Curvature problems
Final Report Abstract
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, we achieved the quantization of gravity for general hyperbolic spacetimes by introducing four new ideas in the process of canonical quantization. The most important one was to eliminate the diffeomorphism constraint by proving that it suffices to consider metrics that split with respect to a fixed global time function and proving that the Euler-Lagrange equations of the Einstein-Hilbert functional derived in this particular class of metrics are still the full Einstein equations. Then after applying the Legendre transformation the only remaining constraint was the Hamilton constraint which was transformed to a hyperbolic equation in a fiber bundle after quantization. We were then able to define a symplectic vector space and apply the techniques of algebraic quantum field theory to obtain a CCR representation. After the quantization of gravity we applied similar ideas to the interaction of gravity with a Yang-Mills field by eliminating the Gauss constraint and finally we succeeded to quantize the interaction of gravity with Yang-Mills and spinor fields thereby achieving a unified quantum theory incorporating the four fundamental forces of nature.
Publications
- A unified quantum theory I: gravity interacting with a Yang-Mills field, 2012, ATMP
Claus Gerhardt
(See online at https://doi.org/10.4310/ATMP.2014.v18.n5.a2) - The quantization of gravity in globally hyperbolic spacetimes, 2012, ATMP
Claus Gerhardt
(See online at https://doi.org/10.4310/ATMP.2013.v17.n6.a5) - A unified quantum theory II: gravity interacting with Yang-Mills and spinor fields, 2013
Claus Gerhardt
