Existence of Particlelike Solutions to Nonlinear Field Theories

Abstract

A pseudovirial theorem is derived for time-independent particlelike solutions of finite energy (singularity-free and spatially localized time-independent solutions) to field theories associated with an action principle. It is shown that a useful necessary condition for the existence of such particlelike solutions is generally obtainable as a corollary to the pseudovirial theorem. This necessary condition is in fact sufficient to preclude existence of any well-localized particlelike solution for all but special field theories with the more common forms of algebraic interaction. On the other hand, strong satisfaction of the necessary condition can lead to model field theories with rigorous closed-form particlelike solutions, as shown by example for a class of Lorentz-covariant theories which feature a real scalar field in interaction with a two-component complex Weyl spinor field. Some of the latter particlelike solutions to the scalar-spinor theory are energetically stable with respect to spatial dilatations, hence likely to be stable in the dynamical sense. A counter example to the more general sufficiency of the strong satisfaction condition is presented, showing that strong satisfaction of the pseudovirial theorem's corollary does not always guarantee the existence of singularity-free particlelike solutions.