Physical wavelets: Lorentz covariant, singularity-free, finite energy, zero action, localized solutions to the wave equation

TL;DR

This paper introduces Lorentz covariant, finite-energy, singularity-free localized solutions to the wave equation, termed 'physical wavelets', which can be extended to various field theories and have implications for classical and quantum physics.

Contribution

It presents a new class of Lorentz covariant, localized wave solutions with zero action and finite energy, applicable to scalar, Maxwell, and Yang-Mills fields.

Findings

Field configurations are finite, nonsingular, and have quadratic falloff in space and time.

Total energy of the wavelets is finite, and total action is zero.

Wavelets can be extended to different field theories, including Maxwell and Yang-Mills.

Abstract

Particle physics has for some time made extensive use of extended field configuations such as solitons, instantons, and sphalerons. However, no direct use has yet been made of the quite extensive literature on ``localized wave'' configurations developed by the engineering, optics, and mathematics communities. In this article I will exhibit a particularly simple ``physical wavelet'' -- it is a Lorentz covariant classical field configuration that lives in physical Minkowski space. The field is everwhere finite and nonsingular, and has quadratic falloff in both space and time. The total energy is finite, the total action is zero, and the field configuration solves the wave equation. These physical wavelets can be constructed for both complex and real scalar fields, and can be extended to the Maxwell and Yang-Mills fields in a straightforward manner. Since these wavelets are finite energy, they are guaranteed to be classically present at finite temperature; since they are zero action, they can contribute to the quantum mechanical path integral at zero ``cost''.