional. Our goal is not to provide a definitive solution to the problem of time, but rather to reinterpret the standard semiclassical approach and to emphasise how the emergent semiclassical time is related to the freedom of choosing coordinates in general relativity; it corresponds to a particular class of canonical gauge choices, and there are more general choices of time coordinate that could be employed.

The semiclassical interpretation can be generalised to what is often referred to as the Born–Oppenheimer (BO) approach to quantum gravity and cosmology [17], [18], [19], [20], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], as it was inspired by the BO approximation in molecular physics [47], [48]. We will use the terms “semiclassical interpretation,” “semiclassical approach,” and “BO approach” interchangeably. As our focus is on the problem of time (in its simplest manifestation) and its solution given by the semiclassical approach, we will not be concerned with field-theoretic issues, such as regularisation of the quantum constraint equations. In the main body of the article, we will restrict ourselves to finite-dimensional models with a single constraint, which are useful for analysing homogeneous and isotropic cosmologies. We include an appendix with the formal generalisation of the results of the article to the field-theoretic case.

Finally, we mention a recent article in which Kamenshchik et al. [40] compared the results of the BO approach and the gauge-fixing approach for a simple minisuperspace model and obtained similar results for both methods. The formalism presented here is complementary to their work. We show that the BO approach is simply a particular choice of gauge for a general homogeneous, isotropic minisuperspace model with nonvanishing potential.

The article is organised as follows. In Section 2, we give a critical overview of the standard semiclassical interpretation and the BO approach to the problem of time. We then describe the particular gauge fixing with which it coincides at the classical level in Section 3. There, we show how the reduced gauge-fixed Hamiltonian can be approximated by the “corrected” Hamiltonian used in the standard semiclassical approach. In Section 4, we analyse the corresponding quantum theory, in particular the question of unitarity in perturbation theory. The archetypical example of the relativistic particle is analysed in Section 5. Finally, in Section 6, we summarise our results and present our conclusions. We include three appendices. Appendices A and B contain formulae that are needed in Section 3 and 4. Appendix C formally extends the results of the article to field theory. We work in units in which c=ℏ=1.

2 The Born–Oppenheimer Approach to the Problem of Time

In this section, we make a critical review of the BO approach to the problem of time in preparation for Section 3 and 4, where we show that the standard results of the BO approach can be obtained by a particular choice of gauge.

2.1 Overview

The Hamiltonian constraint of quantum gravity, referred to as the Wheeler–DeWitt (WDW) equation, does not depend on a time variable, and it is thus analogous to the time-independent Schrödinger equation (TISE) of quantum theory. If some of the variables on which the TISE or the WDW equation depends can be treated semiclassically, one can define a time variable from the phase of the semiclassical part of the wave function. In this way, time emerges from a timeless quantum equation (TISE or WDW) in a semiclassical regime. The basis for such a semiclassical interpretation of time was laid in Mott’s work [49], [50] on α-particle tracks. In [49], [50], Mott analysed the TISE for the composite system of atoms and α particles and showed that, by treating the high-energy α-particle wave function semiclassically, one can derive a TDSE for the atoms. The time parameter is defined from the phase of the α-particle wave function. Mott’s derivation of the TDSE from the TISE was further analysed in [28], [29], [51] and inspired an application to quantum cosmology in [52].

The semiclassical regime in which time emerges can be understood in the context of the BO approximation [17], [18], [19], [20], [37], [38], [39], [40], which is frequently employed in molecular physics [47], [48]. There, one is interested in computing molecular spectra by analysing the quantum dynamics of a system of heavy nuclei and light electrons. In many cases, one can make a WKB approximation to the nuclear wave function and consider an adiabatic approximation in which the electronic wave function follows the semiclassical dynamics of the nuclei. The combination of the WKB expansion for the heavy nuclei and the adiabatic approximation comprises the BO approximation. This can be generalised to any composite system, composed of a “heavy” (or “slowly varying”) part and a “light” (sub)system [28], [29], [32]. In the BO approximation, the evolution of the “light” system follows adiabatically the semiclassical dynamics of the “heavy” part. In the BO approach to the problem of time, the time parameter is defined from the phase of the semiclassical wave function of the “heavy” sector and is sometimes referred to as “WKB time” [53].

Englert [27] and subsequently Briggs and Rost [28], [29] took the position that the (nonrelativistic) quantum mechanics of closed, isolated systems should be fundamentally timeless and thus described by the TISE. They suggested that the BO approximation (or a generalisation thereof) should be used to derive the TDSE from the TISE in a procedure analogous to Mott’s derivation. In this way, Briggs and Rost emphasise that the TDSE would be only an approximation and a mixed classical-quantum equation, as the time parameter is defined in the limit in which the heavy sector becomes classical. Thus, the TISE would be promoted to the fundamental equation of quantum mechanics. This approach was further pursued by Arce [54].

In the context of quantum gravity, Lapchinsky and Rubakov [8] derived the functional TDSE for matter fields propagating in a fixed, vacuum gravitational background by treating the gravitational field semiclassically in the quantum constraint equations. Thus, the gravitational variables served as the heavy sector for the “light” matter fields. Essentially, the same procedure was followed independently by Banks [9] and Banks et al. [10]. Although other separations are possible, the “heavy” variables usually coincide with the gravitational degrees of freedom, whereas the “light” sector consists of the matter variables.

In [9], [10], the semiclassical approximation for the gravitational sector was obtained by formally expanding the quantum constraint equations and their solution in powers of the inverse Planck mass. Such an expansion is valid when all energy scales are much smaller than the Planck scale (weak-coupling expansion), and it is analogous to what was done in Born and Oppenheimer’s original article [47]. In [47], the average mass M of the nuclei was considered to be much larger than the mass m of electrons, such that it was possible to expand the Hamiltonian operator and energy eigenfunctions in powers of (mM)14. To the lowest order, one recovered the dynamics of electrons while the nuclei remained at fixed positions and the nuclear position variables were treated as parameters. Analogously, the gravitational variables enter as parameters in the functional TDSE for matter fields, which propagate in a fixed geometry to the lowest order in the weak coupling expansion [9], [10].

It is possible to include corrections to the (functional) TDSE by computing terms of higher orders in the inverse Planck mass. Such corrections have been computed in [11], [12], [13], [14]. In [12], it was concluded that the corrections include terms that violate unitarity in the matter sector. We shall reexamine this question in Sections 2.2 and 4.5 and find that the dynamics of the gravity-matter system is unitary with a suitable definition of the inner product.

In [8], [9], [10], the backreaction of quantum matter onto the classical gravitational background was not included. In a series of articles, Brout and colleagues [17], [18], [19], [20] refined the method of Banks to include the effect of backreaction of matter in the form of a source term in the Einstein–Hamilton–Jacobi equations for the classical gravitational background. The source term consisted of the expectation value of the matter Hamiltonian (averaged only over matter degrees of freedom) and was also accompanied by Berry connection [55], [56] terms, which is in line with the usual BO approximation used in molecular physics [48], [57], [58], [59]. In [39], [60], [61], it was claimed that the inclusion of backreaction and Berry connection terms leads to a unitary description of the matter-sector dynamics to all orders in the weak-coupling expansion.

However, following an earlier article by Hartle [62], the authors Halliwell [63], D’Eath and Halliwell [64], and Padmanabhan [65] stressed that the backreaction terms were spurious, as they depend on the arbitrary choice of phase for the gravitational wave function, which in turn is related to the freedom associated with the definition of the Berry connection. Halliwell [63] and Singh and Padmanabhan [22], [23] concluded that a semiclassical theory of gravity sourced by the expectation value of the matter Hamiltonian or, in a covariant setting, of the matter energy-momentum tensor is well defined only when the distribution of the matter Hamiltonian is peaked about its average value, or the quantum corrections to the energy-momentum tensor of matter are small in comparison to the classical contribution. This arbitrariness related to the definition of backreaction terms casts doubt on the claim that the inclusion of backreaction guarantees unitarity in the matter sector. In what follows, we will see how this can be resolved.

2.2 A Critical Assessment of the Method. Nonrelativistic Case

To clarify the conceptual points mentioned above and illustrate the BO approach to the problem of time, let us consider a nonrelativistic example, analysed in a different way in [28], [29], [32], [66]. We focus on a composite system of a heavy sector interacting with a “light” subsystem. The heavy sector is associated with a mass scale M and degrees of freedom Q_a, a=1,…n, whereas the “light” system is associated with a scale m≪M and degrees of freedom q_μ, μ=1,…,d. The TISE reads

(1)−12M∑a=1n∂2Ψ∂Qa2+V(Q)Ψ+H^𝒮(Q;∂∂qμ,qμ)Ψ=EΨ,

where V is a potential term for the heavy sector and H^𝒮 is the “light”-system Hamiltonian, which depends only parametrically on the heavy variables Q. In the traditional BO approach, one expands the wave function as the superposition

(2)Ψ(Q;q)=∑kχk(Q)ψk(Q;q),

where ψ_k form a complete system, which is orthonormal with respect to the inner product taken only over the “light” variables q, i.e. ⟨ψk,ψl⟩𝒮(Q):=∫∏μ=1ddqμψ¯k(Q;q)ψl(Q;q)≡δkl. For example, one may choose ψ_k to be the eigenstates of H^𝒮 (if the spectrum is continuous, we replace ∑k→∫dk and δkl→δ(k−l)). For simplicity, we can also rewrite (2) as

(3)Ψ(Q;q)=χ(Q)∑kχk(Q)χ(Q)ψk(Q;q)≡χ(Q)ψ(Q;q).

Such an exact factorisation was considered in [32], [54], [57], [58], [59], [67], [68], and it avoids the complication of having to consider the dynamics of each of the ψ_k states. Evidently, this factorisation is ambiguous, as one can redefine each factor as follows:

(4)χ(Q)=eξ(Q)+iη(Q)χ′(Q),ψ(Q;q)=e−ξ(Q)−iη(Q)ψ′(Q;q),

where ξ and η are smooth functions of the Q variables. Under such a redefinition, the total state remains invariant, Ψ=χψ=χ′ψ′=Ψ′.

The usual procedure is to insert (2) or (3) into (1), multiply the result by ψ¯k or ψ¯, and integrate over the q variables to obtain an equation for χ_k or χ [17], [18], [19], [20], [28], [29], [60], [66]. Such an equation involves the partial averages over the q variables ⟨∂∂Qa⟩𝒮, which are related to the Berry connection, and ⟨H^𝒮⟩𝒮, which is interpreted as a “backreaction” term. One then uses the equation for χ_k or χ to obtain an equation for ψ_k or ψ, which will also involve partial averages over the q variables. The result is a coupled nonlinear system, which can be solved in an iterative, self-consistent way [69]. Here, we decide to take a slightly different but equivalent route. For convenience, we will work with the exact factorisation given in (3). For any choice of χ, we define

(5)𝔍(Q):=12Mχ(Q)∑a=1n∂2χ∂Qa2−V(Q)+E≡J(Q)+iK(Q),

where J and K are the real and imaginary parts of the “source” 𝔍. They can be written in terms of the amplitude and phase of χ as follows:

(6)J(Q)=12M∑a=1n[1R∂2R∂Qa2−(∂φ∂Qa)2]−V(Q)+E,K(Q)=12M∑a=1n(2R∂R∂Qa∂φ∂Qa+∂2φ∂Qa2),

where we used the polar decomposition χ=Reiφ. We note that there is no loss of generality in choosing χ to be a complex wave function (φ ≠ 0), even if Ψ is real, due to the freedom of redefining the states according to (4). If we redefine χ=eξ+iηχ′ [cf. (4)], the “source” 𝔍 changes accordingly,

(7)J(Q)=J′(Q)+12M∑a=1n[2R′∂R′∂Qa∂ξ∂Qa+(∂ξ∂Qa)2+∂2ξ∂Qa2−2∂φ′∂Qa∂η∂Qa−(∂η∂Qa)2],K(Q)=K′(Q)+1M∑a=1n(∂ξ∂Qa∂φ′∂Qa+1R′∂R′∂Qa∂η∂Qa+∂ξ∂Qa∂η∂Qa+12∂2η∂Qa2).

We now insert (3) into (1) to obtain

(8)iM∑a=1n∂φ∂Qa∂ψ∂Qa=(H^𝒮−𝔍)ψ−1M∑a=1n∂logR∂Qa∂ψ∂Qa−12M∑a=1n∂2ψ∂Qa2.

If we define ∂∂t:=1M∑a=1n∂φ∂Qa∂∂Qa, then (8) reads

(9)i∂ψ∂t=(H^𝒮−𝔍)ψ−1M∑a=1n∂logR∂Qa∂ψ∂Qa−12M∑a=1n∂2ψ∂Qa2,

which resembles a TDSE. Nevertheless, the presence of higher derivatives with respect to the Q variables on the right-hand side makes it more akin to a Klein–Gordon equation. Traditionally, the “time” derivative in (9) is defined only when χ (the “heavy”-sector wave function) is approximated by its WKB counterpart [8], [9], [10], such that t is the “WKB time.” However, we stress that this is not necessary. Indeed, we have defined t from the general phase^[1] of χ and used the exact polar decomposition χ=Reiφ. Moreover, we note that φ (and, hence, t) is freely specifiable because of the freedom of performing the redefinitions given in (4). Finally, let us mention that we can reinterpret (5) as a definition of χ given 𝔍, instead of as a definition of 𝔍 given χ. In this way, (5) and (9) can be seen as a coupled system for χ and ψ. We will see in what follows how this is related to the treatment involving the Berry connection and backreaction terms.

If the terms proportional to 1M can be neglected (e.g. by considering that all energy scales related to the “light”-sector are much smaller than M, or by performing a semiclassical expansion), then (9) reduces to a TDSE

(10)i∂ψ∂t=(H^𝒮(t)−𝔍)ψ+𝒪(1M).

In the literature [9], [10], [24], [25], [28], [29], [32], [54], (10) is often interpreted as the Schrödinger equation for the “light” system alone, in which the heavy variables provide the clock that parametrises the evolution of the “light” degrees of freedom. The (real part of the) source term J in (10) can be removed by a phase transformation of ψ [32], [60]. However, such a phase transformation corresponds to a redefinition given in (4), which would lead to a redefinition of φ and t, unless one defines time only from a part of the phase of χ or in some other way. By taking into account the terms of order 1M, one obtains from (9) a “corrected” Schrödinger equation [11], [12], [13], [14]. We will see in what follows under what circumstances can one interpret such an equation as a dynamical equation for the light sector alone.

2.2.1 Partial Averages. Berry Connection

Given an operator Ô, we define its light-sector partial average as

(11)⟨O^⟩𝒮(Q):=∫∏μdqμψ¯(Q;q)O^ψ(Q;q)∫∏νdqνψ¯(Q;q)ψ(Q;q),

provided the integrals converge. The partial averages

(12)⟨∂∂Qa⟩𝒮(Q)=:Va(Q)+iAa(Q),

where A_a and V_a are real, are of particular interest. The one-form with components A_a will be referred to as “Berry connection,” in analogy to the usual Berry connections that appear in adiabatic quantum mechanics [55], [56]. We can write V_a and A_a explicitly as

(13)Va(Q):=12∂∂Qalog(∫∏μdqμψ¯(Q;q)ψ(Q;q)),

(14)Aa(Q):=−i2(∫∏μdqμψ¯(Q;q)∂ψ(Q;q)∂Qa∫∏νdqνψ¯(Q;q)ψ(Q;q)−∫∏μdqμψ(Q;q)∂ψ¯(Q;q)∂Qa∫∏νdqνψ¯(Q;q)ψ(Q;q)).

Under the redefinition given in (4), we obtain the transformation laws

(15)Va=V′a−∂ξ∂Qa,Aa=A′a−∂η∂Qa.

We will also be interested in the partial average ⟨∂2∂Qa2⟩𝒮. Under the redefinition given in (4), we find

(16)ℜ𝔢⟨∂2∂Qa2⟩𝒮=ℜ𝔢⟨∂2∂Qa2⟩𝒮′−∂2ξ∂Qa2+(∂ξ∂Qa)2−(∂η∂Qa)2+2∂η∂QaA′−a2∂ξ∂QaV′,aℑ𝔪⟨∂2∂Qa2⟩𝒮=ℑ𝔪⟨∂2∂Qa2⟩𝒮′−∂2η∂Qa2+2(∂ξ∂Qa∂η∂Qa−∂ξ∂QaA′−a∂η∂QaV′)a.

By multiplying (8) by ψ¯ and integrating over the q variables, we find^[2]

(17)−𝔍=1Mχ∑a=1n∂χ∂Qa⟨∂∂Qa⟩𝒮−⟨H^𝒮⟩𝒮+12M∑a=1n⟨∂2∂Qa2⟩𝒮.

The real and imaginary parts of (17) form a coupled system for V_a and A_a, which can be used to eliminate two of the 2n real components of the partial average ⟨∂∂Qa⟩𝒮. By using the transformation laws of χ and 𝔍 given in (4) and (7) and using (15) and (16), one may verify that (17) is invariant under the state redefinitions given in (4), as it should be.

2.2.2 Backreaction

Let us now relate (5) and (8) to the nonlinear system of equations with backreaction terms, which was analysed in [17], [18], [19], [20]. We first define the “covariant” derivatives [17], [18], [19], [20], [60], [76]

(18)Da±:=∂∂Qa±⟨∂∂Qa⟩𝒮,

which, under the state redefinition given in (4), transform as follows:

(19)Da+χ=eξ+iη∂χ′∂Qa+eξ+iη⟨∂∂Qa⟩𝒮′χ′=eξ+iηDa′+χ′,Da−ψ=e−ξ−iη∂ψ′∂Qa−e−ξ−iη⟨∂∂Qa⟩𝒮′ψ′=e−ξ−iηDa′−ψ′.

We now substitute (17) in (5) and (8) to find the system

(20)−12M∑a=1n[(Da+)2+⟨(Da−)2⟩𝒮]χ+Vχ=(E−⟨H^𝒮⟩𝒮)χ,

(21)−1Mχ∑a=1nDa+χDa−ψ−12M[(Da−)2−⟨(Da−)2⟩𝒮]ψ+(H^𝒮−⟨H^𝒮⟩𝒮)ψ=0.

Due to (19), one can immediately verify that (20) and (21) are invariant under the state redefinitions given in (4). Although (20) and (21) form a nonlinear system due to the presence of the partial averages, they are equivalent to (5) and (8), respectively, which form a linear system if 𝔍 is considered as an independent function.

Equations (20) and (21) were analysed in [17], [18], [19], [20], [39], [60], [76] as equations incorporating the nonlinear effects of the backreaction of the light sector (which usually corresponds to matter in quantum cosmology) onto the heavy sector (normally gravity in quantum cosmology). Backreaction is here understood as the collection of terms involving the light-sector partial averages, in particular the term ⟨H^𝒮⟩𝒮. The presence of the partial averages in (20) and (21), especially of the Berry connection terms, is in line with the usual BO approximation performed in molecular physics [48], [57], [58], [59]. In [17], [18], [19], [20], [60], [76], (20) was interpreted as an equation determining the heavy-sector wave function χ. From the above construction, we see that this is equivalent to interpreting (5) as defining χ given 𝔍. Nonetheless, as we have already noted, it is also possible to interpret (5) as defining 𝔍 given χ and to consider χ as arbitrary due to (4).

Halliwell [63] and Padmanabhan [65] already stressed that the arbitrariness of χ implies that the backreaction terms in (20) and (21) are also arbitrary. This can be understood from the fact that, although ⟨H^𝒮(Q;p^,q)⟩𝒮 is invariant under the state redefinitions of (4), the Berry connection terms, which appear implicitly in (20) and (21), are not [cf. (15)]. We can interpret a choice of χ (or 𝔍), as a particular fixation of (some components of) the partial averages ⟨∂∂Qa⟩𝒮 via (17). Thus, the arbitrariness of the partial-average terms is equivalent to the arbitrariness of χ (or 𝔍).

Furthermore, the physical meaning of different choices of χ is most clearly seen from (9). As we define the time variable from phase φ, which is a particular function of the heavy variables Q, changing φ via (4) corresponds to changing what one means by time. Equivalently, a transformation of the Berry connection A_a [cf. (15)] entails a redefinition of the time variable in the BO approach to the problem of time. This has not been emphasised in the literature so far. In Section 3 and 4, we will analyse how this is related to a gauge choice of the time coordinate. In Section 5, we examine the question of backreaction and partial averages for the archetypical example of the relativistic particle.

2.2.3 “Light”-Sector Unitarity

In the BO approach of [11], [12], [17], [18], [19], [20], [60], [61], [76], the question of whether the dynamics of the light sector is unitary arose due to the interpretation of ψ as the light-sector wave function and of (9) as a “corrected” Schrödinger equation for the light sector.^[3] By light-sector unitarity, we mean the condition

∂∂t∫∏μdqμψ¯(Q;q)ψ(Q;q)=0,

which is equivalent to

(22)0=ℜ𝔢⟨∂∂t⟩𝒮=1M∑a=1n∂φ∂Qaℜ𝔢⟨∂∂Qa⟩𝒮=1M∑a=1n∂φ∂QaVa.

As V_a is not necessarily zero, (22) can only be enforced by a particular choice of χ. Indeed, we see from (13) that Va=0 if and only if ⟨ψ,ψ⟩𝒮=∫∏μdqμψ¯(Q;q)ψ(Q;q) is a constant. For a general factorisation given in (3), this will not be the case, and ⟨ψ,ψ⟩𝒮 will be a function of the heavy variables Q. Nevertheless, we are free to perform a state redefinition as in (4) and fix ξ as follows. Starting from an arbitrary initial factorisation of the total state Ψ(Q;q)=χ′(Q)ψ′(Q;q), we demand

1=⟨ψ,ψ⟩𝒮(Q)=∫∏μdqμψ¯(Q;q)ψ(Q;q)=e−2ξ∫∏μdqμψ¯′(Q;q)ψ′(Q;q),

which implies ξ(Q)=12log(∫∏μdqμψ¯′(Q;q)ψ′(Q;q)). For this choice of ξ(Q), we obtain [cf. (13) and (15)]

Va=V′−a∂ξ∂Qa=0,ψ(Q;q)=ψ′(Q;q)⟨ψ′,ψ′⟩𝒮(Q).

Equation (22) does not follow from the equations with “backreaction” terms. Indeed, by multiplying (21) by ψ¯ and integrating over q_μ, one obtains the trivial result 0 = 0, and no information is gained on the value of V_a. Thus, even with the inclusion of backreaction and Berry connection terms (which are arbitrary), one still needs to enforce the light-sector unitarity by a choice of χ. This was emphasised, in a somewhat different way, in [61].

2.2.4 Marginal and Conditional Wave Functions

Let us assume that we are able to choose a factorisation Ψ=χψn in which light-sector unitarity holds. We obtain

⟨ψn,ψn⟩𝒮(Q)=∫∏μdqμψ¯n(Q;q)ψn(Q;q)=1,⟨χ,χ⟩ℰ=∫∏adQaχ¯(Q)χ(Q)=cℰ,

where cℰ is a finite constant if χ is normalisable. We then define the normalised state χn=1cℰχ, such that ⟨χn,χn⟩ℰ=1. The total state Ψn=χnψn=1cℰΨ is a solution to (1) and is normalised^[4] with respect to the inner product over all variables,

⟨Ψn,Ψn⟩=∫∏adQa∏μdqμΨ¯nΨn=∫∏adQaχ¯nχn⟨ψn,ψn⟩𝒮(Q)=⟨χn,χn⟩ℰ=1.

In this case, it is possible to interpret |Ψn(Q;q)|2 as a joint probability density for the “light” system and the heavy sector to be in the (Q, q) configuration, whereas χn(Q) and ψn(Q;q) are interpreted as a marginal wave function and a conditional wave function, respectively. Indeed, |χn|2 can be seen as the marginal probability density for the heavy variables to be in the Q-configuration regardless of the configuration of the light variables. Analogously, |ψn|2 is then interpreted as the conditional probability density to find the “light” system in the q configuration given that the heavy sector is in the Q configuration. In the context of molecular physics and the BO approximation, this interpretation was used in [57], [58], [59], [67], [68]. Such an interpretation was also adopted by Arce [54] in the context of the problem of time in (nonrelativistic) quantum mechanics. In [54], Arce referred to the partial averages defined in (11) as conditional expectation values. Such an interpretation is possible only when the light-sector unitarity is enforced [58], [59]. If one is able to enforce light-sector unitarity together with a normalisable χ at the exact level of (9), then one may interpret (9) as a “corrected” Schrödinger equation for the conditional light-sector wave function ψ(Q;q).

Alternatively, one may choose χ(Q)=eiφ(Q) and interpret ψ(Q;q) as the wave function for the full system that comprised “heavy” and “light” degrees of freedom. In this way, the factorisation Ψ=χψ=eiφψ is merely a phase transformation of the full system. This is the interpretation that we will adopt in this article (except in Section 4.4), which does not require that light-sector unitarity be enforced. We recall that this phase transformation is needed in order to rewrite the constraint equation (1) in the form of (9), which leads to the TDSE (10) at the lowest order in an expansion in powers of 1M.

At the nonperturbative level (without resorting to such an expansion), the choice of factorisation Ψ=eiφψ evades some of the problems mentioned by Kuchař in his critique of the semiclassical interpretation [4], [5]. Indeed, all the states in the Hilbert space can be transformed according to the same phase factor eiφ. The time variable is defined from only one congruence of trajectories associated with φ, and we need not consider how time “emerges” if the state is a superposition of factors such as Ψ=eiφ1ψ1+eiφ2ψ2 [or more generally (2)], as this state can be rewritten as Ψ=eiφ[ei(φ1−φ)ψ1+ei(φ2−φ)ψ2]≡eiφΨφ for any choice of φ. Thus, it is not necessary to consider the interference of states with different phase prefactors, and there is no “superposition or interference problem” in the definition (choice) of the (phase) time variable. However, the general superposition Ψφ=ei(φ1−φ)ψ1+ei(φ2−φ)ψ2 may not admit an expansion in powers of 1M, and in this case, it is necessary to invoke the decoherence [24], [25], [26], [79] of Ψ into separate factors eiφiψi, which can be treated perturbatively and thus lead to the Schrödinger equation (10) at the lowest order. In general, decoherence is relevant to the study of the classical limit of (a subset of) the quantised variables [80], [81].

3 The BO Approach as a Choice of Gauge. Classical Theory

We now illustrate how the results of the BO approach to the problem of time discussed in the previous section can be obtained by a choice of gauge in the classical theory. We focus on cosmological minisuperspace models and consider for simplicity that the heavy sector coincides with the gravitational sector, while the light variables are given by the matter degrees of freedom, although more general separations are possible [21], [24], [25].

The gravitational-sector configuration space is endowed with local coordinates Qa,a=1,…,n and an indefinite metric G. Indices a,b,… are lowered and raised with the components G_ab and Gab of the metric and its inverse, respectively. We choose local coordinates qμ,μ=1,…,d for the matter-sector configuration space with positive-definite metric h(Q). Indices μ,ν,… are raised and lowered with hμν and hμν, respectively. We consider the action functional

(23)S=∫dt(PaQ˙a+pμq˙μ−NH),

where summation over repeated indices is implied. A dot over a variable indicates differentiation with respect to the parameter t. The lapse N is taken to be an arbitrary multiplier^[5] and the Hamiltonian constraint is

(24)H=Hg(Q,P)+Hm(Q;p,q)=0,Hg(Q,P)=12MGab(Q)PaPb+MV(Q),Hm(Q;p,q)=12hμν(Q;q)pμpν+Vm(Q;q),

where M=132πG and G is Newton’s constant. This is a time-reparametrisation invariant system. We assume that the gravitational potential term V(Q) is nonvanishing, which is achieved if, for example, the cosmological constant term is not zero. The matter degrees of freedom pμ,qμ can be assumed to be associated with a typical mass scale m≪M. The matter-sector Hamiltonian H_m depends only parametrically on the Q coordinates, and we assume V_m is a smooth, nonnegative real function. The equations of motion of this system read

(25)Q˙a=NMGab(Q)Pb,P˙a=−N(12M∂Gcd∂QaPcPd+M∂V∂Qa)−N∂Hm∂Qa,q˙μ=N∂Hm∂pμ,p˙μ=−N∂Hm∂qμ,H=Hg+Hm=0.

One may solve these equations after making a choice of lapse. Alternatively, we can perform a canonical transformation, with generating function F=W(Q,P′;q,p′)−Q′aP′−aq′μp′μ, such that the momenta are substituted by the gradient components of W, Pa=∂W∂Qa, and pμ=∂W∂qμ, and the constraint equation becomes the differential equation

(26)12MGab(Q)∂W∂Qa∂W∂Qb+MV(Q)+Hm(Q;∂W∂q,q)=0,

which will be referred to simply as the Hamilton–Jacobi equation, while W will be called the Hamilton characteristic function. Given a solution of (26), one may pass to the new canonical frame described by the variables Q′,P′,q′,p′, with respect to which the dynamics is trivial. Equivalently, one may still work with the old coordinates Q and q, with respect to which the dynamics is described by the reduced set of equations

(27)Q˙a=NMGab(Q)∂W∂Qb,q˙μ=Nhμν(Q;q)∂W∂qν,

for a given choice of lapse.

3.1 Canonical Variables Adapted to a Choice of Background

The presence of the matter-sector Hamiltonian Hm(Q;p,q) affects the dynamics of the gravitational sector. If Hm(Q;p,q) can be approximated (in a sense which will be discussed in Section 3.4) by a function J(Q) solely of the gravitational configuration variables, we can consider that the dynamics of the gravitational field is approximately dictated by the Hamilton–Jacobi equation [cf. (26)] for the gravitational system in the presence of a source

(28)12MGab(Q)∂φ∂Qa∂φ∂Qb+MV(Q)=−J(Q).

At this stage, however, we can consider J(Q) as arbitrary.^[6] The solution φ will be referred to as the background Hamilton function, and it is analogous to the phase used in (8). It is convenient to define the quantities

(29)Φa(Q)=∂φ∂Qa,

which will be called background momenta. Equations (28) and (29) imply the background momenta are normalised to

(30)Gab(Q)ΦaΦb=−2M(J(Q)+MV(Q)).

As in (5), we note that (28) may be regarded either as a definition of φ given J or as a definition of J given φ. If we change the background Hamilton function, φ(Q)=φ′(Q)+η(Q), we may change the background momenta and source accordingly,

(31)Φa(Q)=Φ′a+∂η∂Qa,J(Q)=J′(Q)−1MGab(Q)Φ′∂η∂Qba−12MGab(Q)∂η∂Qa∂η∂Qb.

If φ is chosen to be a nonconstant function of the Q coordinates, then the background momenta Φ_a will be nontrivial. In this case, we assume that one may define a holonomic vector-field basis in the tangent bundle composed of the vector fields {B1=𝒩M𝚽,Bi},i=2,…n, where 𝚽=GabΦa∂∂Qb and 𝒩≡𝒩(Q) are an arbitrary normalisation function, which can be interpreted as a “background lapse.” The calculations are somewhat simplified if we assume that B_i is orthogonal to B₁.^[7] The basis vectors are then normalised as follows [cf. (30)]:

(32)GabB1aB1b=−2𝒩2(Q)(J(Q)M+V(Q))=G~11,GabB1aBib=0=G~1i,GabBiaBjb=G~ij≡gij.

We then define new coordinates x=(x1,xi) via the integral curves of the basis fields,

(33)B1a=𝒩MGabΦb=∂Qa∂x1,Bia=∂Qa∂xi.

In this way, the gravitational-sector configuration space is foliated by surfaces of constant x¹, on which gij=G~ij is the induced metric. We will denote its inverse by gij. The first of (33) can be interpreted as a (fictitious) “background” equation of motion for Q^a, which depends on the background momenta Φ_a and the background lapse 𝒩 and for which x1(Q) plays the role of a “background time” function [compare the first of (33) to the first of (27)]. We also obtain the useful identities

(34)∂φ∂x1=𝒩MGabΦaΦb=−2𝒩(J+MV),∂φ∂xi=BiaΦa=0.

In Appendix A, we collect formulae related to the change of coordinates given in (33).

The change of coordinates Q↦x induces a canonical transformation. The momenta conjugate to the x coordinates read

(35)P~1=B1aPa=𝒩MGabΦaPb=𝒩MGabΦa∂W∂Qb=∂W∂x1,P~i=BiaPa=Bia∂W∂Qa=∂W∂xi.

We can now use the above variables adapted to the background Hamilton function φ to rewrite the Hamiltonian constraint of the full theory, in which Hm(Q;p,q) is present. In terms of the new canonical variables, (24) reads

(36)H=12MG~11(x)(P~1)2+12Mgij(x)P~iP~j+MV(x)+Hm(x;p,q)=0.

3.2 Gauge Fixing. Reduced Phase Space

Due to time-reparametrisation invariance, we are free to choose a parametrisation in which the following gauge fixing condition holds,

(37)τ(Q(t))=t,

where τ is some smooth function of the Q coordinates.^[8] Such a choice of time parametrisation leads to the following equation for the lapse

(38)1N=1MGab(Q)Pa∂τ∂Qb=1MGab(Q)∂W∂Qa∂τ∂Qb.

We can choose^[9] the parametrisation in which the “background time” function x¹ defines the time coordinate, i.e.

(39)τ(Q(t))=x1(Q(t))=t,

which leads to the fixation of the lapse

(40)1N=1MGab(Q)∂x1∂QaPb=1MG~11(x)P~1=−P~12𝒩2(x)(J(x)+MV(x)).

The momentum conjugate to τ=x1 is Pτ=P~1. We can now solve the Hamiltonian constraint of (36) for Pτ=P~1 to obtain

(41)−Pτ=±2M|𝒩 |[(JM+V)(V+1MHm+12M2gijP~iP~j)]12≡Hred±,

where we used the fact that G~11(x)=[G~11(x)]−1 and (32). The function Hred± is the reduced Hamiltonian for the gauge-fixed system. The corresponding reduced phase-space action is

(42)Sred=∫dτ(P~i∂xi∂τ+pμ∂qμ∂τ−Hred±).

The reduced phase space thus comprised the degrees of freedom P~i,xi,pμ,qμ.

3.3 Background Transformations

In the presence of matter, the Hamilton characteristic function W(Q,P′;q,p′) will in general not coincide with the background Hamilton function φ(Q). Let their difference be S=W−φ. If S is not trivial, we may still interpret it as the Hamilton characteristic function for the system described by the action given in (23), provided we perform the canonical transformation:^[10]

(43)Qa↦Qa,Pa=∂W∂Qa↦Πa=Pa−∂φ∂Qa=∂S∂Qa,qμ↦qμ,pμ=∂W∂qμ↦pμ=∂S∂qμ.

If we change the background Hamilton function, φ=φ′+η, S transforms as S=S′−η so as to leave W=S+φ invariant. Equivalently, the Π-momenta transform as Πa=Π′a−∂η∂Qa so as to keep Pa=Πa+Φa invariant. The matter-sector momenta p_μ are also invariant, as the background Hamilton function does not depend on matter degrees of freedom. Such a change of background Hamilton function amounts to performing a new canonical transformation of the same type as the one given in (43). We will refer to such canonical transformations as background transformations. They will be useful in perturbation theory. The invariance of Pa,pμ implies that the constraint equation is invariant under background transformations and independent of the choice of φ, as it should be.

If we now change to the x coordinates, we find

(44)Π~1=P~1−∂φ∂x1=P~1+2𝒩(x)(J(x)+MV(x)),Π~i=P~i,

where we used (34). Using (32) and (44) and the fact that G~11=(G~11)−1, we may write the constraint equation for an arbitrary choice of φ in the x coordinates as

(45)H=Π~1𝒩−(Π~1)24𝒩2(J+MV)+12Mgij(x)Π~iΠ~j+Hm−J.

Let us now choose the parametrisation τ(Q(t))=x1(Q(t))=t. Due to (34) and (44), we find

(46)∂φ∂x1≡dφdτ,

(47)Pτ=Π~1+dφdτ.

We may rewrite the gauge-fixed lapse of (40) as

(48)N=𝒩1−Π~12𝒩(J+MV),

where we used (44). Using (47), we can also rewrite (41) as

(49)−Π~1=−2𝒩(J+MV)±2M|𝒩 |[(JM+V)(V+1MHm+12M2gijΠ~iΠ~j)]12≡Hred′±,

which is the solution of the transformed constraint equation (45). The function Hred′± differs from Hred± by a total τ derivative and is the reduced Hamiltonian for the gauge-fixed system that comprised the degrees of freedom Π~i,xi,pμ,qμ, with the associated reduced action [cf. (42)]

(50)S′red=∫dτ(Π~i∂xi∂τ+pμ∂qμ∂τ−Hred′±)=Sred−∫dτdφdτ.

Therefore, the reduced canonical theories described by (42) and (50) are equivalent.

3.4 Perturbation Theory

3.4.1 Expansion of the Reduced Hamiltonian

If we assume that all energy scales involved are much smaller than the heavy scale M, it is possible to develop a formal perturbative expansion of the above reduced Hamiltonian in powers of 1M, which corresponds to a weak coupling expansion. We consider that the normalisation function 𝒩 is independent of M, and the source J can be expanded as

(51)J(Q)=∑k=0∞J(k)(Q)Mk.

We further assume that J is chosen such that the inequality |Hm−J|2≪M holds on-shell. Equations (28) and (51) together imply that the background Hamilton function should be expanded as follows:

(52)φ(Q)=Mφ(−1)(Q)+∑k=0∞φ(k)(Q)Mk.

The term proportional to M is needed if the potential V is nonzero [cf. (28)], and we assume this is the case. We then expand the square root in (49) to obtain

(53)Hred′κ=−2𝒩(J+MV)+κ[2M|𝒩V|+|𝒩 |σ(Hm+J)−|𝒩 |4M|V|(Hm−J)2+|𝒩 |σ2MgijΠ~iΠ~j]+𝒪(1M2),

where κ=±1 labels the two solutions of the constraint equation and σ=sgn(V). We refrained from expanding the source J. Equation (53) should be interpreted as the solution of the Hamiltonian constraint for the momentum Π~1, which is conjugate to the coordinate x¹ defined as in (33), when the expansion of φ given in (52) is truncated at order 1M. The reduced equations of motion read

(54)∂xi∂τ=κσ|𝒩 |MgijΠ~j+𝒪(1M2),−∂Π~i∂τ=−2∂∂xi(𝒩J+M𝒩V)+κ∂∂xi[2M|𝒩V|+|𝒩 |σ(Hm+J)−|𝒩 |4M|V|(Hm−J)2+|𝒩 |σ2MgijΠ~iΠ~j]+𝒪(1M2),∂qμ∂τ=κσ|𝒩 |[1−(Hm−J)2MV]∂Hm∂pμ+𝒪(1M2),−∂pμ∂τ=κσ|𝒩 |[1−(Hm−J)2MV]∂Hm∂qμ+𝒪(1M2).

If we choose κ=−σsgn(𝒩)=−sgn(𝒩V), we find from (53) and (54) that all on-shell gravitational-sector momenta (including the on-shell reduced Hamiltonian) exhibit poles in the perturbative parameter, which are the terms proportional to M, whereas xi(τ),qμ(τ) and pμ(τ) are analytic functions of 1M. In particular, we find the lowest-order equations for the gravitational sector

(55)∂xi∂τ=−𝒩gij(Π~jM)+𝒪(1M),∂∂τ(Π~iM)=4∂∂xi(𝒩V)+𝒪(1M),

which imply that, to the lowest order, Πi(τ) are proportional to M and xi(τ) are time-dependent functions of order M⁰. The gravitational-sector trajectory xi(τ) given in (55) is independent of the matter-sector trajectory at this order.

Alternatively, if we choose κ=+σsgn(𝒩)=+sgn(𝒩V), we find that all dynamical solutions (including the on-shell reduced Hamiltonian) are analytic functions of 1M. The lowest-order gravitational-sector equations read

(56)∂xi∂τ=0+𝒪(1M),∂Πi∂τ=−∂∂xi[𝒩(Hm−J)]+𝒪(1M),

which imply that, to the lowest order, both Πi(τ) and xⁱ are of order M⁰, and xⁱ are constants,^[11] i.e. xi(τ)=xi(0)+𝒪(1M). Thus, we see that both sign choices yield solutions xi(τ), which are independent of the matter-sector dynamics at the lowest order. Evidently, sgn(𝒩V) can vary in different regions of configuration space. Therefore, the above perturbative conclusions for a fixed choice of κ only hold when the dynamical trajectory is restricted to a region of phase space in which sgn(𝒩V) is constant.

It is also useful to expand the gauge-fixed lapse given in (48). Using (53), we find

1N=1𝒩−Π~12𝒩2(J+MV)=1𝒩+Hred′κ2𝒩2(J+MV)=1𝒩−1𝒩+κσ|𝒩 |−κJM|𝒩V|+κ(Hm+J)2M|𝒩V|+𝒪(1M2)=κσ|𝒩 |+κ(Hm−J)2M|𝒩V|+𝒪(1M2),

which yields

3.4.2 Propagation of Matter in a Fixed Gravitational Background

We have seen that for both choices of κ the trajectory of the gravitational configuration variables is independent of the matter-sector dynamics (i.e. there is no backreaction from the matter sector onto the gravitational configuration variables) at the lowest order of the weak-coupling expansion. This implies that the clock defined from the heavy variables is not affected by the dynamics of the light variables at this order and thus provides an “external” notion of time for their evolution. The lowest-order equations of motion for the matter sector read [cf. (54)]

(58)∂qμ∂τ=κσ|𝒩(τ,xi(τ))|∂∂pμHm(τ,xi(τ);p,q)+𝒪(1M),∂pμ∂τ=−κσ|𝒩(τ,xi(τ))|∂∂qμHm(τ,xi(τ);p,q)+𝒪(1M),

which are the equations of motion for matter propagating in the fixed gravitational background characterised by the “lapse” κσ|𝒩(τ,x(τ))| and the functions τ=x1,xi(τ). Note that the “lapse” κσ|𝒩(τ,x(τ))| is simply the lowest-order term in the expansion of the gauge-fixed lapse given in (48), as shown in (57). Higher orders in 1M represent corrections from the full, time-reparametrisation invariant theory to the description where the gravitational background is given by a fixed trajectory independent from the matter sector, i.e. the (classical) backreaction of matter is taken into account at higher orders.

3.4.3 Iterative Procedure

The choice κ=+σsgn(𝒩)≡κ+ yields the simple expression for the reduced Hamiltonian

(59)Hred′κ+=𝒩(Hm−J)−𝒩4MV(Hm−J)2+𝒩2MgijΠ~iΠ~j+𝒪(1M2).

This expression may be obtained directly from the transformed Hamiltonian constraint given in (45) in a self-consistent, iterative fashion. We first rewrite (45) as

(60)−Π~1=𝒩(Hm−J)+𝒩2MgijΠ~iΠ~j−(Π~1)24𝒩(J+MV).

By neglecting terms of order 1M in the equation above, we obtain

(61)−Π~1=𝒩(Hm−J),

which is the zeroth-order part of (59). We then substitute (61) in the right-hand side of (60), with the result

−Π~1=𝒩(Hm−J)+𝒩2MgijΠ~iΠ~j−𝒩4(J+MV)(Hm−J)2=𝒩(Hm−J)+𝒩2MgijΠ~iΠ~j−𝒩4MV(Hm−J)2+𝒪(1M2),

which coincides with (59). This iterative solution is essentially the one found in the BO approach in the quantum theory. Indeed, the term proportional to (Hm−J)2 is one of the correction terms obtained by Kiefer and Singh in [12] for a vacuum background (J = 0). Additional correction terms involving the time derivatives of H_m and V were also found by Kiefer and Singh, and we will see how they arise in the quantum theory in Section 4. The term proportional to gijΠ~iΠ~j was neglected in [12]. Here we see that it arises naturally from the expansion of the reduced Hamiltonian, even at the classical level.

The terms of (59) and (60), which are of order 1M (and higher), comprise the gravitational kinetic term 12MgijΠ~iΠ~j−(Π~1)24𝒩2(J+MV). Such terms were referred to in [32] as “corrections” to Hamilton’s equations for the “light” subsystem (here, the matter sector), given that its interaction with a heavy sector (here, the gravitational sector) provides the notion of time. We see from the above construction that such an interpretation is not entirely adequate. Hamilton’s equations (25) are not corrected or altered in any way for the full time-reparametrisation invariant system. They follow, as usual, from the extremisation of the action. The so-called corrections for the “light” subsystem stem from a formal perturbative treatment of the reduced Hamiltonian obtained after a particular choice of time parametrisation has been made. It is not appropriate to interpret such corrections as modifications to the dynamics of matter alone, as we see from the equations of motion (54) that the matter-gravity system is coupled at order 1M; i.e. the gravitational trajectory depends on the matter-sector dynamics. Indeed, for the choice κ=+σsgn(𝒩) in the reduced Hamiltonian, the coordinates xi(τ) are no longer constants (“comoving”) at order 1M, and their evolution follows the lowest-order momenta Π~i, which in turn depend on the matter-sector Hamiltonian H_m.

3.4.4 Hamilton–Jacobi Theory

We may rewrite (60) as the Hamilton–Jacobi equation^[12]

(62)−∂S∂τ=𝒩(x)[Hm(x;∂S∂q,q)−J(x)]+𝒩(x)2Mgij(x)∂S∂xi∂S∂xj−(∂S∂τ)24𝒩(x)(J(x)+MV(x)),

and solve it iteratively, as before. To the lowest order, one finds the TDHJE:

(63)−∂S∂τ=𝒩(x)[Hm(x;∂S∂q,q)−J(x)],

which may be interpreted as the ordinary Hamilton–Jacobi equation associated to (58) when κ=+σsgn(𝒩) is chosen. Thus, we may interpret (63) as the TDHJE for matter propagating in a fixed gravitational background, as x=(τ,xi(0)) to the lowest order for this choice of κ (which corresponds to the iterative procedure). In this case, the arbitrary source term J may be removed by redefining S(τ,xi,q)↦S(τ,xi,q)+∫τ𝒩(λ,xi)J(λ,xi)dλ. At the next order, we obtain the corrected Hamilton–Jacobi equation

(64)−∂S∂τ=𝒩(x)[Hm(x;∂S∂q,q)−J(x)]−𝒩(x)4MV[Hm(x;∂S∂q,q)−J(x)]2+𝒩(x)2Mgij(x)∂S∂xi∂S∂xj+𝒪(1M2),

which corresponds to (54) when κ=+σsgn(𝒩) is chosen. At this order, the dynamics of xi(τ) is taken into account. Therefore, S is most appropriately interpreted as the (τ dependent) Hamilton principal function for the reduced system that comprised the degrees of freedom Π~i,xi,pμ,qμ and not as the “corrected” Hamilton principal function of a system spanned by pμ,qμ alone. In this point, we differ from Briggs [32], who interpreted the equation W=S+φ as a decomposition of the total Hamilton (characteristic) function into a Hamilton (characteristic) function φ for the heavy sector and a Hamilton (principal) function S for the “light” degrees of freedom. We interpret the decomposition W=S+φ as a canonical transformation. We will now construct the quantum theory in analogy to the Hamilton–Jacobi theory.

4 The BO Approach as a Choice of Gauge. Quantum Theory

The main challenge in quantising the constrained system associated with the action given in (23) is to define the Hilbert space of physical states. A state is defined to be “physical” if it is annihilated by the constraint operator.^[13] In Section 4.2, we will choose a tentative definition of the inner product that is conserved with respect to our chosen time variable. To determine the quantum version of the constraint equation (24), we adopt the Laplace–Beltrami factor ordering for both the gravitational and matter-sector Hamiltonians, which yields^[14]

(65)H^gΨ=−12M|Gh|∂∂Qa(|Gh|Gab∂Ψ∂Qb)+MV(Q)Ψ,

(66)H^mΨ=−12h∂∂qμ(hhμν∂Ψ∂qν)+Vm(Q;q)Ψ.

For simplicity of notation, we define the gravitational-sector Laplace–Beltrami operator to be

(67)∇2=1|Gh|∂∂Qa(|Gh|Gab∂∂Qb).

The quantum constraint equation then reads

(68)H^Ψ=(−12M∇2+MV(Q)+H^m(Q;p^,q))Ψ(Q,q)=0,

which will be referred to as the WDW equation. The factor ordering in (65), (66), and (67) was chosen so as to guarantee that the WDW equation is covariant under arbitrary coordinate transformations in the configuration space of both gravitational and matter degrees of freedom [4], [5].

4.1 Quantum Background Transformations

In the classical theory, the Hamilton characteristic function W can be decomposed into W=S+φ for a given choice of background Hamilton function φ. Analogously, given a choice of (classical) φ, we consider the following phase transformation in the quantum theory

(69)Ψ(Q,q)=eiφ(Q)Ψφ(Q,q),O^(P^,Q;p^,q)=eiφ(Q)O^φ(P^φ,Q;p^,q)e−iφ(Q),

for any (physical) state Ψ and any operator Ô. If we change the background Hamilton function, φ=φ′+η, the state Ψ_φ and the operator O^φ change according to the quantum background transformations

(70)Ψφ=e−iηΨφ′,O^φ=e−iηO^φ′eiη,

so as to keep Ψ and Ô invariant. We will fix φ by demanding that it be a solution to the (classical) equation (28), after a choice of source J is made. As before, we define the (real-valued) background momenta to be Φa=∂φ∂Qa. We now define

(71)𝔍(Q,q):=e−iφ(Q)(12M∇2−MV(Q))eiφ(Q)=J(Q)+iK(Q,q),

(72)K(Q,q):=12M∇2φ.

We note that 𝔍 is not the operator −H^g,φ, but rather the complex function obtained by acting with −H^g on the complex exponential eiφ and subsequently multiplying by the conjugate exponential. The dependence of K on the q coordinates comes from the Laplace–Beltrami factor ordering in (67). Under a quantum background transformation, we obtain

(73)K(Q,q)=K′(Q,q)+12M∇2η,

in addition to the transformation laws given in (31). From the full WDW equation (68) and from (71), we obtain an equation for Ψ_φ [cf. (8)],

(74)iGab(Q)ΦaM∂Ψφ∂Qb=(H^m(Q;p^,q)−𝔍(Q,q))Ψφ−12M∇2Ψφ.

One may verify that this equation is invariant under quantum background transformations by using (31), (70), and (73). This is also understood from the fact that (74) is simply H^φΨφ=0. In (127) and (128) of Appendix A, we find that the function K may be rewritten in the x coordinates as

(75)K(Q,q)=12|G~h|∂∂x1(|G~h|𝒩),

for the Laplace–Beltrami factor ordering. By performing the coordinate transformation given in (33) (see also Appendix A), we can rewrite (74) as

(76)i∂Ψφ∂x1=𝒩(x)(H^m(x;p^,q)−𝔍(x,q))Ψφ−𝒩(x)2Mh|G~|∂∂xi(h|G~|gij∂∂xjΨφ)−𝒩(x)2Mh|G~|∂∂x1(h|G~|G~11∂∂x1Ψφ),

which is a quantisation of the corresponding classical (60) and (62). The solution to (76) is the wave function of the gauge-fixed system that comprised the degrees of freedom p^μ,q^μ,Π^i,x^i, whereas x¹ plays the role of the time parameter. Therefore, the evolution of Ψ_φ accounts for the (coupled) dynamics of both matter and gravitational degrees of freedom, contrary to what is usually assumed in a BO context, which is that Ψ_φ is only the matter-sector wave function. The interpretation of Ψ_φ as the wave function for both gravitational and matter degrees of freedom is analogous to the interpretation of the Hamilton (characteristic) function S=W−φ as describing the dynamics of the composite system in the classical theory. We will see in Section 4.4 how to recover the description of the dynamics of quantum matter in a fixed gravitational background.

It is worthwhile to mention the “complex structure problem” [4], [5], which in the formalism presented here can be understood as follows. The factor of i in (76) leads to the coupling of the real and imaginary parts of Ψ_φ. On the other hand, the WDW equation (68) is real, and no such coupling occurs for the real and imaginary parts of Ψ. In fact, we can take Ψ to be real. The complex structure of (76) originates solely from the phase prefactor in Ψ=eiφΨφ, which may seem ad hoc. This has been criticised by Barbour [82] and Kuchař [4] in the context of the WKB approximation. However, one may invoke decoherence [24], [25], [79] to justify the appearance of the phase prefactor. In the present formalism, we assume from the start that the Hilbert space is complex, and in this context, the phase transformation given in (69) is the quantum analogue of the canonical transformation given in (43) in the classical theory. Thus, the phase transformation Ψφ=Ψe−iφ can be employed without loss of generality, even if Ψ is real.

4.2 Inner Product and Unitarity

Given a solution φ to the (classical) equation (28), we may define the coordinate x adapted to the background Hamilton function φ as in (33) (and in Appendix A). Due to the Laplace–Beltrami factor ordering, we may change coordinates Q↦x in the WDW equation (68) to obtain

(77)0=−12M|G~h|∂∂xA(|G~h|G~AB∂Ψ∂xB)+MV(x)Ψ−12h∂∂qμ(hhμν∂Ψ∂qν)+Vm(x;q)Ψ.

Equation (77) leads to the continuity equation

(78)1|G~h|∂∂xA(|G~h|jA)+1h∂∂qμ(hjμ)=0,

where the Klein–Gordon current is defined as [4], [5], [21]

(79)jA(x,q)=ifG~AB2M(Ψ¯1∂Ψ2∂xB−Ψ2∂Ψ¯1∂xB),jμ(x,q)=ifhμν2(Ψ¯1∂Ψ2∂qν−Ψ2∂Ψ¯1∂qν),

for any two solutions Ψ1,Ψ2 of the WDW equation (77). The parameter f is a real constant that will be fixed in what follows. Given φ, the continuity equation (78) implies that the quantity^[15]

(80)(Ψ1,Ψ2)KG:=∫∏i=2ndxi|G~|∏μ=1ddqμhj1(x,q)=∫∏idxi|G~|∏μdqμhifG~112M(Ψ¯1∂Ψ2∂x1−Ψ2∂Ψ¯1∂x1)

is conserved with respect to the x¹ coordinate,

(81)∂∂x1(Ψ1,Ψ2)KG=0.

The conserved charge given in (80) is the Klein–Gordon inner product. If x¹ is considered to be the time parameter, then (81) implies that the dynamics based on the Klein–Gordon inner product is unitary with respect to x¹ evolution. As is well known, the Klein–Gordon inner product is indefinite. Nevertheless, we will see in Sections 4.3 and 4.5 that this inner product is of a definite sign in the perturbative regime, i.e. for solutions of the WDW equation found via the iterative procedure. Thus, a probability interpretation is possible in perturbation theory.

If we perform a quantum background transformation, Ψ1=eiφΨ1,φ,Ψ2=eiφΨ2,φ, the inner product in (80) can be rewritten as

(82)(Ψ1,Ψ2)KG=∫∏idxi|G~|∏μdqμh(−f)G~11M∂φ∂x1Ψ¯1,φΨ2,φ+(Ψ1,φ,Ψ2,φ)KG

Using the first of (34), which implies ∂φ∂x1=M𝒩G~11, we obtain

(83)(Ψ1,Ψ2)KG=∫∏idxi|G~|∏μdqμh(−f)𝒩Ψ¯1,φΨ2,φ+(Ψ1,φ,Ψ2,φ)KG,

which we can rewrite as

(84)(Ψ1,Ψ2)KG=∫∏idxi|G~|∏μdqμh×Ψ¯1,φ[(−f)𝒩+G~112M(if∂∂⁡x1→)−(if∂∂⁡x1←)G~112M]Ψ2,φ.

This form of the Klein–Gordon inner product will be useful in perturbation theory.

4.3 Perturbation Theory I

As in the classical theory, if we restrict ourselves to a regime in which all energy scales are much smaller than the heavy scale M, we can develop a formal perturbative expansion in powers of 1M. We assume that the states Ψ_φ are analytic functions of 1M and admit the formal expansion

(85)Ψφ=∑k=0∞1MkΨφ(k).

Together with (52), (85) implies that the states Ψ=eiφΨφ can be expanded as

(86)Ψ=eiMφ(−1)∑k=0∞1Mkξ(k),

where ξ(k) are coefficients that can be computed from the expansions in (52) and (85). The expansion of (86) is the one usually performed in the semiclassical approach [9], [10], [11], [12].

We now set out to solve the constraint equation (76) in a self-consistent, iterative fashion in analogy to what was done in the Hamilton–Jacobi theory in Section 3.4.4. Let us at first keep only terms to the lowest order in 1M. The inner product in (84) becomes

(87)(Ψ1,Ψ2)KG=(−f)∫∏idxi∏μdqμsgn(𝒩)2|V(x)g(x)|h(x;q)Ψ¯1,φ(x,q)Ψ2,φ(x,q)+𝒪(1M).

Let us restrict ourselves to a region of configuration space where sgn(𝒩) is constant. Then, we see from (87) that the lowest-order approximation to the Klein–Gordon inner product is positive-definitive if we set f=−sgn(𝒩). To the lowest order, (76) reads

i∂Ψφ∂x1=𝒩(x)(H^m(x;p^,q)−𝔍(x,q))Ψφ(x,q)+𝒪(1M),

which can be rewritten as

(88)i∂Ψφ∂x1=𝒩(x)(H^m(x;p^,q)−J(x))Ψφ−i[∂∂x1log(|2Vgh|14)]Ψφ+𝒪(1M),

for constant sgn(𝒩). Equation (88) is a quantum version of the corresponding classical (61) and (63). Evidently, the solutions of (88) are valid only up to order M⁰. The imaginary term on the right-hand side corresponds to the truncation of i𝒩K to the lowest order in 1M. Such a term is present due to the dependence of measure given in (87) on the coordinate x¹, which plays the role of a time parameter^[16] in the gauge-fixed theory. Indeed, if Ψ1,φ and Ψ2,φ are solutions of (88), then the inner product given in (87) is conserved with respect to x¹ up to order M⁰. Although this approximate conservation is a consequence of the exact equation (81), we find it instructive to explicitly verify that (87) is conserved using the solutions to (88), and we have registered this computation in Appendix B, where we keep terms up to order 1M (cf. Section 4.5).

4.4 Propagation of Quantum Matter in a Fixed Gravitational Background

4.4.1 Partial Ehrenfest Equations

As before, we define the matter-sector partial averages of an operator Ô as

(89)⟨O^⟩m(x):=∫∏μdqμhΨ¯φ(x,q)O^Ψφ(x,q)∫∏μdqμhΨ¯φ(x,q)Ψφ(x,q),

provided the integrals converge. We note that ⟨O^⟩m(x) is a function of the remaining gravitational variables x1,xi, or Q^a. We find that self-adjoint matter-sector operators, O^m≡O^m(x;p^,q^), obey the partial Ehrenfest equation

(90)∂∂x1⟨O^m⟩m(x)=⟨∂O^m∂x1⟩m+i𝒩(x)⟨[H^m,O^m]⟩m+⟨[∂∂x1log|h|14,O^m]⟩m+𝒪(1M).

We note that (90) holds despite the fact that the dynamics of Ψ_φ is not unitary in the matter sector. Equation (90) is the Ehrenfest equation for a self-adjoint matter-sector operator defined in the gravitational background corresponding to the time parameter x¹, its associated lapse function 𝒩(x), and the “comoving” coordinates xⁱ.

4.4.2 Matter-Sector Unitarity

We can impose unitarity in the matter sector by considering the factorisation Ψφ(x,q)=χ(x)ψ(x,q), where we demand that the factor χ obeys the equation

(91)i∂χ∂x1=−𝒩(x)J(x)χ(x)−i[∂∂x1log|2V(x)g(x)|14]χ(x)+𝒪(1M).

By inserting Ψφ=χψ into (88) and using (91), we obtain

(92)i∂ψ∂x1=𝒩(x)H^m(x;p^,q)ψ(x,q)−i[∂∂x1log|h|14]ψ(x,q)+𝒪(1M).

Using (91) and (92), one may explicitly verify that

(93)∂∂x1∫∏idxi2|Vg|χ¯(x)χ(x)=0+𝒪(1M),∂∂x1∫∏μdqμhψ¯(x,q)ψ(x,q)=0+𝒪(1M).

Thus, unitarity is enforced separately in each sector. The solution to (91) is

(94)χ(x1,xi)=|2V(x1,xi)g(x1,xi)|−14γ(xi)exp(i∫x1dλ𝒩(λ,xi)J(λ,xi))+𝒪(1M),

where J is understood as its lowest-order approximation, and γ(xi) is an arbitrary factor that satisfies the normalisation condition in the “flat” inner product ∫∏idxiγ¯(xi)γ(xi)=1. Similarly, we can normalise ψ(x,q) in the matter-sector inner product. The dynamics of ψ(x,q) is identical to the one usually studied in quantum theory in a fixed gravitational background.

The original total state Ψ, which is a solution to the WDW equation (68), can thus be written as

(95)Ψ(x,q)=eiφ(x)Ψφ(x,q)=eiφ(x)χ(x)ψ(x,q)+𝒪(1M),

which is just the BO exact factorisation for the total state Ψ [cf. (3)]. We note that we have defined the time variable from φ, which is only part of the phase of the “gravitational factor” eiφ(x)χ(x). The form of (95) was used in [48] as the BO ansatz for the total state of nuclei and electrons in the context of molecular physics.

4.5 Perturbation Theory II

Let us now continue with the iterative procedure and keep terms only up to order 1M. Using the lowest-order equation (88), we can rewrite the inner product given in (84) as^[17]

(96)(Ψ1,Ψ2)KG=∫∏idxi∏μdqμΨ¯1,φ(x,q)ℳ^(x;p^,q)Ψ2,φ(x,q)+𝒪(1M2),

(97)ℳ^(x;p^,q):=−fsgn(𝒩(x))2|V(x)g(x)|h(x;q)[1+12MV(x)H^m(x;p^,q)],

where we used the fact that H^m(x;p^,q) is symmetric with respect to the matter-sector inner product. We now restrict ourselves to a region of configuration space where sgn(𝒩) is constant and set f=−sgn(𝒩). The potential V(x) can have a positive or negative sign. Thus, the inner product given in (96) is positive-definite if the following condition holds^[18]

(98)∫∏idxi∏μdqμ2|Vg|hΨ¯φΨφ≫1M∫∏idxi∏μdqμ12|gV|hΨ¯φH^mΨφ.

The inequality (98) should be satisfied in the regime of validity of perturbation theory. To continue the iterative procedure, we use (88) to eliminate the x¹ derivatives in the right-hand side of (76). After some algebra, we obtain

(99)i∂Ψφ∂x1+iΓ^Ψφ=𝒩(H^m−J)Ψφ−𝒩4MV(H^m−J)2Ψφ−12M2|Vgh|∂∂xi(2|Vgh|𝒩gij∂Ψφ∂xj)+1M𝒱Ψφ+𝒪(1M2),

where we defined^[19]

(100)Γ^:=∂∂x1log|2Vgh|14+12M2|Vgh|∂∂x1(σh2|gV|H^m)−14MVH^m∂∂x1log|2Vgh|14−14MV(∂∂x1log|2Vgh|14)H^m,

(101)𝒱:=132𝒩V|Vgh|(∂∂x12|Vgh|)2−122|Vgh|∂∂x1(14𝒩V∂∂x12|Vgh|),

and σ=sgn(V). Equation (99) is a quantum version of the corresponding classical (59) and (64). The term 1M𝒱 can be interpreted as a “quantum correction,” which is present due to the fact that we quantised the constraint equation (45) rather than the reduced Hamiltonian given in (49) and subsequently adopted an iterative procedure to find the solution to the quantum constraint equation (76). We refrain from factorising Ψφ=χψ to enforce unitarity in the matter sector by a suitable choice of χ, as it is sufficient to interpret Ψ_φ as the wave function for both gravitational and matter degrees of freedom, which is a solution to (99).

Equation (99) was computed in [12] for a vacuum background (J = 0), and the terms involving the derivatives with respect to the xⁱ variables as well as the term proportional to 𝒱 were absent.^[20] Here, such terms arise from the iterative solution of the constraint equation (76). Moreover, the x¹ derivatives of the matter Hamiltonian H^m and of the potential V present in the term iΓ^ were interpreted in [12] as unitarity-violating terms induced by gravity. One is led to this interpretation if one regards Ψ_φ as the matter-sector wave function and (99) as a “corrected” Schrödinger equation for the matter sector. However, as we have argued above, the state Ψ_φ is most appropriately interpreted as the wave function for the coupled system of both gravity and matter that comprised the configurational degrees of freedom xⁱ and q^μ when x¹ is regarded as the time parameter. In this way, the inner product involves an integration not only over matter variables, but also over the xⁱ variables [cf. (96)]. Thus, rather than introducing a violation of unitarity, the term iΓ^ is the factor that guarantees unitarity with respect to the inner product given in (96). Indeed, we explicitly verify that unitarity holds due to this term in Appendix B.

4.6 Backreaction

We now show how the formalism of [17], [18], [19], [20], [39], [60], [76] can be recovered from the above construction. Following what was done in Section 2.2, we compute the equations with “backreaction” terms from (71) and (74). Upon taking the matter-sector partial average^[21] of (74), we find

(102)−J(Q)=iGab(Q)ΦaM⟨∂∂Qb⟩m−⟨H^m(Q;p^,q)⟩m+i⟨K(Q,q)⟩m+12M⟨∇2⟩m.

Using (33) and (75), we can write

iGab(Q)ΦaM⟨∂∂Qb⟩m+i⟨K(Q,q)⟩m=iGab(Q)ΦaM⟨∂∂Qb+𝒩2|G~h|∂∂Qb(|G~h|𝒩)⟩m.

If we now insert the real part of (102) into (28), we obtain

(103)12MGab(Q)(∂φ∂Qa+Aa)(∂φ∂Qb+Ab)+MV(Q)=−⟨H^m(Q;p^,q)⟩m+12M(ℜ𝔢⟨∇2⟩m+GabAaAb),

where we defined the “Berry connection” as Aa=ℑ𝔪⟨∂∂Qa+𝒩2|G~h|∂∂Qa(|G~h|𝒩)⟩m [cf. (12)]. Equation (103) can be seen as a Hamilton–Jacobi equation with quantum backreaction terms. Using the expansion of φ given in (52), we may solve (103) at each order of 1M. The lowest orders read

(104)𝒪(M1):12Gab(Q)∂φ(−1)∂Qa∂φ(−1)∂Qb+V(Q)=0,

(105)𝒪(M0):Gab∂φ(−1)∂Qa∂φ(0)∂Qb=−Gab∂φ(−1)∂QaAb(0)−⟨H^m(Q;p^,q)⟩m(0),

where ⟨O^⟩m(0) denotes the lowest-order approximation to the partial average ⟨O^⟩m, which is the matter-sector expectation value of the operator Ô taken with respect to a solution of (88). Equation (104) is the vacuum Hamilton–Jacobi equation for the gravitational sector. Thus, we see that the effects of backreaction terms enter only in (105), at order M⁰. This is consistent with the expansion of J in (51). The conclusion that backreaction effects are not found to the lowest order was also reached in [12], [24], [25], [39] in the context of the usual semiclassical (BO) interpretation of quantum gravity.

The background Hamilton function φ in (103) is sourced by ⟨H^m(Q;p^,q)⟩m and the Berry connection terms. As (103) is by construction equivalent to (28), we see that the arbitrariness of the Berry connection terms corresponds to the freedom in choosing J. A given choice of J defines a background Hamilton function φ and thus a background gravitational trajectory with respect to which the weak coupling expansion is to be performed. This was also argued in a different way by Parentani in [31], where he emphasised that this procedure corresponds to a background field approximation. The classical (quantum) dynamics of the composite system of gravitational and matter degrees of freedom is encoded in the Hamilton (wave) function W(Q,P′;q,p′) (Ψ(Q,q)), and W (the phase of Ψ) coincides with φ only up to order M¹ in the weak coupling expansion [cf. (104)].

If we insert (102) back into (71) and (74), we obtain

(106)−12M(∇2+⟨∇2⟩m)χφ−GabM∂χφ∂Qa⟨∂∂Qb⟩m+i(K−⟨K⟩m)χφ+MVχφ+⟨H^m⟩mχφ=0,

(107)GabMχφ∂χφ∂Qa(∂∂Qb−⟨∂∂Qb⟩m)Ψφ=(H^m−⟨H^m⟩m)Ψφ−i(K−⟨K⟩m)Ψφ−12M(∇2−⟨∇2⟩m)Ψφ,

where we defined χφ(Q)=eiφ(Q). Equations (106) and (107) are equivalent to (71) and (74), respectively, and are the analogues of the nonrelativistic (20) and (21). We refrain from rewriting (106) and (107) in terms of “covariant” derivatives in analogy to (18).

Nonlinear coupled equations such as (106) and (107) were used in [17], [18], [19], [20], [39], [60], [76] to describe the dynamics of the composite gravity-matter system, and in particular, Ψ_φ was interpreted as the matter-sector wave function. We stress that such an interpretation requires the additional choice of matter-sector unitarity and the ensuing interpretation of χ_φ and Ψ_φ as marginal and conditional wave functions, respectively. Alternatively, Ψ_φ may be interpreted as the wave function for both gravitational and matter degrees of freedom, obtained from the solution Ψ of the WDW equation by a phase transformation. “Semiclassical gravity” emerges from the BO approach to quantum gravity in the sense that (106) is equivalent to (71), which leads to the Hamilton–Jacobi equation with backreaction terms [cf. (103)].

5 A Simple Example: The Relativistic Particle

As a simple example of the above formalism, let us consider the action for a massive relativistic particle in two spacetime dimensions,

(108)S=∫dt(p1q˙1+p2q˙2−NH),H=−1c2(p1)2+(p2)2+m2c2,

where we have restored factors of the speed of light c, and m is the mass of the particle. We will develop an expansion in powers of 1c2, such that c is plays the role of the “heavy” scale M in the formalism we have presented. The heavy sector thus consists of the degrees of freedom (p1,q1), whereas the light sector comprised the (p2,q2) variables. This example was also considered in [12] as an analogy to the semiclassical interpretation of quantum gravity. Here, we examine it to clarify the fact that the results of the semiclassical and BO approaches coincide with a particular choice of gauge both at the classical and quantum levels.

We will choose a gauge adapted to a given background Hamilton function φ. We take φ(q)=mc2q1, which solves the “vacuum” (J = 0) Hamilton–Jacobi equation for the heavy sector,

−1c2(∂φ∂q1)2+m2c2=0.

Let us now define the configuration-space coordinate adapted to this choice of φ. Let us set 𝒩=12m. We then choose the basis vector [cf. (33)]

(109)∂∂x1=𝒩c2(−2)∂φ∂q1∂∂q1=−1mc2∂φ∂q1∂∂q1=−∂∂q1,

which leads to x1=−q1, which is canonically conjugate to p~1=−p1. We now fix the gauge x1(t)=t, which determines the lapse to be

(110)N=−c22p~1.

Solving the constraint equation for p~1 yields the reduced Hamiltonian

(111)−p~1=±c2(p2c)2+m2.

Inserting (111) into (110) yields [cf. (57)]

(112)N=±12(p2c)2+m2=±12m+𝒪(1c2).

Upon performing the canonical transformation p~1=Π~1+∂φ∂x1=Π~1−mc2, we find

(113)−Π~1=−mc2±mc21+(p2mc)2,

which is the solution to the transformed constraint equation [cf. (45)]

(114)2mΠ~1−1c2(Π~1)2+(p2)2=0.

The solution of (114) found in the iterative procedure is the one with the positive sign in front of the square root and reads

(115)−Π~1=12m(p2)2+𝒪(1c2).

In the quantum theory, we promote the variables to operators q^1Ψ=q1Ψ,p^1Ψ=−i∂Ψ∂q1,q^iΨ=q2Ψ,p^2Ψ=−i∂Ψ∂q2. The quantum constraint equation reads

1c2∂2Ψ∂(x1)2−∂2Ψ∂(q2)2+m2c2Ψ=0.

The conserved Klein–Gordon inner product with a suitably chosen constant prefactor is

(116)(Ψ1,Ψ2)KG=∫dq2i2mc2(Ψ¯1∂Ψ2∂x1−Ψ2∂Ψ¯1∂x1).

We perform the phase transformation Ψ=eiφΨφ to obtain the transformed quantum constraint equation

(117)i∂Ψφ∂x1=−12m∂2Ψφ∂(q2)2+12mc2∂2Ψφ∂(x1)2=−12m∂2Ψφ∂(q2)2+𝒪(1c2).

For solutions of (117) found in the iterative procedure, the transformed inner product reads

(118)(Ψ1,Ψ2)KG=∫dq2[Ψ¯1,φΨ2,φ+i2mc2(Ψ¯1,φ∂Ψ2,φ∂x1−Ψ2,φ∂Ψ¯1,φ∂x1)]=∫dq2Ψ¯1,φΨ2,φ+𝒪(1c2).

Equation (106) reads

1c2∂2χφ∂(q1)2+1c2⟨∂2∂(q1)2⟩mχφ+2c2∂χφ∂q1⟨∂∂q1⟩m+m2c2χφ−⟨∂2∂(q2)2⟩mχφ=0.

If we substitute χφ=eiφ=eimc2q1 into the above equation, we find

i⟨∂∂q1⟩m−12m⟨∂2∂(q2)2⟩m+12mc2⟨∂2∂(q1)2⟩m=0,

which is simply the partial average of the constraint equation (117), as x1=−q1. Therefore, χφ=eiφ=eimc2q1 is a solution to (106), which is considered to be the equation for the heavy-sector wave function in the BO approach of [17], [18], [19], [20], [39], [60], [76].

6 Conclusions

The problem of time in canonical quantum gravity has many facets and has inspired various interpretations of the theory [4], [5], [6]. In this article, we have reinterpreted the usual results of the semiclassical interpretation of quantum cosmology based on the view that the independence of the wave function on the time parameter does not conceal the dynamical content of the quantum theory and that it is unnecessary to restrict some of the fields to the (semi)classical regime to recover a notion of dynamics. We have followed a conservative route, in which the interpretation of the quantum dynamics closely follows that of the canonical classical theory, which is far less controversial. The diffeomorphism-induced symmetry of the theory implies that the choice of time parameter is not unique. At the classical level, this implies that Hamilton’s equations can only be solved once the arbitrary Lagrange multipliers associated with the gauge symmetry are fixed. This corresponds to a choice of gauge, i.e. a choice of coordinate system. In a closed, isolated universe, it is natural to fix the gauge by functions of the canonical variables. In this way, the coordinates are fixed by the contents of the universe [15]. In particular, the time variable is given by (the level sets of) a particular function of the canonical variables.

We took the position that the same is true in the quantum theory, which we constructed in analogy to the Hamilton–Jacobi theory. As in the classical case, the quantum dynamics must be understood with respect to a nonunique choice of time as a function of configuration variables and momenta. Such a time function is measured by physical clocks composed of the quantised variables themselves, as Singh [11] noted for the case of “WKB time.” Indeed, the “WKB time” used in the semiclassical interpretation (and its generalisation known as the BO approach) is a function of the canonical variables, and thus, it must correspond to a particular gauge fixing. We have shown that this is the case and that the usual results of the BO approach are obtained by a particular class of gauge choices, which can be made both at the classical and quantum levels. The (WKB) time parameter x¹ is chosen from a congruence of (classical) trajectories associated with a background Hamilton function, which solves the Hamilton–Jacobi equation with arbitrary sources J. Its interpretation is that of the standard of time measured by clocks that travel along the background trajectories defined from the background Hamilton function.

At the quantum level, the chosen time parameter x¹ appears in the quantum constraint equation by a change of coordinates in configuration space, and the usual BO factorisation of the wave function can be replaced by a phase transformation determined by the background Hamilton function. The ambiguity in the usual BO factorisation corresponds to the ambiguity in the choice of phase factor, which, in turn, is related to the freedom to choose different time variables. Thus, there is no need to perform a WKB approximation to recover the concept of time. The inner product and the dynamical interpretation of the theory here constructed, although provisional, are independent of the semiclassical limit. Nevertheless, some approximation method is needed to perform practical computations. In the formalism we have presented, the only approximation used was a weak-coupling expansion. The formalism is applicable not only to quantum cosmology, but also to the timeless nonrelativistic quantum mechanics of closed systems, which was studied in [27], [28], [29], [32], [54].

To reproduce the results of the usual semiclassical approach, the background Hamilton function is chosen such that a weak-coupling expansion is possible. Equivalently, the fields are separated into heavy and light variables, such that a perturbative expansion in the “light-to-heavy” ratio of mass scales can be performed. The background trajectories coincide with the trajectories of the heavy variables to the lowest order in perturbation theory. In the case of general relativity, the “heavy” mass scale is the Planck scale. We have shown that, in the perturbative regime, one can expand the reduced Hamiltonian of the gauge-fixed classical theory to obtain some of the corrections found in the semiclassical approach to quantum cosmology. This shows that some of the “corrections from quantum gravity” found in [11], [12] are, in fact, a result of the weak-coupling expansion of the gauge-fixed system and are present even in the classical theory. The correction terms can be obtained by an iterative solution of the constraint equation both at the classical and quantum levels. In the quantum theory with the Laplace–Beltrami factor ordering, additional correction terms are present, which guarantee unitarity of the theory with respect to the (Klein–Gordon) inner product in the perturbative regime.

The arbitrary source J approximates the backreaction of the light-sector Hamiltonian onto the heavy-sector dynamics in the perturbative regime. This is true both in the classical and quantum theories. In particular, we have shown that the arbitrary source J is associated with the usual quantum backreaction terms, which comprised the expectation value of the light-sector Hamiltonian (averaged only over light variables) and Berry connection terms. We have shown that the ambiguity in the choice of J is equivalent to the ambiguity in the definition of the Berry connection terms, which corresponds to the freedom to choose a particular background Hamilton function and its associated time parameter. We refer the reader to [63], [65], [81] for a complementary discussion on the quantum backreaction terms.

As we have seen, a TDSE appears as the lowest-order approximation to the quantum constraint equation in the weak-coupling expansion. We have interpreted the solution of the quantum constraint equation as the wave function of the composite system of gravitational and matter degrees of freedom. This interpretation is motivated by the fact that higher orders in the weak coupling expansion lead to a corrected TDSE, which includes the momenta of gravitational degrees of freedom (here denoted by xⁱ) and which therefore incorporates their dynamics. Nevertheless, we have argued that it is possible to factorise the wave function such that light-sector unitary is guaranteed, and a marginal-conditional interpretation of the factors is warranted, as was advocated in [54]. In this way, the conditional wave function describes the unitary dynamics of the light sector.

Such observations are important if one wants to analyse the phenomenology of the corrected TDSE. Several works in the literature [37], [38], [85], [86], [87], [88], [89], [90], [91], [92] have addressed this topic by applying the semiclassical or BO approaches to compute quantum gravitational corrections to the Cosmic Microwave Background power spectrum of cosmological scalar and tensor perturbations. Kiefer and Krämer [85], [86], Bini et al. [87], and Brizuela et al. [88], [89], [90] used the corrected TDSE found in [12] to compute the corrected power spectrum. Their method can be reinterpreted with the formalism presented in this article as a particular gauge fixing and subsequent weak-coupling expansion. It is also worth mentioning that Kamenshchik et al. [37], [38], [91] found corrections to the power spectrum by considering “nonadiabatic” effects related to the quantum backreaction and Berry connection terms.

The results of this article lead us to the conclusion that time does not “emerge” only when a subset of the fields is (semi)classical, which is the central tenet of the semiclassical interpretation of quantum gravity. Rather, different notions of time are available in the full quantum theory and are associated with the different coordinate systems that one can employ. The semiclassical approach can be reinterpreted as a particular gauge fixing, and the chosen time function x¹ is meaningful beyond the semiclassical level when interpreted as a combination of the quantised heavy variables. Thus, we expect that the semiclassical approach can be superseded by gauge-fixing methods in a more definitive version of the quantum theory. Moreover, it would be desirable to generalise the inner product that was employed in this article to guarantee positive definiteness and unitarity beyond the perturbative regime and such that the interference between perturbative and nonperturbative solutions could be analysed. In addition to this, it would be useful to compare the quantum dynamics with respect to x¹ to the results associated with more general choices of the time variable (i.e. not given by the phase of the heavy part of the wave function). This will be left for future work.

Appendix A: Canonical Variables Adapted to a Choice of Background. Formulae

In this Appendix, we collect useful formulae related to the change of coordinates given in (33), which we repeat as follows:

(119)B1a=𝒩MGabΦb=∂Qa∂x1,Bia=∂Qa∂xi.

The coordinates defined via (119) induce the following transformation between basis vectors in the tangent space,

(120)∂∂x1=𝒩MGabΦb∂∂Qa=B1a∂∂Qa,∂∂xi=Bia∂∂Qa.

The metric tensor in the new coordinates has components [cf. (32)]

(121)G~AB=Gab∂Qa∂xA∂Qb∂xB=GabBAaBBb,G~11=GabB1aB1b=−2𝒩2(JM+V),G~1i=0,G~ij=GabBiaBjb≡gij,

whereas the inverse metric tensor has components G~11=(G~11)−1, G~1i=0, and gij=G~ij, such that gijgjk=δki. The determinants obey |G~|=|G|B, where B=detBAa. The inverse of the jacobian BAa is

(122)∂xA∂Qa=(B−1)aA=G~ABGabBBb.

The old basis vectors can thus be expressed in terms of the new basis as^[22]

(123)∂∂Qa=(B−1)aA∂∂xA=G~11GabB1b∂∂x1+G~ijGabBib∂∂xj=−Φa2𝒩(J+MV)∂∂x1+gijGabBib∂∂xj.

By differentiating the first of (119) with respect to Q^b, we obtain the useful identity

(124)Gac∂Φc∂Qb=M𝒩∂B1a∂Qb−1𝒩∂∂Qb(𝒩Gac)Φc.

We also have the Hessian identities

(125)∂BBa∂xA=∂2Qa∂xA∂xB=∂2Qa∂xB∂xA=∂BAa∂xB,

which lead to

(126)(B−1)aA∂BAa∂xB=(B−1)aA∂BBa∂xA=∂BBa∂Qa.

Due to the identity given in (124), we may rewrite the function K defined in (72) as follows:

(127)K(Q)=12𝒩∂B1a∂Qa−12M𝒩∂∂Qa(𝒩Gac)Φc+Φa2M|Gh|∂∂Qb(|Gh|Gab).

Now using (119), (123), and (126), we can rewrite (127) in the x-coordinate system,

(128)K(Q)=12𝒩∂B1a∂Qa−GabΦa2M𝒩∂𝒩∂Qb+GabΦa2M1|Gh|∂|Gh|∂Qb=12𝒩∂B1a∂Qa+12|Gh|∂∂x1(|Gh|𝒩)=12𝒩∂B1a∂Qa−12𝒩(B−1)aA∂BAa∂x1+12|G~h|∂∂x1(|G~h|𝒩)=12|G~h|∂∂x1(|G~h|𝒩).

Appendix B: Conservation of the Inner Product in Perturbation Theory. Definition of Partial Averages

It is instructive to verify that the approximate inner product given in (96) is conserved for solutions of (99) up to order 1M. Evidently, this is a consequence of the exact equation (81). We assume sgn(𝒩) is constant and set f=−sgn(𝒩). We first note that, given the factor ordering of H^m in (66), we find that it obeys the symmetry condition

(129)∫∏idxi2|Vg|∏μdqμhΨ¯1H^mΨ2=∫∏idxi2|Vg|∏μdqμh(H^mΨ1)¯Ψ2.

Second, recall that we define the derivative of an operator as (∂∂x1O^)ψ:=∂∂x1(O^ψ)−O^∂ψ∂x1. Then, using the Laplace–Beltrami factor ordering of H^m given in (66) and the measure ℳ^ defined in (97), we define the operators

(130)∂∂x1(σ|gh2V|H^m):=−12∂∂qμ[∂∂x1(σ|gh2V|hμν)∂∂qν]+∂∂x1(σ|gh2V|Vm),

(131)∂∂x1ℳ^:=∂∂x12|Vgh|+1M∂∂x1(σ|gh2V|H^m),

(132)ℌ^:=𝒩(H^m−J)−𝒩4MV(H^m−J)2−12M2|Vgh|∂∂xi(2|Vgh|𝒩gij∂∂xj)+1M𝒱.

Using (100), (130), and (132), it is possible to show that Γ^ and ℌ^ obey the same symmetry condition as H^m given in (129). With these definitions, it is straightforward to prove the identities

(133)2|Vg|h[Γ^(1+12MVH^m)+(1+12MVH^m)Γ^]=∂∂x1ℳ^+𝒪(1M2),[ℌ^,12MVH^m]=[𝒩(H^m−J),12MVH^m]+𝒪(1M2)=0+𝒪(1M2).

We can now rewrite (99) as

(134)∂Ψφ∂x1=−iℌ^Ψφ−Γ^Ψφ+𝒪(1M2).

Using (134), we obtain

∂∂x1∫∏idxi∏μdqμΨ¯1,φℳ^Ψ2,φ=∫∏idxi∏μdqμ[i(ℌ^Ψ¯1,φ)ℳ^Ψ2,φ−iΨ¯1,φℳ^ℌ^Ψ2,φ−(Γ^Ψ¯1,φ)ℳ^Ψ2,φ−Ψ¯1,φℳ^Γ^Ψ2,φ+Ψ¯1,φ(∂∂x1ℳ^)Ψ2,φ]+𝒪(1M2)=∫∏idxi2|Vg|∏μdqμh[i(ℌ^Ψ¯1,φ)(1+12MVH^m)Ψ2,φ−iΨ¯1,φ(1+12MVH^m)ℌ^Ψ2,φ−(Γ^Ψ¯1,φ)(1+12MVH^m)Ψ2,φ−Ψ¯1,φ(1+12MVH^m)Γ^Ψ2,φ+12|Vg|hΨ¯1,φ(∂∂x1ℳ^)Ψ2,φ]+𝒪(1M2)=∫∏idxi∏μdqμ{iΨ¯1,φ2|Vg|h[ℌ^,12MVH^m]Ψ2,φ−Ψ¯1,φ2|Vg|h[Γ^(1+12MVH^m)+(1+12MVH^m)Γ^]Ψ2,φ+Ψ¯1,φ(∂∂x1ℳ^)Ψ2,φ}+𝒪(1M2)=0+𝒪(1M2),

where we used the symmetry condition given in (129) for Γ^ and ℌ^, and subsequently, we applied (133). Thus, we arrive at the result

(135)∂∂x1∫∏idxi∏μdqμΨ¯1,φℳ^Ψ2,φ=0+𝒪(1M2),

which is consistent with the exact equation (81) and confirms that the term iΓ^, which appears in (99) and (134), is the term that guarantees that the dynamics is unitary to this order in perturbation theory.

A final comment about matter-sector partial averages is in order. In Section 4.6, the matter-sector inner product was tacitly taken to be ⟨ψ1,ψ2⟩m=∫∏μdqμhψ¯1(x;q)ψ2(x;q), which is associated with partial averages given in (89). However, due to (96), this may be regarded as the lowest-order approximation to the more general matter-sector inner product ⟨ψ1,ψ2⟩m=∫∏μdqμψ¯1(x;q)ℳ^(x;p^,q)ψ2(x;q), where the measure ℳ^ has to be determined from the perturbative expansion of (84). In this case, the partial average of an operator is given by ⟨O^⟩m=(∫∏μdqμψ¯ℳ^ψ)−1∫∏μdqμψ¯ℳ^O^ψ. At order 1M, we can take ℳ^ to be given by (97) (overall factors of V(x) and g(x) can be eliminated from the matter-sector inner product by suitable factorisations of the wave function), and using (100), we can define Γ^=:𝒩MGabΦaΓ^b. The operator Γ^a inherits its symmetry with respect to the lowest-order matter-sector inner product from the symmetry of Γ^. Using (133), one may verify that matter-sector unitarity is then equivalent to ℜ𝔢⟨∂∂Qa+Γ^a⟩m=0 [cf. (22)], and the “Berry connection” can be defined as Aa=ℑ𝔪⟨∂∂Qa+Γ^a⟩m.

Appendix C: Extension to Field Theory

We now comment on how one can formally extend the formalism presented in this article to the field-theoretic case. The canonical approach to General Relativity in 3 + 1 spacetime dimensions involves a foliation of spacetime into a family of space-like hypersurfaces Σ_t defined as the level sets of some scalar function, τ(y)=t. We parametrise each hypersurface by coordinates y, such that (t,y) defines a coordinate system of spacetime. The spacetime line element is

(136)ds2=−N2dt2+Qij(dyi+Nidt)(dyj+Njdt),

where yⁱ are the coordinates used to parametrise each hypersurface, Q_ij are the components of the induced metric on a given hypersurface, N is the lapse function, and Nⁱ is the shift vector. The covariant derivative compatible with the induced metric will be denoted by a semicolon, such that Qij;k=0. The components of the inverse of the induced metric are denoted by Qij, and the determinant of the induced metric is written as Q. Latin indices are raised and lowered with the induced metric and its inverse. The canonical momentum conjugate to Q_ij is Pij. Matter fields and their conjugate momenta are generically written as q and p, respectively. Neglecting boundary terms, we find the Hamiltonian for the gravity-matter system

(137)H=∫Σtd3y(Nℋ⊥+Niℋi),

(138)ℋ⊥=12MGijlmPijPlm−2MQ(R−2Λ)+ℋ⊥m(Q;p,q),

(139)ℋi=−2QikPkl;l+ℋim(Q;p,q),

where M=132πG, R is the Ricci scalar of a given three-dimensional hypersurface, Λ is a cosmological constant term, ℋ⊥m and ℋim are contributions from the matter sector, and

(140)Gijlm=12Q(QilQjm+QimQjl−QijQlm)

is the inverse DeWitt metric. The DeWitt metric reads

(141)Gijlm=Q2(QilQjm+QimQjl−2QijQlm),

(142)GijlmGlmab=12(δaiδbj+δbiδaj)=δ(aiδb)j.

We work only with closed three-manifolds such that no boundary terms are present in the Hamiltonian. The canonical momenta conjugate to the lapse, and shift functions vanish and constitute primary constraints (see, e.g. [3] and references therein), which we have already eliminated. The lapse and shift are thus multipliers. By varying the action with respect to N and Nⁱ, we obtain the constraints ℋ⊥=0 and ℋi=0, which are referred to as the Hamiltonian and diffeomorphism (momentum) constraints, respectively. In the absence of boundary terms, the dynamics of the theory is entirely contained in these constraints. The Einstein–Hamilton–Jacobi equations are as follows:

(143)12MGijlmδWδQijδWδQlm+MV(Q)+ℋ⊥m(Q;δWδq,q)=0,−2Qik(δWδQkl);l+ℋim(Q;δWδq,q)=0,

where W is the Hamilton characteristic functional for the composite system of both gravitational and matter degrees of freedom, and we defined the potential V(Q)=−2Q(R−2Λ). In analogy to the finite-dimensional model considered in this article, we define the background Hamilton functional φ[Q] as the solution to the equations

(144)12MGijlmδφδQijδφδQlm+MV(Q)=−J⊥,−2Qik(δφδQkl);l=−Ji.

A change of background Hamilton functional φ=φ′+η leads to redefinitions of J⊥,Ji [in analogy to (31)]. By defining S=W−φ, we can rewrite (143) as

(145)−1MGijlmΦijδSδQlm=ℋ⊥m(Q;δSδq,q)−J⊥+12MGijlmδSδQijδSδQlm,

(146)2Qik(δSδQkl);l=ℋim(Q;δSδq,q)−Ji,

where we defined the background momenta Φij:=δφ[Q]δQij(y). Equations (145) and (146) were also considered in [31] in the context of a background field approximation, where Parentani emphasised that the arbitrarily chosen sources J⊥,Ji should be compatible with the Bianchi identities.

Given φ, we define the functionals X1(y,Q] and Xr(y,Q] (r=2,3,4,5,6) as follows [4], [5]:

𝒩(y,Q]MGijlm(y,Q]δφ[Q]∂Qij(y)δX1(y′,Q]δQlm(y)=δ(y−y′),Gijlm(y,Q]δφ[Q]∂Qij(y)δXr(y′,Q]δQlm(y)=0,

where 𝒩(y,Q] is an arbitrary normalisation functional. Now, in analogy to the mechanical case, we define^[23]

(147)δδX1:=𝒩(y,Q]MGijlm(y,Q]Φlm(y,Q]δδQij(y)≡B1|ij(y,Q]δδQij(y),δδXr:=Br|ij(y,Q]δδQij(y),

where the B functionals obey^[24] [cf. (144)]

(149)(B−1)A|ij(y,Q]=G~AB(y,Q]Gijlm(y,Q]BB|lm(y,Q].

We can thus write

(150)δδQij=(B−1)A|ijδδXA=−Φij2𝒩(J⊥+MV)δδX1+grsGijlmBr|lmδδXs.

Using (147) and (150), we can rewrite (145) as

(151)−δSδX1=𝒩[ℋ⊥m(Q;δSδq,q)−J⊥]+𝒩2MgrsδSδXrδSδXs−14𝒩(J⊥+MV)(δSδX1)2,

Equation (151) is analogous to (62) and can also be solved iteratively. The result is

(152)−δSδX1=𝒩[ℋ⊥m(Q;δSδq,q)−J⊥]−𝒩4MV[ℋ⊥m(Q;δSδq,q)−J⊥]2+𝒩2MgrsδSδXrδSδXs+𝒪(1M2),

where we have assumed that J_⊥ can be expanded as in (51). If we now fix the arbitrary shift vector Ni(y,Q] by a suitable choice of coordinates yⁱ, we can define the “background time” derivative

(153)∂∂τ=∫d3y{𝒩(y,Q]MGijlm(y,Q]δφ[Q]δQij(y)δδQlm(y)+2Ni;j(y,Q]δδQij(y)}=∫d3y{δδX1(y,Q]+2Ni;j(y,Q]δδQij(y)}.

We can then combine (146) and (152) to obtain

(154)−∂S∂τ=∫d3y{𝒩[ℋ⊥m(Q;δSδq,q)−J⊥]+Ni[ℋim(Q;δSδq,q)−Ji]}+∫d3y{−𝒩4MV[ℋ⊥m(Q;δSδq,q)−J⊥]2+𝒩2MgrsδSδXrδSδXs}+𝒪(1M2),

which is the field-theoretic analogue of (64). As we have argued in the mechanical case, (154) is most appropriately interpreted as an approximation to the Hamilton–Jacobi equation for the reduced gauge-fixed system that comprised both gravitational and matter degrees of freedom. In this way, the solution S of (154) is not the “corrected” Hamilton principal functional of a system composed of the matter fields alone.

To see the how this corresponds to a particular gauge fixing, we define a field 𝒯(y,Q] as follows:

(155)𝒩(y,Q]MGijlm(y,Q]δφ[Q]δQij(y)δ𝒯(y′,Q]δQlm(y)+2Ni;j(y,Q]δ𝒯(y′,Q]δQij(y)=δ(y−y′),

which coincides with X1(y,Q] only if Ni;j=0. Using (153), we find that the field 𝒯 obeys the “background” equation of motion ∂𝒯∂τ=1. Thus, 𝒯 serves as canonical definition of the “background time” functional. In analogy to the derivation of (150), we find that a solution to (155) obeys

(156)δ𝒯(z,Q]δQij(y)=δ(z−y)[𝒩MΦij+2GijlmNl;m−2𝒩2(J⊥M+V)+4𝒩MΦlmNl;m+4GablmNa;bNl;m]y,

where we have used the first of (144).

We now fix the canonical gauge condition 𝒯(y,Q(t)]=t. For simplicity, we will restrict ourselves to the lowest order of the perturbative regime in order to compute the gauge-fixed lapse. Given a suitable choice of the shift vector Nⁱ, we obtain the relation between the arbitrary “background” lapse 𝒩 and the gauge-fixed lapse N as follows. The chosen gauge condition implies that d𝒯dt(y,Q(t)]=1=∂𝒯∂τ(y,Q(t)]. Thus,

0=d𝒯dt(y,Q(t)]−∂𝒯∂τ(y,Q(t)]=∫d3z{N(z,Q]MGijlm(z,Q]δW[Q]δQij(z)δ𝒯(y,Q(t)]δQlm(z)+2Ni;j(z,Q]δ𝒯(y,Q(t)]δQij(z)}−∂𝒯∂τ(y,Q(t)]=1M∫d3zGijlm(z,Q]δ𝒯(y,Q(t)]δQlm(z)(N(z,Q]δW[Q]δQij(z)−𝒩(z,Q]δφ[Q]δQij(z)).

Using (156), we can rewrite the above equation as

0=1M∫d3zδ(z−y)Gijlm(z,Q][(𝒩MΦlm+2GlmnkNn;k)(NδWδQij−𝒩δφδQij)]z=1MGijlm(y,Q][(𝒩MΦlm+2GlmnkNn;k)(NδWδQij−𝒩δφδQij)]y.

Using W=S+φ, we obtain

0=1M[(NM−𝒩M)GijlmΦijΦlm+NMGijlmΦijδSδQlm+2𝒩(N−𝒩)Ni;jΦij+2N𝒩Ni;jδSδQij].

Now, using the first of (144) and the first of (147), the above equation becomes

N[−2(J⊥M+V)+2Ni;jΦijM𝒩+1M𝒩(δSδX1+2Ni;jδSδQij)]=𝒩[−2(J⊥M+V)+2Ni;jΦijM𝒩].

Finally, using (52) and assuming that Ni,S can be expanded as in (51), the above equation implies

(157)N(y,Q]=𝒩(y,Q]+𝒪(1M),

which agrees with (57) at the lowest order for the choice κ=+sgn(𝒩V), which is the branch of solutions obtained via the iterative procedure as we saw in Section 3.4.3. The terms of higher order in 1M in the expansion of N(y,Q] also depend on the chosen value of Ni(y,Q]. If Ni;j=0, it is possible to show that the field-theoretic analogue of the next order in (57) is also recovered (with Hm→ℋ⊥m and J→J⊥) by using (152).

The quantum theory can be constructed in analogy to what was done in Section 4. The formal Laplace–Beltrami-ordered quantum constraint equations read

(158)−12M|Gh|δδQij(|Gh|GijlmδΨδQlm)+MV(Q)Ψ+ℋ^⊥m(Q;−iδδq,q)Ψ=0,2iQik(δΨδQkl);l+ℋ^im(Q;−iδδq,q)Ψ=0,

where h is the determinant of the matter-sector field-space metric. Equations (158) are the quantum analogues of (143). Given a choice of background Hamilton functional φ[Q] [cf. (144)], we perform the phase transformation Ψ=eiφΨφ to obtain the constraint equations

(159)iMGijlmΦijδΨφδQlm=(ℋ^⊥m−𝔍⊥)Ψφ−12M|Gh|δδQij(|Gh|GijlmδΨφδQlm),2iQik(δΨφδQkl);l+(ℋ^im−Ji)Ψφ=0,

where we have neglected the covariant derivatives φ;l and (Ψφ);l, and we defined 𝔍⊥:=J⊥+i2M|Gh|δδQij(|Gh|GijlmΦlm). Equations (159) are the quantum versions of (145) and (146) and the field-theoretic analogues of (74).

Using (150) and (153), we can combine both equations given in (159) into the approximate Schrödinger equation

(160)i∂Ψφ∂τ=∫d3y[𝒩(ℋ^⊥m−𝔍⊥)Ψφ+Ni(ℋ^im−Ji)Ψφ]+𝒪(1M),

which was derived in [8], [9], [10], [12], [24], [25] without the 𝔍⊥,Ji terms. We refrain from computing the corrections of order 1M to the above equation. They should be found in formal analogy to what was done in Section 4.5 for the mechanical case.

Acknowledgement

The author would like to thank David Brizuela, Claus Kiefer, Manuel Krämer, Ward Struyve, and especially Branislav Nikolić for useful discussions and the Bonn-Cologne Graduate School of Physics and Astronomy for financial support.

References

[1] R. L. Arnowitt, S. Deser, and C. W. Misner, Gen. Rel. Grav. 40, 1997 (2008).10.1007/s10714-008-0661-1Search in Google Scholar

[2] P. G. Bergmann and A. Komar, Int. J. Theor. Phys. 5, 15 (1972).10.1007/BF00671650Search in Google Scholar

[3] J. M. Pons, D. C. Salisbury, and K. A. Sundermeyer, J. Phys. Conf. Ser. 222, 012018 (2010).10.1088/1742-6596/222/1/012018Search in Google Scholar

[4] K. V. Kuchař, Int. J. Mod. Phys. D 20, 3 (2011).10.1142/S0218271811019347Search in Google Scholar

[5] C. J. Isham, Canonical Quantum Gravity and the Problem of Time 19th Int. Colloquium on Group Theoretical Methods in Physics, Salamanca, Spain 1992 (arXiv:gr-qc/9210011).Search in Google Scholar

[6] E. Anderson, in: Fundamental Theories of Physics Vol 190, Springer International Publishing, Cham, Switzerland 2017.Search in Google Scholar

[7] U. H. Gerlach, Phys. Rev. 177, 1929 (1969).10.1103/PhysRev.177.1929Search in Google Scholar

[8] V. G. Lapchinsky and V. A. Rubakov, Acta Phys. Polon. B 10, 1041 (1979).Search in Google Scholar

[9] T. Banks, Nucl. Phys. B 249, 332 (1985).10.1016/0550-3213(85)90020-3Search in Google Scholar

[10] T. Banks, W. Fischler, and L. Susskind, Nucl. Phys. B 262, 159 (1985).10.1016/0550-3213(85)90070-7Search in Google Scholar

[11] T. P. Singh, Class. Quant. Grav. 7, L149 (1990).10.1088/0264-9381/7/7/006Search in Google Scholar

[12] C. Kiefer and T. P. Singh, Phys. Rev. D 44, 1067 (1991).10.1103/PhysRevD.44.1067Search in Google Scholar

[13] S. P. Kim, Phys. Rev. D 52, 3382 (1995).10.1103/PhysRevD.52.3382Search in Google Scholar PubMed

[14] A. O. Barvinsky and C. Kiefer, Nucl. Phys. B 526, 509 (1998).10.1016/S0550-3213(98)00349-6Search in Google Scholar

[15] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).10.1103/PhysRev.160.1113Search in Google Scholar

[16] J. J. Halliwell and S. W. Hawking, Phys. Rev. D 31 1777 (1985) [Adv. Ser. Astrophys. Cosmol. 3, 277 (1987)].10.1103/PhysRevD.31.1777Search in Google Scholar

[17] R. Brout, Found. Phys. 17, 603 (1987).10.1007/BF01882790Search in Google Scholar

[18] R. Brout, G. Horwitz G, and D. Weil, Phys. Lett. B 192, 318 (1987).10.1016/0370-2693(87)90114-6Search in Google Scholar

[19] R. Brout, Z. Phys. B Con. Mat. 68, 339 (1987).10.1016/0016-6480(87)90047-5Search in Google Scholar

[20] R. Brout and G. Venturi, Phys. Rev. D 39, 2436 (1989).10.1103/PhysRevD.39.2436Search in Google Scholar

[21] A. Vilenkin, Phys. Rev. D 39, 1116 (1989).10.1103/PhysRevD.39.1116Search in Google Scholar

[22] T. P. Singh and T. Padmanabhan, Ann. Phys. 196, 296 (1989).10.1016/0003-4916(89)90180-2Search in Google Scholar

[23] T. Padmanabhan and T. P. Singh, Class. Quant. Grav. 7, 411 (1990).10.1088/0264-9381/7/3/015Search in Google Scholar

[24] C. Kiefer, Report Freiburg THEP-94/4, Contribution for the Lanczos Conference Proceedings, arXiv:gr-qc/9405039 (1994).Search in Google Scholar

[25] C. Kiefer, The Semiclassical Approximation to Quantum Gravity Canonical Gravity: From Classical to Quantum (Lecture Notes in Physics vol 434) (Eds. J. Ehlers. H. Friedrich), Springer, Berlin 1994.Search in Google Scholar

[26] C. Kiefer, Does Time Exist in Quantum Gravity? Towards a Theory of Spacetime Theories (Einstein Studies vol 13) (Eds. D. Lehmkuhl, G. Schiemann, E. Scholz), Birkhäuser, New York, NY 2017 [arXiv:0909.3767 [gr-qc]].10.1007/978-1-4939-3210-8_10Search in Google Scholar

[27] F. Englert, Phys. Lett. B 228, 111 (1989).10.1016/0370-2693(89)90534-0Search in Google Scholar

[28] J. S. Briggs and J. M. Rost, Eur. Phys. J. D 10, 311 (2000).10.1007/s100530050554Search in Google Scholar

[29] J. S. Briggs and J. M. Rost, Found. Phys. 31, 693 (2001).10.1023/A:1017525227832Search in Google Scholar

[30] Gauge Theories of Gravitation (Eds. M. Blagojević, F. W. Hehl), Imperial College Press 2013 [arXiv:1210.3775 [gr-qc]].Search in Google Scholar

[31] R. Parentani, Class. Quant. Grav. 17, 1527 (2000).10.1088/0264-9381/17/6/314Search in Google Scholar

[32] J. S. Briggs, Phys. Rev. A 91, 052119 (2015).10.1103/PhysRevA.91.052119Search in Google Scholar

[33] R. Parentani, Phys. Rev. D 56, 4618 (1997).10.1103/PhysRevD.56.4618Search in Google Scholar

[34] R. Brout and R. Parentani, Int. J. Mod. Phys. D 8, 1 (1999).10.1142/S0218271899000031Search in Google Scholar

[35] D. Marolf, arXiv:gr-qc/9508015 (1995).Search in Google Scholar

[36] J. B. Hartle and D. Marolf, Phys. Rev. D 56, 6247 (1997).10.1103/PhysRevD.56.6247Search in Google Scholar

[37] A. Y. Kamenshchik, A. Tronconi, and G. Venturi, Phys. Lett. B 726, 518 (2013).10.1016/j.physletb.2013.08.067Search in Google Scholar

[38] A. Y. Kamenshchik, A. Tronconi, and G. Venturi, Phys. Lett. B 734, 72 (2014).10.1016/j.physletb.2014.05.028Search in Google Scholar

[39] A. Y. Kamenshchik, A. Tronconi, and G. Venturi, Class. Quant. Grav. 35, 015012 (2018).10.1088/1361-6382/aa8fb3Search in Google Scholar

[40] A. Y. Kamenshchik, A. Tronconi, T. Vardanyan, and G. Venturi, Int. J. Mod. Phys. D 28, 1950073 (2019).10.1142/S0218271819500731Search in Google Scholar

[41] R. Balbinot, A. Barletta, and G. Venturi, Phys. Rev. D 41, 1848 (1990).10.1103/PhysRevD.41.1848Search in Google Scholar PubMed

[42] E. Anderson, Class. Quant. Grav. 24, 2935 (2007).10.1088/0264-9381/24/11/011Search in Google Scholar

[43] E. Anderson, Class. Quant. Grav. 24, 2979 (2007).10.1088/0264-9381/24/11/012Search in Google Scholar

[44] E. Anderson, Class. Quant. Grav. 28, 185008 (2011).10.1088/0264-9381/28/18/185008Search in Google Scholar

[45] E. Anderson, Class. Quant. Grav. 31, 025006 (2014).10.1088/0264-9381/31/2/025006Search in Google Scholar

[46] E. Anderson, Gen. Rel. Grav. 46, 1708 (2014).10.1007/s10714-014-1708-0Search in Google Scholar

[47] M. Born and R. Oppenheimer, Ann. der Phys. 389, 457 (1927).10.1002/andp.19273892002Search in Google Scholar

[48] L. S. Cederbaum, J. Chem. Phys. 128, 124101 (2008).10.1063/1.2895043Search in Google Scholar PubMed

[49] N. F. Mott, Proc. R. Soc. Lond. A 126, 79 (1929).10.1098/rspa.1929.0205Search in Google Scholar

[50] N. F. Mott, Math. Proc. Cambridge 27, 553 (1931).10.1017/S0305004100009816Search in Google Scholar

[51] J. B. Barbour, Class. Quant. Grav. 11, 2875 (1994).10.1088/0264-9381/11/12/006Search in Google Scholar

[52] J. J. Halliwell, Phys. Rev. D 64, 044008 (2001).10.1103/PhysRevD.64.044008Search in Google Scholar

[53] H. D. Zeh, Phys. Lett. A 126, 311 (1988).10.1016/0375-9601(88)90842-0Search in Google Scholar

[54] J. C. Arce, Phys. Rev. A 85, 042108 (2012).10.1103/PhysRevA.85.042108Search in Google Scholar

[55] C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284 (1979).10.1063/1.437734Search in Google Scholar

[56] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).10.1098/rspa.1984.0023Search in Google Scholar

[57] A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010).10.1103/PhysRevLett.105.123002Search in Google Scholar PubMed

[58] A. Abedi, N. T. Maitra, and E. K. U. Gross, J. Chem. Phys. 137, 22A530 (2012).10.1063/1.4745836Search in Google Scholar PubMed

[59] J. L. Alonso, J. Clemente-Gallardo, P. Echenique-Robba, and J. A. Jover-Galtier, J. Chem. Phys. 139, 087101 (2012).10.1063/1.4818521Search in Google Scholar PubMed

[60] C. Bertoni, F. Finelli, and G. Venturi, Class. Quant. Grav. 13, 2375 (1996).10.1088/0264-9381/13/9/005Search in Google Scholar

[61] C. Kiefer and D. Wichmann, Gen. Rel. Grav. 50, 66 (2018).10.1007/s10714-018-2390-4Search in Google Scholar

[62] J. B. Hartle, ASI Series (Series B: Physics) vol 156 (Eds. B. Carter, J. B. Hartle), Springer, Boston, MA 1987.Search in Google Scholar

[63] J. J. Halliwell, Phys. Rev. D 36, 3626 (1987).10.1103/PhysRevD.36.3626Search in Google Scholar PubMed

[64] P. D. D’Eath and J. J. Halliwell, Phys. Rev. D 35, 1100 (1987).10.1103/PhysRevD.35.1100Search in Google Scholar

[65] T. Padmanabhan, Class. Quant. Grav. 6, 533 (1989).10.1088/0264-9381/6/4/012Search in Google Scholar

[66] C. Kiefer, Quantum Gravity (International Series of Monographs on Physics), 3rd ed., Oxford University Press, Oxford 2012.10.1093/oxfordhb/9780199298204.003.0024Search in Google Scholar

[67] G. Hunter, Int. J. Quantum Chem. 9, 237 (1975).10.1002/qua.560090205Search in Google Scholar

[68] L. S. Cederbaum, J. Chem. Phys. 138, 224110 (2013).10.1063/1.4807115Search in Google Scholar PubMed

[69] E. Anderson, in: XXIXth International Workshop on High Energy Physics: New Results and Actual Problems in Particle & Astroparticle Physics and Cosmology. World Scientific Publishing Co. Pte. Ltd, Singapore 2014, p. 182 (arXiv:1306.5812 [gr-qc]).Search in Google Scholar

[70] A. Schild, Phys. Rev. A 98, 052113 (2018).10.1103/PhysRevA.98.052113Search in Google Scholar

[71] J. Greensite, Nucl. Phys. B 342, 409 (1990).10.1016/0550-3213(90)90196-KSearch in Google Scholar

[72] T. Padmanabhan, Pramana 35, L199 (1990).10.1007/BF02875295Search in Google Scholar

[73] J. Greensite, Nucl. Phys. B 351, 749 (1991).10.1016/S0550-3213(05)80043-4Search in Google Scholar

[74] T. Brotz and C. Kiefer, Nucl. Phys. B 475, 339 (1996).10.1016/0550-3213(96)00304-5Search in Google Scholar

[75] N. Pinto-Neto and W. Struyve, (arXiv:1801.03353 [gr-qc]) (2018).Search in Google Scholar

[76] G. Venturi, Class. Quant. Grav. 7, 1075 (1990).10.1088/0264-9381/7/6/014Search in Google Scholar

[77] S. P. Kim, Phys. Lett. A 205, 359 (1995).10.1016/0375-9601(95)00584-PSearch in Google Scholar

[78] S. Massar and R. Parentani, Phys. Rev. D 59, 123519 (1999).10.1103/PhysRevD.59.123519Search in Google Scholar

[79] C. Kiefer, Phys. Rev. D 47, 5414 (1993).10.1103/PhysRevD.47.5414Search in Google Scholar

[80] C. Kiefer, Class. Quant. Grav. 4, 1369 (1987).10.1088/0264-9381/4/5/031Search in Google Scholar

[81] J. J. Halliwell, Phys. Rev. D 39, 2912 (1989).10.1103/PhysRevD.39.2912Search in Google Scholar

[82] J. B. Barbour, Phys. Rev. D 47, 5422 (1993).10.1103/PhysRevD.47.5422Search in Google Scholar

[83] B. S. DeWitt, Rev. Mod. Phys. 29, 377 (1957).10.1103/RevModPhys.29.377Search in Google Scholar

[84] C. Lämmerzahl, Phys. Lett. A 203, 12 (1995).10.1016/0375-9601(95)00345-4Search in Google Scholar

[85] C. Kiefer and M. Krämer, Phys. Rev. Lett. 108, 021301 (2012).10.1103/PhysRevLett.108.021301Search in Google Scholar PubMed

[86] C. Kiefer, J. Phys. Conf. Ser. 442, 012025 (2013).10.1088/1742-6596/442/1/012025Search in Google Scholar

[87] D. Bini, G. Esposito, C. Kiefer, M. Krämer, and F. Pessina, Phys. Rev. D 87, 104008 (2013).10.1103/PhysRevD.87.104008Search in Google Scholar

[88] D. Brizuela, C. Kiefer, and M. Krämer, Phys. Rev. D 93, 104035 (2016).10.1103/PhysRevD.93.104035Search in Google Scholar

[89] D. Brizuela, C. Kiefer, and M. Krämer, Phys. Rev. D 94, 123527 (2016).10.1103/PhysRevD.94.123527Search in Google Scholar

[90] D. Brizuela and M. Krämer, Galaxies 6, 6 (2018).10.3390/galaxies6010006Search in Google Scholar

[91] A. Y. Kamenshchik, A. Tronconi, and G. Venturi, Phys. Rev. D 94, 123524 (2016).10.1103/PhysRevD.94.123524Search in Google Scholar

[92] A. Y. Kamenshchik, A. Tronconi, T. Vardanyan, and G. Venturi, Phys. Rev. D 97, 123517 (2018).10.1103/PhysRevD.97.123517Search in Google Scholar

[93] D. Giulini and C. Kiefer, Class. Quant. Grav. 12, 403 (1995).10.1088/0264-9381/12/2/009Search in Google Scholar

Received: 2019-06-30

Accepted: 2019-07-10

Published Online: 2019-08-07

Published in Print: 2019-12-18

Articles in the same Issue

Frontmatter
Atomic, Molecular & Chemical Physics
Selenium Zinc Oxide (Se/ZnO) Nanoparticles: Synthesis, Characterization, and Photocatalytic Activity
Dynamical Systems & Nonlinear Phenomena
Fluid Flow and Solute Transfer in a Tube with Variable Wall Permeability
Gravitation & Cosmology
Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology
Hydrodynamics
Formation of the Capillary Ridge on the Free Surface Dynamics of Second-Grade Fluid Over an Inclined Locally Heated Plate
Solid State Physics & Materials Science
Green Zn₃Al₂Ge₂O₁₀: Mn²⁺ Phosphors: Solid-Phase Synthesis, Structure, and Luminescent Properties
First Principles Study of Thermodynamic Properties of Cd_xZn_1−xO (0 ≤ x ≤ 1) Ternary Alloys
Elastic and Ultrasonic Studies on RM (R = Tb, Dy, Ho, Er, Tm; M = Zn, Cu) Compounds
Tailoring of Bandgap to Tune the Optical Properties of Ga_1−xAl_xY (Y = As, Sb) for Solar Cell Applications by Density Functional Theory Approach

https://doi.org/10.1515/zna-2019-0223

Keywords for this article

Born–Oppenheimer Approximation; Gauge Fixing; Quantum Cosmology; Semiclassical Interpretation

Articles in the same Issue

Frontmatter
Atomic, Molecular & Chemical Physics
Selenium Zinc Oxide (Se/ZnO) Nanoparticles: Synthesis, Characterization, and Photocatalytic Activity
Dynamical Systems & Nonlinear Phenomena
Fluid Flow and Solute Transfer in a Tube with Variable Wall Permeability
Gravitation & Cosmology
Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology
Hydrodynamics
Formation of the Capillary Ridge on the Free Surface Dynamics of Second-Grade Fluid Over an Inclined Locally Heated Plate
Solid State Physics & Materials Science
Green Zn₃Al₂Ge₂O₁₀: Mn²⁺ Phosphors: Solid-Phase Synthesis, Structure, and Luminescent Properties
First Principles Study of Thermodynamic Properties of Cd_xZn_1−xO (0 ≤ x ≤ 1) Ternary Alloys
Elastic and Ultrasonic Studies on RM (R = Tb, Dy, Ho, Er, Tm; M = Zn, Cu) Compounds
Tailoring of Bandgap to Tune the Optical Properties of Ga_1−xAl_xY (Y = As, Sb) for Solar Cell Applications by Density Functional Theory Approach