Non-Topological (Dynamical) Approach to Stability of 't Hooft-Polyakov Monopole
At classical level, dynamical derivation of the properties and conservation laws for topologically non-trivial systems from Noether theorem versus the derivation of the system's properties on topological grounds are considered as distinct. We do celebrate any agreements in results derived from these two distinct approaches: i.e. the dynamical versus the topological approach. Here we consider the Corrigan-Olive-Fairlie-Nuyts solution based on which we study the stability of the 't Hooft- Polyakov outer field, known as its Higgs vacuum, and derive its stability, dynamically, from the equations of motion rather than from the familiar topological approach. Then we use our derived result of the preservation of the Higgs vacuum asymptotically to derive the stability of the 't Hooft-Polyakov monopole, even if inner core is perturbed, where we base that on observing that the magnetic charge must be conserved if the Higgs vacuum is preserved asymptotically. We also, alternatively, note stability of 't Hooft-Polyakov monopole and the conservation of its magnetic charge by again using the result of the Higgs vacuum asymptotic preservation to use Eq.(5) to show that no non-Abelian radiation allowed out of the core as long as the Higgs vacuum is preserved and restored, by the equations of motion, if perturbed. We start by deriving the asymptotic equations of motion that are valid for the monopole's field outside its core; next we derive certain constraints from the asymptotic equations of motion of the Corrigan-Olive-Fairlie-Nuyts solution to the 't Hooft-Polyakov monopole using the Lagrangian formalism of singular theories, in particular that of Gitman and Tyutin. The derived constraints will show clearly the stability of the monopole's Higgs vacuum its restoration by the equations of motion of the Higgs vacuum, if disturbed.