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

It was recently shown that Newtonian dynamics of macroscopic particles can be derived from unitary Schr\"odinger evolution under a random-matrix assumption on the system-environment interaction. In that framework, classical phase space is realized geometrically as a manifold of localized equivalence classes in quantum state space, the tangent component of Schr\"odinger evolution reproduces Newtonian motion, and environmental interactions stabilize the state near this manifold. We extend this framework to quantum fields. The field itself is not assumed to become classical. Instead, macroscopic particles stabilized near the classical particle manifold interact with the field through the sector of field state space accessible to localized particle dynamics. The classical field is represented by the corresponding localized sector, and finite probe resolution leads to a quotient description in terms of localized equivalence classes of field states. The tangent component of the quantum-field Schr\"odinger dynamics on this localized quotient sector yields the corresponding classical field equations. Finite-dimensional simulations illustrate the mechanism for scalar and electromagnetic fields. The accessible field coordinates satisfy the sourced Klein--Gordon and Maxwell equations, and a localized test charge responds to the electromagnetic field through the Lorentz force. Thus classical field behavior emerges within unitary Schr\"odinger dynamics, without identifying the classical field with an expectation value, without relying on coherent states as special physical states, and without introducing a nonunitary collapse postulate.