When Fields Are Not Degrees of Freedom

TL;DR

This paper argues that classical electrodynamics should be reformulated as delay differential equations rather than using traditional initial value formulations, impacting the ontological status of electromagnetic fields.

Contribution

It demonstrates that Maxwell-Lorentz theory's initial value problem is ill-posed and proposes a delay differential equations reformulation with philosophical implications.

Findings

Initial conditions often lead to singularities or discontinuities.

Reformulation as delay differential equations avoids these issues.

Electromagnetic fields may not be independent ontological entities.

Abstract

We show that in the Maxwell-Lorentz theory of classical electrodynamics most initial values for fields and particles lead to an ill-defined dynamics, as they exhibit singularities or discontinuities along light-cones. This phenomenon suggests that the Maxwell equations and the Lorentz force law ought rather to be read as a system of delay differential equations, that is, differential equations that relate a function and its derivatives at different times. This mathematical reformulation, however, leads to physical and philosophical consequences for the ontological status of the electromagnetic field. In particular, fields cannot be taken as independent degrees of freedom, which suggests that one should not add them to the ontology.