Quantum geometrodynamics of Einstein and conformal (Weyl-squared) gravity

We discuss the canonical quantization of general relativity and Weyl-squared gravity. We present the classical and quantum constraints and discuss their similarities and differences. We perform a semiclassical expansion and discuss the emergence of time for the two theories. While in the first case semiclassical time has a scale and a shape part, in the second case it only has a shape part.


Introduction
The quantization of gravity is perhaps the most important open problem in theoretical physics [1]. Among the oldest approaches is quantum geometrodynamics ("Wheeler-DeWitt equation"). This is still a promising approach because it is very conservative: its equations are found if one searches for quantum wave equations that directly lead to Einstein's equations in the semiclassical (WKB) limit [2]. In 1926, Erwin Schrödinger found his wave equation by formulating classical mechanics in Hamilton-Jacobi form and "guessing" a wave equation that leads to this form in the limit of geometric optics. We recall what he wrote in 1926 [3]: We know today, in fact, that our classical mechanics fails for very small dimensions of the path and for very great curvatures. Perhaps this failure is in strict analogy with the failure of geometrical optics . . . that becomes evident as soon as the obstacles or apertures are no longer large compared with the real, finite, wavelength. . . . Then it becomes a question of searching for an undulatory mechanics, and the most obvious way is by an elaboration of the Hamiltonian analogy on the lines of undulatory optics. 1 The same procedure can be applied to Einstein's equations [2]. If written into Hamilton-Jacobi form, it can be "translated" into the Wheeler-DeWitt equation and the diffeomorphism (momentum) constraints. These equations are timeless, that is, there is no external time parameter or external spacetime background. One can introduce an intrinsic time by the structure of the Wheeler-DeWitt equation whose kinetic term is locally hyperbolic; this intrinsic time is related to √ h, where h is the determinant of the three-metric h ab . An extrinsic time is sometimes used in the alternative approach of reduced quantization; a typical choice is York's time (see e.g. [1], p. 148), which is related to the trace K of the extrinsic curvature.  Starting with quantum geometrodynamics as the fundamental framework, one can derive the limit of quantum (field) theory in an external background spacetime by a Born-Oppenheimer type of approximation scheme with respect to the Planck mass [1,2]. At leading order, the semiclassical or "WKB time" is recovered as an emergence concept from the timeless quantum equations.
In this contribution, we want to shed more light on the nature of emergent time. We will study the quantum geometrodynamics corresponding to general relativity (GR) and the quantum geometrodynamics corresponding to conformal (Weyl-squared) gravity, briefly called Weyl theory. The latter serves as a model for a gravitational theory devoid of scale. We will discuss the common features of and the differences between those theories.
For this purpose, we will employ a unimodular decomposition of the three-metric. In quantum GR (section 2), we can identify the scale and the shape part of the WKB time; the scale part is related to the intrinsic time in the vacuum Wheeler-DeWitt equation. So the WKB time needs for its "passing" both scale and shape.
In section 3, we present a model of quantum geometrodynamics based on Weyl-squared gravity. We find there equations similar to the equations of quantum GR, but with a different explicit structure.
In section 4, we discuss the semiclassical limit of these equations, which gives rise to a "Weyl-WKB time" (WWKB time). It has the property that the evolution of the scale degree of freedom does not contribute to it; this is because the WWKB time inherits the conformal symmetry of the Weyl-squared action from which the intrinsic time is absent. WWKB time is thus scale-less and conformally invariant; we call it the shape time. In section 5, we present our conclusions. Further details and references can be found in our papers [4,5,6].

Canonical quantum general relativity
Instead of the usual canonical formulation of GR and its quantization, we formulate the quantum geometrodynamics here in variables which manifestly reveal the conformal behaviour of the theory. We are interested, in particular, in identifying the three-metric volume √ h and the trace of the extrinsic curvature K as canonical variables carrying a single degree of freedom. This allows us to keep track of "intrinsic time" and "extrinsic time" by investigating what happens to these canonical degrees of freedom. These variables are called unimodular-conformal variables and are discussed in detail in [4,5,6]. To formulate GR in these variables, we will briefly review here section 2 from [7].
The extrinsic curvature and its trace are given by where D i is the covariant derivative with respect to the three-metric. We decompose these as well as the metric variables into scale and shape part, called unimodular-conformal decomposition: where κ = 8πG (c = 1), and we keep in mind that (3) R can also be decomposed if needed (see the Appendix in [5]). Since the configuration variables are a andh ij , the conjugate momenta will be p a and p ij , respectively, Using the definitions (2)-(4) and recalling that the ADM momentum is given by (6) can be related to the trace and traceless part of the ADM momentum, respectively, by so that p ij Performing a Legendre transformation, we see that the total Hamiltonian is simply a linear combination of first class constraints, The momenta with respect to the lapse densityN and shift N i are the primary constraints pN ≈ 0 and p i ≈ 0, respectively. They give rise to secondary constraints: the Hamiltonian constraint H E ⊥ ≈ 0 and the momentum constraints H E i ≈ 0, which in unimodular-conformal variables have the following form: For completeness, we have added the cosmological constant Λ.
It is important to notice that the constraints explicitly depend on the scale density, which is related to intrinsic time √ h, recall (2). It is this and only this variable that is responsible for the conformal non-invariance of GR [7].
Canonical quantization now proceeds by promoting the canonical variables into operators acting on wave functionals Ψ h ab (x), a(x) : From the classical Poisson brackets, one gets the standard commutators. The constraints are then promoted to operators annihilating the wave functional, leading to the Wheeler-DeWitt equation and the momentum constraints, respectively, 8th We have added the parts of the Hamiltonian and momentum constraintsĤ m ⊥ andĤ m a coming from quantized matter (non-gravitational fields), so that the wave functional now depends on matter fields, too, Ψ ≡ Ψ h ab , a, φ .
It is evident from (13) that the Wheeler-DeWitt equation has an indefinite kinetic term, with a playing the role of "intrinsic time". This intrinsic time is, however, constructed from the three-metric, so the Wheeler-DeWitt equation is timeless in the sense of absence of spacetime. That spacetime has disappeared is in full analogy to the absence of classical trajectories in quantum mechanics. The absence of spacetime is symbolized in Figure 1 by the absence of the pointer in the clock. (But the scale is present, waiting to be measured in the semiclassical limit.) The notion of spacetime can be recovered from full canonical quantum GR by a Born-Oppenheimer type of expansion with respect to the Planck-mass squared, m 2 P = /G. In this limit, we obtain the well known theories of quantum field theory in curved spacetime (see e.g. [1] for all technical details). We write the total wave functional in the form where φ stands for 'matter field', and perform an expansion of S (which is a complex function) with respect to the Planck-mass squared, We insert this expansion into the Wheeler-DeWitt equation and compare different orders of m 2 P . At highest order, m 4 P , we find that S E 0 is independent of φ, At order m 2 P , one obtains an equation for S E 0 : Rewriting this in terms of the original variables and using the chain rule for the functional derivative with respect to the metric, for the Hamilton-Jacobi functional is the inverse of the DeWitt supermetric [1]. Since this equation is equivalent to all Einstein equations [8], one has recovered classical GR at this order of semicalssical expansion.
At the next order, m 0 P , an equation for S E 1 is obtained that can be simplified by writing and demanding for D the standard WKB prefactor equation to hold [1]. After some manipulations, one arrives at the following equation for ψ (1) : The form of this quation reminds one of the Tomonaga-Schwinger equation for ψ (1) . Using again (18), the left-hand side of this equation has been rewritten in terms of the original variables (middle expression in (21)), which is well known from the literature [1].
To make the Tomonaga-Schwinger form explicit, we define thus introducing a local "bubble" (Tomonaga-Schwinger) time functional τ (x). We can thus write (21) in the form also known as Tomonaga-Schwinger equation. Note thatĤ m ⊥ in (23) differs fromĤ m ⊥ in (22) by a factor of a, which also induces the same relative rescaling of the bubble time. This is just a consequence of using unimodular-conformal variables.
It is of interest to have a closer look at the bubble time in unimodular-conformal variables, We see from this that there are two main components, a derivative along the direction (in functional configuration space) of the scale density a and a component along the five directions of the conformal (shape) parth ij . The scale part is the part that corresponds to intrinsic time in the full Wheeler-DeWitt equation (13). We will call the two independent contributions to the bubble time as "scale time" and "shape time", respectively. Figure 2 symbolizes the emergence of time (the pointer) in the semiclassical approximation. While the picture shows only scale time, the full semiclassical time contains, of course, also the shape part. We finally note that τ is not a scalar function, because this would be in contradiction to the commutator of the matter Hamiltonian densities at different space point [9]. A functional Schrödinger equation can be obtained from the Tomonaga-Schwinger equation after choosing a particular foliation and integrating over space. The next order of the Born-Oppenheimer expansion, m −2 P , leads to genuine quantum gravitational effects; for example, corrections to the power spectrum of the CMB anisotropies [10].
where C μνλρ is the Weyl tensor. This action is invariant under the Weyl transformations We have written this action in a form that is appropriate for discussing the semiclassical approximation of its quantum version: α W is a dimensionless constant, and it is the quantity S W / that is relevant for the semiclassical expansion. We do not expect the Weyl theory to be a classical alternative to GR, but it may be of relevance in the early universe and at the Planck scale, where scales may become irrelevant [11]. We note that (25) defines a theory with higher derivatives, which demands an enlargement of the configuration space (see [5] for details and references to earlier work). Here, we have to add the extrinsic curvature K ab as an additional independent canonical variable besides the three-metric h ab . This can be accomplished by adding a constraint to the original Lagrangian which takes care of the definition (1). In unimodular-conformal variables, the new Lagrangian becomes [5] where "T" stands for the traceless part of the corresponding quantity. For the canonical momenta, we find from (27): Note that the momentap ij andP ij are traceless. The quantityC T ab is the traceless part of the "electric" part of the Weyl tensor [5]. The constraint analysis proceeds in the standard way. In addition to the primary constraints with respect to lapse and shift (as known from GR), we have here an additional primary constraint given bȳ The presence of this constraint suggests thatK is arbitrary, in the same manner as pN ≈ 0 and p i ≈ 0 suggest thatN and N i are arbitrary. The secondary constraints follow from the conservation of the primary constraints and read (attention is here restricted to the vacuum case) A combination of the secondary constraint Q W and the primary constraintP generates conformal transformations (of a andK) [12,5]. Due to the enlargement of the configuration space, the structure of these constraints differs from those of GR, see (9) and (10). We note, in particular, that the intrinsic time in (9), which is related to the determinant of the three-metric, is absent here because the scale density is arbitrary due to conformal invariance. This will later be connected with the emergence of pure shape time in the semiclassical limit.
Canonical quantization now proceeds in a manner analogously to (11) and (12), taking into account the enlarged configuration space. Adding a (conformally coupled) matter part symbolized by φ, we arrive at a quantum wave functional Ψ[h ij , a,K T ij ,K, φ], which is subject to the following constraints: The first equation is the "Weyl-Wheeler-DeWitt equation" (WWDW equation) and reads The new quantum momentum constraints are given by As for the classical constraints, the WWDW equation is structurally different from the Wheeler-DeWitt equation (13). Its dynamics is not only determined by the three-metric, but also by the extrinsic curvature (which classically corresponds to the time evolution of the three-metric). But there is no scale a and thus no intrinsic time. This is symbolized in Figure 3. We also note that in the vacuum case drops out of these equations. The third and fourth of the above constraints show that the wave functional does not depend on a and K, δΨ and is thus conformally invariant (apart possibly from a phase). This is the most important difference to GR, especially in the light of the semiclassical approximation which we will discuss in the next section. . Recovery of (shape) time in the semiclassical limit In this section, we will discuss the semiclassical approximation scheme for the WWDW equation [4]. While in quantum GR (section 2) m −2 P was used as the expansion parameter, this role will be played here by α −1 W . Let us consider Weyl gravity with a conformally coupled matter field φ. We restrict ourselves to this case (massless vector or conformally coupled scalar field) because the constraints will remain first class and a andK will remain arbitrary. The general case is discussed in [6]. Writing and performing an expansion of S with respect to the (dimensionless) variable α −1 W , we find the following equations at consecutive orders of α W : • α 1 W : This gives the Weyl-Hamilton-Jacobi equation for S W 0 . We first find  Figure 5. Emergence of full WKB time from a scale-less quantum theory of gravity?

Conclusion
Let us briefly summarize our main points as follows: • The canonical quantization of Weyl gravity can be performed analogously to GR, but the structure of the configuration space is different: it consists of both the three-metric and the extrinsic curvature. • We have performed a consistent decomposition of variables into scale part and conformally invariant part. The constraints can be rewritten in terms of the conformally invariant parts. • We have displayed and discussed the Weyl-Wheeler-DeWitt equation and the Weyl diffeomorphism constraints. There are also new constraints which show that the wave functional is conformally invariant. • We have discussed the semiclassical expansion and the recovery of time for both quantum GR and the quantum Weyl theory. Whereas in the former we get both a scale and a shape time, the latter only leads to a shape time.