Path Integral Quantization of Dual Abelian Gauge Theory

The path integral for 3+1 abelian gauge theory is rewritten in terms of a real antisymmetric field allowing a dual action that couples the electric and magnetic currents to the photon and each other in a gauge invariant manner. Standard perturbative abelian quantum electrodynamics reemerges when the monopole current vanishes. For certain simple relationships between the monopole current and the electric current, the altered photon propagator can exhibit abelian charge confinement or develop mass, modeling effects believed to be present in non-abelian theories. Since the classic work of Dirac [1] magnetic monopoles have been studied in a variety of contexts. Retaining the usual gauge field structure requires singular gauge field configurations in the presence of magnetic charge. Dirac was the first to show that the singularities — Dirac “strings” — are not physical as long as the electric and magnetic charges satisfy the quantization condition eg = 1 2 n and its generalization, where n is an integer. Wu and Yang [2] showed the relationship of the Dirac monopole to a U(1) principal bundle with the base manifold given by a sphere around the monopole, with single-valuedness of the transition function responsible for the quantization condition. Beginning with the work of ’t Hooft and Polyakov [3] much attention has been focused on BPS monopoles in non-abelian gauge theories [4]. An important result regarding the effects of magnetic monopoles was the demonstration by Seiberg andWitten [5] that N = 2 supersymmetric QCD, a field theory exhibiting duality, breaks chiral symmetry and confines through the presence of monopole-antimonopole pairs in the ground state. It has been widely considered that monopoles may play a similar role in standard QCD [6]. The current value of the Parker bound [7] indicates that free magnetic monopoles are rare, if indeed they exist at all. However, their inclusion in quantum processes has many intriguing aspects. Zwanziger [8] addressed this problem by modifying the standard abelian action to include a dual gauge field. Gamsberg and Milton [9] analyzed the current–current interaction term, first proposed by Schwinger [10], using a path integral over the gauge field. In non-abelian theories monopole effects are evaluated by perturbing around the monopole field configuration, in effect treating the monopole as a classical solution [11]. Making this approach quantum mechanically consistent for an arbitrary set of monopoles requires integrating the associated path integral over the monopole moduli space, and much effort has gone into solving this difficult problem [12]. The purpose of this paper is to derive the path integral for self-dual 3+1 abelian gauge theory, allowing both a magnetic current J and an electric current . For the purposes of this paper, self-duality means that the path integral is invariant under the transformation J → − and  → J . When coupled with the transformations of the electric and magnetic fields E → B and B → −E, this is an invariance of the classical Maxwell’s equations in the presence of magnetic charge [13]. For the case that the magnetic current J is globally proportional to the usual electric current , there exists a duality transformation that eliminates the magnetic current, returning Maxwell’s equations to their standard form. However, the standard Maxwell action for the gauge field, 1 2 (E − B), is not invariant under a duality transformation of the electric and magnetic fields. As a result, a path integral quantization employing the standard E-Mail: [email protected]