BRST technique for the cosmological density matrix

TL;DR

This paper develops a BRST-based path integral approach to define the microcanonical density matrix in closed cosmology, addressing challenges in quantizing gravitational systems without global charges.

Contribution

It introduces a novel BRST/BFV formalism application to derive the cosmological density matrix and statistical sum, overcoming issues with noncommutative Dirac constraints.

Findings

Path integral representation for the cosmological projector derived.

Regularization of the physical inner product achieved via BRS gauge fixing.

Framework simplifies quantization in closed gravitational systems.

Abstract

The microcanonical density matrix in closed cosmology has a natural definition as a projector on the space of solutions of Wheeler-DeWitt equations, which is motivated by the absence of global non-vanishing charges and energy in spatially closed gravitational systems. Using the BRST/BFV formalism in relativistic phase space of gauge and ghost variables we derive the path integral representation for this projector and the relevant statistical sum. This derivation circumvents the difficulties associated with the open algebra of noncommutative quantum Dirac constraints and the construction/regularization of the physical inner product in the subspace of BRS singlets. This inner product is achieved via the Batalin-Marnelius gauge fixing in the space of BRS-invariant states, which in its turn is shown to be a result of truncation of the BRST/BFV formalism to the "matter" sector of relativistic phase space.