Topics in Koopman-von Neumann Theory

TL;DR

This thesis explores the operatorial approach to classical mechanics via Koopman-von Neumann theory, analyzing phases, classical-quantum analogies, and extensions to forms and Jacobi fields, revealing fundamental limitations and ongoing quantization research.

Contribution

It provides new insights into the phase role in KvN states, extends the formalism to differential forms, and discusses the impossibility of simultaneous positivity and unitarity in the extended space.

Findings

Analysis of phases in KvN states and classical-quantum comparisons.

Extension of KvN formalism to forms and Jacobi fields using Grassmann variables.

Demonstration that positive definiteness and unitarity cannot coexist in the extended Hilbert space.

Abstract

In this thesis we study several features of the operatorial approach to classical mechanics pionereed by Koopman and von Neumann (KvN) in the Thirties. In particular in the first part we study the role of the phases of the KvN states. We analyze, within the KvN theory, the two-slit experiment and the Aharonov-Bohm effect and we make a comparison between the classical and the quantum case. In the second part of the thesis we study the extension of the KvN formalism to the space of forms and Jacobi fields. We first show that all the standard Cartan calculus on symplectic spaces can be performed via Grassmann variables or via suitable combinations of Pauli matrices. Second we study the extended Hilbert space of KvN which now includes forms and prove that it is impossible to have at the same time a positive definite scalar product and a unitary evolution. Clear physical reasons for this phenomenon are exhibited. We conclude the thesis with some work in progress on the issue of quantization.