Heat Kernel Approach in Quantum Field Theory

TL;DR

This paper reviews the heat kernel method for calculating effective actions in quantum field theory and gravity, focusing on asymptotic expansions for various differential operators on different manifold types.

Contribution

It provides a comprehensive overview of methods for heat kernel expansion applicable to a wide range of operators and boundary conditions in quantum field theory and gravity.

Findings

Summarizes techniques for heat kernel asymptotic expansion.

Analyzes operators on manifolds with and without boundary.

Discusses boundary conditions and their impact on calculations.

Abstract

We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.