Dualities of self-dual nonlinear electrodynamics
Jorge G. Russo, Paul Townsend
Abstract
A bstract For any causal nonlinear electrodynamics theory that is “self-dual” (electromagnetic U(1)-duality invariant), the Legendre-dual pair of Lagrangian and Hamiltonian densities $$ \left\{\mathcal{L},\mathcal{H}\right\} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>L</mml:mi> <mml:mi>H</mml:mi> </mml:mfenced> </mml:math> are constructed from functions $$ \left\{\ell, \mathfrak{h}\right\} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>ℓ</mml:mi> <mml:mi>h</mml:mi> </mml:mfenced> </mml:math> on ℝ + related to a particle-mechanics Lagrangian and Hamiltonian. We show how a ‘duality’ relating ℓ to $$ \mathfrak{h} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> implies that $$ \mathcal{L} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> and $$ \mathcal{H} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> are related by a simple map between appropriate pairs of variables. We also discuss Born’s “Legendre self-duality” and implications of a new “Φ-parity” duality. Our results are illustrated with many examples.