Electric-magnetic Duality and the “loop Representation” in Abelian Gauge Theories

Abelian Gauge Theories are quantized in a geometric representation that generalizes the Loop Representation and treates electric and magnetic operators on the same footing. The usual canonical algebra is turned into a topological algebra of non local operators that resembles the order-disorder dual algebra of ’t Hooft. These dual operators provide a complete description of the physical phase space of the theories. ∗e-mail: [email protected] Postal address: A.P. 47399, Caracas 1041-A, Venezuela 1 Dual symmetry in Electromagnetism has been a source of recent developments in Theoretical Physics. Besides the related dual symmetry in superstrings theory (for a recent review and references see ref 1), the notion of duality appears in supersymmetric Yang-Mills and topological theories [2-6]. There is also a renewed interest in the study of the analogous to electric-magnetic duality in conventional Yang-Mills theory [7-11]. The purpose of this letter is to study, within the geometrical language that the Loop Representation (L.R.) provides [12-15], the topological content underlying the “electro-magnetic” duality that ordinary Abelian gauge theories present. As we shall see, when these theories are quantized in a generalized L.R. that explicitly incorporates the dual symmetry, the canonical Poisson algebra results translated into a non-local algebra of geometric dependent operators. This algebra is topological, in the sense that it is not affected by continuous deformations of the spatial manifold. To begin, let us consider free Maxwell theory in 3+1 dimensions. We shall work in flat space time, with gμν = diag(+,−,−,−). Maxwell equations ∂Fμν = 0 (1.a) ∂ ∗Fμν = 0 (1.b) where ∗Fμν = 1 2 εμνλρF , are invariant under ⇀ E→ − ⇀ B, ⇀ B→ ⇀ E (i.e.: F → F ). Then, it is equally admisible to take for the Maxwell lagrangian, the usual one: L = − 1 4 Fμν(A)F (A) (2) with Fμν(A) = ∂μAν − ∂νAμ, or a “dual” lagrangian L̃ = − 1 4 ∗Fμν(Ã) F (Ã) (3)