RuprechtKarlsUniversität Heidelberg
Institut für

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Institut für Angewandte Mathematik Im Neuenheimer Feld 205 D69120 Heidelberg 
Tel. + 49 (0) 62 21  54 14100 (Sekretariat) email: gerhardt@math.uniheidelberg.de 
Area of Research:  Partial Differential Equations, Differential Geometry and General Relativity 
A unified quantum theory incorporating the four fundamental forces of nature is one of the major open problems in physics. The Standard Model combines electromagnetism, 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 EulerLagrange equation of the EinsteinHilbert functional
(0.1) 
where , , is a globally hyperbolic Lorentzian manifold, the scalar curvature and 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 and considers the foliation of given by the slices
(0.2) 
We may, without loss of generality, assume that the spacetime metric splits
(0.3) 
cf. [2, Theorem 3.2]. Then, the Einstein equations also split into a tangential part
(0.4) 
and a normal part
(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 [1]. The normal Einstein equations can be expressed by the socalled Hamilton condition
(0.6) 
where 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 and the possible quantum solutions of gravity are supposed to satisfy the socalled WheelerDeWitt equation
(0.7) 
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 with base space ,
(0.8) 
and fibers , ,
(0.9) 
the elements of which are the positive definite symmetric tensors of order two, the Riemannian metrics in . The hyperbolic operator is then expressed in the form
(0.10) 
where is the Laplacian of the DeWitt metric given in the fibers, the scalar curvature of the metrics , and is defined by
(0.11) 
where is a fixed metric in such that instead of densities we are considering functions. The WheelerDeWitt equation could be solved in 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 in the quantized equation, which prevents the application of separation of variables, since the metrics are the spatial variables. In the paper [4] we overcame this difficulty by quantizing the Hamilton equations instead of the Hamilton condition.
As a result we obtained the equation
(0.12) 
in , where the Laplacian is the Laplacian in (0.10). The lower order terms of
(0.13) 
were eliminated during the quantization process. However, the equation (0.12) is only valid provided , since the resulting equation actually looks like
(0.14) 
This restriction seems to be acceptable, since is the dimension of the base space which, by general consent, is assumed to be . The fibers add additional dimensions to the quantized problem, namely,
(0.15) 
The fiber metric, the DeWitt metric, which is responsible for the Laplacian in (0.12) can be expressed in the form
(0.16) 
where the coordinate system is
(0.17) 
The , , are coordinates for the hypersurface
(0.18) 
We also assumed that and that the metric in (0.11) is the Euclidean metric . It is wellknown that is a symmetric space
(0.19) 
It is also easily verified that the induced metric of in is isometric to the Riemannian metric of the coset space .
Now, we were in a position to use separation of variables, namely, we wrote a solution of (0.12) in the form
(0.20) 
where is a spatial eigenfunction of the induced Laplacian of
(0.21) 
and is a temporal eigenfunction satisfying the ODE
(0.22) 
with
(0.23) 
The eigenfunctions of the Laplacian in are wellknown and we chose the kernel of the Fourier transform in 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 [5] we quantized the EinsteinHilbert functional combined with the functionals of the other fundamental forces of nature, i.e., we looked at the Lagrangian functional
(0.24) 
where is a positive coupling constant, and a globally hyperbolic spacetime with metric , , where the metric splits as in (0.3).
The functional consists of the EinsteinHilbert functional, the YangMills and Higgs functional and a massive Dirac term.
The YangMills field
(0.25) 
corresponds to the adjoint representation of a compact, semisimple Lie group with Lie algebra . The ,
(0.26) 
are the structural constants of .
We assumed the Higgs field to have complex valued components.
The spinor field has a spinor index , , and a colour index , . Here, we suppose that the Lie group has a unitary representation such that
(0.27) 
are antihermitian matrices acting on . The symbol is now defined by
(0.28) 
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
(0.29) 
where the subscripts refer to gravity, YangMills, Dirac and Higgs. On the lefthand side are the Hamilton functions of the respective fields. They depend on the Riemannian metrics , the YangMills connections and the spinor and Higgs fields. We were not able to quantize the nongravitational Hamiltonians for arbitrary metrics , but we proposed the following model: Choosing the fiber coordinates as in (0.17) the fiber metrics can be written in the form
(0.30) 
where
(0.31) 
cf. [3, Equ. (1.4.103)]. We were able to prove that the nongravitational Hamiltonians could be expressed in the form
(0.32) 
where the embellished Hamiltonians depend on , provided 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
(0.33) 
where the subscript refers to the fields of the Standard Model or to a corresponding subset of fields.
In the quantization process, we quantized for general but only for by the usual methods of QFT. Let resp. be the spatial eigendistributions of the respective Hamilton operators, then, the solutions of the WheelerDeWitt equation are given by , where satisfies an ODE and is evaluated at in the fibers.
The solutions of the WheelerDeWitt equation
(0.34) 
can then be achieved by using separation of variables. We proved:
Theorem 0.1. Let , and let be an eigendistribution of when such that
(0.35) 
(0.36) 
and let be a solution of the ODE
(0.37) 
then
(0.38) 
is a solution of the WheelerDeWitt equation
(0.39) 
where is evaluated at and where we note that .
We referred to and as the spatial eigenfunctions and to as the temporal eigenfunction.
Remark 0.2. We could also apply the respective Fourier transforms to resp. and consider
(0.40) 
as the solution in Fourier space, where would be expressed with the help of the ladder operators.
The temporal eigenfunctions must satisfy an ODE of the form
(0.41) 
where
(0.42) 
For simplicity we shall only state the result when which is tantamount to setting .
Theorem 0.3. Assume and , then the solutions of the ODE (0.41) are generated by
(0.43) 
and
(0.44) 
where is the Bessel function of the first kind.
Lemma 0.4. The solutions in the theorem above diverge to complex infinity if tends to zero and they converge to zero if tends to infinity.
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.
[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:10.1007/9783319773711.
[4] _________ , The quantization of gravity: Quantization of the Hamilton equations, Universe 7 (2021), no. 4, 91, doi:10.3390/universe7040091.
[5] _________ , A unified quantization of gravity and other fundamental forces of nature, Universe 8 (2022), no. 8, 404, doi:10.3390/universe8080404.
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