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 compan-ion paper [1]. 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 ba-sic 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 prin-ciples, 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.
Highlights
Introduction and background1 IntroductionThe Schwinger-Keldysh formalism [2, 3] was developed to enable one to compute temporally ordered correlation functions in arbitrary states of a relativistic QFT
To illustrate the general principles, we focus on Langevin dynamics of a Brownian particle, rephrasing some known results in terms of thermal equivariant cohomology
Rather we describe in some detail the much simpler problem of a Brownian particle undergoing Langevin dynamics using the machinery of thermal equivariant cohomology
Summary
The Schwinger-Keldysh formalism [2, 3] was developed to enable one to compute temporally ordered correlation functions in arbitrary states (either pure or mixed) of a relativistic QFT. The established mathematical framework allows us to give a more precise characterization of the SK-KMS algebraic structures, and we explore basic properties of the thermal translational symmetry that is crucial in this scheme, see section 6 At this point, we are ready for physical applications. In order to keep the material accessible for non-experts, we have chosen to illustrate the general construction using a very simple system of dissipative dynamics: the stochastic motion of a Brownian particle as described by Langevin dynamics This is reviewed in some detail, both from the traditional viewpoint on the topological supersymmetry as presented in standard textbooks and in the language of the equivariant cohomology, amplifying our discussion in the appendix A of [10]. The appendices contain some further elaboration of some of the mathematical results, in particular, the Mathai-Quillen and Kalkman isomorphisms between the various algebraic constructions
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