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Interplay of advanced and retarded solutions in electromagnetic wave equations

Determine the nature of the interplay between advanced and retarded solutions of the inhomogeneous wave equations for the electric scalar potential, magnetic vector potential, and electric Hertz potential, including whether and how advanced solutions should be treated alongside retarded solutions to yield physically meaningful, causal descriptions.

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Background

The paper studies general solutions to the inhomogeneous wave equations for electromagnetic potentials and emphasizes that both retarded and advanced solutions separately satisfy these equations. Historically, advanced solutions are often discarded due to causality, but the precise role and combination of advanced and retarded solutions has been regarded as unresolved.

This open question motivates the author’s derivation of k-space solutions and discussion of physical implications. The issue concerns whether advanced solutions contribute in a meaningful way without violating causality, and how such contributions manifest in the general solutions for scalar, vector, and Hertz potentials.

References

While it is known that the general wave equations are satisfied separately by retarded and advanced solutions - with the latter being usually discarded based on causality considerations -, the interplay between the advanced and retarded solutions have remained open questions 1.

— Derivation and physical interpretation of the general solutions to the wave equations for electromagnetic potentials (2411.03362 - Raicu, 5 Nov 2024) in Section 1 (Introduction)