We use the stress-energy tensor and the associated energy-momentum conservation to study the interactions between two widely separated monopoles (or monopole and antimonopole). By defining a set of minimal conditions to represent the above systems, we show how the problem reduced mathematically to a known electrostatic problem. The force between the monopoles (or monopole and antimonopole) is then, to the leading order, the expected repulsive (attractive) Coulomb force. We also discuss how the Prasad-Sommerfield limit alters the problem, leading to twice the Coulomb force between a monopole and antimonopole and zero force between two monopoles.
It is shown that exact axially symmetric, regular self-dual multi-instanton solutions of the SU(2) Yang-Mills equations can be generated from the vacuum solution by means of singular conformal mappings of a complex plane followed by a singular gauge transformation. The regularizing gauge for the 't Hooft singular solution is found and the corresponding Yang-Mills gauge potential 4, without any singularity and with the correct asymptotic behavior is given. The relations between the parameters in different forms of the solution are explored and the possible implications on the calculations of quantum effects are noted.
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