Heat kernel methods for Lifshitz theories

Andrei O Barvinsky,Mario Herrero-Valea,Diego Blas,Guillem Pérez-Nadal,Dmitry V Nesterov,Christian F Steinwachs

doi:10.1007/jhep06(2017)063

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.

Highlights

Natural situation in condensed matter physics, e.g [1, 2]
Quantum Lifshitz theories are currently being explored in a variety of fields, from condensed matter [1, 2, 35] to quantum gravity [3, 4]. This interest stems from the adequacy of these theories to describe the behaviour close to fixed points with anisotropic properties between space and time, which may be present in different physical situations or even at a fundamental level
In this work we have developed a formalism to efficiently compute the one-loop effective action for such theories in curved backgrounds

Summary

Introduction

Natural situation in condensed matter physics, e.g [1, 2]. It has been suggested that this may happen at a fundamental level, since a renormalizable quantum theory of gravity in 4-dimensions seems possible by imposing anisotropy between space and time, a defining feature of Lifshitz theories [3, 4].1 Lifshitz theories will be the main focus of our work. In this work we provide general methods for the heat kernel approach to the one-loop effective action for Lifshitz theories which can be applied to particular physical models. The heat kernel technique is a powerful tool in quantum field theory to calculate the quantum effective action in an arbitrary field background — a fundamental quantity which contains almost all the information about the quantum system and from which many of the results of quantization can be derived directly This approach, initially originating from asymptotic expansion methods for solutions of partial differential equations, turned out to be very efficient both in physical and mathematical applications [38,39,40] (see [41]).

Objectives

Methods

Conclusion