Stability of the soliton in the broken O ( 3 ) nonlinear σ - model in 2 + 1 dimensions Samuel

In this diploma work we discuss a soliton in the context of quantum field theory. The model considered is the O(3) nonlinear σ-model in 3 dimensions with a symmetry breaking mass term. The stability properties of the circularly symmetric “hedgehog” soliton with topological number 1 is analyzed both classically and quantum mechanically. As far as classical field mechanics is concerned, a numerical code has been developed which integrates the partial differential equation of the problem. The result is that the soliton shrinks to a singular field configuration in finite time. The quantum mechanical calculations use the collective coordinate approach in a dilatation variable. The classically instable soliton is found to be stabilized through quantum effects. The spectrum of the quantum soliton is that of a harmonic oscillator. If the conclusions are correct and “quantum solitons” exist, this could have interesting and novel applications in cosmology.