Schwinger-Keldysh formalism I: BRST symmetries and superspace

TL;DR

This paper reviews the Schwinger-Keldysh formalism for quantum dynamics out-of-equilibrium, emphasizing BRST symmetries and superspace techniques to enhance understanding of low energy dynamics and out-of-time-order correlations.

Contribution

It introduces a superspace formalism revealing BRST symmetries in the Schwinger-Keldysh approach, facilitating analysis of out-of-time-order correlators and complex quantum systems.

Findings

BRST symmetries are inherent and robust in the formalism

Superspace approach makes symmetries manifest and easier to analyze

Framework applicable to chaos, scrambling, and black hole physics

Abstract

We review the Schwinger-Keldysh, or in-in, formalism for studying quantum dynamics of systems out-of-equilibrium. The main motivation is to rephrase well known facts in the subject in a mathematically elegant setting, by exhibiting a set of BRST symmetries inherent in the construction. We show how these fundamental symmetries can be made manifest by working in a superspace formalism. We argue that this rephrasing is extremely efficacious in understanding low energy dynamics following the usual renormalization group approach, for the BRST symmetries are robust under integrating out degrees of freedom. In addition we discuss potential generalizations of the formalism that allow us to compute out-of-time-order correlation functions that have been the focus of recent attention in the context of chaos and scrambling. We also outline a set of problems ranging from stochastic dynamics, hydrodynamics, dynamics of entanglement in QFTs, and the physics of black holes and cosmology, where we believe this framework could play a crucial role in demystifying various confusions. Our companion paper arXiv:1610.01941 describes in greater detail the mathematical framework embodying the topological symmetries we uncover here.