What is the quantity dual to the Lagrangian density?

Abstract

We address the lately discovered Lagrangian density ${\cal L}$ of nonlinear electrodynamics preserving both conformal invariance and electric-magnetic duality to show that the quantity dual to ${\cal L}$ is ${\cal K}=\frac12\sqrt{\Theta_{\mu\nu}\Theta^{\mu\nu}}$, where $\Theta_{\mu\nu}$ is the stress-energy tensor built out of this ${\cal L}$. We point out that ${\cal L}$ and ${\cal K}$ make up a pair of canonically conjugate variables which can be regarded as local coordinates of a two-dimensional symplectic manifold.