Fermion-field nontopological solitons

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.