On the classical central charge

TL;DR

This paper explores how the central charge, a key feature in the symmetry algebra of classical field theories, can be derived from the canonical Poisson bracket structure in the Lagrangian framework.

Contribution

It provides a derivation of the Lagrangian form of the central term starting from the canonical Poisson bracket algebra.

Findings

Derived the Lagrangian form of the central charge from canonical structure

Clarified the relationship between symmetry algebra and Lagrangian formulation

Enhanced understanding of central extensions in classical field theories

Abstract

In the canonical formulation of a classical field theory, symmetry properties are encoded in the Poisson bracket algebra, which may have a central term. Starting from this well understood canonical structure, we derive the related Lagrangian form of the central term.