# An Algebraic Approach to Koopman Classical Mechanics

## TL;DR

This paper reformulates classical mechanics using algebraic operators and constructs a Hilbert space framework, bridging classical and quantum mechanics and offering new insights into the measurement problem.

## Contribution

It introduces a unary operator algebraic formulation of classical mechanics that parallels quantum mechanics, enabling a unified measurement theory and addressing the measurement problem.

## Key findings

- Classical mechanics can be expressed with noncommutative operators.
- A Hilbert space representation of classical mechanics is constructed.
- The approach provides a formal way to reconcile collapse and no-collapse interpretations.

## 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.