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arXiv:2604.03778

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

Alexey A. Kryukov

quant-phSubmitted Apr 4, 2026• Announced Apr 7, 2026• 0 citations
v1 · Apr 4, 2026v2 · Jun 26, 2026
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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.

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Introduction

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In contemporary field theory, the word classical is reserved for an analytical framework in which the local equations of motion provide a complete description of the evolution of the fields. Classical field theory is a differential expression of change in functions of space and time, which summarizes the state of a physical system entirely in terms of smooth fields. The differential (holonomic) structure of field theory, derived from the action principle, implies that field theories are microscopically reversible by design: differential changes experience no significant obstacles in a system and may be trivially undone. Yet, when summed macroscopically, in the context of an environment, such individually reversible changes lead to the well known irreversible behaviours of thermodynamics: the reversal of paths through an environmental landscape would require the full history of the route taken. Classical field theory thus forms a basis for both the microscopic and the macroscopic. When applied to quantum mechanics, the classical framework is sometimes called the first quantization . The first quantization may be considered the first stage of a more complete theory, which goes on to deal with the issues of many–particle symmetries and interacting fields. Quantum mechanics is classical field theory with additional assumptions about measurement. The term quantum mechanics is used as a name for the specific theory of the Schrödinger equation, which one learns about in undergraduate studies, but it is also sometimes used for any fundamental description of physics, which employs the measurement axioms of Schrödinger quantum mechanics, i.e. where change is expressed in terms of fields and groups.

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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.
The Newtonian motion of a macroscopic particle is derived from the linear Schrödinger equation with a Hamiltonian consisting of the free-particle term and a random Hamiltonian drawn from the Gaussian Unitary Ensemble. The random term models interaction with the environment. We show that the parameters governing the resulting state-space random walk, together with the treatment of experimentally indistinguishable states as equivalence classes, explain the contrasting behavior of microscopic and macroscopic systems. The analysis extends previous work deriving the Born rule for microscopic particles when the free-particle term is negligible.
We develop a state-space geometric framework for measurement, classicality, and quantum paradoxes, based on one dynamical conjecture. Classical configuration space and classical phase space for a mechanical system arise as distinguished submanifolds of projective quantum state space. On these submanifolds, the Fubini--Study geometry induces Euclidean classical geometry, and the tangent component of Schrödinger evolution reproduces Newtonian dynamics. Within this framework, interactions with measuring devices and environments are described by random-matrix dynamics on projective state space, generated by matrices drawn from the Gaussian Unitary Ensemble. We show that this random-matrix dynamics yields isotropic diffusion, giving Born-rule transition probabilities in microscopic measurements and stabilizing classical behavior in macroscopic systems. We further argue that the random-matrix conjecture is not an independent ad hoc assumption: under natural translation-invariance assumptions on the distribution of state-space steps originating on the classical submanifold, the unitary lift of homogeneous and isotropic Brownian motion on that submanifold is uniquely given by the Gaussian Unitary Ensemble, up to scale and an irrelevant scalar part. The resulting framework provides a unitary account of measurement and the quantum-to-classical transition and, if accepted, offers a dynamical resolution of standard quantum paradoxes.
We develop a dynamical framework for quantum measurement based on stochastic but unitary evolution in projective state space. Random Hamiltonians drawn from the Gaussian Unitary Ensemble generate stochastic unitary dynamics of the quantum state, while equivalence classes reflecting finite detector resolution define classical observables as well as classical configuration-space and phase-space submanifolds. When the evolution is constrained to the phase-space submanifold, free Schrödinger dynamics reduces to Newtonian motion, while stochastic motion constrained to the classical configuration-space submanifold yields ordinary Brownian motion in classical space. Transition probabilities under the stochastic dynamics satisfy the Born rule, whereas the constrained classical evolution produces the normal probability distributions characteristic of classical measurements. We show that, in this setting, measurement, state reduction, and the quantum-classical transition emerge from unitary dynamics alone, without invoking nonunitary collapse or additional postulates. Entanglement and EPR correlations arise geometrically from the evolution of joint states in composite state space, preserving locality in spacetime. The framework provides a unified dynamical account of measurement and classicality compatible with the structure of quantum mechanics.
It was recently shown that the Born rule for transition probabilities of a microscopic particle can be derived from the linear Schr{\"o}dinger equation with a Hamiltonian represented by random matrices from the Gaussian Unitary Ensemble, taken to be independent at different time points. Under such an evolution, the particle's state undergoes an isotropic random walk on the projective state space, and the relative frequencies of reaching different regions obey the Born rule. In this work, we extend these results to demonstrate how the same framework accounts for the emergence of Newtonian dynamics for macroscopic bodies on a submanifold of state space, thereby providing a unified geometric account of the quantum-classical transition.
In contemporary field theory, the word classical is reserved for an analytical framework in which the local equations of motion provide a complete description of the evolution of the fields. Classical field theory is a differential expression of change in functions of space and time, which summarizes the state of a physical system entirely in terms of smooth fields. The differential (holonomic) structure of field theory, derived from the action principle, implies that field theories are microscopically reversible by design: differential changes experience no significant obstacles in a system and may be trivially undone. Yet, when summed macroscopically, in the context of an environment, such individually reversible changes lead to the well known irreversible behaviours of thermodynamics: the reversal of paths through an environmental landscape would require the full history of the route taken. Classical field theory thus forms a basis for both the microscopic and the macroscopic. When applied to quantum mechanics, the classical framework is sometimes called the first quantization . The first quantization may be considered the first stage of a more complete theory, which goes on to deal with the issues of many–particle symmetries and interacting fields. Quantum mechanics is classical field theory with additional assumptions about measurement. The term quantum mechanics is used as a name for the specific theory of the Schrödinger equation, which one learns about in undergraduate studies, but it is also sometimes used for any fundamental description of physics, which employs the measurement axioms of Schrödinger quantum mechanics, i.e. where change is expressed in terms of fields and groups.
Recently, a geometric embedding of the classical space and classical phase space of an n-particle system into the space of states of the system was constructed and shown to be physically meaningful. Namely, the Newtonian dynamics of the particles was recovered from the Schroedinger dynamics by constraining the state of the system to the classical phase space submanifold of the space of states. A series of theorems related to the embedding and the Schroedinger evolution with a random Hamiltonian was proven and shown to be applicable to the process of measurement in classical and quantum mechanics. Here, these results are applied to have a fresh look at the main quantum-mechanical thought experiments and paradoxes and to provide a new insight into the process of collapse and the motion of macroscopic bodies in quantum mechanics.