Space Time Covariance of Canonical Quantization of Gravity: A (formal) general result and the (rigorous) explicit case of 2+1 Quantum Cosmology

A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity (using the Canonical Equations of motion), become weakly vanishing functions of the Quadratic and Linear Constraints. The implication of this result is that (formal) Dirac consistency of the Quantum Operator Constraints (annihilating the wave Function) suffices to guarantee space time covariance f the ensuing quantum theory: An ordering for each invariant relation will always exist such that the emanating operator has an eigenvalue identical to the classical value. The example of 2+1 Quantum Cosmology is explicitly considered: The four possible ``Cosmological Solutions'' --two for pure Einstein's equations plus two more when a $\Lambda$ term is present- are exhibited and the corresponding models are quantized. The invariant relations describing the geometries are explicitly calculated and promoted to operators whose eigenvalues are their corresponding classical values.


Introduction
The problem of space time covariance of a Quantum Theory of Gravity [1] within the context of Canonical Quantization can essentially be described as follows: Classically, Einstein's Field equations are known to be (not manifestly but explicitly) equivalent to the Hamiltonian and Momentum constraints plus the Canonical Equations of motion. This is understandable since, although the canonical analysis uses objects defined on the hypersurface, the momenta involve the extrinsic curvature and thus carry the information of the embedding of the hypersurface in space time. When we canonically quantize, the momenta become functional derivatives with respect to the spatial metric (choice of Polarization in the phase space) and thus the fate of space time covariance is somehow obscure. Of course if one takes special care so that the geometrical meaning of the classical constraints is maintained at the quantum level, one is justified to expect space time covariance of the ensuing theory. Indeed as we shall see in the first section, when we suitably define the equivalence between two set of state vectors corresponding to the different foliations of the same space time, then space time covariance is indeed achieved.
In the next sections we present the example of 2+1 spatially homogeneous cosmological models in the absence as well as in the presence of Λ term. The classical solutions i.e. the different "cosmological" disguises of Minkowski space time or the maximally symmetric space are explicitly given. The corresponding models are quantized and the classical observables are turned into operators in such a way that their eigenvalues coincide with the classical values. In this sense it is proven that the two pairs of state vectors are equivalent.

A Classical Theorem and its Quantum Implications
As is well known [2], the canonical analysis of pure Gravity consists in the following statements: H 0 (g ij , π ij ) ≈ 0 (2.1a) H k (g ij , π ij ) ≈ 0 (2.1b) g ij = {g ij , H} (2.1c) π ij = {π ij , H} (2.1d) are equivalent to the Einstein Equations. If we adopt the notion that classical observables, are all the geometrical objects that do not depend on the gauge, i.e. the coordinate system, then we are led to identify these observables with invariant relations among space time scalars. These scalar can be either curvature or higher derivative or metric scalars. The scalars themselves do not describe the space time in an invariant manner since their functional form in term of the coordinates changes when the coordinate system is altered. A way to generate invariant relations among these scalars, albeit not the most efficient one, is to take a base of 4 scalars (say Q 1 , . . . Q 4 ) and solve for the coordinates. Any other scalar (say Q 5 ) is now expressible in terms of the 4 scalars chosen (Q 5 = f (Q 1 , . . . , Q 4 )). Now this relation is characteristic of the geometry i.e. does not change form under coordinate transformations. In this sense a geometry is complectly characterized by a set of relations: The index i is at most countable. Turning these relations into function on the phase space we notice that they become weakly vanishing quantities: In implementing this step use has been made of canonical equations of motion in order to substitute all higher time derivatives of the metric of the slice. But at this point, we invoke a known theorem of constrained dynamics [2]: Theorem "Every weakly vanishing function in Phase Space is strongly equal to some expression containing the Constraints (which define a surface in Phase Space)" Moreover the expression under discussion ought to vanish on-mass shell (i.e. when the constraints are set to zero). Thus: The translation of the above result in the velocity phase space reads as: Take any invariant relation and substitute for the higher time derivatives of the metric the relations given by solving the spatial Einstein equations for the accelerations. The result will be that each and every invariant relation will become a function of the G 0 0 , G 0 k quantities. The implications of this for the quantum theory are obvious and important: Adopting the point of view that the quantum observables are to be the operator analogues of the quantum observables, we are assured that the will exist a factor ordering such that these operators, when acting on the appropriate states defined by: will have vanishing eigenvalues: This establishes the (formal) space time covariance of our theory.

Preliminaries on Spatially Homogeneous Spacetimes
Before proceeding, it is deemed as appropriate to exhibit some basic assumptions. Spacetime, is assumed to be the pair (M, g) where M is a 3-dimensional, Hausdorff, connected, time-oriented and C ∞ manifold, and g is a (0, 2) tensor field, globally defined, C ∞ , non degenerate and Lorentzian i.e. it has signature (−, +, +). In the spirit of 2+1 analysis, M = R × Σ t , where the 2-dimensional orientable submanifolds Σ t (surfaces of simultaneity), are spacelike surfaces of constant time; their foliation in time, results in the entire spacetime. When the assumption of spatial homogeneity is imposed, this corresponds to imposition of the action of a symmetry group of transformations G upon the manifolds Σ t . Usually the group G is not only continuous, but also a Lie group -thus denoted by G r , where r is the dimension of the space of its parameters. Avoiding the details on these issues -these matters can be easily found in every standard reference see e.g. [5]-we simply state that spatially homogeneous models are described (apart the topology of Σ t ) by an invariant basis one forms σ α i (x)dx i (i.e. their Lie Derivative with respect to the generators of the Lie Group G r , are zero) -when such exists; a counter example is that of Groups which act multiply transitively-the Lie Algebra of the generators of the corresponding Lie Group which is the realization of the Lie Group G r , and a line element of the form: where γ αβ (t) is the metric induced on the surfaces Σ t (and constant on them), N(t) is the lapse function, N α (t) is the shift vector (N α (t) = γ αβ (t)N β (t), γ αβ (t) being the inverse of γ αβ (t)), and C α µν (x) are the structure functions(constants) for the corresponding Open(Closed) Lie Algebra. Greek indices are world indices (i.e. count the one-forms), while the Latin indices count the spatial dimensions which are 2. In this work we consider the following assumptions: A 1 Lie Groups G r are to correspond to Closed Lie Algebras. Thus the quantities C α µν (x) will be space independent (constants). A 2 Lie Groups G r are to act simply transitively. A Lie Group G r acts transitively if: (ii) the rank of the matrix constructed by the generators (seen as vector fields) should be equal to the dimension of Σ t (= 2).
Geometrically, the above demands mean that two different points in a given domain of Σ t can be interchanged by a transformation-member of the Lie Group. Simply transitive action, corresponds to the case r = dimension of Σ t (= 2).
Under these two assumptions, one can see that in 2 dimensions, there are only two distinct Closed Lie Algebras; the Abelian, where all the structure constant vanish C α µν = 0 and the Non Abelian, where there is only one non vanishing structure constant -say C 1 12 = 1; every other choice for the non vanishing structure constants will fall into this, after a linear mixing of the initial set i.e. after the usage of a new set of the form C α µν = Υ α β ∆ κ µ ∆ λ ν C β κλ , where ∆ α β ∈ GL(2, R), and Υ α β its inverse. In this way the two models emerge; each one corresponds to a spatially homogeneous cosmological model.
At this point, a question arises; is there any particular class of General Coordinate Transformations (G.C.T.'s) which can serve to simplify the form of Einstein's Field Equations (E.F.E.'s)? The answer is positive and a thorough investigation of this problem and its consequences, is given in [3]; indeed, not only there is a class of G.C.T.'s which preserve the manifest spatial homogeneity of the line element (3.1), but also forms a continuous (and virtually, Lie) group. This group is closely related to the symmetries of the symmetry Lie Group G r ; it is its automorphism group. In the spirit of the 2+1 analysis, the most general G.C.T.'s, modulo time reparameterization, are: After insertion of (3.3) into (3.1), the restriction of preservation of manifest homogeneity of the latter, leads to the allocations: and: x) ditto for the vector P α (t, x), with σ i α (x) being the inverses of σ α i (x) -quantities which exist in simply transitive cases. In order for the transformations (3.3) to have a well defined, non trivial action, it is pertinent for the quantities Λ α β and P α to be space independent. So: and: Thus (3.7) from allocations, turn to a set of highly non linear, partial differential equations. Integrability conditions for this system i.e. Frobenious' Theorem, results in the system (the dot, whenever used, denotes differentiation with respect to time): and, "Time-Dependent Automorphisms Inducing Diffeomorphisms" emerge. The automorphisms of a Lie Group G r consist a continuous group. Those members of that group which are continuously connected to the identity element, form a Lie Group as well -though the topology of the latter might be different from that of the former. If one considers parametric families of the automorphic matrices, characterized by the parameters τ i , Λ α β (t; τ i ) and, as usual, demands: where λ α β(i) , are the generators with respect to the parameter τ i , of the Lie Algebra of the Automorphisms, then from the first of (3.9), after a differentiation with respect to τ i , will have: For an extensive treatment on these issues see [4], while for the relation and usage of these generators with conditional symmetries [6] see [7]. In n+1 analysis (here n=2), the E.F.E.'s in vacuum, when the cosmological constant Λ c exists, take the form: where, δ α β is the Kronecker's Delta. Since G.C.T.'s are symmetries of the E.F.E.'s, the same holds for the automorphisms i.e. under such a G.C.T.: where S α β is the inverse of Λ α β . The effect of a time reparameterization is trivial and thus omitted. Finally some terminology is needed; (3.12a) is called "Quadratic Constraint", (3.12b) are called "Linear Constraints", and (3.12c) are simply the "Equations of Motions".

The Abelian Model -Classical Consideration
This model is characterized by the vanishing of all the structure constants of the corresponding Lie Algebra. Since C α µν = 0, it is also a Class A model; it trivially holds that C α αν = 0. A choice for the basis one-forms, in the spatial coordinate basis (x 1 , x 2 ), is σ α i (x) = δ α i i.e. the Kronecker's Delta. In matrix notation: The solution to the system (3.9) is: While, a basis (unique only modulo linear transformations) which spans the space of solutions to (3.11) is: Any solution to (3.11) is a linear combination of these quantities.
Obviously, vector P α (t) inherits the entire freedom carried by the G.C.T.'s; modulo time reparameterizations, two arbitrary functions of time, were expected. Indeed, two appear in this vector and none in the matrix Λ α β . Thus, according to (3.8), the only possible usage of this freedom, is to make the shift vector, to vanish. Then, the E.F.E.'s (3.12) assume a mush simpler form, in terms of the new variables γ αβ , and N : Linear Constraints are Identically Satisfied (4.4b) A form, which can be more simplified at the "gauge" Thus, dropping the tildes for the sake of simplicity: By virtue of (3.13d), and since C α µν = 0, the E.F.E.'s become:

Abelian Model with vanishing Cosmological Constant
The previous set of equations, assume the form: where ϑ α β is a constant matrix, and the Einstein's summation convention, is in use. The integration of (4.7) is straightforward. The quadratic constraint, simply states that the determinant of the matrix ϑ α β vanishes; thus the initial number of the independent components, is at most 3. But the equations of motion, can be rewritten as: and a consistency requirement, emerges: An exhaustive consideration of all the cases concerning a matrix ϑ α β , with vanishing determinant, plus the consistency requirement, results in only the following two distinct -at first sight-solutions to the previous system: and: Corresponding to the full solutions (at the gauge N(t) = √ γ): and: The different notation in spatial coordinates has a meaning; these two solutions, are not independent. The transformation (t, takes the line element (ds 1 ) 2 to (ds 2 ) 2 . Thus the only solution for the case under consideration, is the Minkowskian spacetime with the usual topology of R 3 -since the second solution is the Minkowskian in "peculiar" coordinates:

Abelian Model with non vanishing Cosmological Constant
In this case, the E.F.E.'s have a slightly more altered form: Contraction (i.e. setting α = β) of the equations of motion results in the integral of motion: Under the conformal transformation γ αβ = γ αβ √ γ the equations of motion become: and contraction of this set, results in (since γ = 1) ϑ α α = 0. Substitution of the quadratic constraint, and usage of the integral of motion (4.17), yield: since ϑ α α = 0. Again, consideration of equations of motion, in terms of γ αβ for all the traceless matrices ϑ α β , plus the consistency requirement (i.e.γ αβ =γ βα ) deduce the following two solutions: and: Since, for all the cases, w ≥ 0 the integral of motion (4.17) dictates that Λ c > 0. Inversion of the conformal transformation and integration of (4.17) for each of the two solutions, yield: and: Corresponding to the full solutions (at the gauge N(t) = √ γ): and: because ϑ 1 1 is not an essential constant [3], and thus it can be absorbed. For once more, the different notation in spatial coordinates has a meaning; these two solutions, are not independent. The transformation (t, takes the line element(ds 1 ) 2 to (ds 2 ) 2 . Thus the only solution for the case under consideration, is the Minkowskian spacetime with the usual topology of R 3 -since the second solution is the first in "peculiar" coordinates:

The Non Abelian Model -Classical Consideration
This model is characterized by the fact that there is only one non vanishing structure constant -say C 1 12 = 1; any other choice will fall into this (see Introduction). It is also a Class B model, since C α αν = 1. A choice for the basis one-forms, in the spatial coordinate basis (x 1 , x 2 ) is: Now, the solution to the system (3.9) is: While now, a basis (unique only modulo linear transformations) which spans the space of solutions to (3.11) is: Any solution to (3.11) is a linear combination of these quantities -for this model. This time, both the automorphic matrices Λ α β (t) and the vector P α (t) have the freedom carried by the G.C.T.'s. It is wiser (the reason for this, will be revealed in the next lines) to exploit this freedom contained in Λ α β (t) in order to simplify the initial, though most general, form for the scale factor matrix γ αβ (t) rather than using the same freedom contained in P α (t) in order to set the shift vector equal to vanish. In this wise, an initial full scale factor matrix, can be brought to the form: Thus the initial set of dynamical variables, consists of a scale factor matrix of the previous form, plus a shift vector (N 1 (t), N 2 (t)). Insertion of this set into both (3.12b) and (3.13d) results in respectively: N a (t) = 0 (5.5) and: Thus, it can be easily seen that the rest of E.F.E.'s have the form (4.5). That is reason for using the matrix Λ α β (t), instead of P α (t) (in order to have a zero shift initially); the wanted vanishing was deduced, and not produced at the cost of wasting on valuable freedom. The E.F.E.'s (4.5) with: and: can be most easily integrated. Indeed, equations of motion admit the integral of motion: The quadratic constraint assumes the form: and thus not only sets the constant w equal to zero, but also demands Λ c > 0 -if this term exists. The integration of (5.9) with w = 0, is a trivial matter, the result being: when Λ c = 0 the corresponding full solution being: with the usual topology of R 3 , and: when Λ c = 0 the corresponding full solution being: ] and the usual topology of R 3 again.

Quantum Description of the Models
In trying to quantize gravity, one faces the problem of quantizing a constrained system. The main steps one has to follow are: (i) define the basic operators g µν and π µν and the canonical commutation relation they satisfy.
(ii) define quantum operators H µ whose classical counterparts are the constraint functions H µ .
(iii) define the quantum states Ψ[g] as the common null eigenvector of H µ , i.e. those satisfying H µ Ψ[g] = 0. (As a consequence, one has to check that H µ , form a closed algebra under the basic Canonical Commutation Relations (CCR).) (iv) find the states and define the inner product in the space of these states.
Concerning point (iii) it is pertinent to clarify the meaning of the imposition of the quantum constraints upon Ψ[g]. A straightforward (modulo regularization prescriptions) but tedious calculation shows that any functional which is not a scalar functional of the curvature invariants (see [8]) does not solve the linear constraints. Therefore, the imposition of the linear constraints, ensures that the wave functional will be a (scalar) functional of the n-geometry (in n spatial dimensions) and not of the coordinate system. Then, the dynamical evolution is provided by the quadratic constraint; the consistency of the quantum algebra, guarantees that the final wave functional, will be independent of the 3 dimensional coordinate system. In the absence of a full solution to the problem, a partial solution, generally known as quantum cosmology, has been employed. This is an approximation to quantum gravity in which one freezes out all but a finite number of degrees of freedom, and quantizes the rest. In this way one is left with a much more manageable problem that is essentially quantum mechanics with constraints. The gravitational field is represented by no more than 1 degree of freedom (the one scale factor of some anisotropic Bianchi-like Type model). In principle, the dynamical variables are the scale factors γ αβ (t)'s, the lapse function N(t) and the shift vector N α (t). The presence of the linear constraints -along with the conditional symmetries of the corresponding Hamiltonian-enabled a reduction of the initial configuration space to a lower dimensional one, spanned by the curvature invariant characterizing the 2-geometry. The ultimate justification of this reduction is the fact that -from the point of view of the 2-geometry-the omitted degrees of freedom, are not physical but gauge [4]. It is true that at the classical level, the scale factor matrix, can be diagonalized on mass-shell -through a constant matrix e.g. [3] for the 3dimensional analogous-while the shift can be set equal to zero. However, if one intends to give weight to all states, one has to start with the most general form which is described by the 3 scale factors γ αβ (t)'s and the lapse function N(t). The absence of H α 's due to the vanishing of the C α βγ 's, implies that in principle all γ αβ 's are candidates as arguments for the wave function which solves the quadratic constraint (Wheeler-DeWitt equation). This is in contrast to what happens in the Non Abelian case, where one combination of γ αβ 's and C α βγ 's, parameterize the reduced configuration space.

Quantization of the Abelian Model
In this section, we present a complete reduction of the initial configuration space for the Abelian geometry -by extracting as many gauge degrees of freedom, as possible. Two separate cases are considered; when the cosmological constant is present and when is not. In either case, a wave function which depends on one degree of freedom is found, by imposing on it, the quantum versions of all classical integrals of motion as additional conditions. The Hamiltonian of the a Class A, spatially homogeneous cosmological system is H = N(t)H 0 + N α (t)H α , where: is the quadratic constraint, with: γ being the determinant of γ αβ , and: are the linear constraints. For all Class A Types, the canonical equations of motion, following from (6.1), are equivalent to Einstein's equations derived from line element -see [9] for the 3+1 dimensional analogous.
In the Schrödinger representation: with the relevant operators, satisfying the basic Canonical Commutation Relations (CCR's) which correspond to the classical ones: In the Abelian case, C α µν = 0, thus the only operator which must annihilate the wave function, is H 0 ; and the Wheeler-DeWitt equation H 0 Ψ = 0, will produce a wave function, initially residing on a 3-dimensional configuration space -spanned by γ αβ 's. If the linear constraints existed, a first reduction of the initial configuration space, would take place [6]. New variables, instead of the 3 scale factors, would emerge -say q i , with i < 3. Then a new "physical" metric would be induced: According to Kuchař's and Hajicek's [6] prescription, the "kinetic" part of H 0 would have to be realized as the conformal Laplacian (in order for the equation to respect the conformal covariance of the classical action), based on the physical metric (6.7). In the presence of conditional symmetries, further reduction can take place, a new physical metric would then be defined similarly, and the above mentioned prescription, would have to be used after the final reduction [11]. The Abelian case, is an extreme example in which all the linear constraints, vanish identically; thus no initial physical metric, exists -another peculiarity reflecting the high spatial symmetry of the model under consideration. In compensation, a lot of integrals of motion exist ant the problem of reduction, finds its solution through the notion of "Conditional Symmetries". These linear in momenta integrals of motion, if seen as vector field on the configuration space spanned by γ αβ 's, induce -through their integral curvesmotions of the form γ αβ = Λ µ α Λ ν β γ µν , Λ ∈ GL(2, ℜ) (see section 2 of [4]) which not only are identical to the action of spatial diffeomorphisms, but also describe the action of the automorphism group -since GL(2, ℜ) is the Aut(G) which corresponds to the Abelian models.
The generators of this automorphism group, are (in a collective form and matrix notation) the following 4 -one for each parameter: with the defining property: Exponentiating all these matrices, one obtains the outer automorphism group of the Abelian model, since there is not Inner Automorphism subgroup (all structure constants vanish).
For full pure gravity, Kuchař [11] has shown that there are no other first-class functions, homogeneous and linear in the momenta, except the linear constraints -ditto in 2+1 analysis. If however, we impose extra symmetries, such quantities may emerge -as it will be shown. We are therefore -according to Dirac [10]-justified to seek the generators of these extra symmetries; their quantum-operator analogues will be imposed as additional conditions on the wave function. The justification for such an action, is obvious since these generators correspond to spatial diffeomorphisms -which are the covariance of the theory. Thus, these additional conditions are expected to lead us to the final reduction, by revealing the true degrees of freedom. Such quantities are, generally, called in the literature "Conditional Symmetries" [11].
From matrices (6.8), we can construct the linear -in momenta-quantities: , . . . , 4} (6.10) In order to write analytically these quantities, the following base is chosen: It is straightforward to calculate the Poisson Brackets between E (I) and H 0 : We therefore conclude that, when the cosmological constants is non vanishing, only the first three E (I) , are first-class and thus integrals of motion. If Λ vanishes, all the four quantities, are first-class. Out of the three quantities E (I) , only two are functionally independent (i.e. linearly independent, if we allow for the coefficients of the linear combination, to be functions of the γ αβ 's); numerically, they are all independent. The algebra of E (I) can be easily seen to be: The non vanishing structure constants of the algebra (6.16), are found to be: At this point, in order to achieve the desired reduction, we propose that the quantities E (I) -with I ∈ {1, . . . , 3}-must be promoted to operational conditions acting on the requested wave function Ψ -since they are first class quantities and thus integrals of motion (see (6.14)). In the Schrödinger representation: In general, systems of equations of this type, must satisfy consistency conditions decreed by the Frobenious Theorem: Subtraction of these two and usage of (6.15), results in: This relation constitutes a selection rule for the numerical values of the integrals of motion. In view of the Lie Algebra (6.17), selection rule (6.20) sets K 1 = K 2 = K 3 = 0. This fact restores the action of the diffeomorphisms as covariances of the quantum theory, in the sense that now, we have conditions of the form E (I) Ψ = 0. Instead, if we also had E (4) (as is the case Λ = 0) then K 4 would remain arbitrary. With this outcome, and using the method of characteristics [12], the system of the two functionally independent P.D.E.'s (6.18), can be integrated. The result is: i.e. an arbitrary (but well behaved) function of γ -the determinant of the scale factor matrix.
A note is pertinent here; from basic abstract algebra, is well known that the basis of a linear vector space, is unique -modulo linear mixtures. Thus, although the form of the system (6.18) is base dependent, its solution (6.21), is base independent.
The next step, is to construct the Wheeler-DeWitt equation which is to be solved by the wave function (6.21). The degree of freedom, is 1; the q = γ. According to Kuchař's proposal [6], upon quantization, the kinetic part of Hamiltonian is to be realized as the conformal Beltrami operator -based on the induced physical metric -according to (6.7), with q = γ: In the Schrödinger representation: where: is the 1-dimensional Laplacian based on g 11 (g 11 g 11 = 1). Note that in 1-dimension the conformal group is totally contained in the G.C.T. group, in the sense that any conformal transformation of the metric can not produce any change in the -trivial-geometry and is thus reachable by some G.C.T. Therefore, no extra term in needed in (6.24), as it can also formally be seen by taking the limit n = 1, R = 0 in the general definition: Thus: So, the Wheeler-DeWitt equation -by virtue of (6.21)-, reads: The general solution to this equation, is: where J n and Y n , are the Bessel Functions of the first and second kind respectively -both of zero order-and c 1 , c 2 , arbitrary constants. Some comments on this wave function. Indeed, at first sight, the fact that Ψ depends only on one argument and particularly on γ, seems to point to some undesirable degeneracy regarding anisotropy; classically γ can be gauged to e t and thus it seems as though the anisotropy parameter does not enter Ψ at all. If, however, we reflect thoroughly, we will realize that this objection rests strongly on a -not generally accepted-mingling of the classical notion of anisotropy and the interpretation of the wave function. Indeed if we adopt the interpretation that the wave function Ψ (along with a suitable measure), is to give weight to all configurations parameterized by γ αβ , then the anisotropic configuration will, in general, acquire different probabilities. The degeneracy occurs only between two different anisotropic configurations with the same determinant γ. In compensation the scheme proposed here, avoids the gauge degrees of freedom as much as possible. The final probabilistic interpretation must await the selection of a proper measure. If the cosmological constant Λ is zero, some changes will take place. The first concerns the obvious alteration to the potential in the Hamiltonian; it vanishes. This consequently, causes an alteration to the Poisson Bracket (6.12), which takes the form: while, (6.15) still holds. Thus in the present case, there are four integrals of motion -instead of three. Also, the P.D.E. system (6.18) consists of four members (instead of three), but now out of the four quantities E (I) , only three are functionally independent; the previous two, plus E (4) . Again, using the method of characteristics [12], the system of the three functionally independent P.D.E.'s (6.18), can be integrated. The result is: where γ is the determinant of the scale factor and K 4 , the remaining constant -according to selection rule (6.20). The fact that this wave function does not depend on any combination of γ αβ 's in an arbitrary manner (i.e. Ψ is not an arbitrary function of γ αβ 's), might be taken as an indication that no reduced Wheeler-DeWitt equation can be written. On the other hand, this wave function does contain an arbitrary (essential) constant, which ought to be fixed by the dynamics. The puzzle can be solved by the following compromise; the initial configuration space, should be the mini-superspace i.e. we should write the Wheeler-DeWitt equation, based on the supermetric L αβµν .
In the Schrödinger representation: Thus using (A.11), (A.13), (A.14) for n = 2 and D = 3, one may find respectively -see appendix: R = 2 (6.32) L αβµν Γ αβµν κλ = −γ κλ (6.33) and: Then Kuchař's proposal for the Hamiltonian reads: Substitution of the wave function (6.30) in the Wheeler-DeWitt equation H 0 Ψ = 0, with H 0 given by the previous relation, determines the constant K 4 . The outcome is: The constants c 1 , c 2 , remain arbitrary and may be fixed after the selection of a proper measure via normalizability requirements.

Quantization of the Non Abelian Model
In the present section, the quantization of the Non Abelian model, is exhibited. This model is a Class B since for the corresponding -to the underlying group of isometryalgebra, holds that the only non vanishing structure constant is C 1 12 = 1 (first paragraph of section 3). Class B Cosmological Models, have a peculiarity; if the simplifying hypothesis of homogeneity, is inserted in the Lagrangian of the full theory the reduced Lagrangian and/or Hamiltonian obtained, will result in equations of motion which are not always equivalent to the equations one would gets by the imposition of the same hypothesis, directly on the full Einstein's Field Equations -plus matter, possibly. This situation does not occur for the case of Class A models. Suppose that one adopts the canonical analysis in the framework of the Hamiltonian description. Then, the problem of the existence of a "valid" Hamiltonian (i.e. of a Hamiltonian which produces equations of motion equivalent to the corresponding Einstein equations), arises. A great many of works have dealt with the problem ( [13] and the references therein). The conclusion was that for Class B spacetimes, with a general scale factor matrix γ αβ (t), a valid Hamiltonian is not known -a serious drawback since one major aim of the Hamiltonian approach, is the quantization of the system under discussion. Though it is extremely difficult to attack this problem, partial solutions have been given in [13]. Indeed, in that work, a Hamiltonian constructed out of the scale factor matrix γ αβ (t) and the structure constants C α µν , which resembles in form the Hamiltonian for the Class A models, is constructed and sufficient conditions on the various parts of that Hamiltonian are given, in order for the last, to be a valid one. The set of these conditions is large, the conditions themselves a little complicated and auxiliary quantities enter the scheme. But, in 2+1 analysis, not all are needed; the equations of motion, are nothing but the linear mixing of the time development of the constraints -a fact which simplifies the system of the conditions to be satisfied and the procedure of identifications.
Thus, following the entire procedure described in [13], the following results are obtained: The valid Hamiltonian for the Non Abelian Model, is: where: The quantity q is scalar under the action of the automorphism group, corresponds to the unique curvature invariant of the spatial surfaces of simultaneity Σ t (since R is the only curvature invariant and R = 8q), and thus exhibits the only true degree of freedom -relative to the geometry. The corresponding scalar (under the action of the automorphism group) Lagrangian is: where Θ αβµν is the inverse of Θ αβµν : given by: (γ αµ γ βν + γ αν γ βµ ) + 9 5 γ αβ γ µν (6.41f) S αβ = ∆ αβµν Σ µν (6.41g) G αβ = ∆ αβµν γ µν (6.41h) S = ∆ αβµν Σ αβ Σ µν (6.41i) G = ∆ αβµν γ αβ γ µν (6.41j) Γ = ∆ αβµν γ αβ Σ µν (6.41k) and: Again upon quantization, following Kuchař's proposal, in the Schrödinger representation: where: is the 1-dimensional Laplacian based on the "physical metric" g 11 : with g 11 g 11 = 1 -similarly to the Abelian case. It has been mentioned that in 1-dimension the conformal group is totally contained in the G.C.T. group, in the sense that any conformal transformation of the metric can not produce any change in the -trivialgeometry and is thus reachable by some G.C.T. Therefore, no extra term in needed in (6.46), as it can also formally be seen by taking the limit n = 1, R = 0 in the general definition: Thus: So, the Wheeler-DeWitt equation now, reads: The general solution to this equation, is: where J n , is the Bessel Function of the first kind and c 1 , c 2 , arbitrary constants. If Λ vanishes, the solution is: where Y n is the Bessel Function of the second kind, and c 3 , c 4 , arbitrary constants.

A Appendix
In this appendix, we give some useful formulae, concerning the mini-superspace.