Origin of the tail in Green's functions in odd dimensional space-times

TL;DR

This paper investigates the physical origin of the tail in Green's functions in odd-dimensional space-times, revealing that a hidden scalar charge imprinted on the light cone shell causes the violation of Huygens' principle.

Contribution

It introduces a regularization approach that uncovers a hidden scalar charge on the light cone shell, explaining the tail phenomenon in odd-dimensional Green's functions.

Findings

The tail arises from a charge imprinted on the light cone shell.

Regularization reveals a non-zero contribution outside the source region.

Scalar charge's relativistic contraction affects the tail formation.

Abstract

It is well known that the scalar field Green's function in odd dimensions has a tail, i.e. a non-zero support inside the light cone, which in turn implies that the Huygens' principle is violated. However, the reason behind this behavior is still not quite clear. In this paper we shed more light on the physical origin of the tail by regularizing the term which is usually ignored in the literature since it vanishes due to the action of the delta function. With this extra term the Green's function does not satisfy the source-free wave equation (in the region outside of the source). We show that this term corresponds to a charge imprinted on the light cone shell. Unlike the vector field charge, a moving scalar field charge is not Lorentz invariant and is contracted by the relativistic $\sqrt{1-v^2}$ factor. If a scalar charge is moving at the speed of light, it appears to be zero in the static (with respect to the original physical charge) observer's frame. However, the field it sources is not entirely on the light cone. Thus, it is likely that this hidden charge sources the mysterious tail in odd dimensions.