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Noncommutative Unification of General Relativity with Quantum Mechanics and Canonical Gravity Quantization

The groupoid approach to noncommutative unification of general relativity with quantum mechanics is compared with the canonical gravity quantization. It is shown that by restricting the corresponding noncommutative algebra to its (commutative) subalgebra, which determines the space-time slicing, an algebraic counterpart of superspace (space of 3-metrics) can be obtained. It turns out that when this space-time slicing emerges the universe is already in its commutative regime. We explore the consequences of this result.


Introduction
In recent years a new approach has appeared to the quantization of gravity, the one based on noncommutative geometry. The idea is to make spacetime a noncommutative space (which is essentially nonlocal) with the hope that in this way at least some major obstacles to the gravity quantization could eventually be overcome. There are many attempts in this direction [1]. In [2] we have followed Connes [3, p. 99] who, in order to make a space X noncommutative defines a noncommutative algebra not directly on X but rather on a groupoid over X. This approach, which has been further developed in the series of works [4], will be called a groupoid approach to the unification of general relativity and quantum mechanics.
The aim of the present paper is to compare the groupoid approach with the canonical gravity quantization [5], which can be thought of as a "reference point" for other methods of quantizing gravity. The groupoid approach is "more radical" in the sense that in this approach the noncommutative counterpart of the differential structure is quantized whereas in the canonical method three-metrics play the role of "quantization variables". We show that in spite of this difference the superspace formulation of general relativity (which could be regarded as a prerequisite of the canonical quantization) can be obtained from the groupoid approach if the corresponding noncommutative algebra is restricted to its commutative subalgebra which determines a suitable slicing of space-time. Consequently, in the groupoid approach when the space-time slicing appears gravity is already in its "classical (non-quantum) regime". However, this conclusion could follow from a simplification inherent in our model, and could eventually be avoided if one considers a more general module of the noncommutative counterpart of vector fields (the module of derivations of a given algebra).
We organize our material in the following way. To make the paper self-contained and to fix our notation, in Section 2, we give a summary of the groupoid approach to noncommutative unification of general relativity with quantum mechanics. In Section 3, we define the noncommutative algebraic counterpart of the standard concept of superspace (the space of three metrics). The comparison of the canonical gravity quantization with the groupoid approach is done in Section 4, and some conclusions and comments are collected in Section 5.

Basic ideas of the model
The main idea of the groupoid approach to the unification of general relativity and quantum mechanics is to forget, in the very beginning, the concept of space-time and start with the abstract space G = E × Γ, where E is the total space of a principal fibre bundle, and Γ its structural group such that the orbits of the action of Γ on E form a smooth manifold M interpreted as spacetime (this construction can eventually be generalized to the category of differential spaces of constant dimension, see [6]). We endow G with the groupoid structure. In the present paper, for the sake of concreteness, we shall assume that E is the total space of the frame bundle over a space-time manifold M, and Γ the group SO (3,1). Of course, M = (G/SO(3, 1))/SO(3, 1)). Then one defines the algebra as the (intrinsic) direct sum where A const = pr * (C ∞ (M, C)), and C ∞ c (G, C) is the family of smooth compactly supported complex valued functions on G. The multiplication in the algebra A is defined in the following way: with γ, γ 1 , γ 2 ∈ G p , G p being the fiber in G over p ∈ E; integration is with respect to the Haar measure; (2) if a, b ∈ A const they are multiplied in the usual way, i. e., a * b = a · b; (3) if a ∈ A const and b ∈ C ∞ c (G, C), one evidently a noncommutative algebra. We also define the involution of a ∈ A by a * (γ) = a(γ −1 ) where γ = (p, g), p ∈ E, g ∈ Γ. 1 Let us also define the subalgebra A proj = π * M C ∞ (M, C) ⊂ A const . It plays the important role in our model since by restricting the algebra A to the subalgebra A proj we recover the space-time manifold of general relativity. 1 One should notice that we have corrected the definition of the algebra A as compared with our previous works (see [2,4]). This corection does not change our previous results.
Let us consider the set DerA of all derivations of the algebra A. DerA is a Z(A)-module, where Z(A) denotes the center of A, and can be regarded as a noncommutative counterpart of vector fields. In the following, we shall consider a noncommutative differential geometry as defined by the Z(A)- of A parallel to E and Γ, respectively (this is only a simplifying assumption which in the general case should be relaxed).
First, we define a metric on the Z(A)-submodule V as a Z(A)-bilinear non-degenerate symmetric mapping g : V × V → A, and for our model we choose the following metric adapted to the product structure of V where g E and g Γ are metrics on E and Γ, respectively, and pr E and pr Γ are the obvious projections. It turns out that the "vertical component" pr * Γ g Γ of the metric g is essentially unique (this is true for a broad class of derivation based noncommutative differential calculi, see [7]), whereas the "parallel component" pr * E g E of g is a lifting of the Lorentz metric in spacetime M (see also [8]). Now, with the help of the Koszul formula, we define the linear connection; then the curvature and the usual Ricci operator R : V → V which is the counterpart of the Ricci tensor with one index up and one index down (for details see [2]). In this way, we have all quantities needed to write the noncommutative Einstein equation derivations v ∈ V Γ satisfy it trivially, see [8]).
Let us consider the representation of the algebra A in the Hilbert space where B(H) denotes an algebra of bounded operators on H and G q is the fiber of G over q ∈ E, given by the formula The integral is taken with respect to the Haar measure. The completion of A with respect to the norm a = sup q∈E π q (a) is a C * -algebra (see [3, p. 102]). We shall denote this algebra by E.
We assume (as a separate axiom) that the dynamics of a quantum gravitational system is described by the following equation for every q ∈ E, where v ∈ kerG, and (F v ) v∈kerG is a one-parameter family We shall also assume that [F v , π q (a)] is a bounded operator.
The fact that v ∈ kerG makes of eqs.
(2) and (4) a "noncommutative dynamical system". We could also say that noncommutative Einstein equa-tion (2) plays the role of a "boundary condition" for quantum dynamical equation (4). To solve this system means to find the set It can be easily verified that it is a subalgebra of E.
LetĒ G be the smallest closed involutive subalgebra of the algebra E containing E G .Ē G is said to be generated by E G . Since E is a C * -algebra and every closed involutive subalgebra of a C * -algebra is a C * -algebra (see [10, Sec. 1.3.3]),Ē G is also a C * -algebra; it will be called Einstein C * -algebra or simply Einstein algebra, and the pair (Ē G , kerG) -Einstein differential algebra.
Now, the idea is to perform quantization with the help the usual C *algebraic method (see, for instance, [11], [12, chapter 9]) with the Einstein algebraĒ G as our basic C * -algebra. According to this method, a quantum gravitational system is represented byĒ G , and its observables by Hermitian elements ofĒ G . If a is a Hermitian element ofĒ G , and φ a state onĒ G then φ(a) is the expectation value of the observable a when the system is in the state φ.
It can be shown that this gravity quantization scheme correctly repro-duces the usual general relativity (on space-time) and quantum mechanics (in the Heisenberg picture) when the algebra A is restricted to its center Z(A) (or to some subset of Z(A)) (see [2,8]). given by

Algebraic version of superspace
The quotient space S(S) = Riem(S) Diff(S) is called superspace. Its global properties were studied by Fischer [13] (see also [14]).
In a particular coordinate system any metric h ∈ Riem(S) can be represented as a covariant metric tensor h ij (x) or as a contravariant metric tensor h ij (x), x ∈ S. Then, as shown by DeWitt [15], there exists a metric on S(S), called the Wheeler-DeWitt metric, which assumes the form It has the signature (− + + + ++) for each point of the 3-geometry.
Let us now consider a slicing (S t ) t∈T of M such that S t is diffeomorphic if there is g ∈ Γ such that q = pg. Now, it can be easily seen that A S ⊂ A proj ⊂ Z(A). Indeed, pr −1 (x) ⊂ pr −1 (S) for every x ∈ S. Consequently, the differential algebra (A S , V S ) is commutative. We denote the set of all metrics in the module V S by Riem(A S ). As an analogue of Diff(S) we should take the set IsoA S of all isomorphisms of A S into itself. We have the action Any isomorphism f : A S → A S induces the mapping (which is also an isomorphism) where v ∈ V S , α ∈ A. Therefore, one has

Noncommutative gravity and canonical quantization
We now briefly recollect the canonical method of quantizing gravity to compare it with our approach. Any space-time metric can be locally written in the form where h ij , i, j = 1, 2, 3 is the metric tensor on the spacelike hypersurface S =const, N is called lapse function; it measures the proper time separation between hypersurfaces t =const. The so-called shift vector N i measures the deviation of curves x i =const from the normal to S (in the following we use units such that c =h = 1). The extrinsic curvature of S can be written as where the stroke "|" denotes covariant differentiation with respect to the 3-metric h ij . The momentum canonically conjugated to h ij is given by

The classical Hamiltonian is
where with 3 R being the scalar curvature of h ij and Λ the cosmological constant.
By making the standard substitution: h ij → h ij , π ij → −i δ δh ij (δ is the functional derivative) one obtains the counterpart of the Schrödinger equation TheĤ 0 -part of this equation is the celebrated Wheeler-DeWitt equation. This is the fundamental equation for the "wave function of the universe" Ψ[h ij ] which is the functional of the 3-metric (we do not take into account any matter fields).
We should emphasize that in the Wheeler-DeWitt approach it is the 3metric that is quantized (and the momentum canonically conjugated to it), whereas in our approach the "quantization variables" are elements of the Einstein C * -algebraĒ G . However, we can ask the question: what would happen to the equations of our theory (eqs. (2) and (4)) if we restrictĒ G to (Ē G ) S , i. e. if we go to the "superspace limit"?
Since (Ē G ) S ⊂ Z(Ē G ) eq. (4) reduces to the trivial identity (0 ≡ 0) and hence it becomes insignificant. We are left with eq. (2) which, in this case, is reduced to the usual Einstein equations. In this way, gravity decouples from quantum mechanics. This is an important conclusion: if we go to the superspace limit quantum gravity effects become negligible. In this process, the slicing of space-time emerges, and consequently the concepts of time and instantaneous spaces become meaningful. This means that we are well beyond the Planck threshold in the non-quantum gravity regime (see [16] where the emergence of time from the noncommutative era has been studied).
As it is well known, the Wheeler-DeWitt equation corresponds to the sta-tionary Schrödinger equation. Eq. (4) plays the similar role in our approach since, for weak gravitational fields it reduces to the Schrödinger equation (in the Heisenber picture of quantum mechanics) [8]. However, one should not forget that the Wheeler-DeWitt equation is the equation for three-metrics, whereas eq. (4) is the equation for elements of the algebraĒ G .

Concluding remarks
We have demonstrated that if in the groupoid approach to the unification of general relativity and quantum mechanics, proposed in [2], the algebra A = A proj ⊕ C ∞ c (G, C) is restricted to its subalgebra A S , consisting of functions constant on pr −1 (S t ) t∈T , where (S t ) t∈T is a time slicing of space-time M, one obtains the superspace formulation of general relativity.
The important point is that our approach shows that at the level where time slicing of space-time appears, quantum gravity effects are already insignificant (i. e., gravity is too weak to exhibit quantum effects, see above Section 4). This seems reasonable since in the quantum gravity regime we would expect some kind of "foamy mixture" of space and time which is excluded by the well defined time slicing of space-time. This conclusion could be the consequence of a simplifying assumption incorporated into our model, namely that our noncommutative differential algebra is based on the submodules of derivations parallel to E and Γ, respectively. In this model "geometry parallel to E" is, in principle, responsible for gravity effects and "geometry parallel to Γ" is responsible for quantum effects. The fact that we have neglected "mixed terms" (those coming both from V E and V Γ ) means that in our model gravity is "weakly coupled" to quantum effects. Consequently, if we restrict the algebra A to its subalgebra A S (this restricting essentially means that slicing of space-time enters the scene) all terms parallel to Γ automatically are switched off. Such terms would be responsible for a "fluctuating slicing" of space-time which could be enough for an approximate validity of the canonical quantization of gravity. The decisive step in checking this hypothesis would be to construct a counterpart of our model based on a more general module of derivations.
The analogous situation occurs in the canonical quantization approach.
One begins with the sliced classical space-time (with no quantum effects).
Then one performs the canonical quantization, as the result of which 3geometries begin to fluctuate, and the sliced regime of space-time becomes "fuzzy".
As it is well known, when Einstein's equations are formulated as a constrained Hamiltonian system, the Hamiltonian constraint and the equations of motion determine the evolution of three-metrics in superspace, and the momentum constraint implies that the Hamiltonian flow is orthogonal (in the Wheeler-DeWitt metric) to the orbits of the diffeomorphism group (although these two directions need not be disjoint [14]). Since in our algebraic approach the submodule V S corresponds to the family of vector fields on the superspace S(A) the above mentioned regularities should be reflected in the structure of this submodule.