Absence of classical lumps

Abstract

Solutions of the equations of classical Yang-Mills theory in four dimensional Minkowski space are studied. It is proved (Theorem 1) that there is no finite energy (nonsingular) solution of the Yang-Mills equations having the property that there exists ɛ,R,t ₀>0 such that

$$E_R (t) = \int\limits_{|\bar x| \leqq R} {\theta _{00} (t,\bar x)d^3 \bar x \geqq \varepsilon foreveryt > t_0 ,} $$

$\theta _{00} (\bar x,t)$ being the energy density. Previously known theorems on the absence of finite energy nonsingular solutions that radiate no energy out to spatial infinity are particular cases of Theorem 1. The result stated in Theorem 1 is not restricted to the Yang-Mills equations. In fact, it extends to a large class of relativistic equations (Theorem 2).