Stationary phase approximation and quantum soliton families
It is shown that solitons in the anisotropic \(\lambda (\phi^*\phi)^ 2_ 2\) model correspond to critical behaviour in a related dynamical system: they are present in the separatrix dividing the phase space of a completely integrable system into parts of bound and unbound motion. The topology of the configuration space is fully described in terms of the topology of the space of paths in \(S^ 2\), whose geodesics with fixed endpoints give rise to the same Morse series as the family of solitons of the model, according to their asymptotic behaviour. When computed by the Stationary Phase Approximation to the Functional Integral, quantum observables, such as energy and wavefunctions, of quantum states corresponding to the different kinds of solitons exhibit properties which are directly related to the topological structure previously discovered. Using an appropriate coordinate system the model is equivalent to one with two nearly independent fields; once a ground state has been chosen the second field behaves exactly as though it were governed by the sine- Gordon Lagrangian. It is thus discovered that the model contains time- dependent solutions which are multisoliton scattering states.
scientific article; zbMATH DE number 4080472
scientific article