What is the quantity dual to the Lagrangian density?

TL;DR

This paper identifies the quantity dual to the Lagrangian density in nonlinear electrodynamics, showing it forms a canonical pair with the Lagrangian and can be viewed as coordinates on a symplectic manifold.

Contribution

It demonstrates that the dual quantity to the Lagrangian density is a specific scalar derived from the stress-energy tensor, establishing a canonical conjugate relationship.

Findings

The dual quantity to the Lagrangian density is ${f K} = rac{1}{2}\sqrt{\Theta_{\mu u}\Theta^{\mu u}}$.

${f L}$ and ${f K}$ form a pair of canonically conjugate variables.

${f L}$ and ${f K}$ serve as local coordinates on a two-dimensional symplectic manifold.

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.