Path integrals in curved space and the worldline formalism
Fiorenzo Bastianelli
TL;DR
The paper develops and applies the worldline formalism to quantum field theories in curved space, addressing how to define unambiguous path integrals for particles moving in a curved background and how to use them to compute one-loop effective actions and amplitudes in the presence of external gravity. It extends the framework to vector and antisymmetric tensor fields via the N=2 spinning-particle model, enabling the extraction of Seeley-DeWitt coefficients for all p-forms and the examination of exact duality relations between p-forms and their duals. It also analyzes regularization schemes—mode regularization, time slicing, and dimensional regularization—and shows how suitable counterterms render the results regulator-independent, with concrete demonstrations on curvature-related calculations. Together, these results provide a practical, regulator-insensitive toolkit for one-loop QFT in curved backgrounds and illuminate topological and duality aspects of higher-form fields in external gravity.
Abstract
We describe how to construct and compute unambiguously path integrals for particles moving in a curved space, and how these path integrals can be used to calculate Feynman graphs and effective actions for various quantum field theories with external gravity in the framework of the worldline formalism. In particular, we review a recent application of this worldline approach and discuss vector and antisymmetric tensor fields coupled to gravity. This requires the construction of a path integral for the N=2 spinning particle, which is used to compute the first three Seeley-DeWitt coefficients for all p-form gauge fields in all dimensions and to derive exact duality relations.