Hilbert Space Theory of Classical Electrodynamics

TL;DR

This paper reformulates classical electrodynamics using Hilbert space wave functions, clarifying the quantum-like features in classical optics and exploring how classical processes can mimic quantum information protocols.

Contribution

It introduces a Hilbert space framework for classical electrodynamics, enabling analysis of quantum-like phenomena and classical implementations of quantum processes.

Findings

Classical electrodynamics can be represented in Hilbert space without interference restrictions.

Quantum processes preserving Wigner function positivity can be realized with classical optics.

The framework clarifies the connection between classical optics and quantum information concepts.

Abstract

Classical electrodynamics is reformulated in terms of wave functions in the classical phase space of electrodynamics, following the Koopman-von Neumann-Sudarshan prescription for classical mechanics on Hilbert spaces {\em sans} the superselection rule which prohibits interference effects in classical mechanics. This is accomplished by transforming from a set of commuting observables in one Hilbert space to another set of commuting observables in a larger Hilbert space. This is necessary to clarify the theoretical basis of much recent work on quantum-like features exhibited by classical optics. Furthermore, following Bondar et al ({\em Phys.Rev. A} {\bf 88}, 052108, (2013)), it is pointed out that quantum processes that preserve the positivity or nonpositivity of the Wigner function can be implemented by classical optics. This may be useful in interpreting quantum information processing in terms of classical optics.