The quantization of gravity: Quantization of the Hamilton equations

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form $-\D u=0$ in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler-DeWitt metric provided $n\not=4$. Using then separation of variables the solutions $u$ can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space $SL(n,\R[])/SO(n)$. Since one can define a Schwartz space and tempered distributions in $SL(n,\R[])/SO(n)$ as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.

where N = N n+1 , n ≥ 3, is a globally hyperbolic Lorentzian manifold, R the scalar curvature and Λ a cosmological constant. We also omitted the integration density in the integral. In order to apply a Hamiltonian description of general relativity, one usually defines a time function x 0 and considers the foliation of N given by the slices We may, without loss of generality, assume that the spacetime metric splits cf. [6,Theorem 3.2]. Then, the Einstein equations also split into a tangential part (1.4) G ij + Λg ij = 0 and a normal part where the naming refers to the given foliation. For the tangential Einstein equations one can define equivalent Hamilton equations due to the groundbreaking paper by Arnowitt, Deser and Misner [1]. The normal Einstein equations can be expressed by the so-called Hamilton condition where H is the Hamiltonian used in defining the Hamilton equations. In the canonical quantization of gravity the Hamiltonian is transformed to a partial differential operator of hyperbolic typeĤ and the possible quantum solutions of gravity are supposed to satisfy the so-called Wheeler-DeWitt equation (1.7)Ĥu = 0 in an appropriate setting, i.e., only the Hamilton condition (1.6) has been quantized, or equivalently, the normal Einstein equation, while the tangential Einstein equations have been ignored. In [6] we solved the equation (1.7) in a fiber bundle E with base space S 0 , where ∆ is the Laplacian of the Wheeler-DeWitt metric given in the fibers, R the scalar curvature of the metrics g ij (x) ∈ F (x), and ϕ is defined by (1.11) where χ ij is a fixed metric in S 0 such that instead of densities we are considering functions. The Wheeler-DeWitt equation could be solved in E but only as an abstract hyperbolic equation. The solutions could not be split in corresponding spatial and temporal eigenfunctions. Therefore, we discarded the Wheeler-DeWitt equation in [7], see also [8,Chapter 1], and looked at the evolution equations given by the second Hamilton equation. The left-hand side, a time derivative, we replaced with the help of the Poisson brackets. On the right-hand side we implemented the Hamilton condition, equation (1.6). After canonical quantization the Poisson brackets became a commutator and we applied both sides of the equation to smooth functions with compact support defined in the fiber bundle. The resulting equation we evaluated for a particular metric which we considered important to the problem and then obtained a hyperbolic equation in the base space, which happened to be identical to the Wheeler-DeWitt equation obtained as a result of a canonical quantization of a Friedman universe, if we only looked at functions that did not depend on x but only on the scale factor, which now acted as a time variable. Evidently, this result can not be regarded as the solution to the problem of quantizing gravity in a general setting.
The underlying mathematical reason for the difficulty was the presence of the term R in the quantized equation, which prevents the application of separation of variables, since the metrics g ij are the spatial variables. In this paper we overcome this difficulty by quantizing the Hamilton equations without alterations, i.e., we completely discard the Hamilton condition. From a logical point of view this approach is as justified as the prior procedure by quantizing only the normal Einstein equation and discarding the tangential Einstein equations-despite the fact that the tangential Einstein equations are equivalent to the Hamilton equations. This equivalence is considered to be an essential prerequisite for canonical quantization, which is the quantization of the Hamilton equations.
During quantization the transformed Hamiltonian is acting on smooth functions u which are only defined in the fibers, i.e., they only depend on the metrics g ij and not explicitly on x ∈ S 0 . As result we obtain the equation This restriction seems to be acceptable, since n is the dimension of the base space S 0 which, by general consent, is assumed to be n = 3. The fibers add additional dimensions to the quantized problem, namely, The fiber metric, the Wheeler-DeWitt metric, which is responsible for the Laplacian in (1.12) can be expressed in the form where the coordinate system is The (ξ A ), 1 ≤ A ≤ m, are coordinates for the hypersurface We also assume that S 0 = R n and that the metric χ ij in (1.11) is the Euclidean metric δ ij . It is well-known that M is a symmetric space It is also easily verified that the induced metric of M in E is identical to the Riemannian metric of the coset space G/K. Now, we are in a position to use separation of variables, namely, we write a solution of (1.12) in the form where v is a spatial eigenfunction of the induced Laplacian of M and w is a temporal eigenfunction satisfying the ODE The eigenfunctions of the Laplacian in G/K are well-known and we choose the kernel of the Fourier transform in G/K in order to define the eigenfunctions. This choice also allows us to use Fourier quantization similar to the Euclidean case such that the eigenfunctions are transformed to Dirac measures and the Laplacian to a multiplication operator in Fourier space.
Here is a more detailed overview of the main results. Let N AK be an Iwasawa decomposition of G and (1.24) g = n + a + k be the corresponding direct sum of the Lie algebras. Let a * be the dual space of a, then the Fourier kernel is defined by the eigenfunctions with λ ∈ a * , x = gK ∈ G/K, and b ∈ B, where B is the Furstenberg boundary, see Section 5 and Section 6 for detailed definitions and references. We then pick a particular b 0 ∈ B and use e λ,b0 as eigenfunctions of −∆ The Fourier transform of e λ,b0 is and of −∆f The elementary gravitons correspond to special characters in a * , namely, for the off-diagonal gravitons and for the diagonal gravitons. Note, that only (n − 1) diagonal elements g ii can be freely chosen because of the condition (1.18).
To define the temporal eigenfunctions, we shall here only consider the case 3 ≤ n ≤ 16, then all temporal eigenfunctions are generated by the two real eigenfunctions contained in where µ > 0 is chosen appropriately. These eigenfunctions become unbounded if the big bang (t=0) is approached and they vanish if t goes to infinity.

Definitions and notations
The main objective of this section is to state the equations of Gauß, Codazzi, and Weingarten for spacelike hypersurfaces M in a (n+1)-dimensional Lorentzian manifold N . Geometric quantities in N will be denoted by (ḡ αβ ), (R αβγδ ), etc., and those in M by (g ij ), (R ijkl ), etc.. Greek indices range from 0 to n and Latin from 1 to n; the summation convention is always used. Generic coordinate systems in N resp. M will be denoted by (x α ) resp. (ξ i ). Covariant differentiation will simply be indicated by indices, only in case of possible ambiguity they will be preceded by a semicolon, i.e., for a function u in N , (u α ) will be the gradient and (u αβ ) the Hessian, but e.g., the covariant derivative of the curvature tensor will be abbreviated byR αβγδ; . We also point out that with obvious generalizations to other quantities.
Let M be a spacelike hypersurface, i.e., the induced metric is Riemannian, with a differentiable normal ν which is timelike.
In local coordinates, (x α ) and (ξ i ), the geometric quantities of the spacelike hypersurface M are connected through the following equations (2.2) x α ij = h ij ν α the so-called Gauß formula. Here, and also in the sequel, a covariant derivative is always a full tensor, i.e.
The comma indicates ordinary partial derivatives.
In this implicit definition the second fundamental form (h ij ) is taken with respect to ν.
The second equation is the Weingarten equation where we remember that ν α i is a full tensor. Finally, we have the Codazzi equation and the Gauß equation Now, let us assume that N is a globally hyperbolic Lorentzian manifold with a Cauchy surface. N is then a topological product I × S 0 , where I is an open interval, S 0 is a Riemannian manifold, and there exists a Gaussian coordinate system (x α ), such that the metric in N has the form where σ ij is a Riemannian metric, ψ a function on N , and x an abbreviation for the spacelike components (x i ). We also assume that the coordinate system is future oriented, i.e., the time coordinate x 0 increases on future directed curves. Hence, the contravariant timelike vector (ξ α ) = (1, 0, . . . , 0) is future directed as is its covariant version (ξ α ) = e 2ψ (−1, 0, . . . , 0).
Let M = graph u | S 0 be a spacelike hypersurface then the induced metric has the form where σ ij is evaluated at (u, x), and its inverse (g ij ) = (g ij ) −1 can be expressed as Hence, graph u is spacelike if and only if |Du| < 1.
The covariant form of a normal vector of a graph looks like and the contravariant version is

Thus, we have
Remark 2.1. Let M be spacelike graph in a future oriented coordinate system. Then the contravariant future directed normal vector has the form (2.14) and the past directed In the Gauß formula (2.2) we are free to choose the future or past directed normal, but we stipulate that we always use the past directed normal. Look at the component α = 0 in (2.2) and obtain in view of (2.15)

The Hamiltonian approach to general relativity
The Einstein equations with a cosmological constant Λ in a Lorentzian manifold N = N n=1 , n ≥ 3, with metricḡ αβ , 0 ≤ α, β ≤ n, are the Euler-Lagrange equations of the functional whereR is the scalar curvature of the metric and where we omitted the density |ḡ|. The Euler-Lagrange equations are where G αβ is the Einstein tensor. We proved in [6,Theorem 3.2], see also [8,Theorem 1.3.2], that it suffices to consider only metrics that split, i.e., metrics that are of the form where (x i ) are spatial coordinates, x 0 is a time coordinate, g ij are Riemannian metrics defined on the slices is an arbitrary smooth function in N .
A stationary metric in that restricted class is also stationary with respect to arbitrary compact variations and, hence, satisfies the full Einstein equations.
Following Arnowitt, Deser and Misner [1] the functional in (3.1) can be expressed in the form is the square of the second fundamental form of the slices M (t) R the scalar curvature of the slices M (t), the interval (a, b) is compactly contained in (3.10) and Ω is a bounded open subset of the fixed slice where we assume Here, we also assume N to be globally hyperbolic such that there exists a global time function and N can be written as a topological product and let In physics is known as the DeWitt metric, or more precisely, a conformal metric, where the conformal factor is even a density, is known as the DeWitt metric, but we prefer the above definition. Combining (3.8) and (3.16) J can be expressed in the form The Lagrangian density L is a regular Lagrangian with respect to the variables g ij . Define the conjugate momenta (3.20) and the Hamiltonian density Since the Lagrangian is regular with respect to the variables g ij , the tangential Einstein equations are equivalent to the Hamilton equations where the differentials on the right-hand side of these equations are variational or functional derivatives.
The mixed Einstein equations vanish and the normal equation which is also known as the Hamilton condition. We define the Poisson brackets Then, the second Hamilton equation can also be expressed as In the next section we want to quantize the Hamilton equations or, more precisely, , where∆ is the Laplacian with respect to the metric g ij (t, ·).

The quantization
For the quantization of the Hamiltonian setting we first replace all densities by tensors by choosing a fixed Riemannian metric in S 0 and, for a given metric g = (g ij (t, x)), we define such that the Einstein-Hilbert functional J in (3.19) on page 9 can be written in the form The Hamilton density H is then replaced by the function where now The effective Hamiltonian is of course Fortunately, we can, at least locally, assume The Lorentzian metric of the ambient space then has the form (4.11) Secondly, we use the same model as in [6, Section 3]: The Riemannian metrics g ij (t, ·) are elements of the bundle T 0,2 (S 0 ). Denote by E the fiber bundle with base S 0 where the fibers consist of the Riemannian metrics (g ij ).
We shall consider each fiber to be a Lorentzian manifold equipped with the DeWitt metric. Each fiber F has dimension Let (ξ a ), 0 ≤ a ≤ m, be coordinates for a local trivialization such that is a local embedding. The DeWitt metric is then expressed as (4.14) where a comma indicates partial differentiation. In the new coordinate system the curves can be written in the form and we infer Hence, we can express (3.6) as where we now refrain from writing down the density √ χ explicitly, since it does not depend on (g ij ) and therefore should not be part of the Legendre transformation. We also emphasize that we are now working in the gauge w = 1. Denoting the Lagrangian function in (4.18) by L, we define and we obtain for the Hamiltonian function H (4.20) where G ab is the inverse metric. The fibers equipped with the metric The Lorentzian metric in the fibers can then be expressed in the form where we note that in that reference is a misprint, namely, the spatial part of the metric has an additional factor 4(n−1) n which should be omitted. Defining a new time variable ξ 0 = t by setting we infer The new metric G AB is independent of t. When we work in a local trivialization of the bundle E the coordinates ξ A are independent of x as well as the time coordinate t, cf. [8,Lemma 1.4.4].
We can now quantize the Hamiltonian setting using the original variables g ij and π ij . We consider the bundle E equipped with the metric (4.24), or equivalently, which is the covariant form, in the fibers and with the Riemannian metric χ in S 0 . Furthermore, let (4.28) C ∞ c (E) be the space of real valued smooth functions with compact support in E.
In the quantization process, where we choose = 1, the variables g ij and π ij are then replaced by operatorsĝ ij andπ ij acting in C ∞ c (E) satisfying the commutation relations (4.29) [ĝ ij ,π kl ] = iδ kl ij , while all the other commutators vanish. These operators are realized by definingĝ ij to be the multiplication operator (4.30)ĝ ij u = g ij u andπ ij to be the functional differentiation is the Euler-Lagrange operator of the functional Hence, if u only depends on (x, g ij ) and not on derivatives of the metric, then Therefore, the transformed HamiltonianĤ can be looked at as the hyperbolic differential operator where ∆ is the Laplacian of the metric in (4.27) acting on functions We used this approach in [6] to transform the Hamilton constraint to the Wheeler-DeWitt equation which can be solved with suitable Cauchy conditions. However, the above hyperbolic equation can only be solved abstractly because of the scalar curvature term R, which makes any attempt to apply separation of variables techniques impossible. Therefore, we discard the Wheeler-DeWitt equation by ignoring the Hamilton constraint and quantize the Hamilton equations instead. This approach is certainly as justified as quantizing the Hamilton constraint, which takes only the normal Einstein equations into account, whereas the Hamilton equations are equivalent to the tangential Einstein equations. Furthermore, the resulting hyperbolic equation will be independent of R and we can apply separation of variables. Dropping the hats in the following to improve the readability equation (3.34) is then transformed to where we note that now w = 1. We have cf. (4.35). We apply both sides to functions u ∈ C ∞ c (E), where we additionally require i.e., u does not explicitly depend on x ∈ S 0 . Hence, we deduce For the second commutator on the right-hand side of (4.40) we obtain where the last term is the Euler-Lagrange operator of the functional where the semicolon indicates covariant differentiation in S 0 with respect to the metric g ij ,∆ is the corresponding Laplacian. We also note that (4.48) in Riemannian normal coordinates. Hence, we conclude that the operator on the left hand-side of equation (4.39) applied to u is equal to equ. (1.4.89)]. On the other hand, applying the right-hand side of (4.39) to u we obtain where the Laplacian is the Laplacian in the fibers, since Thus, we conclude  To solve the equation (4.54) we first choose the Gaussian coordinate system (ξ a ) = (t, ξ A ) such that the metric has form as in (4.26). Then, the hyperbolic equation can be expressed as where∆ is the Laplacian of the hypersurface We shall try to use separation of variables by considering solutions u which are products where v is a spatial eigenfunction, or eigendistribution, of the Laplacian∆ which can be looked at as an implicit eigenvalue equation. The function u in (4.57) will then be a solution of (4.54).
In the next sections we shall determine spatial and temporal eigendistributions by assuming

Spatial eigenfunctions in M
The hypersurface can be considered to be a sub bundle of E, where each fiber M (x) is a hypersurface in the fiber F (x) of E. We shall use the same notation M for the sub bundle as well as for the hypersurface and in general we shall omit the reference to the base point x ∈ S 0 . Furthermore, we specify the metric χ ij ∈ T 0,2 (S 0 ), which we used to define ϕ, to be equal to the Euclidean metric such that in Euclidean coordinates Then, it is well-known that each M (x) with the induced metric (G AB ) is a symmetric space, namely, it is isometric to the coset space which has to be satisfied by the elements of M . But before we can define the eigenfunctions and analyze their properties, we have to recall some basic definitions and results of the theory of symmetric spaces. We shall mainly consider the coset space in (5.3) which will be the relevant space for our purpose. Its so-called quadratic model, the naming of which will be obvious in the following, is the space of symmetric positive definite matrices in R n with determinant equal to 1, i.e., the quadratic model of G/K is identical to an arbitrary fiber M (x) of the sub bundle M of E. Since the symmetric space G/K, as a Riemannian space, is isometric to its quadratic model, the eigenfunctions of the Laplacian in the respective spaces can be identified via the isometry.
Unless otherwise noted the symbol X should denote the coset space G/K, where G is the Lie group SL(n, R) and K = SO(n). The elements of G will be referred to by g, h, . . ., we shall also express the elements in X by x, y, . . ., and by a slight abuse of notation the elements of M will also occasionally be referred to by the symbol g, but always in the form g ij .
The canonical isometry between the quadratic model M and X is given by the map where the star denotes the transpose, hence, the name quadratic model. For fixed (g ij ) ∈ M , the action is an isometry in M , where M is equipped with the metric and where The Iwasawa decomposition is unique. When (5.12) g = nak we define the maps n, A, k by (5.13) g = n(g)A(g)k(g).
We also use the expression log A(g), where log is the matrix logarithm. In case of diagonal matrices (5.14) a = diag(a 1 , . . . , a n ) with positive entries Note that the functions we define in G should also be defined in G/K, i.e., we would want that which is indeed the case. If we used the Iwasawa decomposition G = KAN , then would be valid which would be useful if we considered the right coset space K \ G.
Remark 5.1. (i) The Lie algebra a is a (n-1)-dimensional real algebra, which, as a vector space, is equipped with a natural real, symmetric scalar product, namely, the trace form (ii) Let a * be the dual space of a. Its elements will be denoted by Greek symbols, some of which have a special meaning in the literature. The linear forms are also called additive characters.
(iii) Let λ ∈ a * , then there exists a unique matrix H λ ∈ a such that This definition allows to define a dual trace form in a * by setting for λ, µ ∈ a * (5.25) λ, µ = H λ , H µ .
(iv) The Lie algebra g is a direct sum (5.26) g = n + a + k.
Let E ij , 1 ≤ i < j ≤ n, be the matrices with component 1 in the entry (i, j) and other components zero, then these matrices form a basis of n. For H ∈ a, H = diag(x i ), the Lie bracket in g, which is simply the commutator, applied to H and E ij yields Hence, the E ij are the eigenvectors for the characters α ij ∈ a * defined by Here, E ij is said to be an eigenvector of α ij , if The eigenspace of α ij is one-dimensional. The characters α ij are called the relevant characters, or the (a, n) characters. They are also called the positive restricted roots. The set of these characters will be denoted by Σ + . We define on the other hand, where C i ∈ a has 1 in the first i entries of the diagonal, −i in the (i + 1)-th entry and zero in the other entries. Furthermore, cf. [15, p. 266]. Hence, we conclude Remark 5.3. The eigenfunctions of the Laplacian will depend on the additive characters. The above characters α ij , 1 ≤ i < j ≤ n, will represent the elementary gravitons stemming from the degrees of freedom in choosing the coordinates of a metric tensor. The diagonal elements offer in general additional n degrees of freedom, but in our case, where we consider metrics satisfying only (n − 1) diagonal components can be freely chosen, and we shall choose the first (n − 1) entries, namely, The corresponding additive characters are named α i , 1 ≤ i ≤ n − 1, and are defined by The characters α i , 1 ≤ i ≤ n − 1, and α ij 1 ≤ i < j ≤ n, will represent the Definition 5.4. Let λ ∈ a * , then we define the spherical function where the Haar measure dk is normalized such that K has measure 1, and where G = N AK.
Observe, that i.e., ϕ λ can be lifted to X = G/K. The Weyl chambers are the connected components of the set They consist of diagonal matrices having distinct eigenvalues. The Weyl chamber a + , defined by is called the positive Weyl chamber and the elements H ∈ a + , H = diag(h i ), satisfy Let M resp. M be the centralizer resp. normalizer of a in K, then We are now ready to describe the Fourier theory and Plancherel formula, due to Harish-Chandra for K-bi-invariant functions, cf. [9,10] and [11, p. 48], and by Helgason for arbitrary functions in L 1 (X) and L 2 (X), cf. [12] and [13,Theorem 2.6]. The extension of the Fourier transform to the Schwartz space S (X) and its inversion is due to Eguchi and Okamato [4], this paper is only an announcement without proofs; the proofs are given in [3]. We follow the presentation in Helgason's book [14,Chapter III].
To simplify the expressions in the coming formulas the measures are normalized such that the total measures of compact spaces are 1 and the Lebesgue measure in Euclidean space is normalized such that the Fourier transform and its inverse can be expressed by the simple formulas The Fourier transform for functions f ∈ C ∞ c (X, C) is then defined by for λ ∈ a and b ∈ B, or, if we define The We also denote the Fourier transform by F such that  (λ, b), defined by (5.62), extends to an isometry of L 2 (X) onto L 2 (a * + ×B) (with measure |c(λ)| −2 dλdb on a * + × B). Moreover We shall consider the eigenfunctions e λ,b as tempered distributions of the Schwartz space S (X) and shall use their Fourier transforms as the spatial eigenfunctions of which is a multiplication operator, in the next section.

Fourier quantization
The Fourier theory in X = G/K which we described at the end of the preceding section uses the eigenfunctions as the Fourier kernel. The Fourier quantization in Euclidean space uses the Fourier transform of the Hamilton operator, or only the spatial part of Hamilton operator, which in our case is and the Fourier transforms of the corresponding physically relevant eigenfunctions. If the Hamilton operator is the Euclidean Laplacian in R n , then the spatial eigenfunctions would be Therefore, we consider the eigenfunctions e λ,b as a starting point. As in the Euclidean case the e λ,b are tempered distributions. We first need to extend the Fourier theory to the corresponding Schwartz space S (X) and its dual space S (X), the space of tempered distributions. Let D(G) be the algebra of left invariant differential operators in G and D(G) be the algebra of right invariant differential operators. Furthermore, let (6.4) ϕ 0 = ϕ λ| λ=0 be the spherical function with parameter λ = 0. Then, ϕ 0 satisfies the following estimates where (6.7) d = card Σ + the cardinality of the set of positive restricted roots. Here, we used the following definitions, for g = k 1 ak 2 (a ∈ A, k 1 , k 2 ∈ K), cf. The Fourier transform for f ∈ S (X) is then well defined Integrating over B we obtain (6.11) cf. [17, equ. (1.8)] for the last inequality. Hence, we deduce Lemma 6.2. F (λ) satisfies Proof. The spherical function ϕ λ has this property, cf. [15,Theorem 5.2,p. 100].
Next, we define the Schwartz space S (a * ×B). Note that a * is a Euclidean space, in our case a * = R n−1 , and B = K/M is a compact Riemannian space. Hence, we define the Schwartz space S (a * × B) as follows, cf. [4, Def. 2, p. 240]: Definition 6.3. Let S (a * × B) denote the set of all functions F ∈ C ∞ (a * ×B, C) which satisfy the following condition: for any natural numbers l, m, q (6.13) p l,m,q (F ) = sup where α = (α 1 , . . . , α r ), r = dim a * , is a multi-index (6.14) D α F = D α1 1 · · · D αr r F are partial derivatives with respect to λ ∈ a * , and ∆ B is the Laplacian in B.
The semi-norms p l,m,q define a topology on S (a * × B) with respect to which it is a Fréchet space.  We can now define the Fourier quantization. Let S (X) resp.Ŝ (a * × B) be the dual spaces of tempered distributions, then (6.20) is continuous. Let (F −1 ) * be the dual operator defined by for ω ∈ S (X) and F ∈Ŝ (a * × B). Let Hence, we deduce Lemma 6.6. Let ω ∈ S (X) then we may call (F −1 ) * ω to be the Fourier transform of ω Proof. S(X) can be embedded in S (X) be defining for ω ∈ S (X) ω is obviously an element of S (X) and the embedding is antilinear. On the other hand, in view of the Plancherel formula, we have and thus, because of (6.24), Looking at the Fourier transformed eigenfunctions it is obvious that the dependence on b has to be eliminated, since there is neither a physical nor a mathematical motivation to distinguish between e λ,b and e λ,b . The first ansatz would be to integrate over B, i.e., we would consider the Fourier transform of (6.32) B e λ,b db = ϕ λ which is equal to the Fourier transform of the spherical function ϕ λ , i.e., (6.33)φ λ = δ λ and it would act on the functions These functions satisfy the relation (6.12) which in turn implies if λ was allowed to range in all of a * . Hence, we would have to restrict λ to the positive Weyl chamber a * + , but then, we would not be able to define the eigenfunctions corresponding to the elementary gravitons g ii , 2 ≤ i ≤ n − 1, since the corresponding λ belong to different Weyl chambers, cf. Remark 5.3 on page 21.
Therefore, we pick a distinguished b ∈ B, namely, and only consider the eigenfunctions e λ,b0 with corresponding Fourier transforms Then we can prove: Lemma 6.7. Let δ λ be defined as above, then for any s ∈ W satisfying s · λ = λ, there exists F ∈Ŝ (a * × B) such that and satisfies (6.38).
The Fourier transform of the Laplacian is a multiplication operator similar to the Euclidean case. Lemma 6.8. (i) Let f ∈ S (X), then (ii) Let ω ∈ S (X), then −∆ω is defined as usual Proof. "(i)" The result follows immediately by partial integration. "(ii)" From (6.24) and (6.45) we deduce where λ ∈ a * is fixed, then In Remark 5.3 on page 21 we already identified the additive characters corresponding to the elementary gravitons, namely, the characters (6.52) α ij , 1 ≤ i < j ≤ n and (6.53) We shall now define the corresponding forms in a * with arbitrary energy levels: Definition 6.9. Let λ ∈ R + be arbitrary. Then we consider the characters where we recall that the terms embellished by a tilde refer to the corresponding unit vectors. Then the eigenfunctions representing the elementary gravitons are e λαi,b0 and e λαij ,b0 .

Temporal eigenfunctions
The temporal eigenfunctions w = w(t) have to satisfy the ODE (4.59) on page 16. or equivalently, where µ 0 should be equal to and where (|λ| 2 + |ρ| 2 ) is the eigenvalue of a spatial eigenfunction.
To solve (7.1) we make the ansatz to obtain In order to choose µ such that the term in the braces vanishes, we have to ensure that Now, the estimate ρ 2 = (n − 1) 2 n 12 and (7.8) m = (n − 1)(n + 2) 2 .
One can easily check that In case n ≥ 17 and we obtain the solution while for Remark 7.1. In all three cases (7.5), (7.10) and (7.12) we obtain two real independent solutions, which become unbounded, if the big bang (t = 0) is approached and vanish, if t goes to infinity. The two real solutions contained in (7.3), which generate all possible temporal eigenfunctions, if 3 ≤ n ≤ 16, seem to be the physically relevant solutions.

Conclusions
Quantizing the Hamilton equations instead of the Hamilton constraint we obtained the simple equation Expressing then the fiber metric as in (4.26) on page 13 we can use separation of variables and write the solutions u as products where g ij (x, ξ A ) is a local trivialization of the sub bundle M the fibers of which consists of the metrics g ij with unit determinant, or more precisely, where δ ij is the Euclidean metric. Using Euclidean coordinates in S 0 we can identify the fibers M (x) with the symmetric space (8.5) G/K = SL(n, R n )/SO(n).
The Riemannian metric in G/K is identical to the induced fiber metric of M (x) such that the spatial eigenfunctions of the corresponding (spatial) Laplacians can also be identified. Due to the well-known Fourier theory in G/K we choose the Fourier kernel elements ) v(g ij (x, ξ A )) = f (π −1 (g ij (x, ξ A )), i.e., are the spatial eigenfunctions with eigenvalues (|λ| 2 + |ρ| 2 ). The eigenfunctions corresponding to the elementary gravitons we defined in Definition 6.9 on page 29. They are characterized by special characters α i , 1 ≤ i ≤ n−1, for the diagonal gravitons and α ij , 1 ≤ i < j ≤ n, for the off-diagonal gravitons. The temporal eigenfunctions w(t), which we defined in the previous section, have the properties that they become unbounded if t → 0 and they vanish, together with all derivatives, if t → ∞.
Furthermore, if we consider t < 0, then the functions (8.13)w(t) = w(−t), t < 0, also satisfy the ODE (7.1) on page 30 for t < 0, i.e., they are also temporal eigenfunctions if the light cone in E is flipped. Thus, we conclude Theorem 8.1. The quantum model we derived for gravity can be described by products of spatial and temporal eigenfunctions of corresponding self-adjoint operators with a continuous spectrum. The spatial eigenfunctions can be expressed as Dirac measures in Fourier space and the spatial Laplacian as a multiplication operator. The spatial eigenvalues are strictly positive (8.14) |λ| 2 + |ρ| 2 ≥ |ρ| 2 ≥ |ρ(3)| 2 = 1.
Choosing λ = 0 we have a common ground state with smallest eigenvalue |ρ| 2 which could be considered to be the source of the dark energy. Furthermore, we have a big bang singularity in t = 0. Since the same quantum model is also valid by switching from t > 0 to t < 0, with appropriate changes to the temporal eigenfunctions, one could argue that at the big bang two universes with different time orientations could have been created such that, in view of the CPT theorem, one was filled with matter and the other with anti-matter.