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 Koopmanvon Neumann approach. In this form, the measurement theory for unary classical mechanics can be the same 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 "nocollapse" interpretations of quantum mechanics.
 Publication:

Annals of Physics
 Pub Date:
 March 2020
 DOI:
 10.1016/j.aop.2020.168090
 arXiv:
 arXiv:1901.00526
 Bibcode:
 2020AnPhy.41468090M
 Keywords:

 Classical mechanics;
 Koopmanvon Neumann formalism;
 Quantum mechanics;
 Quantum Physics;
 Physics  Classical Physics
 EPrint:
 v5: new title, abstract, introduction, ..., for resubmission (major revision requested by referee)