Energy Renormalization in Ordinary Wave Mechanics

A very simple, exactly soluble compound-particle model, proposed by Wigner and Weisskopf in 1930, is briefly re-examined from the standpoint of renormalization. It consists of postulating, in the center-of-mass system, the wave equations $\left[i\right(\frac{\partial }{\partial t})+(\frac{1}{2m}\left){\nabla }^2\right]\psi (\mathrm{x}, t)=F\left(\mathrm{x}\right)\chi \mathrm{}\left(t\right),$ $\left[i\right(\frac{d}{\mathrm{dt}})-\mu ]\chi \mathrm{}\left(t\right)=\int d^{3}xF\left(\mathrm{x}\right)\psi (\mathrm{x}, t)$ for two particles of separation x and reduced mass $m$, interacting through the formation and decay of an intermediate particle with a real form factor $F$. The analytic behavior of the $S$ matrix is discussed in the local case $F\left(\mathrm{x}\right)=C\delta \left(\mathrm{x}\right)$.