Proof of a Symmetrized Trace Conjecture for the Abelian Born-Infeld Lagrangian

TL;DR

This paper proves a conjecture that the Abelian Born-Infeld Lagrangian can be expressed as a symmetrized trace of Lorentz invariant bilinears, providing a new mathematical formulation for gauge theories.

Contribution

It establishes that the Born-Infeld Lagrangian for a U(1)^2n gauge group can be written as a symmetrized trace after eliminating auxiliary fields, and proves a related theorem on matrix equations.

Findings

Lagrangian expressed as a symmetrized trace of bilinears

Theorem on solutions of unilateral matrix equations

Perturbative solutions are sums of symmetrized and commutator terms

Abstract

In this paper we prove a conjecture regarding the form of the Born-Infeld Lagrangian with a U(1)^2n gauge group after the elimination of the auxiliary fields. We show that the Lagrangian can be written as a symmetrized trace of Lorentz invariant bilinears in the field strength. More generally we prove a theorem regarding certain solutions of unilateral matrix equations of arbitrary order. For solutions which have perturbative expansions in the matrix coefficients, the solution and all its positive powers are sums of terms which are symmetrized in all the matrix coefficients and of terms which are commutators.