Dirac-von Neumann Type Axiomatic Structure for Classical Electromagnetism

TL;DR

This paper establishes a Hilbert space framework for classical electromagnetism using Dirac-von Neumann axioms, revealing classical analogues of quantum concepts like commutation relations and uncertainty principles.

Contribution

It introduces a Hilbert space formulation for classical electromagnetism, paralleling quantum mechanics, and derives classical analogues of quantum operators, commutation relations, and uncertainty principles.

Findings

Existence of a classical Hilbert space with Hermitian operators.

Classical analogue of the Heisenberg Uncertainty Principle.

Relativistic proof of Maxwell's equations without Galilean relativity.

Abstract

We demonstrate the existence of a complex Hilbert Space with Hermitian operators for calculations in \textit{classical} electromagnetism that parallels the Hilbert Space of quantum mechanics. The axioms of this classical theory are the so-called Dirac-von Neumann axioms, however, with classical potentials in place of the wavefunction and the indeterministic collapse postulate removed. This approach lets us derive a variety of fundamental expressions for electromagnetism using minimal mathematics and a calculation sequence well-known for traditional quantum mechanics. We also demonstrate the existence of the wave commutation relationship $[\hat{x},\hat{k}]=i$, which is a unique classical analogue to the canonical commutator $[\hat{x},\hat{p}]=i\hbar$. The difference between classical and quantum mechanics lies in the presence of $\hbar$. The noncommutativity of observables for a classical theory simply reflects its wavenature. A classical analogue of the Heisenberg Uncertainty Principle is developed for electromagnetic waves, and its implications discussed. Further comparisons between electromagnetism, Koopman-von Neumann-Sudarshan (KvNS) classical mechanics (for point particles), and quantum mechanics are made. Finally, supplementing the analysis presented, we additionally demonstrate an elegant, completely relativistic version of Feynman's proof of Maxwell's equations \citep{Dyson}. Unlike what \citet{Dyson} indicated, there is no need for Galilean relativity for the proof to work. This fits parsimoniously with our usage of classical Lie commutators for electromagnetism.