Pseudo-orthogonal Yang-Mills theories and connections to gravity

Abstract

The goal of this thesis is to study pseudo-orthogonal Yang-Mills actions and understand under which conditions they contain the Hilbert-Einstein action free of instabilities. The physical motivation is to construct a renormalizable theory of quantum gravity. We first provide the mathematical background necessary to introduce gauge theories in curved spacetime and General Relativity. Subsequently, we develop the concept of geometrical Yang-Mills theory, i.e. YangMills theories for which part of the gauge connection takes also the role of the cotetrad fields. The resulting theory retains only a part of the original gauge group as its group of symmetry. We show that for some pseudo-orthogonal groups one obtains the Hilbert action (and consequently Einstein’s equations) as part of the theory. Then, specializing to a coordinate system, we derive the Hamiltonian of such theories and we analyze the constraints that arise in phase space due to the redundancy of the Yang-Mills Lagrangian. We study the class of such constraints and it turns out that the theory possesses both first and second class constraints. Finally, we establish the conditions under which the constraints are preserved by the evolution. The next steps – left for future work – would include the study of dynamics, stability and symmetry breaking to the theory that at low energies would be equivalent to Einstein’s general relativity.