Electric-magnetic duality as a secondary symmetry

Abstract

In both the abelian and non-abelian classical point magnetic monopole theories, electric current conservation is a consequence of gauge invariance, but, since there is no magnetic gauge group, magnetic current conservation is not a Noether-type conservation law. In the abelian models, the equations of motion (but not the lagrangian) are invariant to the duality rotations in electric-magnetic charge space, but this is not the case in the non-abelian models. In an attempt to understand these and related points, we introduce a generalization of Noether's theorem. Consider a physical system described by a set of variables Φ and characterized by a lagrangian density L (Φ). A transformation law Φ → GΦ which leaves L invariant leads to a conserved current J μ ( Φ). We then call G a primary symmetry. A second transformation law Φ → DΦ which leaves the equations of motion, but not L , invariant then leads to another conserved current J μ ( DΦ). We then call D a secondary symmetry. Our main point is that J μ ( DΦ) may be conserved even if the equations of motion are not invariant under D. All that is required is that the change of the equations of motion under D is perpendicular (in the field space) to the change of the fields under G. Then we call D an incomplete secondary symmetry. We show that in both the abelian and non-abelian monopole theories, duality is an incomplete secondary symmetry whose associated conservation law is magnetic current conservation. Thus it is the interpretation of duality as a secondary symmetry which explains magnetic current conservation and which generalizes from the abelian theories to the non-abelian ones. This suggests that magnetic current conservation may remain valid in quantum field theory.