Schwinger-Keldysh formalism II: Thermal equivariant cohomology

TL;DR

This work presents a rigorous mathematical framing of the Schwinger-Keldysh SK-KMS algebra as an extended (two-generator) equivariant cohomology, casting the causality Ward identities of near-thermal quantum dynamics into cohomological language. By embedding SK and KMS symmetries into a superspace with two Grassmann directions, the authors derive a consistent $\mathcal{N}_{T}=2$ algebra that organizes BRST-like charges and thermal translations, and then gauge this thermal translation symmetry into a $U(1)_{T}$ connection. The Langevin (Brownian) dynamics example demonstrates how a dissipative effective action emerges as a gauge-invariant, cohomological construction, with fluctuation-dissipation relations and Jarzynski work relations appearing as Ward identities in the CPT-broken phase. The framework provides a robust, renormalization-group-friendly approach to low-energy near-thermal physics and offers a path to hydrodynamic effective actions and potential insights into black-hole physics through thermal equivariant cohomology. Overall, the work unifies SK thermal physics with extended equivariant cohomology, enabling systematic, symmetry-protected statements about dissipative dynamics and non-equilibrium processes in quantum systems.

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.

Figures (2)

• Figure 1: An illustration of the dependencies of various ideas presented here and in Haehl:2016pec. The left hand side shows how basic ingredients of Schwinger-Keldysh formalism lead to an equivariant cohomology algebra; these are largely the content of Haehl:2016pec and reviewed here only briefly. The present paper follows the logic on the right hand side. Part \ref{['part:maths']} reviews the mathematical structure of equivariant cohomology and derives from there the algebras uncovered in Haehl:2016pec as a special case of the general formalism when the group acts by thermal translations. For a considerably more extensive list of applications, see §11 of Haehl:2016pec.
• Figure 2: Illustration of the spacetime picture as it emerges from the proposed KMS gauge invariance. We upgrade the spacetime manifold on which our quantum system resides to a thermal fibre bundle. The grey manifold represents a Lorentzian spacetime with a typical Cauchy slice indicated in red. We assume local thermal equilibrium (as in, e.g., hydrodynamics) at each spacetime point which guarantees a thermal vector ${\bm \beta}^\mu$. Geometrically we encode this vector field as a circle fibration with a thermal circle whose size is set by the local temperature. The KMS transformations we seek implement equivariance with respect to thermal translations along this local imaginary time circle. Restricting to a gauge slice corresponds picking a Lorentzian section of this fibration. Note that the size of the thermal circle is exaggerated; our arguments are clean in the high temperature limit where the size of the thermal circle is much smaller than the fluctuation scale.