Schwinger-Keldysh formalism II: Thermal equivariant cohomology

TL;DR

This paper explores the mathematical structure of the Schwinger-Keldysh formalism for near-thermal quantum systems, revealing a connection to equivariant cohomology that helps understand symmetries in low-energy effective theories.

Contribution

It introduces a thermal equivariant cohomology framework for the Schwinger-Keldysh formalism, providing a new algebraic perspective on symmetries in near-thermal quantum systems.

Findings

Reformulation of Schwinger-Keldysh algebra as thermal equivariant cohomology.

Application to Langevin dynamics of Brownian particles.

Potential implications for effective actions in dissipative hydrodynamics and black hole physics.

Abstract

Causally ordered correlation functions of local operators in near-thermal quantum systems computed using the Schwinger-Keldysh formalism obey a set of Ward identities. These can be understood rather simply as the consequence of a topological (BRST) algebra, called the universal Schwinger-Keldysh superalgebra, as explained in our companion paper arXiv:1610.01940. In the present paper we provide a mathematical discussion of this topological algebra. In particular, we argue that the structures can be understood in the language of extended equivariant cohomology. To keep the discussion self-contained, we provide a basic review of the algebraic construction of equivariant cohomology and explain how it can be understood in familiar terms as a superspace gauge algebra. We demonstrate how the Schwinger-Keldysh construction can be succinctly encoded in terms a thermal equivariant cohomology algebra which naturally acts on the operator (super)-algebra of the quantum system. The main rationale behind this exploration is to extract symmetry statements which are robust under renormalization group flow and can hence be used to understand low-energy effective field theory of near-thermal physics. To illustrate the general principles, we focus on Langevin dynamics of a Brownian particle, rephrasing some known results in terms of thermal equivariant cohomology. As described elsewhere, the general framework enables construction of effective actions for dissipative hydrodynamics and could potentially illumine our understanding of black holes.