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

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''.