Schr\"{o}dinger-picture formulation of a scalar quantum field driven by white noise

TL;DR

This paper introduces a Schr"{o}dinger-picture approach to a scalar quantum field influenced by white noise, providing exact solutions and analyzing the statistical and energetic properties of the field in a stochastic framework.

Contribution

It develops an exact, solvable stochastic Schr"{o}dinger equation formulation for a white-noise-driven quantum field, preserving Gaussian structure and linking quantum and classical stochastic dynamics.

Findings

Exact solutions for the wave functional kernel functions.

Expectation value of the field obeys classical stochastic equations.

Energy production rate matches Lindblad equation results.

Abstract

We develop a Schr\"{o}dinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schr\"{o}dinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler-Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.