Fermion-field nontopological solitons

TLDR

The existence of a strong‑coupling limit in relativistic quantum field theory remains unresolved. The study investigates the interaction between a scalar field and n fermion fields in three dimensions by decomposing the Hamiltonian into a quasiclassical part and a quantum‑correction term. The authors examine the effects of the quantum‑correction term on the system after this decomposition. They prove general theorems guaranteeing soliton solutions, show the quasiclassical solution closely approximates the full quantum solution across a wide range of coupling constants, and find this approximation improves with large fermion number, remains good for small N, and is again accurate in the strong‑coupling regime when fermions are nonrelativistic.

Abstract

The interaction between a scalar field and a set of $n$ fermion fields in three space dimensions is investigated by decomposing the total Hamiltonian $H$ into a sum of two terms: $H = {H}_{\mathrm{qcl}}+{H}_{\mathrm{corr}}$, where ${H}_{\mathrm{qcl}}$ denotes the quasiclassical part and ${H}_{\mathrm{corr}}$ the quantum correction. General theorems are given for ${H}_{\mathrm{qcl}}$ concerning the existence of soliton solutions, the general properties of such solutions, and the condition under which the lowest energy state of ${H}_{\mathrm{qcl}}$ is a soliton solution, not the usual plane-wave solution. The effects of the quantum-correction term ${H}_{\mathrm{corr}}$ are examined. It is shown that the quasiclassical solution is a good approximation to the quantum solution over a wide range of the coupling constant. The approximation becomes very good when the fermion number $N$ is large. Even for small $N$ (2 or 3) and weak coupling, the quasiclassical solution remains a fairly good approximation. In the strong-coupling region and for arbitrary $N$, the quasiclassical approximation becomes again very good, at least when the fermions are nonrelativistic. The question whether the relativistic quantum field theory has a strong-coupling limit or not is not resolved.