An exact solution in Poincaré gauge theory: Monopole solution

Abstract

Poincaré gauge theory is treated such that a translational gauge field is actually identified with the Einstein’s gravitational field and the Lorentz gauge field is thought of as a pair of Yang–Mills-like field and its partner. An exact solution is investigated in this model, which is similar to the Wu–Yang solution in Yang–Mills theory. As a result, a solution is found. According to Kalb’s ansatz, the ‘‘Yang–Mills parts’’ of the solution are interpreted as ‘‘electric’’ and ‘‘magnetic’’ fields that are created by a point source with a unit ‘‘magnetic’’ charge and a certain ‘‘electric’’ charge. The translational gauge field then gives the Reissner–Nordström space-time.

Interaction of Massless Dirac Field with a Poincare Gauge Field

“…And we have derived again a solution which can be interpreted as the "Yang-Mills" fields generated by the gauge charge Q 2 and the spacetime curved by Reissner-Nordström type black-hole. However, the solution is special one of more general set of solutions [7] which are derived from the universal idea following to 'tHooft [8]. Accordingly, we can expect that the investigations will give us the qualitative but universal results.…”

“…In this paper we make slightly more loose choice than in Ref. [7] as follows: α + 2 3 a = β − 2 3 a = γ + 3 2 a = 0 and 4a 1 = 3a 2 = 2a 4 = 24a 6 , a 5 = 2a 3 . In this choice action integral can be rewritten as…”

“…In the next section some special solutions of PGT-monopoles are derived again in a different way from Ref. [7]. In Sec.…”

“…In those solutions there seems to exist such ones that the property depends strongly on the structure of the gauge group and therefore have little dependence on the choice of parameters, i.e., on a peculiarity of the models. For example, in a few years ago [7] one of the authors has obtained exact solutions (monopole solutions) in a model where PGT can be actually identified with complex Einstein-Yang=Mills theory. The idea is based on an universal property which is an extension of 't Hooft's original idea [8]: any non-Abelian theory can have a monopole solution if it has a compact covering group in its static limit.…”

“…In this paper we shall re-derive some special solutions of above ones, but in the different method from Ref [7]. For, owing to this method we can more naturally interpret two kinds of integral (vector) constants, Q 2 , Q 1 as the "gauge charges" which creat the "Yang-Mills" field and its partner field, respectively.…”