Classical states, quantum field measurement

TL;DR

This paper constructs Lorentz-invariant classical state spaces for quantum fields using classical random fields, revealing that classical fields can exhibit quantum-like properties such as entanglement and noncommutativity.

Contribution

It introduces a classical Hilbert space framework for quantum fields, generalizing quantum measurement theories and showing classical fields can mimic quantum properties.

Findings

Classical state spaces support quantum field actions.

Classical observables exhibit noncommutativity similar to quantum mechanics.

Entangled states can be distinguished from mixed states in classical fields.

Abstract

Classical Koopman--von Neumann Hilbert spaces of states are constructed here by the action of classical random fields on a vacuum state in ways that support an action of the quantized electromagnetic field and of the $U(1)$--invariant observables of the quantized Dirac spinor field, allowing a manifestly Lorentz invariant classical understanding of the state spaces of the two field theories, generalizing the Quantum--Mechanics--Free Systems of Tsang&Caves and Quantum Non-Demolition measurements. The algebra of functions on a classical phase space is commutative but the algebra of classical observables associated with coordinate transformations is noncommutative, so that, for example, we can as much ask whether a classical state is an eigenstate of a rotation as we can in quantum mechanics and so that entangled states can be distinguished from mixed states, making classical random fields as weird as quantum fields.