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A unified quantum theory incorporating the four fundamental forces of nature is one of the major open problems in physics. The Standard Model combines electro-magnetism, the strong force and the weak force, but ignores gravity. The quantization of gravity is therefore a necessary first step to achieve a unified quantum theory.

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

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 $x0$ and considers the foliation of $N$ given by the slices

cf. [2, Theorem 3.2]. Then, the Einstein equations also split into a tangential 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 type $Hˆ$ and the possible quantum solutions of gravity are supposed to satisfy the so-called Wheeler-DeWitt equation

in an appropriate setting, i.e., only the Hamilton condition (0.6) has been quantized, or equivalently, the normal Einstein equation, while the tangential Einstein equations have been ignored.

In [2] we solved the equation (0.7) in a fiber bundle $E$ with base space $S0$ ,

the elements of which are the positive definite symmetric tensors of order two, the Riemannian metrics in $S 0$ . The hyperbolic operator $ˆH$ is then expressed in the form

where $Δ$ is the Laplacian of the DeWitt metric given in the fibers, $R$ the scalar curvature of the metrics $gij(x) ∈ F (x)$ , and $φ$ is defined by

where $ρij$ 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.

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 $gij$ are the spatial variables. In the paper [5] we overcame this difficulty by quantizing the Hamilton equations instead of the Hamilton condition.

in $E$ , where the Laplacian is the Laplacian in (0.10). The lower order terms of $ˆ H$

were eliminated during the quantization process. However, the equation (0.12) is only valid provided $n ⁄= 4$ , since the resulting equation actually looks like

This restriction seems to be acceptable, since $n$ is the dimension of the base space $S0$ which, by general consent, is assumed to be $n = 3$ . The fibers add additional dimensions to the quantized problem, namely,

The fiber metric, the DeWitt metric, which is responsible for the Laplacian in (0.12) can be expressed in the form

We also assumed that $S0 = ℝn$ and that the metric $ρij$ in (0.11) is the Euclidean metric $δij$ . It is well-known that $M$ is a symmetric space

It is also easily verified that the induced metric of $M$ in $E$ is isometric to the Riemannian metric of the coset space $G ∕K$ .

Now, we were in a position to use separation of variables, namely, we wrote a solution of (0.12) in the form

The eigenfunctions of the Laplacian in $G ∕K$ are well-known and we chose the kernel of the Fourier transform in $G ∕K$ in order to define the eigenfunctions. This choice also allowed 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.

In [6] we to quantized the Einstein-Hilbert functional combined with the functionals of the other fundamental forces of nature, i.e., we looked at the Lagrangian functional

where $αN$ is a positive coupling constant, $˜Ω ⋐ N = N n+1$ and $N$ a globally hyperbolic spacetime with metric $¯gαβ$ , $0 ≤ α,β ≤ n$ , where the metric splits as in (0.3).

The functional $J$ consists of the Einstein-Hilbert functional, the Yang-Mills and Higgs functional and a massive Dirac term.

corresponds to the adjoint representation of a compact, semi-simple Lie group $G$ with Lie algebra $𝔤$ . The $f¯c$ ,

The spinor field $ψ = (ψI) A$ has a spinor index $A$ , $1 ≤ A ≤ n 1$ , and a colour index $I$ , $1 ≤ I ≤ n 2$ . Here, we suppose that the Lie group has a unitary representation $R$ such that

are antihermitian matrices acting on $ℂn2$ . The symbol $A μψ$ is now defined by

There are some major difficulties in achieving a quantization of the functional in (0.24). First we were unable to quantize the corresponding Hamilton equations, hence, we quantized the Hamilton condition which has the form

where the subscripts refer to gravity, Yang-Mills, Dirac and Higgs. On the left-hand side are the Hamilton functionals of the respective fields. They depend on the Riemannian metrics $gij$ , the Yang-Mills connections and the spinor and Higgs fields. We were not able to quantize the non-gravitational Hamiltons for arbitrary metrics $gij$ , but we proposed the following model: Choosing the fiber coordinates as in (0.17) the fiber metrics $gij$ can be written in the form

cf. [3, Equ. (1.4.103)]. We were able to prove that the non-gravitational Hamiltonians could be expressed in the form

where the embellished Hamiltonians depend on $σij$ , provided $n = 3$ and provided that the mass term in the Dirac Lagrangian and the Higgs Lagrangian are slightly modified. The embellished Hamiltonians are then standard Hamiltonians without any modifications. The Hamilton constraint then has the form

where the subscript $SM$ refers to the fields of the Standard Model or to a corresponding subset of fields.

In the quantization process, we quantized $HG$ for general $σij$ but $˜HSM$ only for $σij = δij$ by the usual methods of QFT. Let $v$ resp. $ψ$ be the spatial eigendistributions of the respective Hamilton operators, then, the solutions $u$ of the Wheeler-DeWitt equation are given by $u = wvψ$ , where $w$ satisfies an ODE and $u$ is evaluated at $(t,δij)$ in the fibers.

Theorem 0.1. Let $n = 3$ , $v = eλ,b0$ and let $ψ$ be an eigendistribution of $H˜SM$ when $σij = δij$ such that

2 - ΔM eλ,b0 = (|λ| + 1)eλ,b0,

(0.35)

˜ HSM ψ = λ1ψ, λ1 ≥ 0,

(0.36)

and let $w$ be a solution of the ODE

-m-∂ m ∂w- 32 2 -2 32 -1 - 23 t ∂t (t ∂t ) + 3 (|λ| + 1)t w + 3 α N λ1t w 64 -2 2 + 3 αN Λt w = 0

(0.37)

then

u = weλ,b0ψ

(0.38)

is a solution of the Wheeler-DeWitt equation

Hˆu = 0,

(0.39)

where $eλ,b0$ is evaluated at $σij = δij$ and where we note that $m = 5$ .

We referred to $eλ,b0$ and $ψ$ as the spatial eigenfunctions and to $w$ as the temporal eigenfunction.

Remark 0.2. We could also apply the respective Fourier transforms to $˜ - Δeλ,b0$ resp. $˜ HSM ψ$ and consider

ˆ w ˆeλ,b0ψ

(0.40)

as the solution in Fourier space, where $ˆ ψ$ would be expressed with the help of the ladder operators.

For simplicity we shall only state the result when $m3 = 0$ which is tantamount to setting $Λ = 0$ .

Theorem 0.3. Assume $m3 = 0$ and $m2 > 0$ , then the solutions of the ODE (0.41) are generated by

J(3√m-----4i, 3m t23)t-2 2 1 2 2

(0.43)

and

3√------ 3 23 -2 J(- 2 m1 - 4i,2m2t )t ,

(0.44)

where $J(λ,t)$ is the Bessel function of the first kind.

For details of the monograph The Quantization of Gravity [3] published by Springer International click here and for a review of the book by Paulo Moniz for the Mathematical Reviews here.

In [4] and [3] we also quantized the full Einstein equations, however, the resulting hyperbolic equation in $E$ could only be solved abstractly, since the elliptic parts of the hyperbolic operator acted both in the fibers as well as in the base space and we were not able to find solutions that could be expressed as products of spatial and temporal eigenfunctions of self-adjoint operators. Recently, we solved this problem, cf. [7].

References

[1] R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gravitation: an introduction to current research (Louis Witten, ed.), John Wiley, New York, 1962, pp. 227–265.

[2] Claus Gerhardt, The quantization of gravity in globally hyperbolic spacetimes, Adv. Theor. Math. Phys. 17 (2013), no. 6, 1357–1391, arXiv:1205.1427, doi:10.4310/ATMP.2013.v17.n6.a5.

[3] _________ , The Quantization of Gravity, 1st ed., Fundamental Theories of Physics, vol. 194, Springer, Cham, 2018, doi:10.1007/978-3-319-77371-1.

[4] _________ , The quantization of gravity, Adv. Theor. Math. Phys. 22 (2018), no. 3, 709–757, arXiv:1501.01205, doi:10.4310/ATMP.2018.v22.n3.a4.

[5] _________ , The quantization of gravity: Quantization of the Hamilton equations, Universe 7 (2021), no. 4, 91, doi:10.3390/universe7040091.

[6] _________ , A unified quantization of gravity and other fundamental forces of nature, Universe 8 (2022), no. 8, 404, doi:10.3390/universe8080404.

[7] _________ , The quantization of gravity: The quantization of the full Einstein equations, Symmetry 15 (2023), no. 8, 1599, doi:10.3390/sym15081599.

Ruprecht-Karls-Universität Heidelberg

Institut für
Mathematik

Prof. Dr. Claus Gerhardt

A UNIFIED QUANTIZATION OF GRAVITY AND OTHER FUNDAMENTAL FORCES OF NATURE

References

Links

Ruprecht-Karls-Universität Heidelberg

Institut für Mathematik

Prof. Dr. Claus Gerhardt

A UNIFIED QUANTIZATION OF GRAVITY AND OTHER FUNDAMENTAL FORCES OF NATURE

References

Links

Institut für
Mathematik