Ruprecht-Karls-Universität Heidelberg

Institut für
Angewandte Mathematik

Prof. Dr. Claus Gerhardt

Institut für Angewandte Mathematik
Im Neuenheimer Feld 205
D-69120 Heidelberg

Tel. + 49 (0) 62 21 - 54 14100 (Sekretariat)
email: gerhardt@math.uni-heidelberg.de


Area of Research: Partial Differential Equations, Differential Geometry and General Relativity


The quantization of gravity interacting with a Yang-Mills and Higgs field

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 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 and the diffeomorphism constraint. Dirac proved how to quantize a constrained Hamiltonian system---at least in principle---and his method has been applied to the Hamiltonian setting of gravity. 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.
However, in a series of papers we achieved the quantization of gravity for general hyperbolic spacetimes and developed a mathematical model which describes the quantized interaction of gravity with a Yang-Mills and Higgs field. An overview of the results can be found here The quantization of gravity interacting with a Yang-Mills and Higgs field (pdf).


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