Nonclassical analogs of solitons in quantum field theory (Q1326814)

A long-standing problem is whether solitons, i.e. particle-like solutions of classical nonlinear field equations, have quantum analogues, at least in a two-dimensional spacetime. Starting from a Lagrangian density \({1 \over 2} (\partial \varphi)^ 2 - \lambda^{-2} V (\lambda \varphi)\) and canonical commutation relations, the author shows by a scaling argument that \(\gamma = \lambda^ 2\) acts like an effective Planck constant. He then explores the semiclassical regime (expansion in \(\lambda)\) and finds a discrete series ``quantum solitons'', characterized by an integer \(N\). These states disappear when \(\lambda \to 0\): they have total energy \(O(\lambda)\). Still, they cannot be shown to exist in a rigorous sense for the quantum regime. The author also provides calculations of the profile functions and masses, and investigates the stability of these states in the \(\varphi^ 4\) model.