β Scribed by Daksh Lohiya
No coin nor oath required. For personal study only.
The stabilities of Yang-Mills magnetic and Coulomb charges have been studied separately before, without laying much emphasis on the singular nature of the effective potential in the resulting radial equation. We investigate an analysis that is valid simultaneously for both Yang-Mills magnetic monopoles and dyons in flat as well as curved spacetime. We find the instability modes with their qualitative behaviour determined completely by the large distance behaviour of the radial equation. One of the results for the dyon is that we could get unstable modes even if there were no such modes for the constituent Yang-Mills "electric" and "magnetic" charges separately.
Coulomb-like solutions to the classical Yang-Mills theories give rise to long range forces between quarks. We shall study the classical stability of such solutions in this paper. A typical instability would provide a "screening" of the classical source [ 1 ].
The usual way to analyse the stability of a solution is by linearizing small perturbations around it and examining the temporal behaviour of the resulting linear system. In the case we encounter, the time and radial dependence of these perturbations are separable as !P = exp(--iwt) R(r) g(0, ~0). The (in)stability search reduces to finding the fluctuation eigenfrequencies consistent with the boundary conditions required of a finite localized perturbation. If all the frequencies turn out to be real then the solution is stable and if any of them are complex then some fluctuations would grow exponentially with time-indicating instability. We shall restrict ourselves to the study of Yang-Mills Coulomb-like and monopole-like solutions in flat spacetime and also look for corresponding properties of spherically symmetric solutions of Einstein-Yang-Mills systems in general relativity. In flat spacetime the problem was studied by Mandula [2]. One of the stability arguments was based on the analytic properties of the perturbations as a function of the source strength in an "effective Schrodinger equation." However, it is known [3] that solutions for a singular potential gV(r) are not analytic functions of the strength parameter g at g = 0. The instability of the Yang-Mills monopoles was studied by Brandt and Neri 104
π SIMILAR VOLUMES
Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations and under the analogues of Lorentz and general coordinate transformations. The case treated in some detail, recovering the Wilson ac