At classical level, dynamical derivation of the properties and conservation laws for topologically non-trivial systems from Noether theorem versus the derivation of the systems properties on topological grounds are considered as distinct. We do celeb
rate 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 monopoles 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 monopoles Higgs vacuum its restoration by the equations of motion of the Higgs vacuum, if disturbed.
We show that based on the general solution, given by Corrigan, Olive, Fairlie and Nuyts, in the region outside the monopoles core; the equations of motion in the Higgs vacuum (i.e. outside the monopoles core) will not allow asymptotically non-singula
r extended non-trivial non-Dyonic (including, also, all static) solutions of the t Hooft-Polyakov monopole. In other words, unless the monopoles magnetic charge is shielded (by some mechanism), the Dirac string is inevitable asymptotically, in the region outside the monopoles core, for all non-Dyonic solutions that are admissible by the equations of motion. That we show that the non-dyonic solutions (based on Corrigan et al) will include all admissible static solutions and their gauge transform might be interpreted as that all admissible dyonic solutions (based on Corrigan et al) are composite solutions.
In [7] we proposed a non-generational conjectural derivation of all first class constraints (involving, only, variables compatible with canonical Poisson brackets) for realistic gauge (singular) field theories; and we verified the conjecture in cases
of electromagnetic field, Yang Mills fields interacting with scalar and spinor fields, and the gravitational field. Here we will further verify our conjecture for the case of t Hooft- Polyakov (HP) monopoles field (i.e. in the Higgs Vacuum); and show that we will reproduce the results in Ref.[6], which we reached at using Diracs standard multi-generational algorithm.
We propose a single-step non-generational conjecture of all first class constraints,(involving only variables compatible with canonical Poisson brackets), for a realistic gauge singular field theory. We verify our proposal for the free electromagneti
c field, Yang-Mills fields in interaction with spinor and scalar fields, and we also verify our proposal in the case gravitational field. We show that the first class constraints which were reached at using the standard Diracs multi-generational algorithm will be reproduced using the proposed conjecture. We make no claim that our conjecture will be valid for all mathematically plausible Lagrangians; but, nevertheless the examples we consider here show that this conjecture is valid for wide range or much of realistic fields of physical interest that are know to exist and are manifested in nature
