Heat kernel and worldline path integrals for the Proca theory

In this thesis we study the heat kernel, a useful tool to analyze various properties of different quantum field theories. In particular, we focus on the study of the one-loop effective action and the application of worldline path integrals to derive perturbatively the heat kernel coefficients for the Proca theory of massive vector fields. It turns out that the worldline path integral method encounters some difficulties if the differential operator of the heat kernel is of non-minimal kind. More precisely, a direct recasting of the differential operator in terms of worldline path integrals, produces in the classical action a non-perturbative vertex and the path integral cannot be solved. In this work we wish to find ways to circumvent this issue and to give a suggestion to solve similar problems in other contexts.