Asymptotic conservation law with Feynman boundary condition

TL;DR

This paper derives a new asymptotic conservation law in classical electromagnetism under Feynman boundary conditions and explores novel imaginary modes in the Feynman solution that differ from retarded solutions.

Contribution

It introduces the analogue of a known asymptotic conservation law with Feynman boundary conditions and analyzes the emergence of purely imaginary modes in the Feynman solution at higher orders.

Findings

Derived the asymptotic conservation law with Feynman boundary condition.

Identified purely imaginary modes in the Feynman solution at order e^3.

Found that these modes violate Ashtekar-Struebel fall offs and suggest new modes at higher orders.

Abstract

Recently it was shown that classical electromagnetism admits new asymptotic conservation laws \cite{2007.03627}. In this paper we derive the analogue of the first of these asymptotic conservation laws upon imposing Feynman boundary condition on the radiative field. We also show that the Feynman solution at $\mathcal{O}(e^3)$ contains purely imaginary modes falling off as $\lbrace \frac{\log u}{u^nr}, n\geq 0\rbrace$ which are absent in the retarded solution. The $\log u$ mode has also appeared in \cite{1903.09133,1912.10229} and violates the Ashtekar-Struebel fall offs for the radiative field\cite{AS}. We expect that new $\frac{(\log u)^m}{u^mr}$-modes would appear in the Feynman solution at order $\mathcal{O}(e^{2m+1})$. Thus, all the other modes are expected to preserve the Ashtekar-Struebel fall offs.