Classical states, quantum field measurement

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.

v6: As accepted by Physica Scripta. v5: As submitted to Physica Scripta. Accepted Manuscript embargoed for 12 months. v4: 16 pages. Koopman-von Neumann given its proper dues; more reorganization of the text, math unchanged. v3: 13 pages. Reorganized. v2: 6 pages. Additional calculations

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Cited by in corpus (3)

• Entanglement: Quantum or Classical?
• An Algebraic Approach to Koopman Classical Mechanics
• Quantum dynamics of the classical harmonic oscillator