A rigorous Keldysh functional integral for fermions

TL;DR

This paper establishes a rigorous mathematical framework for fermionic quantum field theories using the Keldysh functional integral, proving convergence, analyticity, and decay properties under general conditions.

Contribution

It introduces a mathematically rigorous Keldysh functional integral for fermions, with convergence proofs and bounds applicable to a broad class of dissipative quantum systems.

Findings

Proves convergence of Grassmann Gaussian integrals in the continuum limit.

Establishes bounds for the effective action and expectation values.

Demonstrates decay and clustering properties in the thermodynamic limit.

Abstract

We provide a mathematically rigorous Keldysh functional integral for fermionic quantum field theories. We show convergence of a discrete-time Grassmann Gaussian integral representation in the time-continuum limit under very general hypotheses. We also prove analyticity of the effective action and explicit bounds for the truncated (connected) expectation values of the non-equilibrium system. These bounds imply clustering with a summable decay in the thermodynamic limit, provided these properties hold at time zero, and provided that the determinant bound and decay constant of the fermionic Keldysh covariance are bounded uniformly in the volume. We then give bounds for these constants and show that uniformity in the volume indeed holds for a general class of systems. Finally we show that in the setting of dissipative quantum systems, these bounds are not necessarily restricted to short times.