Abstract
Long ago Weinberg showed, from first principles, that the amplitude for a single photon exchange between an electric current and a magnetic current violates Lorentz invariance. The obvious conclusion at the time was that monopoles were not allowed in quantum field theory. Since the discovery of topological monopoles there has thus been a paradox. On the one hand, topological monopoles are constructed in Lorentz invariant quantum field theories, while on the other hand, the low-energy effective theory for such monopoles will reproduce Weinberg’s result. We examine a toy model where both electric and magnetic charges are perturbatively coupled and show how soft-photon resummation for hard scattering exponentiates the Lorentz violating pieces to a phase that is the covariant form of the Aharonov-Bohm phase due to the Dirac string. The modulus of the scattering amplitudes (and hence observables) are Lorentz invariant, and when Dirac charge quantization is imposed the amplitude itself is also Lorentz invariant. For closed paths there is a topological component of the phase that relates to aspects of 4D topological quantum field theory.
Highlights
Topological monopoles are constructed in Lorentz invariant quantum field theories, while on the other hand, the low-energy effective theory for such monopoles will reproduce Weinberg’s result
For closed paths there is a topological component of the phase that relates to aspects of 4D topological quantum field theory
We show how the resummation of soft photons removes the nμ dependence once Dirac charge quantization is imposed
Summary
Most QED calculations make no mention of topology, and many physicists find the jargon and results unfamiliar. Linking numbers may trace their genesis to Gauss’ study of magnetism, giving a certain poetry to its appearance in the modern approach to magnetic monopoles Gauss recorded his discovery of the linking number in his diary/logbook in 1833, but the result was not published until 1867 when it was included in his collected work on electrodynamics [23]. Note that the linking number (2.3) counts the signed crossings of the curve C with an arbitrary Stokes surface bounded by C. Gauss’ linking number plays an important role in 3D topological quantum field theories It turns up in the first order term of the expectation value of the Wilson line in SU(2) ChernSimons theory [31, 32], which is the poster child for topological quantum field theories.
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