Quantum fluctuations in aφ4field theory. II. One-mode approximation

We study the effect of quantum fluctuations on one-particle states in a ${\phi }^4$ field theory, using a self-consistent Hartree-type approximation. For one-particle states, the presence of quantum fluctuations implies that perturbation theory should not be done about the $c$-number constant field associated with the vacuum, but rather one should start with a space-dependent classical field specially chosen for the single-particle state. The classical field no longer obeys the classical field equations; it emerges from a self-consistent calculation including the effect of quantum fluctuations. In this paper, we make a further approximation by keeping only the lowest quantum mode in the fluctuation (i.e., the one-mode approximation). We demonstrate that the inclusion of a single mode in the quantum fluctuations is enough to produce many physically interesting results which are qualitatively different from those obtained in a purely classical system. In particular, we find that a stable "baglike" solution emerges naturally. We determine the particle wave functions, the space-dependent classical fields, and the energies of these baglike one-particle states explicitly in both the weak- and the strong-coupling limits. We exhibit the numerical solutions for intermediate coupling strengths. Finally, we discuss briefly the connections of our solutions and the kink-antikink bound-state solutions obtained by Dashen, Hasslacher, and Neveu.