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On Koopman-von Neumann Waves
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In this paper we study the classical Hilbert space introduced by Koopman and von Neumann in their operatorial formulation of classical mechanics. In particular we show that the states of this Hilbert space do not spread, differently than what happens in quantum mechanics. The role of the phases associated to these classical "wave functions" is analyzed in details. In this framework we also perform the analog of the two-slit experiment and compare it with the quantum case.
Forward citations
Cited by 7 Pith papers
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The History of Hilbert-Space Formulations of Classical Physics
Classical wave functions with ρ = |ψ|² were introduced after Koopman and von Neumann, probably by Schönberg, so calling them 'KvN waves' is a misattribution.
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Quantum observables for probabilistic classical particles
Solutions of the Liouville equation can be rewritten as a Schrödinger equation whose observables are non-commuting 'quantum' operators, reproducing the harmonic oscillator and hydrogen atom spectra as special subsystems.
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Classical probabilistic realisation of quantum double-slit interference
Classical probability distributions over complex scalar fields realize Schrödinger dynamics and double-slit interference for a quantum particle via conserved-charge subsystems.
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Quantum mechanics for classical transport equations
Classical probabilistic transport equations are reformulated as quantum systems whose wave function obeys Schrödinger evolution and whose observables include non-commuting operators for statistical quantities.
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Cosmic Large-Scale Structure Formation from Newtonian Particle Dynamics
The authors present tree-level and one-loop power spectra from their particle-based field theory, matching simulations on large scales but deviating by 10-20% at intermediate scales.
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Quantum field theory for classical fields
A classical Klein-Gordon field with random initial conditions, described through specially defined fluctuating observables, obeys the functional-integral rules of a quantum field theory.
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Note About Koopman-von Neumann Theory and Density Matrix
The note argues that the classical N-particle distribution equals the diagonal of the density matrix operator in coordinate representation and derives a generalized BBGKY hierarchy for reduced density matrices.
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