Path integral approach to heat kernel in massive gravity

In this thesis, we consider the Fierz-Pauli theory of linearized massive gravity in an Einstein spacetime. The aim is to study the one-loop effective action of this theory employing the heat kernel method, which consists in a variety of perturbation methods applied to minimal second order operators on manifolds, which allow us to study asymptotic expansions and singularities of Green functions. It is a powerful technique in mathematical physics, with applications ranging from black hole entropy to mathematical finance. In the Fierz-Pauli model of massive gravity, the operator entering the heat kernel is non-minimal, so we need ways to relate it to minimal operators in order to avoid a rather tedious treatment of the heat kernel expansion in the presence of non-minimal operators, which can be analyzed either by means of covariant projectors or by employing the reduction method suggested by Barvinsky and Vilkovisky. Our approach is based on computing the path integral of the Fierz-Pauli action with the Faddeev-Popov procedure, using appropriate gauge-fixing functions. In fact, the addition of a mass term to the action for massless gravity breaks the gauge symmetry of the theory, which is the general coordinate invariance. Because of this, the Fierz-Pauli theory of massive gravity is not a gauge theory and, of course, a gauge-fixing cannot be performed. Nevertheless, by first introducing new fields in the theory with the so-called Stückelberg trick, we can restore a gauge symmetry to the theory. These manipulations allow us to perform the computation of the path integral and the evaluation of the heat kernel coefficients by using the well-known Seeley-DeWitt method.