Noether currents and generators of local gauge transformations in the covariant canonical formalism

TL;DR

This paper explores how Noether currents serve as generators of local gauge transformations within the covariant canonical formalism, emphasizing the equal treatment of space and time through differential forms.

Contribution

It demonstrates that Noether currents generate local gauge transformations in the covariant canonical formalism, extending the understanding of symmetries in this framework.

Findings

Noether current forms generate infinitesimal transformations.

The formalism treats space and time equally using differential forms.

Application to gauge fields and gravity shows the formalism's versatility.

Abstract

We investigate generators of local transformations in the covariant canonical formalism (CCF). The CCF treats space and time on an equal footing regarding the differential forms as the basic variables. The conjugate forms $\pi_A$ are defined as derivatives of the Lagrangian $d$-form $L(\psi^A, d\psi^A)$ with respect to $d\psi^A$, namely $\pi_A := \partial L/\partial d\psi^A$, where $\psi^A $ are $p$-form dynamical fields. The form-canonical equations are derived from the form-Legendre transformation of the Lagrangian form $H:=d\psi^A \wedge \pi_A - L$. We show that the Noether current form is the generator of an infinitesimal transformation $\psi^A \to \psi^A + \delta \psi^A$ if the transformation of the Lagrangian form is given by $\delta L=dl$ and $\delta \psi^A$ and $l$ depend on only $\psi^A$ and the parameters. As an instance, we study the local gauge transformation for the gauge field and the local Lorentz transformation for the second order formalism of gravity.