Generating Functionals for Quantum Field Theories with Random Potentials

TL;DR

This paper develops a formalism using generating functionals and the replica trick to compute correlators in quantum field theories with random potentials, relevant for disordered condensed matter and cosmological models.

Contribution

It introduces two novel generating functionals employing the replica trick for averaging over random potentials in quantum field theories, enabling calculations of both in-out and in-in correlators.

Findings

Formalism for single scalar field with random mass and interactions

Two types of generating functionals for different correlator calculations

Application to Gaussian and Euclidean random potentials

Abstract

We consider generating functionals for computing correlators in quantum field theories with random potentials. Examples of such theories include condensed matter systems with quenched disorder (e.g. spin glass) or cosmological systems in context of the string theory landscape (e.g. cosmic inflation). We use the so-called replica trick to define two different generating functionals for calculating correlators of the quantum fields averaged over a given distribution of random potentials. The first generating functional is appropriate for calculating averaged (in-out) amplitudes and involves a single replica of fields, but the replica limit is taken to an (unphysical) negative one number of fields outside of the path integral. When the number of replicas is doubled the generating functional can also be used for calculating averaged probabilities (squared amplitudes) using the in-in construction. The second generating functional involves an infinite number of replicas, but can be used for calculating both in-out and in-in correlators and the replica limits are taken to only a zero number of fields. We discuss the formalism in details for a single real scalar field, but the generalization to more fields or to different types of fields is straightforward. We work out three examples: one where the mass of scalar field is treated as a random variable and two where the functional form of interactions is random, one described by a Gaussian random field and the other by a Euclidean action in the field configuration space.