An Algebraic Approach to Koopman Classical Mechanics

Abstract

Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving a noncommutative operator algebraic variant of the Koopman-von Neumann approach. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical signal analysis approach that can also be successfully applied to the measurement problem of quantum mechanics. The development offers elementary mathematics that allows a formal reconciliation of "collapse" and "no-collapse" interpretations of quantum mechanics.

v5: new title, abstract, introduction, ..., for resubmission (major revision requested by referee)

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Cited by in corpus (11)

• Entanglement: Quantum or Classical?
• Eisenhart lift of Koopman-von Neumann mechanics
• The collapse of a quantum state as a joint probability construction
• Quantum dynamics of the classical harmonic oscillator
• Contextuality and the fundamental theorems of quantum mechanics
• Operational classical mechanics: Holonomic Systems
• Unitary representation of the Poincaré group for classical relativistic dynamics
• Lévy-Leblond Equation and Eisenhart-Duval lift in Koopman-von Neumann Mechanics
• Incompatible observables in classical physics: A closer look at measurement in Hamiltonian mechanics
• Connecting Commutativity and Classicality for Multi-Time Quantum Processes
• Evading Quantum Mechanics à la Sudarshan: quantum-mechanics-free subsystem as a realization of Koopman-von Neumann mechanics