  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.
