Ruprecht-Karls-Universität Heidelberg Institut für Angewandte Mathematik Prof. Dr. Claus Gerhardt

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.

The Einstein equations are the Euler-Lagrange equations of the Einstein-Hilbert functional and quantization of a Lagrangian theory requires to switch from a Lagrangian view to a Hamiltonian view. In a ground breaking paper, Arnowitt, Deser and Misner [1] expressed the Einstein-Hilbert Lagrangian in a form which allowed to derive a corresponding Hamilton function by applying the Legendre transformation. However, since the Einstein-Hilbert Lagrangian is singular, the Hamiltonian description of gravity is only correct if two additional constraints are satisfied, namely, the Hamilton constraint, which is expressed by the equation , where is the Hamilton function, and the diffeomorphism constraint. Dirac [3] proved how to quantize a constrained Hamiltonian system—at least in principle—and his method has been applied to the Hamiltonian setting of gravity, cf. the paper of DeWitt [2] and the monographs by Kiefer [15] and Thiemann [16]. In the general case, when arbitrary globally hyperbolic spacetime metrics are allowed, the problem turned out to be extremely difficult and solutions could only be found by assuming a high degree of symmetry, cf., e.g., [4].

However, in [6, 7, 5] we proposed a model for the quantization of gravity for general hyperbolic spacetimes, in which we eliminated the diffeomorphism constraint by reducing the number of variables and proving that the Euler-Lagrange equations for this special class of metrics were still the full Einstein equations. The Hamiltonian description of the Einstein-Hilbert functional then allowed a canonical quantization. We quantized the action by looking at the Wheeler-DeWitt equation in a fiber bundle , where the base space is a Cauchy hypersurface of the spacetime which has been quantized and the elements of the fibers are Riemannian metrics. The fibers of are equipped with a Lorentzian metric such that they are globally hyperbolic and the transformed Hamiltonian, which is now a hyperbolic operator, , is a normally hyperbolic operator acting only in the fibers. The Wheeler-DeWitt equation has the form with and we defined with the help of the Green’s operator a symplectic vector space and a corresponding Weyl system.

The Wheeler-DeWitt equation seems to be the obvious quantization of the Hamilton condition. However, acts only in the fibers and not in the base space which is due to the fact that the derivatives are only ordinary covariant derivatives and not functional derivatives, though they are supposed to be functional derivatives, but this property is not really invoked when a functional derivative is applied to , since the result is the same as applying a partial derivative.

Therefore, we discarded the Wheeler-DeWitt equation in [12], see also [13, Chapter 1], and expressed the Hamilton condition differently by looking at the evolution equation of the mean curvature of the foliation hypersurfaces and implementing the Hamilton condition on the right-hand side of this evolution equation. The left-hand side, a time derivative, we replaced with the help of the Poisson brackets. After canonical quantization the Poisson brackets became a commutator and now we could employ the fact that the derivatives are functional derivatives, since we had to differentiate the scalar curvature of a metric. As a result we obtained an elliptic differential operator in the base space, the main part of which was the Laplacian of the metric.

On the right-hand side of the evolution equation the interesting term was

the square of the mean curvature. It transformed to a second time derivative, the only remaining derivative with respect to a fiber variable, since the differentiations with respect to the other variables canceled each other. The resulting quantized equation is then a wave equation in a globally hyperbolic spacetime

where is the Cauchy hypersurface. When is a space of constant curvature then the wave equation, considered only for functions which do not depend on , is identical to the equation obtained by quantizing the Hamilton constraint in a Friedman universe without matter but including a cosmological constant.

There also exist temporal and spatial self-adjoint operators resp. such that the hyperbolic equation is equivalent to

where , and has a pure point spectrum with eigenvalues while, for , it is possible to find corresponding eigendistributions for each of the eigenvalues , if is asymptotically Euclidean or if the quantized spacetime is a Friedmann universe or a black hole with a negative cosmological constant, cf. [9, 14, 8, 11], see also [13, Chapters 3–6]. The hyperbolic equation then has a sequence of smooth solutions which are products of temporal eigenfunctions and spatial eigendistributions. Due to this „spectral resolution“ of the wave equation we were also able to apply quantum statistics to the quantized systems, cf. [10], see also [13, Chapter 6].

Recently, we applied these quantum gravitational results to spatially unbounded Friedmann universes in order to answer some questions related to dark energy, dark matter, inflation and the missing antimatter. cf. [14].

An overview of the results can be found here pdf file.

For details of the monograph The Quantization of Gravity [13] published by Springer International click here and for a review of the book by Paulo Moniz for the Mathematical Reviews here.

References

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