Hilbert Space Theory of Classical Electrodynamics

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.