Schwinger-Keldysh superspace in quantum mechanics

TL;DR

This paper explores the Hilbert space representation of BRST symmetry in Schwinger-Keldysh quantum mechanics, clarifying the role of ghost fields and operator algebra in the superspace formalism.

Contribution

It explicitly constructs the Hilbert space structure of the Schwinger-Keldysh superspace, revealing the natural emergence of ghost insertions and their impact on operator algebra.

Findings

Background ghost insertions arise naturally in the formalism.

Schwinger-Keldysh difference operators are dressed by ghost bilinears.

The final state is BRST closed and ghost fields have specific boundary conditions.

Abstract

We examine, in a quantum mechanical setting, the Hilbert space representation of the BRST symmetry associated with Schwinger-Keldysh path integrals. This structure had been postulated to encode important constraints on influence functionals in coarse-grained systems with dissipation, or in open quantum systems. Operationally, this entails uplifting the standard Schwinger-Keldysh two-copy formalism into superspace by appending BRST ghost degrees of freedom. These statements were previously argued at the level of the correlation functions. We provide herein a complementary perspective by working out the Hilbert space structure explicitly. Our analysis clarifies two crucial issues not evident in earlier works: firstly, certain background ghost insertions necessary to reproduce the correct Schwinger-Keldysh correlators arise naturally. Secondly, the Schwinger-Keldysh difference operators are systematically dressed by the ghost bilinears, which turn out to be necessary to give rise to a consistent operator algebra. We also elaborate on the structure of the final state (which is BRST closed) and the future boundary condition of the ghost fields.