New representation for Lagrangians of self-dual nonlinear electrodynamics

Abstract

We elaborate on a new representation of Lagrangians of 4D nonlinear electrodynamics including the Born-Infeld theory as a particular case. In this new formulation, in parallel with the standard Maxwell field strength $F_{\alpha \beta },{\bar{F}}_{\dot{\alpha }\dot{\beta }}$, an auxiliary bispinor field $V_{\alpha \beta },{\bar{V}}_{\dot{\alpha }\dot{\beta }}$ is introduced. The gauge field strength appears only in bilinear terms of the full Lagrangian, while the interaction Lagrangian $E$ depends on the auxiliary fields, $E=E(V^2,{\bar{V}}^2)$. The generic nonlinear Lagrangian depending on $F,\bar{F}$ emerges as a result of eliminating the auxiliary fields. Two types of self-duality inherent in the nonlinear electrodynamics models admit a simple characterization in terms of the function $E$. The continuous SO(2) duality symmetry between nonlinear equations of motion and Bianchi identities amounts to requiring $E$ to be a function of the SO(2) invariant quartic combination $V^2{\bar{V}}^2$, which explicitly solves the well-known self-duality condition for nonlinear Lagrangians. The discrete self-duality (or self-duality under Legendre transformation) amounts to a weaker condition $E(V^2,{\bar{V}}^2)=E(-V^2,-{\bar{V}}^2)$. We show how to generalize this approach to a system of $n$ Abelian gauge fields exhibiting U(n) duality. The corresponding interaction Lagrangian should be U(n) invariant function of $n$ bispinor auxiliary fields.

Latex file, 11 pages, Based on the talk on the simposium "Supersymmetries and quantum symmetries", Karpacz, Poland, September 21-25, 2001; References and note added, acknowledgements are corrected