# Heat kernel methods for Lifshitz theories

## TL;DR

This paper develops a systematic heat kernel approach to compute the one-loop effective action in Lifshitz theories, which feature anisotropic scaling and broken Lorentz invariance, providing new tools for quantum field theory in such backgrounds.

## Contribution

It introduces a method to reduce Lifshitz operator calculations to relativistic cases and simplifies computations for special operators, advancing quantum field theory techniques.

## Key findings

- Efficient algorithms for Lifshitz effective action calculations
- Simplified techniques for special Lifshitz operators
- Explicit applications demonstrating method effectiveness

## Abstract

We study the one-loop covariant effective action of Lifshitz theories using the heat kernel technique. The characteristic feature of Lifshitz theories is an anisotropic scaling between space and time. This is enforced by the existence of a preferred foliation of space-time, which breaks Lorentz invariance. In contrast to the relativistic case, covariant Lifshitz theories are only invariant under diffeomorphisms preserving the foliation structure. We develop a systematic method to reduce the calculation of the effective action for a generic Lifshitz operator to an algorithm acting on known results for relativistic operators. In addition, we present techniques that drastically simplify the calculation for operators with special properties. We demonstrate the efficiency of these methods by explicit applications.