New representation for Lagrangians of self-dual nonlinear electrodynamics

TL;DR

This paper introduces a new auxiliary-field based representation for 4D nonlinear electrodynamics Lagrangians, including Born-Infeld, simplifying duality symmetry analysis and generalizing to multiple gauge fields with U(n) symmetry.

Contribution

It presents a novel auxiliary bispinor field formulation for nonlinear electrodynamics, clarifies duality symmetries, and extends the framework to multiple gauge fields with U(n) invariance.

Findings

Simplifies the characterization of duality symmetries in nonlinear electrodynamics.

Provides a method to derive nonlinear Lagrangians from auxiliary fields.

Generalizes the approach to systems with multiple gauge fields and U(n) symmetry.

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.