ASYMPTOTIC BEHAVIOR AND OPERATOR MASS RENORMALIZATION IN PHI4 MODEL

As is well known in the theory of nonlinear differential equations, straightforward perturbation theory leads to approximate solutions with the wrong asymptotic behavior. We consider a real scalar field satifying the Klein-Gordon equation with a φ{symbol}3 self-current, in both the classical and second-quantized versions. We demonstrate that the usual, approximate solutions are asymptotically unbounded in both cases. We present a solution up to a normalization factor for the function or Heisenberg operator φ{symbol} to first order in the coupling with the correct asymptotic behavior. In the quantum case this forces one to introduce a q-number renormalized mass. The discussion is limited to solutions periodic in a finite box. A discussion of the wave-function renormalization and the limit of infinite box volume is deferred here. The method of approximation used is a generalization of standard techniques in the theory of nonlinear differential equations, and does not require adiabatic switching of the interaction. © 1969 Società Italiana di Fisica.