Doi-Peliti Methods for Non-commuting Observables

The quantum-field methods derived by Doi and Peliti to treat reaction-diffusion problems were originally based on a classical probability description and on Markovian evolution. The fully-decoherent limit of an underlying quantum transport process, which these assumptions entail, restricts the classical description to the commuting observables of the quantum formulation, and coarse-grains the entropy to the equilibrium Boltzmann form. In this paper the Doi-Peliti formulation is generalized to describe driven quantum ensembles, for which the classical state variables arise as expectations of non-commuting quantum observables. The resulting classical description remains of Martin-Siggia-Rose form, but preserves the exact entropy of the quantum density matrix, including its symmetries under unitary transformation of the state variables and the resulting dependence on classical currents required by these symmetries. The Fokker-Planck equation defines the universality class of driven quantum ensembles, and within this class the classical state-variable field theory is invertible to the underlying Schwinger-Keldysh time-loop quantum formulation.