The Quantization of Gravity: Quantization of the Hamilton Equations

Abstract

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form $-\Delta u=0$ in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided $n\ne 4$. 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 $SL(n,{\mathbb{R}}^{})/SO\left(n\right)$. Since one can define a Schwartz space and tempered distributions in $SL(n,{\mathbb{R}}^{})/SO\left(n\right)$ 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.

1. Introduction

General relativity is a Lagrangian theory, i.e., the Einstein equations are derived as the Euler–Lagrange equation of the Einstein–Hilbert functional

$${\int }_N(\overline{R}-2\Lambda ),$$

(1)

where $N=N^{n+1}$, $n\ge 3$, is a globally hyperbolic Lorentzian manifold, $\overline{R}$ is the scalar curvature, and $\Lambda$ is 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

$$M\left(t\right)=\{x^0=t\}.$$

(2)

We may, without loss of generality, assume that the spacetime metric splits

$$d{\overline{s}}^2=-w^2{\left(d{x}^0\right)}^2+g_{ij}(x^0,x)d{x}^{i}d{x}^j,$$

(3)

cf. ([1], Theorem 3.2). Then, the Einstein equations also split into a tangential part

$$G_{ij}+\Lambda g_{ij}=0$$

(4)

and a normal part

$$G_{\alpha \beta }{\nu }^{\alpha }{\nu }^{\beta }-\Lambda =0,$$

(5)

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 [2]. The normal Einstein equations can be expressed by the so-called Hamilton condition

$$\mathcal{H}=0,$$

(6)

where $\mathcal{H}$ 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 type $\widehat{\mathcal{H}}$ and the possible quantum solutions of gravity are supposed to satisfy the so-called Wheeler–DeWitt equation

$$\widehat{\mathcal{H}}u=0$$

(7)

in an appropriate setting, i.e., only the Hamilton condition (6) has been quantized, or equivalently, the normal Einstein equation, while the tangential Einstein equations have been ignored.

In [1], we solved Equation (7) in a fiber bundle E with base space ${\mathcal{S}}_0$,

$${\mathcal{S}}_0=\{x^0=0\}\equiv M\left(0\right),$$

(8)

and fibers $F\left(x\right)$, $x\in {\mathcal{S}}_0$,

$$F\left(x\right)\subset T_x^{0,2}\left({\mathcal{S}}_0\right),$$

(9)

the elements of which are the positive definite symmetric tensors of order two, the Riemannian metrics in ${\mathcal{S}}_0$. The hyperbolic operator $\widehat{\mathcal{H}}$ is then expressed in the form

$$\widehat{\mathcal{H}}=-\Delta -(R-2\Lambda )\phi ,$$

(10)

where $\Delta$ is the Laplacian of the Wheeler–DeWitt metric given in the fibers, R the scalar curvature of the metrics $g_{ij}\left(x\right)\in F\left(x\right)$, and φ is defined by

$${\phi }^2=\frac{det{g}_{ij}}{det{\chi }_{ij}},$$

(11)

where ${\chi }_{ij}$ is a fixed metric in S₀ 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 [3] (see also [4], Chapter 1) and looked at the evolution equations given by the second Hamilton equation. We replaced the left-hand side, a time derivative, with the help of the Poisson brackets. On the right-hand side, we implemented the Hamilton condition, Equation (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. We evaluated the resulting equation for a particular metric that 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. 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 $x\in {\mathcal{S}}_0$. As a result, we obtain the equation

$$-\Delta u=0$$

(12)

in E, where the Laplacian is the Laplacian in Equation (10). The lower order terms of $\widehat{\mathcal{H}}$

$$(R-2\Lambda )\phi$$

(13)

present on both sides of the equation cancel each other. However, Equation (12) is only valid provided $n\ne 4,$ since the resulting equation actually looks like

$$-(\frac{n}{2}-2)\Delta u=0.$$

(14)

This restriction seems to be acceptable, since n is the dimension of the base space ${\mathcal{S}}_0$ which, by general consent, is assumed to be $n=3$. The fibers add additional dimensions to the quantized problem, namely,

$$dimF=\frac{n(n+1)}{2}\equiv m+1.$$

(15)

The fiber metric, the Wheeler–DeWitt metric, which is responsible for the Laplacian in Equation (12), can be expressed in the form

$$d{s}^2=-\frac{16(n-1)}{n}d{t}^2+\phi G_{AB}d{\xi }^{A}d{\xi }^B,$$

(16)

where the coordinate system is

$$\left({\xi }^a\right)=({\xi }^0,{\xi }^A)\equiv (t,{\xi }^A).$$

(17)

The $\left({\xi }^A\right)$, $1\le A\le m$, are coordinates for the hypersurface

$$M\equiv M\left(x\right)=\{\left(g_{ij}\right):t^4=det{g}_{ij}\left(x\right)=1,\forall x\in {\mathcal{S}}_0\}.$$

(18)

We also assume that ${\mathcal{S}}_0={\mathbb{R}}^n$ and that the metric ${\chi }_{ij}$ in Equation (11) is the Euclidean metric ${\delta }_{ij}$. It is well-known that M is a symmetric space

$$M=SL(n,{\mathbb{R}}^{})/SO\left(n\right)\equiv G/K.$$

(19)

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 Equation (12) in the form

$$u=w\left(t\right)v\left({\xi }^A\right),$$

(20)

where v is a spatial eigenfunction of the induced Laplacian of M

$$-{\Delta }_{M}v\equiv -\Delta v={\left(\right|\lambda |}^2+{\left|\rho \right|}^2)v$$

(21)

and w is a temporal eigenfunction satisfying the ODE

$$\ddot{w}+m{t}^{-1}\dot{w}+{\mu }_0{t}^{-2}w=0$$

(22)

with

$${\mu }_0=\frac{16(n-1)}{n}{\left(\right|\lambda |}^2+{\left|\rho \right|}^2).$$

(23)

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 $NAK$ be an Iwasawa decomposition of G and

$$\mathfrak{g}=\mathfrak{n}+\mathfrak{a}+\mathfrak{k}$$

(24)

be the corresponding direct sum of the Lie algebras. Let ${\mathfrak{a}}^{*}$ be the dual space of $\mathfrak{a}$; then, the Fourier kernel is defined by the eigenfunctions

$$e_{\lambda ,b}\left(x\right)=e^{(i\lambda +\rho )logA(x,b)}$$

(25)

with $\lambda \in {\mathfrak{a}}^{*}$, $x=gK\in G/K$, and $b\in B$, where B is the Furstenberg boundary (see Section 5 and Section 6 for detailed definitions and references). We then pick a particular $b_0\in B$ and use $e_{\lambda ,b_0}$ as eigenfunctions of $-\Delta$

$$-\Delta e_{\lambda ,b_0}={\left(\right|\lambda |}^2+{\left|\rho \right|}^2)e_{\lambda ,b_0}.$$

(26)

The Fourier transform of $e_{\lambda ,b_0}$ is

$${\widehat{e}}_{\lambda ,b_0}={\delta }_{\lambda }\otimes {\delta }_{b_0}$$

(27)

and of $-\Delta f$

$$\mathcal{F}(-\Delta f)={\left(\right|\lambda |}^2+{\left|\rho \right|}^2)\widehat{f},\lambda \in {\mathfrak{a}}^{*},f\in \mathcal{S}(G/K).$$

(28)

The elementary gravitons correspond to special characters in ${\mathfrak{a}}^{*}$, namely,

$${\alpha }_{ij},1\le i<j\le n,$$

(29)

for the off-diagonal gravitons and

$${\alpha }_i,1\le i\le n-1$$

(30)

for the diagonal gravitons. Note that only $(n-1)$ diagonal elements $g_{ii}$ can be freely chosen because of the condition (18).

To define the temporal eigenfunctions, we shall here only consider the case $3\le n\le 16$; then, all temporal eigenfunctions are generated by the two real eigenfunctions contained in

$$w\left(t\right)=t^{-\frac{m-1}{2}}{e}^{i\mu logt},$$

(31)

where $\mu >0$ is chosen appropriately. These eigenfunctions become unbounded if the big bang (t = 0) is approached, and they vanish if t goes to infinity.

2. 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 $\left({\overline{g}}_{\alpha \beta }\right),\left({\overline{R}}_{\alpha \beta \gamma \delta }\right)$, etc., and those in M by $\left(g_{ij}\right),\left(R_{ijkl}\right)$, 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 $\left(x^{\alpha }\right)$ resp. $\left({\xi }^i\right)$. Covariant differentiation will simply be indicated by indices; only in the case of possible ambiguity will they be preceded by a semicolon, i.e., for a function u in N, $\left(u_{\alpha }\right)$ will be the gradient, and $\left(u_{\alpha \beta }\right)$ the Hessian, but, e.g., the covariant derivative of the curvature tensor will be abbreviated by ${\overline{R}}_{\alpha \beta \gamma \delta ;ϵ}$. We also point out that

$${\overline{R}}_{\alpha \beta \gamma \delta ;i}={\overline{R}}_{\alpha \beta \gamma \delta ;ϵ}{x}_i^{ϵ}$$

(32)

with obvious generalizations to other quantities.

Let M be a spacelike hypersurface, i.e., the induced metric is Riemannian, with a differentiable normal $\nu$ that is timelike.

In local coordinates, $\left(x^{\alpha }\right)$ and $\left({\xi }^i\right)$, the geometric quantities of the spacelike hypersurface M are connected through the following equations

$$x_{ij}^{\alpha }=h_{ij}{\nu }^{\alpha }$$

(33)

the so-called Gauß formula. Here, and also in the sequel, a covariant derivative is always a full tensor, i.e.,

$$x_{ij}^{\alpha }=x_{,ij}^{\alpha }-{\Gamma }_{ij}^k{x}_k^{\alpha }+{\overline{\Gamma }}_{\beta \gamma }^{\alpha }{x}_i^{\beta }{x}_j^{\gamma }.$$

(34)

The comma indicates ordinary partial derivatives.

In this implicit definition, the second fundamental form $\left(h_{ij}\right)$ is taken with respect to $\nu$.

The second equation is the Weingarten equation

$${\nu }_i^{\alpha }=h_i^k{x}_k^{\alpha },$$

(35)

where we remember that ${\nu }_i^{\alpha }$ is a full tensor.

Finally, we have the Codazzi equation

$$h_{ij;k}-h_{ik;j}={\overline{R}}_{\alpha \beta \gamma \delta }{\nu }^{\alpha }{x}_i^{\beta }{x}_j^{\gamma }{x}_k^{\delta }$$

(36)

and the Gauß equation

$$R_{ijkl}=-\{h_{ik}{h}_{jl}-h_{il}{h}_{jk}\}+{\overline{R}}_{\alpha \beta \gamma \delta }{x}_i^{\alpha }{x}_j^{\beta }{x}_k^{\gamma }{x}_l^{\delta }.$$

(37)

Now, let us assume that N is a globally hyperbolic Lorentzian manifold with a Cauchy surface. N is then a topological product $I\times {\mathcal{S}}_0$, where I is an open interval, ${\mathcal{S}}_0$ is a Riemannian manifold, and there exists a Gaussian coordinate system $\left(x^{\alpha }\right)$, such that the metric in N has the form

$$d{\overline{s}}_N^2=e^{2\psi }\{-{d{x}^0}^2+{\sigma }_{ij}(x^0,x)d{x}^{i}d{x}^j\},$$

(38)

where ${\sigma }_{ij}$ is a Riemannian metric, $\psi$ is a function on N, and x is an abbreviation for the spacelike components $\left(x^i\right)$. 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 $\left({\xi }^{\alpha }\right)=(1,0,\dots ,0)$ is future-directed, as is its covariant version $\left({\xi }_{\alpha }\right)=e^{2\psi }(-1,0,\dots ,0)$.

Let $M=graphu{}_{{|}_{{\mathcal{S}}_0}}$ be a spacelike hypersurface

$$M=\{(x^0,x):x^0=u\left(x\right),x\in {\mathcal{S}}_0\};$$

(39)

then, the induced metric has the form

$$g_{ij}=e^{2\psi }\{-u_i{u}_j+{\sigma }_{ij}\}$$

(40)

where ${\sigma }_{ij}$ is evaluated at $(u,x)$, and its inverse $\left(g^{ij}\right)={\left(g_{ij}\right)}^{-1}$ can be expressed as

$$g^{ij}=e^{-2\psi }\{{\sigma }^{ij}+\frac{u^i}{v}\frac{u^j}{v}\},$$

(41)

where $\left({\sigma }^{ij}\right)={\left({\sigma }_{ij}\right)}^{-1}$ and

$$\begin{array}{cc}\hfill u^i& ={\sigma }^{ij}{u}_j\hfill \\ \hfill v^2& =1-{\sigma }^{ij}{u}_i{u}_j\equiv 1-{\left|Du\right|}^2.\hfill \end{array}$$

(42)

Hence, $graphu$ is spacelike if and only if $\left|Du\right|<1$.

The covariant form of a normal vector of a graph looks like

$$\left({\nu }_{\alpha }\right)=\pm v^{-1}{e}^{\psi }(1,-u_i).$$

(43)

and the contravariant version is

$$\left({\nu }^{\alpha }\right)=\mp v^{-1}{e}^{-\psi }(1,u^i).$$

(44)

Thus, we have

Remark 1.

Let M be spacelike graph in a future-oriented coordinate system. Then, the contravariant future-directed normal vector has the form

$$\left({\nu }^{\alpha }\right)=v^{-1}{e}^{-\psi }(1,u^i)$$

(45)

and the past directed

$$\left({\nu }^{\alpha }\right)=-v^{-1}{e}^{-\psi }(1,u^i).$$

(46)

In the Gauß Formula (33), 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 $\alpha =0$ in (33) and obtain in view of (46)

$$e^{-\psi }{v}^{-1}{h}_{ij}=-u_{ij}-{\overline{\Gamma }}_{00}^0{u}_i{u}_j-{\overline{\Gamma }}_{0j}^0{u}_i-{\overline{\Gamma }}_{0i}^0{u}_j-{\overline{\Gamma }}_{ij}^0.$$

(47)

Here, the covariant derivatives are taken with respect to the induced metric of M, and

$$-{\overline{\Gamma }}_{ij}^0=e^{-\psi }{\overline{h}}_{ij},$$

(48)

where $\left({\overline{h}}_{ij}\right)$ is the second fundamental form of the hypersurfaces $\{x^0=\mathrm{const}\}$.

An easy calculation shows

$${\overline{h}}_{ij}{e}^{-\psi }=-\frac{1}{2}{\dot{\sigma }}_{ij}-\dot{\psi }{\sigma }_{ij},$$

(49)

where the dot indicates differentiation with respect to $x^0$.

3. The Hamiltonian Approach to General Relativity

The Einstein equations with a cosmological constant $\Lambda$ in a Lorentzian manifold $N=N^{n=1}$, $n\ge 3$, with metric ${\overline{g}}_{\alpha \beta }$, $0\le \alpha , \beta \le n$, are the Euler–Lagrange equations of the functional

$$J={\int }_N(\overline{R}-2\Lambda ),$$

(50)

where $\overline{R}$ is the scalar curvature of the metric and where we omitted the density $\sqrt{|\overline{g}|}$. The Euler–Lagrange equations are

$$G_{\alpha \beta }+\Lambda {\overline{g}}_{\alpha \beta }=0,$$

(51)

where $G_{\alpha \beta }$ is the Einstein tensor. We proved in ([1], Theorem 3.2) (see also [4], Theorem 1.3.2) that it suffices to consider only metrics that split, i.e., metrics that are of the form

$$d{\overline{s}}^2=-w^2{\left(d{x}^0\right)}^2+g_{ij}(x^0,x)d{x}^{i}d{x}^j,$$

(52)

where $\left(x^i\right)$ are spatial coordinates, $x^0$ is a time coordinate, $g_{ij}$ are Riemannian metrics defined on the slices

$$M\left(t\right)=\{x^0=t\},t\in (a,b),$$

(53)

and

$$0<w=w(x^0,x)$$

(54)

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 [2], the functional in (50) can be expressed in the form

$$J={\int }_a^b{\int }_{\Omega }{\left\{\right|A|}^2-H^2+R-2\Lambda \}w\sqrt{g},$$

(55)

cf. ([4], Equation (1.3.37)), where

$${\left|A\right|}^2=h^{ij}{h}_{ij}$$

(56)

is the square of the second fundamental form of the slices $M\left(t\right)$

$$h_{ij}=-\frac{1}{2}{w}^{-1}{\dot{g}}_{ij},$$

(57)

$H^2$ is the square of the mean curvature

$$H=g^{ij}{h}_{ij},$$

(58)

R the scalar curvature of the slices $M\left(t\right)$, the interval $(a,b)$ is compactly contained in

$$I=x^0\left(N\right),$$

(59)

and $\Omega$ is a bounded open subset of the fixed slice

$${\mathcal{S}}_0\equiv M\left(0\right),$$

(60)

where we assume

$$0\in I.$$

(61)

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

$$N=I\times {\mathcal{S}}_0.$$

(62)

Let $F=F\left(h_{ij}\right)$ be the scalar curvature operator

$$F=\frac{1}{2}(H^2-{\left|A\right|}^2)$$

(63)

and let

$$F^{ij,kl}=g^{ij}{g}^{kl}-\frac{1}{2}\{g^{ik}{g}^{jl}+g^{il}{g}^{jk}\}$$

(64)

be its Hessian, then

$$F^{ij,kl}{h}_{ij}{h}_{kl}=2F=H^2-{\left|A\right|}^2$$

(65)

and

$$F^{ij}=F^{ij,kl}{h}_{kl}=H{g}^{ij}-h^{ij}.$$

(66)

In physics,

$$G^{ij,kl}=-F^{ij,kl}$$

(67)

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 (57) and (65), J can be expressed in the form

$$J={\int }_a^b{\int }_{\Omega }\{\frac{1}{4}{G}^{ij,kl}{\dot{g}}_{ij}{\dot{g}}_{kl}{w}^{-2}+(R-2\Lambda )\}w\sqrt{g}.$$

(68)

The Lagrangian density $\mathcal{L}$ is a regular Lagrangian with respect to the variables $g_{ij}$. Define the conjugate momenta

$$\begin{array}{cc}\hfill {\pi }^{ij}=\frac{\partial \mathcal{L}}{\partial {\dot{g}}_{ij}}& =\frac{1}{2}{G}^{ij,kl}{\dot{g}}_{kl}{w}^{-1}\sqrt{g}\hfill \\ \hfill & =-G^{ij,kl}{h}_{kl}\sqrt{g}\hfill \end{array}$$

(69)

and the Hamiltonian density

$$\begin{array}{cc}\hfill \mathcal{H}& ={\pi }^{ij}{\dot{g}}_{ij}-\mathcal{L}\hfill \\ \hfill & =\frac{1}{\sqrt{g}}w{G}_{ij,kl}{\pi }^{ij}{\pi }^{kl}-(R-2\Lambda )w\sqrt{g},\hfill \end{array}$$

(70)

where

$$G_{ij,kl}=\frac{1}{2}\{g_{ik}{g}_{jk}+g_{il}{g}_{jk}\}-\frac{1}{n-1}{g}_{ij}{g}_{kl}$$

(71)

is the inverse of $G^{ij,kl}$.

Since the Lagrangian is regular with respect to the variables $g_{ij}$, the tangential Einstein equations

$$G_{ij}+\Lambda g_{ij}=0$$

(72)

are equivalent to the Hamilton equations

$${\dot{g}}_{ij}=\frac{\delta \mathcal{H}}{\delta {\pi }^{ij}}$$

(73)

and

$${\dot{\pi }}^{ij}=-\frac{\delta \mathcal{H}}{\delta g_{ij}},$$

(74)

where the differentials on the right-hand side of these equations are variational or functional derivatives.

The mixed Einstein equations vanish

$$G_{0j}+\Lambda {\overline{g}}_{0j}=0,1\le j\le n,$$

(75)

and the normal equation

$$G_{\alpha \beta }{\nu }^{\alpha }{\nu }^{\beta }-\Lambda =0$$

(76)

is equivalent to

$${\left|A\right|}^2-H^2=R-2\Lambda ,$$

(77)

cf. ([5], Equation (1.1.43)), which in turn is equivalent to

$$\mathcal{H}=0,$$

(78)

which is also known as the Hamilton condition.

We define the Poisson brackets

$$\{u,v\}=\frac{\delta u}{\delta g_{kl}}\frac{\delta v}{\delta {\pi }^{kl}}-\frac{\delta u}{\delta {\pi }^{kl}}\frac{\delta v}{\delta g_{kl}}$$

(79)

and obtain

$$\{g_{ij},{\pi }^{kl}\}={\delta }_{ij}^{kl},$$

(80)

where

$${\delta }_{ij}^{kl}=\frac{1}{2}\{{\delta }_i^k{\delta }_j^l+{\delta }_i^l{\delta }_j^k\}.$$

(81)

Then, the second Hamilton equation can also be expressed as

$${\dot{\pi }}^{ij}=\{{\pi }^{ij},\mathcal{H}\}.$$

(82)

In the next section, we want to quantize the Hamilton equations or, more precisely,

$$\begin{array}{cc}\hfill g_{ij}\{{\pi }^{ij},\mathcal{H}\}& =-g_{ij}\frac{\delta \mathcal{H}}{\delta g_{ij}}\hfill \\ \hfill & =(\frac{n}{2}-2){\left(\right|A|}^2-H^2)w\sqrt{g}+\frac{n}{2}(R-2\Lambda )w\sqrt{g}\hfill \\ \hfill & -Rw\sqrt{g}-(n-1)\tilde{\Delta }w\sqrt{g},\hfill \end{array}$$

(83)

cf. ([4], Equations (1.3.64) and (1.3.65)), where $\tilde{\Delta }$ is the Laplacian with respect to the metric $g_{ij}(t,·)$.

4. The Quantization

For the quantization of the Hamiltonian setting, we first replace all densities by tensors by choosing a fixed Riemannian metric in ${\mathcal{S}}_0$

$$\chi =\left({\chi }_{ij}\left(x\right)\right),$$

(84)

and, for a given metric $g=\left(g_{ij}(t,x)\right)$, we define

$$\phi =\phi (x,g_{ij})={\left(\frac{det{g}_{ij}}{det{\chi }_{ij}}\right)}^{\frac{1}{2}}$$

(85)

such that the Einstein–Hilbert functional J in (68) on page 8 can be written in the form

$$J={\int }_a^b{\int }_{\Omega }\{\frac{1}{4}{G}^{ij,kl}{\dot{g}}_{ij}{\dot{g}}_{kl}{w}^{-2}+(R-2\Lambda )\}w\phi \sqrt{\chi }.$$

(86)

The Hamilton density $\mathcal{H}$ is then replaced by the function

$$H=\{{\phi }^{-1}{G}_{ij,kl}{\pi }^{ij}{\pi }^{kl}-(R-2\Lambda )\phi \}w,$$

(87)

where now

$${\pi }^{ij}=-\phi G^{ij,kl}{h}_{kl}$$

(88)

and

$$h_{ij}=-{\phi }^{-1}{G}_{ij,kl}{\pi }^{kl}.$$

(89)

The effective Hamiltonian is, of course,

$$w^{-1}H.$$

(90)

Fortunately, we can, at least locally, assume

$$w=1$$

(91)

by choosing an appropriate coordinate system: Let $(t_0,x_0)\in N$ be an arbitrary point, then consider the Cauchy hypersurface

$$M\left(t_0\right)=\left\{t_0\right\}\times {\mathcal{S}}_0$$

(92)

and look at a tubular neighborhood of $M\left(t_0\right)$, i.e., we define new coordinates $(t,x^i)$, where $\left(x^i\right)$ are coordinate for ${\mathcal{S}}_0$ near $x_0$ and t is the signed Lorentzian distance to $M\left(t_0\right)$ such that the points

$$(0,x^i)\in M\left(t_0\right).$$

(93)

The Lorentzian metric of the ambient space then has the form

$$d{\overline{s}}^2=-d{t}^2+g_{ij}d{x}^{i}d{x}^j.$$

(94)

Secondly, we use the same model as in ([1], Section 3): The Riemannian metrics $g_{ij}(t,·)$ are elements of the bundle $T^{0,2}\left({\mathcal{S}}_0\right)$. Denote by E the fiber bundle with base ${\mathcal{S}}_0$ where the fibers consist of the Riemannian metrics $\left(g_{ij}\right)$. We shall consider each fiber to be a Lorentzian manifold equipped with the DeWitt metric. Each fiber F has dimension

$$dimF=\frac{n(n+1)}{2}\equiv m+1.$$

(95)

Let $\left({\xi }^a\right)$, $0\le a\le m$, be coordinates for a local trivialization such that

$$g_{ij}(x,{\xi }^a)$$

(96)

is a local embedding. The DeWitt metric is then expressed as

$$G_{ab}=G^{ij,kl}{g}_{ij,a}{g}_{kl,b},$$

(97)

where a comma indicates partial differentiation. In the new coordinate system, the curves

$$t\to g_{ij}(t,x)$$

(98)

can be written in the form

$$t\to {\xi }^a(t,x)$$

(99)

and we infer

$$G^{ij,kl}{\dot{g}}_{ij}{\dot{g}}_{kl}=G_{ab}{\dot{\xi }}^a{\dot{\xi }}^b.$$

(100)

Hence, we can express (55) as

$$J={\int }_a^b{\int }_{\Omega }\{\frac{1}{4}{G}_{ab}{\dot{\xi }}^a{\dot{\xi }}^b\phi +(R-2\Lambda )\phi \},$$

(101)

where we now refrain from writing down the density $\sqrt{\chi }$ explicitly, since it does not depend on $\left(g_{ij}\right)$ 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 (101) by L, we define

$${\pi }_a=\frac{\partial L}{\partial {\dot{\xi }}^a}=\phi G_{ab}\frac{1}{2}{\dot{\xi }}^b$$

(102)

and we obtain for the Hamiltonian function H

$$\begin{array}{cc}\hfill H& ={\dot{\xi }}^a\frac{\partial L}{\partial {\dot{\xi }}^a}-L\hfill \\ \hfill & =\phi G_{ab}\left(\frac{1}{2}{\dot{\xi }}^a\right)\left(\frac{1}{2}{\dot{\xi }}^b\right)-(R-2\Lambda )\phi \hfill \\ \hfill & \hfill ={\phi }^{-1}{G}^{ab}{\pi }_a{\pi }_b-(R-2\Lambda )\phi \hfill \end{array}$$

(103)

where $G^{ab}$ is the inverse metric.

The fibers equipped with the metric

$$\left(\phi G_{ab}\right)$$

(104)

are then globally hyperbolic Lorentzian manifolds as we proved in ([4], Theorem 1.4.2). In the fibers, we can introduce new coordinates $\left({\xi }^a\right)=({\xi }^0,{\xi }^A)$, $0\le a\le m$, and $1\le A\le m$, such that

$$\tau \equiv {\xi }^0=log\phi$$

(105)

and $\left({\xi }^A\right)$ are coordinates for the hypersurface

$$M=\{\phi =1\}=\{\tau =0\}.$$

(106)

The Lorentzian metric in the fibers can then be expressed in the form

$$d{s}^2=-\frac{4(n-1)}{n}\phi d{\tau }^2+\phi G_{AB}d{\xi }^{A}d{\xi }^B,$$

(107)

cf. ([4], Equation (1.4.28)), where we note that there is a misprint in that reference, namely, the spatial part of the metric has an additional factor $\frac{4(n-1)}{n}$ that should be omitted. Defining a new time variable ${\xi }^0=t$ by setting

$$\phi =t^2,$$

(108)

we infer

$$d{s}^2=-\frac{16(n-1)}{n}d{t}^2+\phi G_{AB}d{\xi }^{A}d{\xi }^B.$$

(109)

The new metric $G_{AB}$ is independent of t. When we work in a local trivialization of the bundle E, the coordinates ${\xi }^A$ are independent of x as well as the time coordinate t, cf. ([4], Lemma 1.4.4).

We can now quantize the Hamiltonian setting using the original variables $g_{ij}$ and ${\pi }^{ij}$. We consider the bundle E equipped with the metric (107), or equivalently,

$$\left(\phi G^{ij,kl}\right),$$

(110)

which is the covariant form, in the fibers and with the Riemannian metric $\chi$ in ${\mathcal{S}}_0$. Furthermore, let

$$C_c^{\infty }\left(E\right)$$

(111)

be the space of real valued smooth functions with compact support in E.

In the quantization process, where we choose $\hslash =1$, the variables $g_{ij}$ and ${\pi }^{ij}$ are then replaced by operators ${\widehat{g}}_{ij}$ and ${\widehat{\pi }}^{ij}$ acting in $C_c^{\infty }\left(E\right)$, satisfying the commutation relations

$$[{\widehat{g}}_{ij},{\widehat{\pi }}^{kl}]=i{\delta }_{ij}^{kl},$$

(112)

while all the other commutators vanish. These operators are realized by defining ${\widehat{g}}_{ij}$ to be the multiplication operator

$${\widehat{g}}_{ij}u=g_{ij}u$$

(113)

and ${\widehat{\pi }}^{ij}$ to be the functional differentiation

$${\widehat{\pi }}^{ij}=\frac{1}{i}\frac{\delta }{\delta g_{ij}},$$

(114)

i.e., if $u\in C_c^{\infty }\left(E\right)$, then

$$\frac{\delta u}{\delta g_{ij}}$$

(115)

is the Euler–Lagrange operator of the functional

$${\int }_{{\mathcal{S}}_0}u\sqrt{\chi }\equiv {\int }_{{\mathcal{S}}_0}u.$$

(116)

Hence, if u only depends on $(x,g_{ij})$ and not on derivatives of the metric, then

$$\frac{\delta u}{\delta g_{ij}}=\frac{\partial u}{\partial g_{ij}}.$$

(117)

Therefore, the transformed Hamiltonian $\widehat{H}$ can be looked at as the hyperbolic differential operator

$$\widehat{H}=-\Delta -(R-2\Lambda )\phi ,$$

(118)

where $\Delta$ is the Laplacian of the metric in (110) acting on functions

$$u=u(x,g_{ij}).$$

(119)

We used this approach in [1] to transform the Hamilton constraint to the Wheeler–DeWitt equation

$$\widehat{H}u=0\mathrm{in}E,$$

(120)

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.

Following Dirac, the Poisson brackets in (82) on page 9 are replaced by $\frac{1}{i}$ times the commutators in the quantization process since $\hslash =1$, i.e., we obtain

$$\{{\pi }^{ij},H\}\to i[\widehat{H},{\widehat{\pi }}^{ij}].$$

(121)

Dropping the hats in the following to improve the readability, Equation (83) is then transformed to

$$\begin{array}{c}\hfill i{g}_{ij}[H,{\pi }^{ij}]=(\frac{n}{2}-2){\left(\right|A|}^2-H^2)\phi +\frac{n}{2}(R-2\Lambda )\phi -R\phi ,\end{array}$$

(122)

where we note that $w=1$ now. We have

$$\begin{array}{cc}\hfill i[H,{\pi }^{ij}]& =[H,\frac{\delta }{\delta g_{ij}}]\hfill \\ \hfill & =[-\Delta ,\frac{\delta }{\delta g_{ij}}]-[(R-2\Lambda )\phi ,\frac{\delta }{\delta g_{ij}}],\hfill \end{array}$$

(123)

cf. (118). We apply both sides to functions $u\in C_c^{\infty }\left(E\right)$, where we additionally require

$$u=u\left(g_{ij}\right),$$

(124)

i.e., u does not explicitly depend on $x\in {\mathcal{S}}_0$. Hence, we deduce

$$[-\Delta ,\frac{\delta }{\delta g_{ij}}]u=[-\Delta ,\frac{\partial }{\partial g_{ij}}]u=-R_{\phantom{ij},kl}^{ij}{u}^{kl},$$

(125)

because of the Ricci identities, where

$$R_{\phantom{ij},kl}^{ij}$$

(126)

is the Ricci tensor of the fiber metric (110) and

$$u^{kl}=\frac{\partial u}{\partial g_{kl}}$$

(127)

is the gradient of u.

For the second commutator on the right-hand side of (123), we obtain

$$\begin{array}{c}\hfill -[(R-2\Lambda )\phi ,\frac{\delta }{\delta g_{ij}}]u=-(R-2\Lambda )\phi \frac{\partial u}{\partial g_{ij}}+\frac{\delta }{\delta g_{ij}}\left\{(R-2\Lambda )u\phi \right\},\end{array}$$

(128)

where the last term is the Euler–Lagrange operator of the functional

$$\begin{array}{cc}\hfill {\int }_{{\mathcal{S}}_0}(R-2\Lambda )u\phi \equiv & {\int }_{{\mathcal{S}}_0}(R-2\Lambda )u\phi \sqrt{\chi }\hfill \\ \hfill =& {\int }_{{\mathcal{S}}_0}(R-2\Lambda )u\sqrt{g}\hfill \end{array}$$

(129)

with respect to the variable $g_{ij}$, since the scalar curvature R depends on the derivatives of $g_{ij}$. In view of ([4], Equation (1.4.84)), we have

$$\begin{array}{cc}\hfill \frac{\delta }{\delta g_{ij}}\left\{(R-2\Lambda )u\phi \right\}=& \frac{1}{2}(R-2\Lambda )g^{ij}u\phi -R^{ij}u\phi \hfill \\ \hfill +& \phi \{u_{;}^{\phantom{;}ij}-\tilde{\Delta }u{g}^{ij}\}+(R-2\Lambda )\phi \frac{\partial u}{\partial g_{ij}},\hfill \end{array}$$

(130)

where the semicolon indicates covariant differentiation in ${\mathcal{S}}_0$ with respect to the metric $g_{ij}$, $\tilde{\Delta }$ is the corresponding Laplacian. We also note that

$$\begin{array}{cc}\hfill D_{k}u& =\frac{\partial u}{\partial x^k}+\frac{\partial u}{\partial g_{ij}}\frac{\partial g_{ij}}{\partial x^k}\hfill \\ \hfill & =\frac{\partial u}{\partial x^k}=0.\hfill \end{array}$$

(131)

in Riemannian normal coordinates. Hence, we conclude that the operator on the left hand-side of Equation (122) applied to u is equal to

$$\frac{n}{2}(R-2\Lambda )\phi u-R\phi u$$

(132)

in E, since

$$g_{ij}{R}_{\phantom{ij},kl}^{ij}=0,$$

(133)

cf. ([4], Equation (1.4.89)). On the other hand, applying the right-hand side of (122) to u, we obtain

$$-(\frac{n}{2}-2)\Delta u+\frac{n}{2}(R-2\Lambda )\phi u-R\phi u,$$

(134)

where the Laplacian is the Laplacian in the fibers, since

$${\left(\right|A|}^2-H^2)\phi ={\phi }^{-1}{G}_{ij,kl}{\pi }^{ij}{\pi }^{kl}\to -\Delta .$$

(135)

Thus, we conclude

$$-(\frac{n}{2}-2)\Delta u=0$$

(136)

in E, and we have proved the following theorem:

Theorem 1.

The quantization of Equation (83) on page 9 leads to the hyperbolic equation

$$-\Delta u=0$$

(137)

in E provided $n\ne 4$ and $u\in C_c^{\infty }\left(E\right)$ only depends on the fiber elements $g_{ij}$.

To solve Equation (137), we first choose the Gaussian coordinate system $\left({\xi }^a\right)=(t,{\xi }^A)$ such that the metric has form as in (109). Then, the hyperbolic equation can be expressed as

$$\frac{n}{16(n-1)}{t}^{-m}\frac{\partial }{\partial t}\left(t^m\frac{\partial u}{\partial t}\right)-t^{-2}\overline{\Delta }u=0,$$

(138)

where $\overline{\Delta }$ is the Laplacian of the hypersurface

$$M=\{t=1\}.$$

(139)

We shall try to use separation of variables by considering solutions u that are products

$$u(t,{\xi }^A)=w\left(t\right)v\left({\xi }^A\right),$$

(140)

where v is a spatial eigenfunction, or eigendistribution, of the Laplacian $\overline{\Delta }$

$$-\overline{\Delta }v=\lambda v$$

(141)

and w a temporal eigenfunction satisfying the ODE

$$\frac{n}{16(n-1)}{t}^{-m}\frac{\partial }{\partial t}\left(t^m\frac{\partial w}{\partial t}\right)+\lambda t^{-2}w=0,$$

(142)

which can be looked at as an implicit eigenvalue equation. The function u in (140) will then be a solution of (137).

In the next sections, we shall determine spatial and temporal eigendistributions by assuming

$${\mathcal{S}}_0={\mathbb{R}}^n$$

(143)

equipped with the Euclidean metric. The dimension n is then merely supposed to satisfy $n\ge 3$, though, of course, Equation (137) additionally requires $n\ne 4$.

5. Spatial Eigenfunctions in $\mathit{M}$

The hypersurface

$$M=\{\phi =1\}$$

(144)

can be considered to be a sub-bundle of E, where each fiber $M\left(x\right)$ is a hypersurface in the fiber $F\left(x\right)$ 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 $x\in {\mathcal{S}}_0$. Furthermore, we specify the metric ${\chi }_{ij}\in T^{0,2}\left({\mathcal{S}}_0\right)$, which we used to define φ, to be equal to the Euclidean metric such that in Euclidean coordinates,

$${\phi }^2=\frac{det{g}_{ij}}{det{\delta }_{ij}}=det{g}_{ij}.$$

(145)

Then, it is well known that each $M\left(x\right)$ with the induced metric $\left(G_{AB}\right)$ is a symmetric space, namely, it is isometric to the coset space

$$G/K=SL(n,{\mathbb{R}}^{})/SO\left(n\right),$$

(146)

cf. ([6], equ.(5.17), p. 1123) and ([7], p. 3). The eigenfunctions in symmetric spaces, and especially of the coset space in (146), are well known; they are the so-called spherical functions. One can also define a Fourier transformation for functions in $L^2(G/K)$ and prove a Plancherel formula, similar to the Euclidean case, cf. ([8], Chapter III). Similar to the Euclidean case, we shall use the Fourier kernel to define the eigenfunctions, or eigendistributions, since the spherical functions, because of their symmetry properties, are not specific enough to represent the elementary gravitons corresponding to the diagonal metric variables $g_{ii}$, $1\le i\le n-1$. Recall that from the n diagonal coefficients of a metric, only $n-1$ are independent because of the assumption

$$det{g}_{ij}=1,$$

(147)

which has to be satisfied by the elements of M.

However, 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 (146), 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 ${\mathbb{R}}^n$ with determinant equal to 1, i.e., the quadratic model of $G/K$ is identical to an arbitrary fiber $M\left(x\right)$ 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 $SL(n,{\mathbb{R}}^{})$ and $K=SO\left(n\right)$. The elements of G will be referred to by $g,h,\dots$; we shall also express the elements in X by $x,y,\dots$, 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

$$\begin{array}{cc}\hfill \pi :G/K& \to M\hfill \\ \hfill x=gk\in gK& \to \pi \left(x\right)=gk{\left(gk\right)}^{*}=g{g}^{*},\hfill \end{array}$$

(148)

where the star denotes the transpose, hence the name quadratic model. For fixed $\left(g_{ij}\right)\in M$, the action

$$\left[g\right]\left(g_{ij}\right)=g\left(g_{ij}\right)g^{*},g\in G,$$

(149)

is an isometry in M, where M is equipped with the metric

$${\tilde{G}}^{ij,kl}=\frac{1}{2}\{g^{ik}{g}^{jl}+g^{il}{g}^{jk}\},$$

(150)

and where

$$\left(g^{ij}\right)={\left(g_{ij}\right)}^{-1},$$

(151)

cf. ([4], Equation (1.4.46), p. 22).

Let

$$G=NAK$$

(152)

be an Iwasawa decomposition of G, where N is the subgroup of unit upper triangle matrices, and let A be the abelian subgroup of diagonal matrices with strictly positive diagonal components and $K=SO\left(n\right)$. The corresponding Lie algebras are denoted by

$$\mathfrak{g},\mathfrak{n},\mathfrak{a}\mathrm{and}\mathfrak{k}.$$

(153)

Here,

$$\begin{array}{cc}\hfill \mathfrak{g}& = \mathrm{real} \mathrm{matrices} \mathrm{with} \mathrm{zero} \mathrm{trace}\hfill \\ \hfill \mathfrak{n}& = \mathrm{subspace} \mathrm{of} \mathrm{strictly} \mathrm{upper} \mathrm{triangle} \mathrm{matrices} \mathrm{with} \mathrm{zero} \mathrm{diagonal}\hfill \\ \hfill \mathfrak{a}& = \mathrm{subspace} \mathrm{of} \mathrm{diagonal} \mathrm{matricesc} \mathrm{with} \mathrm{zero} \mathrm{trace}\hfill \\ \hfill \mathfrak{k}& = \mathrm{subspace} \mathrm{of} \mathrm{skew-symmetric} \mathrm{matrices}.\hfill \end{array}$$

(154)

The Iwasawa decomposition is unique. When

$$g=nak,$$

(155)

we define the maps $n,A,k$ by

$$g=n\left(g\right)A\left(g\right)k\left(g\right).$$

(156)

We also use the expression $logA\left(g\right)$, where log is the matrix logarithm. In case of diagonal matrices

$$a=diag(a_1,\dots ,a_n)$$

(157)

with positive entries

$$loga=diag(log{a}_i),$$

(158)

hence

$$A\left(g\right)=e^{logA\left(g\right)}.$$

(159)

Helgason uses the symbol $A\left(g\right)$ if G decomposed as in (152) but uses the symbol $H\left(g\right)$ if

$$G=KAN,$$

(160)

which can be obtained by applying the isomorphism

$$g\to g^{-1}.$$

(161)

Because of the uniqueness

$$H\left(g\right)=A{\left(g\right)}^{-1},$$

(162)

hence

$$logH\left(g\right)=-logA\left(g\right),$$

(163)

cf. ([8], Equations (2) and (3), p. 198).

Note that the functions we define in G should also be defined in $G/K$, i.e., we would want that

$$A\left(g\right)=A\left(gK\right),$$

(164)

which is indeed the case. If we used the Iwasawa decomposition $G=KAN$, then

$$H\left(g\right)=H\left(Kg\right)$$

(165)

would be valid, which would be useful if we considered the right coset space K\G.

Remark 2.

(i) The Lie algebra $\mathfrak{a}$ 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

$$\langle H_1,H_2\rangle =tr\left(H_1{H}_2\right),H_i\in \mathfrak{a}.$$

(166)

(ii) Let ${\mathfrak{a}}^{*}$ be the dual space of $\mathfrak{a}$. 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 $\lambda \in {\mathfrak{a}}^{*}$; then, there exists a unique matrix $H_{\lambda }\in \mathfrak{a}$ such that

$$\lambda \left(H\right)=\langle H_{\lambda },H\rangle \forall H\in \mathfrak{a}.$$

(167)

This definition allows to define a dual trace form in ${\mathfrak{a}}^{*}$ by setting for $\lambda ,\mu \in {\mathfrak{a}}^{*}$

$$\langle \lambda ,\mu \rangle =\langle H_{\lambda },H_{\mu }\rangle .$$

(168)

(iv) The Lie algebra $\mathfrak{g}$ is a direct sum

$$\mathfrak{g}=\mathfrak{n}+\mathfrak{a}+\mathfrak{k}.$$

(169)

Let $E_{ij}$, $1\le i<j\le n$, be the matrices with component 1 in the entry $(i,j)$ and other components zero; then, these matrices form a basis of $\mathfrak{n}$. For $H\in \mathfrak{a}$, $H=diag\left(x_i\right)$, the Lie bracket in $\mathfrak{g}$, which is simply the commutator, applied to H and $E_{ij}$ yields

$$[H,E_{ij}]=(x_i-x_j)E_{ij}\forall H\in \mathfrak{a}.$$

(170)

Hence, the $E_{ij}$ are the eigenvectors for the characters ${\alpha }_{ij}\in {\mathfrak{a}}^{*}$ defined by

$${\alpha }_{ij}\left(H\right)=x_i-x_j.$$

(171)

Here, $E_{ij}$ is said to be an eigenvector of ${\alpha }_{ij}$, if

$$[H,E_{ij}]={\alpha }_{ij}\left(H\right)E_{ij}\forall H\in \mathfrak{a}.$$

(172)

The eigenspace of ${\alpha }_{ij}$ is one-dimensional. The characters ${\alpha }_{ij}$ are called the relevant characters, or the $(\mathfrak{a},\mathfrak{n})$ characters. They are also called the positive restricted roots. The set of these characters will be denoted by ${\Sigma }^{+}$. We define

$$\tau =\sum _{\alpha \in {\Sigma }^{+}}\alpha$$

(173)

and

$$\rho =\frac{1}{2}\tau .$$

(174)

Lemma 1.

Let $H=diag\left(x_i\right)\in \mathfrak{a}$ and define

$${\lambda }_i\left(H\right)=\sum _{k=1}^i{x}_k,\mathrm{for}1\le i\le n-1,$$

(175)

then

$$\rho =\sum _{i=1}^{n-1}{\lambda }_i.$$

(176)

Furthermore,

$$\langle \rho ,\rho \rangle =\frac{1}{12}{(n-1)}^{2}n.$$

(177)

Proof. 

“(176)” follows from the definition of $\rho$ and $\tau$. For details, see ([7], p. 84).

“(177)” From (168), we obtain

$$\langle \rho ,\rho \rangle =\langle H_{\rho },H_{\rho }\rangle$$

(178)

and the definition of $\rho$ implies

$$H_{\rho }=\frac{1}{2}{H}_{\tau }.$$

(179)

On the other hand,

$$H_{\tau }=\sum _{i=1}^{n-1}{C}_i,$$

(180)

where $C_i\in \mathfrak{a}$ has 1 in the first i entries of the diagonal, $-i$ in the $(i+1)$-th entry, and zero in the other entries. Furthermore,

$$\langle C_i,C_j\rangle =0,i\ne j,$$

(181)

and

$$\langle C_i,C_i\rangle =i^2+i,$$

(182)

cf. ([7], p. 266). Hence, we conclude

$$\langle \rho ,\rho \rangle =\frac{1}{4}\sum _{i=1}^{n-1}(i^2+i)=\frac{1}{12}{(n-1)}^{2}n.$$

(183)

□

Remark 3.

The eigenfunctions of the Laplacian will depend on the additive characters. The above characters ${\alpha }_{ij}$, $1\le i<j\le n$, will represent the elementary gravitons stemming from the degrees of freedom in choosing the coordinates

$$g_{ij},1\le i<j\le n,$$

(184)

of a metric tensor. The diagonal elements offer in general additional n degrees of freedom, but in our case, where we consider metrics satisfying

$$det{g}_{ij}=1,$$

(185)

only $(n-1)$ diagonal components can be freely chosen, and we shall choose the first $(n-1)$ entries, namely,

$$g_{ii},1\le i\le n-1.$$

(186)

The corresponding additive characters are named ${\alpha }_i,1\le i\le n-1$, and are defined by

$${\alpha }_i\left(H\right)=h_i,$$

(187)

if

$$H=diag(h_1,\dots ,h_n).$$

(188)

The characters ${\alpha }_i$, $1\le i\le n-1$, and ${\alpha }_{ij}$$1\le i<j\le n$ will represent the $\frac{(n+2)(n-1)}{2}$ elementary gravitons at the character level. We shall normalize the characters by defining

$${\tilde{\alpha }}_i={\parallel H_{{\alpha }_i}\parallel }^{-1}{\alpha }_i$$

(189)

and

$${\tilde{\alpha }}_{ij}={\parallel H_{{\alpha }_{ij}}\parallel }^{-1}{\alpha }_{ij}$$

(190)

such that the normalized characters have unit norm, cf. (168).

Definition 1.

Let $\lambda \in {\mathfrak{a}}^{*}$; then, we define the spherical function

$${\phi }_{\lambda }\left(g\right)={\int }_K{e}^{(i\lambda +\rho )logA\left(kg\right)}dk,g\in G,$$

(191)

where the Haar measure $dk$ is normalized such that K has measure 1, and where $G=NAK$.

Observe that

$${\phi }_{\lambda }\left(g\right)={\phi }_{\lambda }\left(gK\right),$$

(192)

i.e., ${\phi }_{\lambda }$ can be lifted to $X=G/K$.

The Weyl chambers are the connected components of the set

$$\mathfrak{a}\setminus \bigcup _{1\le i<j\le n}{\alpha }_{ij}^{-1}\left(0\right).$$

(193)

They consist of diagonal matrices having distinct eigenvalues. The Weyl chamber ${\mathfrak{a}}_{+}$, defined by

$${\mathfrak{a}}_{+}=\{H\in \mathfrak{a}:{\alpha }_{ij}\left(h\right)>0,1\le i<j\le n\},$$

(194)

is called the positive Weyl chamber, and the elements $H\in {\mathfrak{a}}_{+}$, $H=diag\left(h_i\right)$ satisfy

$$h_1>h_2>\dots >h_n.$$

(195)

Let M resp. $M^{\prime }$ be the centralizer resp. normalizer of $\mathfrak{a}$ in K; then,

$$W=M^{\prime }/M$$

(196)

is the Weyl group that acts simply transitive on the Weyl chambers. The Weyl group can be identified with the group $S_n$ of permutations in our case, i.e., if $s\in W$ and $H=diag\left(h_i\right)\in \mathfrak{a}$, then

$$s·H=diag\left(h_{s\left(i\right)}\right).$$

(197)

The subgroup M consists of the diagonal matrices $diag\left({ϵ}_i\right)$ with $|{ϵ}_i|=1$.

Let B be the homogeneous space

$$B=K/M;$$

(198)

then, B is a compact Riemannian space with a K-invariant Riemannian metric, cf. ([9], Theorem 3.5, p. 203). B is known as the Furstenberg boundary of X and the map

$$\begin{array}{cc}\hfill \phi :B\times A& \to X\hfill \\ \hfill (kM,a)& \to kaK\hfill \end{array}$$

(199)

is a differentiable, surjective map, while the restriction of φ to

$$B\times A^{+},A^{+}=exp{\mathfrak{a}}_{+},$$

(200)

is a diffeomorphism with an open, dense image; additionally,

$$X=K\overline{A^{+}}eK$$

(201)

cf. ([8], Prop. 1.4, p. 62). If x = gK, b = kM and G = NAK, we define

$$A(x,b)=A(gK,kM)=A\left(k^{-1}g\right),$$

(202)

cf. (156).

We are now ready to describe the Fourier theory and Plancherel formula due to Harish-Chandra for K-bi-invariant functions, cf. [10,11] and ([12], p. 48), and by Helgason for arbitrary functions in $L^1\left(X\right)$ and $L^2\left(X\right)$, cf. [13] and ([14], Theorem 2.6). The extension of the Fourier transform to the Schwartz space $\mathcal{S}\left(X\right)$ and its inversion is due to Eguchi and Okamato [15]. This paper is only an announcement without proofs; the proofs are given in [16]. We follow the presentation in Helgason’s book ([8], 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

$$\widehat{f}\left(\xi \right)={\int }_{{\mathbb{R}}^n}f\left(x\right)e^{-i\langle \xi ,x\rangle }dx$$

(203)

and

$$f\left(x\right)={\int }_{{\mathbb{R}}^n}\widehat{f}\left(\xi \right)e^{i\langle \xi ,x\rangle }d\xi .$$

(204)

The Fourier transform for functions $f\in C_c^{\infty }(X,\mathbb{C})$ is then defined by

$$\widehat{f}(\lambda ,b)={\int }_{X}f\left(x\right)e^{(-i\lambda +\rho )logA(x,b)}dx$$

(205)

for $\lambda \in \mathfrak{a}$ and $b\in B$, or, if we define

$$e_{\lambda ,b}\left(x\right)=e^{(i\lambda +\rho )logA(x,b)},$$

(206)

by

$$\widehat{f}(\lambda ,b)={\int }_{X}f\left(x\right){\overline{e}}_{\lambda ,b}\left(x\right)dx.$$

(207)

The functions $e_{\lambda ,b}$ are real analytic in x and are eigenfunctions of the Laplacian, cf. ([8], Prop. 3.14, p. 99),

$$-\Delta e_{\lambda ,b}={\left(\right|\lambda |}^2+{\left|\rho \right|}^2)e_{\lambda ,b},$$

(208)

where

$${\left|\lambda \right|}^2=\langle \lambda ,\lambda \rangle ,$$

(209)

cf. (168), and similarly for ${\left|\rho \right|}^2$. We also denote the Fourier transform by $\mathcal{F}$ such that

$$\mathcal{F}\left(f\right)=\widehat{f}.$$

(210)

Its inverse ${\mathcal{F}}^{-1}$ is defined in $R\left(\mathcal{F}\right)$ by

$$f\left(x\right)=\frac{1}{\left|W\right|}{\int }_B{\int }_{{\mathfrak{a}}^{*}}\widehat{f}(\lambda ,b){\left|\mathfrak{c}\left(\lambda \right)\right|}^{-2}d\lambda db,$$

(211)

where $\mathfrak{c}\left(\lambda \right)$ is Harish-Chandra’s $\mathfrak{c}$-function and

$$\left|W\right|=cardW,$$

(212)

the number of elements in W, in our case $\left|W\right|=n!$.

As in the Euclidean case, a Plancherel formula is valid, namely, citing from ([8], Theorem 1.5, p. 202):

Theorem 2.

The Fourier transform $f\left(x\right)\to \widehat{f}(\lambda ,b)$, defined by (205), extends to an isometry of $L^2\left(X\right)$ onto $L^2({\mathfrak{a}}_{+}^{*}\times B)$ (with measure ${\left|\mathfrak{c}\left(\lambda \right)\right|}^{-2}d\lambda db$ on ${\mathfrak{a}}_{+}^{*}\times B$). Moreover,

$${\int }_X{f}_1\left(x\right){\overline{f}}_2\left(x\right)dx=\frac{1}{\left|W\right|}{\int }_{{\mathfrak{a}}^{*}\times B}{\widehat{f}}_1(\lambda ,b)\overline{{\widehat{f}}_2}(\lambda ,b){\left|\mathfrak{c}\left(\lambda \right)\right|}^{-2}d\lambda db.$$

(213)

We shall consider the eigenfunctions $e_{\lambda ,b}$ as tempered distributions of the Schwartz space $\mathcal{S}\left(X\right)$ and shall use their Fourier transforms

$${\widehat{e}}_{\lambda ,b}={\delta }_{(\lambda ,b)}={\delta }_{\lambda }\otimes {\delta }_b$$

(214)

as the spatial eigenfunctions of

$$\mathcal{F}(-\Delta )=m\left(\mu \right)=\left(\right|\mu {|}^2+\left|\rho {|}^2\right),$$

(215)

which is a multiplication operator, in the next section.

6. Fourier Quantization

The Fourier theory in $X=G/K$, which we described at the end of the preceding section, uses the eigenfunctions

$$e_{\lambda ,b}\left(x\right)=e^{(i\lambda +\rho )logA(x,b)},(\lambda ,b)\in {\mathfrak{a}}^{*}\times B,$$

(216)

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

$$-\Delta =-{\Delta }_M=-{\Delta }_X,$$

(217)

and the Fourier transforms of the corresponding physically relevant eigenfunctions. If the Hamilton operator is the Euclidean Laplacian in ${\mathbb{R}}^n$, then the spatial eigenfunctions would be

$$e^{i\langle \xi ,x\rangle }.$$

(218)

Therefore, we consider the eigenfunctions $e_{\lambda ,b}$ as a starting point. As in the Euclidean case, the $e_{\lambda ,b}$ are tempered distributions. We first need to extend the Fourier theory to the corresponding Schwartz space $\mathcal{S}\left(X\right)$ and its dual space ${\mathcal{S}}^{\prime }\left(X\right)$, the space of tempered distributions.

Let $D\left(G\right)$ be the algebra of left invariant differential operators in G and $\overline{D}\left(G\right)$ be the algebra of right invariant differential operators. Furthermore, let

$${\phi }_0={\phi }_{\lambda }{}_{{|}_{\lambda =0}}$$

(219)

be the spherical function with parameter $\lambda =0$. Then, φ₀ satisfies the following estimates

$$0<{\phi }_0\left(a\right)\le {\phi }_0\left(e\right)=1\forall a\in A,$$

(220)

and

$${\phi }_0\left(a\right)\le c{(1+|a\left|\right)}^d{e}^{-\rho loga},a\in A^{+},$$

(221)

where

$$d=card{\Sigma }^{+}$$

(222)

the cardinality of the set of positive restricted roots. Here, we used the following definitions, for $g=k_{1}a{k}_2$ ($a\in A,k_1,k_2\in K$), cf. (201) on page 19,

$$\left|g\right|=\left|a\right|=\left|loga\right|,$$

(223)

and c is a positive constant.

The Schwartz space $\mathcal{S}\left(X\right)$ is then defined by

Definition 2.

The Schwartz space $\mathcal{S}\left(G\right)$ is defined as the subspace of $C^{\infty }(G,\mathbb{C})$ the topology of which is given by the seminorms

$$p_{l,D,E}\left(f\right)=\underset{g\in G}{sup}{(1+|g\left|\right)}^l{\phi }_0{\left(g\right)}^{-1}\left|\left(DEf\right)\left(g\right)\right|<\infty$$

(224)

for arbitrary $l\in \mathbb{N}$, $D\in D\left(G\right)$ and $E\in \overline{D}\left(G\right)$. The Schwartz space $\mathcal{S}\left(X\right)$ consists of those functions in $\mathcal{S}\left(G\right)$ that are right invariant under K.

The Fourier transform for $f\in \mathcal{S}\left(X\right)$ is then well defined

$$\widehat{f}(\lambda ,b)={\int }_{X}f\left(x\right){\overline{e}}_{\lambda ,b}\left(x\right)dx.$$

(225)

Integrating over B, we obtain

$$\begin{array}{cc}\hfill F\left(\lambda \right)& ={\int }_B\widehat{f}(\lambda ,b)db\hfill \\ \hfill & ={\int }_{X}f\left(x\right){\int }_B{e}^{(-i\lambda +\rho )logA(x,b)}dbdx\hfill \\ \hfill & ={\int }_{X}f\left(x\right){\phi }_{-\lambda }\left(x\right)dx,\hfill \end{array}$$

(226)

cf. ([17], Equation (1.8)) for the last inequality. Hence, we deduce

Lemma 2.

$F\left(\lambda \right)$ satisfies

$$F(s·\lambda )=F\left(\lambda \right)\forall s\in W.$$

(227)

Proof. 

The spherical function ${\phi }_{\lambda }$ has this property, cf. ([7], Theorem 5.2, p. 100). □

Next, we define the Schwartz space $\mathcal{S}({\mathfrak{a}}^{*}\times B)$. Note that ${\mathfrak{a}}^{*}$ is a Euclidean space; in our case, ${\mathfrak{a}}^{*}={\mathbb{R}}^{n-1}$, and $B=K/M$ is a compact Riemannian space. Hence, we define the Schwartz space $\mathcal{S}({\mathfrak{a}}^{*}\times B)$ as follows, cf. ([15], Def. 2, p. 240):

Definition 3.

Let $\mathcal{S}({\mathfrak{a}}^{*}\times B)$ denote the set of all functions $F\in C^{\infty }({\mathfrak{a}}^{*}\times B,\mathbb{C})$ that satisfy the following condition: for any natural numbers $l,m,q$

$$p_{l,m,q}\left(F\right)=\underset{(\lambda ,b)\in {\mathfrak{a}}^{*}\times B}{sup}{(1+|\lambda |}^2{)}^l\sum _{\left|\alpha \right|\le m}\left|{(-{\Delta }_B+1)}^q{D}^{\alpha }F\right|<\infty ,$$

(228)

where $\alpha =({\alpha }_1,\dots ,{\alpha }_r)$, $r=dim{\mathfrak{a}}^{*}$, is a multi-index,

$$D^{\alpha }F=D_1^{{\alpha }_1}\dots D_r^{{\alpha }_r}F$$

(229)

are partial derivatives with respect to $\lambda \in {\mathfrak{a}}^{*}$, and ${\Delta }_B$ is the Laplacian in B.

The seminorms $p_{l,m,q}$ define a topology on $\mathcal{S}({\mathfrak{a}}^{*}\times B)$ with respect to which it is a Fréchet space.

Theorem 3.

The Fourier transform $\mathcal{F}$

$$\mathcal{F}:\mathcal{S}\left(X\right)\to \mathcal{S}({\mathfrak{a}}^{*}\times B)$$

(230)

is continuous, and if we define

$$\widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)=\{F\in \mathcal{S}({\mathfrak{a}}^{*}\times B):F\left(\lambda \right)={\int }_{B}F(\lambda ,b)db \mathrm{satisfies}\left(227\right)\},$$

(231)

then

$$\mathcal{F}:\mathcal{S}\left(X\right)\to \widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)$$

(232)

is a linear topological isomorphism.

Proof. 

Confer ([15], Theorem 4) and ([16], Lemma 4.8.2 & Theorem 4.8.3, p. 212) □

Remark 4.

Note that the measure in $\widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)$ is defined by

$$d\mu (\lambda ,b)=\frac{1}{\left|W\right|}{\left|\mathfrak{c}\left(\lambda \right)\right|}^{-2}d\lambda db$$

(233)

and that the function

$$\lambda \to {\left|\mathfrak{c}\left(\lambda \right)\right|}^{-1}$$

(234)

has slow growth, cf. ([8], Lemma 3.5, p. 91).

We can now define the Fourier quantization. Let ${\mathcal{S}}^{\prime }\left(X\right)$ resp. $\widehat{{\mathcal{S}}^{\prime }}({\mathfrak{a}}^{*}\times B)$ be the dual spaces of tempered distributions; then,

$${\mathcal{F}}^{-1}:\widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)\to \mathcal{S}\left(X\right)$$

(235)

is continuous. Let ${\left({\mathcal{F}}^{-1}\right)}^{*}$ be the dual operator

$${\left({\mathcal{F}}^{-1}\right)}^{*}:{\mathcal{S}}^{\prime }\left(X\right)\to \widehat{{\mathcal{S}}^{\prime }}({\mathfrak{a}}^{*}\times B)$$

(236)

defined by

$$\langle \omega ,{\mathcal{F}}^{-1}\left(F(\lambda ,b)\right)\rangle =\langle ({{\mathcal{F}}^{-1})}^{*}\omega ,F(\lambda ,b)\rangle$$

(237)

for $\omega \in {\mathcal{S}}^{\prime }\left(X\right)$ and $F\in \widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)$. Let

$$F(\lambda ,b)=\widehat{f}(\lambda ,b),f\in \mathcal{S}\left(X\right);$$

(238)

then,

$$\langle \omega ,f\rangle =\langle ({{\mathcal{F}}^{-1})}^{*}\omega ,\widehat{f}\rangle .$$

(239)

Now, choose $\omega =e_{\lambda ,b}$, where $(\lambda ,b)\in {\mathfrak{a}}^{*}\times B$ is arbitrary but fixed; then,

$$\langle \omega ,f\rangle ={\int }_{X}f\left(x\right){\overline{e}}_{\lambda ,b}\left(x\right)dx=\widehat{f}(\lambda ,b).$$

(240)

Hence, we deduce

$${\left({\mathcal{F}}^{-1}\right)}^{*}\omega ={\delta }_{(\lambda ,b)}={\delta }_{\lambda }\otimes {\delta }_b.$$

(241)

Lemma 3.

Let $\omega \in {\mathcal{S}}^{\prime }\left(X\right)$; then, we may call ${\left({\mathcal{F}}^{-1}\right)}^{*}\omega$ to be the Fourier transform of ω

$${\left({\mathcal{F}}^{-1}\right)}^{*}\omega =\widehat{\omega }.$$

(242)

Proof. 

$\mathcal{S}\left(X\right)$ can be embedded in ${\mathcal{S}}^{\prime }\left(X\right)$ by defining for $\omega \in \mathcal{S}\left(X\right)$

$$\langle \omega ,f\rangle ={\int }_{X}f\overline{\omega }dx,\forall f\in \mathcal{S}\left(X\right).$$

(243)

$\omega$ is obviously an element of ${\mathcal{S}}^{\prime }\left(X\right)$ and the embedding is antilinear. On the other hand, in view of the Plancherel formula, we have

$${\int }_{X}f\overline{\omega }dx={\int }_{{\mathfrak{a}}^{*}\times B}\widehat{f}(\lambda ,b)\overline{\widehat{\omega }}(\lambda ,b)d\mu (\lambda ,b)$$

(244)

and thus, because of (239),

$$\langle {\left({\mathcal{F}}^{-1}\right)}^{*}\omega ,\widehat{f}\rangle ={\int }_{{\mathfrak{a}}^{*}\times B}\widehat{f}\overline{\widehat{\omega }}d\mu (\lambda ,b).$$

(245)

□

Looking at the Fourier transformed eigenfunctions

$${\widehat{e}}_{\lambda ,b}={\delta }_{\lambda }\otimes {\delta }_b,$$

(246)

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 $e_{\lambda ,b}$ and $e_{\lambda ,b^{\prime }}$. The first ansatz would be to integrate over B, i.e., we would consider the Fourier transform of

$${\int }_B{e}_{\lambda ,b}db={\phi }_{\lambda },$$

(247)

which is equal to the Fourier transform of the spherical function ${\phi }_{\lambda }$, i.e.,

$${\widehat{\phi }}_{\lambda }={\delta }_{\lambda },$$

(248)

and it would act on the functions

$$F\left(\mu \right)={\int }_B\widehat{f}(\mu ,b)db,f\in \mathcal{S}\left(X\right).$$

(249)

These functions satisfy the relation (227), which in turn implies

$$s·{\delta }_{\lambda }={\delta }_{s^{-1}·\lambda }={\delta }_{\lambda }\forall s\in W$$

(250)

if $\lambda$ was allowed to range in all of ${\mathfrak{a}}^{*}$. Hence, we would have to restrict $\lambda$ to the positive Weyl chamber ${\mathfrak{a}}_{+}^{*}$, but then, we would not be able to define the eigenfunctions corresponding to the elementary gravitons $g_{ii}$, $2\le i\le n-1$, since the corresponding $\lambda$ belong to different Weyl chambers, cf. Remark 3 on page 18.

Therefore, we pick a distinguished $b\in B$, namely,

$$b_0=eM,e=id\in K,$$

(251)

and only consider the eigenfunctions $e_{\lambda ,b_0}$ with corresponding Fourier transforms

$${\delta }_{\lambda }\equiv {\delta }_{\lambda }\otimes {\delta }_{b_0}={\widehat{e}}_{\lambda ,b_0},\lambda \in {\mathfrak{a}}^{*}.$$

(252)

Then, we can prove:

Lemma 4.

Let ${\delta }_{\lambda }$ be defined as above; then, for any $s\in W$ satisfying $s·\lambda \ne \lambda$, there exists $F\in \widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)$ such that

$$\langle {\delta }_{\lambda },F\rangle =F(\lambda ,b_0)\ne F(s·\lambda ,b_0)=\langle {\delta }_{s·\lambda },F\rangle .$$

(253)

Proof. 

Let $\psi \in C_c^{\infty }\left({\mathfrak{a}}^{*}\right)$ be a function satisfying

$$\psi \left(\lambda \right)\ne \psi (s·\lambda )$$

(254)

and choose $\eta \in C^{\infty }\left(B\right)$ with the properties

$$\eta \left(b_0\right)=1$$

(255)

and

$${\int }_B\eta db=0,$$

(256)

then

$$F=\psi \eta \in \widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B)$$

(257)

and satisfies (253). □

The Fourier transform of the Laplacian is a multiplication operator similar to the Euclidean case.

Lemma 5.

(i) Let $f\in \mathcal{S}\left(X\right)$; then,

$$\mathcal{F}(-\Delta f)=m\left(\lambda \right)\widehat{f}(\lambda ,b)\in \widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B),$$

(258)

where

$$m\left(\lambda \right)={\left|\lambda \right|}^2+{\left|\rho \right|}^2,\lambda \in {\mathfrak{a}}^{*}.$$

(259)

(ii) Let $\omega \in {\mathcal{S}}^{\prime }\left(X\right)$; then, $-\Delta \omega$ is defined as usual

$$\langle -\Delta \omega ,f\rangle =\langle \omega ,-\Delta f\rangle$$

(260)

and

$$\mathcal{F}(-\Delta \omega )=m\left(\lambda \right)\widehat{\omega }\in \widehat{{\mathcal{S}}^{\prime }}({\mathfrak{a}}^{*}\times B),$$

(261)

where

$$\langle m\left(\lambda \right)\widehat{\omega },F(\lambda ,b)\rangle =\langle \widehat{\omega },m\left(\lambda \right)F(\lambda ,b)\rangle \forall F\in \widehat{\mathcal{S}}({\mathfrak{a}}^{*}\times B).$$

(262)

Proof. 

“(i)” The result follows immediately by partial integration.

“(ii)” From (239) and (260), we deduce

$$\begin{array}{cc}\hfill \langle -\Delta \omega ,f\rangle & =\langle \widehat{\omega },\mathcal{F}(-\Delta f)\rangle \hfill \\ \hfill & =\langle \widehat{\omega },m\left(\lambda \right)\widehat{f}(\lambda ,b)\rangle \hfill \\ \hfill & =\langle m\left(\lambda \right)\widehat{\omega },\widehat{f}(\lambda ,b)\rangle \hfill \\ \hfill & =\langle \mathcal{F}(-\Delta \omega ),\widehat{f}(\lambda ,b)\rangle .\hfill \end{array}$$

(263)

□

Now, choosing

$$\omega =e_{\lambda ,b_0},$$

(264)

where $\lambda \in {\mathfrak{a}}^{*}$ is fixed, then

$$\mathcal{F}(-\Delta \omega )=m\left(\mu \right)\widehat{\omega }=m\left(\mu \right){\delta }_{\lambda }=m\left(\lambda \right){\delta }_{\lambda },$$

(265)

since

$$\langle m\left(\mu \right){\delta }_{\lambda },F(\mu ,b)\rangle =\langle {\delta }_{\lambda },m\left(\mu \right)F(\mu ,b)\rangle =m\left(\lambda \right)F(\lambda ,b_0).$$

(266)

In Remark 3 on page 18, we already identified the additive characters corresponding to the elementary gravitons, namely, the characters

$${\alpha }_{ij},1\le i<j\le n$$

(267)

and

$${\alpha }_i,1\le i\le n-1.$$

(268)

We shall now define the corresponding forms in ${\mathfrak{a}}^{*}$ with arbitrary energy levels:

Definition 4.

Let $\lambda \in {\mathbb{R}}_{+}^{}$ be arbitrary. Then, we consider the characters

$$\lambda {\tilde{\alpha }}_i\wedge \lambda {\tilde{\alpha }}_{ij},$$

(269)

where we recall that the terms embellished by a tilde refer to the corresponding unit vectors. Then, the eigenfunctions representing the elementary gravitons are $e_{\lambda {\tilde{\alpha }}_i,b_0}$ and $e_{\lambda {\tilde{\alpha }}_{ij},b_0}$.

The corresponding eigenvalue with respect to $-\Delta$ is ${\left|\lambda \right|}^2+{\left|\rho \right|}^2$, where, by a slight abuse of notation, ${\left|\lambda \right|}^2={\lambda }^2$ and ${\left|\rho \right|}^2=\langle \rho ,\rho \rangle$. Note that ${\left|\rho \right|}^2$ is always strictly positive; indeed,

$${\left|\rho \left(n\right)\right|}^2\ge {\left|\rho \left(3\right)\right|}^2=1,$$

(270)

if $X=SL(n,{\mathbb{R}}^{})/SO\left(n\right)$ and $n\ge 3$, cf. (183) on page 18.

7. Temporal Eigenfunctions

The temporal eigenfunctions $w=w\left(t\right)$ have to satisfy the ODE (142) on page 14, or equivalently,

$$\ddot{w}+m{t}^{-1}\dot{w}+{\mu }_0{t}^{-2}w=0,$$

(271)

where ${\mu }_0$ should be equal to

$${\mu }_0=\frac{16(n-1)}{n}{\left(\right|\lambda |}^2+{\left|\rho \right|}^2),$$

(272)

and where $\left(\right|\lambda {|}^2+\left|\rho {|}^2\right)$ is the eigenvalue of a spatial eigenfunction.

To solve (271), we make the ansatz

$$w\left(t\right)=t^{-\frac{(m-1)}{2}}{e}^{i\mu logt},\mu >0,$$

(273)

to obtain

$$\ddot{w}+m{t}^{-1}\dot{w}+{\mu }_0{t}^{-2}w=\{-\frac{{(m-1)}^2}{4}+{\mu }_0-{\mu }^2\}w.$$

(274)

In order to choose $\mu$ such that the term in the braces vanishes, we have to ensure that

$${\mu }_0-\frac{{(m-1)}^2}{4}>0.$$

(275)

Now, the estimate

$${\mu }_0-\frac{{(m-1)}^2}{4}\ge \frac{16(n-1)}{n}{\rho }^2-\frac{{(m-1)}^2}{4}$$

(276)

is valid, where

$${\rho }^2=\frac{{(n-1)}^{2}n}{12}$$

(277)

and

$$m=\frac{(n-1)(n+2)}{2}.$$

(278)

One can easily check that

$$\frac{16(n-1)}{n}{\rho }^2-\frac{{(m-1)}^2}{4}=\left\{\begin{array}{cc}>0,\hfill & 3\le n\le 16,\hfill \\ <0,\hfill & 17\le n.\hfill \end{array}\right.$$

(279)

In case $n\ge 17$ and

$${\mu }_0-\frac{{(m-1)}^2}{4}<0,$$

(280)

we obtain the solution

$$w=c_1{t}^{-\frac{m-1}{2}+\sqrt{\frac{{(m-1)}^2}{4}-{\mu }_0}}+c_2{t}^{-\frac{m-1}{2}-\sqrt{\frac{{(m-1)}^2}{4}-{\mu }_0}},$$

(281)

while for

$${\mu }_0-\frac{{(m-1)}^2}{4}=0,$$

(282)

we get

$$w=c_1{t}^{-\frac{m-1}{2}}+c_2{t}^{-\frac{m-1}{2}}logt.$$

(283)

Remark 5.

In all three cases ((275), (280), and (282)), 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 (273), which generate all possible temporal eigenfunctions, if $3\le n\le 16$, seem to be the physically relevant solutions.

8. Conclusions

Quantizing the Hamilton equations instead of the Hamilton constraint, we obtained the simple equation

$$-\Delta u=0$$

(284)

in the fiber bundle E provided $n\ne 4$, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric in the fibers and where u is a smooth function that is only defined in the fibers of E

$$u=u\left(g_{ij}\left(x\right)\right),x\in {\mathcal{S}}_0={\mathbb{R}}^n.$$

(285)

Expressing then the fiber metric as in (109) on page 11, we can use separation of variables and write the solutions u as products

$$u=w\left(t\right)v\left(g_{ij}(x,{\xi }^A)\right),$$

(286)

where $g_{ij}(x,{\xi }^A)$ 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,

$$\frac{det{g}_{ij}\left(x\right)}{det{\delta }_{ij}\left(x\right)}=1,$$

(287)

where ${\delta }_{ij}$ is the Euclidean metric. Using Euclidean coordinates in ${\mathcal{S}}_0$, we can identify the fibers $M\left(x\right)$ with the symmetric space

$$G/K=SL(n,{\mathbb{R}}^n)/SO\left(n\right).$$

(288)

The Riemannian metric in $G/K$ is identical to the induced fiber metric of $M\left(x\right)$ 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

$$e_{\lambda ,b_0}\left(y\right)=e^{(i\lambda +\rho )logA(y,b_0)},\lambda \in {\mathfrak{a}}^{*},$$

(289)

where we used the Iwasawa decomposition $G=NAK$ and where $b_0$ is the distinguished point specified in (251) on page 24. These smooth functions are tempered distributions and are eigenfunctions of the Laplacian

$$-\Delta e_{\lambda ,b_0}={\left(\right|\lambda |}^2+{\left|\rho \right|}^2)e_{\lambda ,b_0}.$$

(290)

Their Fourier transforms are Dirac measures

$${\widehat{e}}_{\lambda ,b_0}={\delta }_{\lambda }\otimes {\delta }_{b_0}.$$

(291)

In Fourier space, the Laplacian is a multiplication operator

$$\mathcal{F}(-\Delta f)={\left(\right|\lambda |}^2+{\left|\rho \right|}^2)\widehat{f}(\lambda ,b)\forall f\in \mathcal{S}(G/K),$$

(292)

where $\lambda$ ranges in ${\mathfrak{a}}^{*}$ and b in the Furstenberg boundary B.

Let

$$\pi :G/K\to M$$

(293)

be the canonical isometry defined in (148) on page 15; then, the eigenfunctions f in $G/K$ can be transformed to be eigenfunctions in the fibers of the sub-bundle M by defining

$$v\left(g_{ij}(x,{\xi }^A)\right)=f({\pi }^{-1}\left(g_{ij}(x,{\xi }^A)\right),$$

(294)

i.e.,

$$e_{\lambda ,b_0}\circ {\pi }^{-1}\circ g_{ij}(x,{\xi }^A),\lambda \in {\mathfrak{a}}^{*},$$

(295)

are the spatial eigenfunctions with eigenvalues $\left(\right|\lambda {|}^2+\left|\rho {|}^2\right)$. The eigenfunctions corresponding to the elementary gravitons we defined in Definition 4 on page 25. They are characterized by special characters ${\alpha }_i$, $1\le i\le n-1$, for the diagonal gravitons and ${\alpha }_{ij}$, $1\le i<j\le n$, for the off-diagonal gravitons.

The temporal eigenfunctions $w\left(t\right)$, which we defined in the previous section, have the properties that they become unbounded if $t\to 0$ and they vanish, together with all derivatives, if $t\to \infty$.

Furthermore, if we consider $t<0$, then the functions

$$\tilde{w}\left(t\right)=w(-t),t<0,$$

(296)

also satisfy the ODE (271) on page 25 for $t<0$, i.e., they are also temporal eigenfunctions if the light cone in E is flipped.

Thus, we conclude

Theorem 4.

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

$${\left|\lambda \right|}^2+{\left|\rho \right|}^2\ge {\left|\rho \right|}^2\ge {\left|\rho \left(3\right)\right|}^2=1.$$

(297)

Choosing $\lambda =0$, we have a common ground state with smallest eigenvalue ${\left|\rho \right|}^2$, 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 antimatter.