Interaction with the Environment via Random Matrices and the Emergence of Classical Field Theory

Abstract

It was recently shown that Newtonian dynamics of macroscopic particles can be derived from unitary Schrödinger evolution under an assumption on the system-environment interaction, namely that the interaction Hamiltonian effectively exhibits a random-matrix structure, leading to stochastic yet unitary evolution on state space. The derivation is geometric: classical phase space is realized as a submanifold of quantum state space, and Schrödinger evolution, when restricted to the corresponding tangent bundle, reproduces Newtonian motion, while environmental interactions ensure localization near this submanifold. In the present work, this framework is extended to quantum fields. We construct manifolds of states localized near classical field configurations and show that classical fields arise as coordinates on these manifolds. The extension is achieved by embedding both particle and field degrees of freedom into a joint state-space geometry and analyzing the induced evolution on the tangent bundle of localized states. Within this setting, the unitary Schrödinger dynamics, combined with the random-matrix model of system-environment interaction, yields effective diffusion in state space together with repeated localization due to environmental recording. As a result, although field states are not themselves confined near classical configurations, the interaction constrains the particle to probe only a restricted sector of the field, corresponding to a tubular neighborhood of localized field states. The resulting dynamics reproduces classical field equations, including the sourced Klein-Gordon equation and the corresponding force law. Classical field behavior thus emerges from unitary quantum dynamics without recourse to coherent states or modifications of the Schrödinger equation, and the formulation extends naturally to other fields, including the electromagnetic field.