Generators of local gauge transformations in the covariant canonical formalism of fields

TL;DR

This paper explores the generators of local gauge transformations within the covariant canonical formalism, applying it to matter, gauge, and gravity fields, and clarifies their algebraic structure and relation to Noether currents.

Contribution

It introduces a formalism for gauge transformation generators in the covariant canonical framework, including their explicit form and algebraic properties for various field types.

Findings

Generators are expressed as combinations of Noether currents and their derivatives.

The gauge algebra closes with structure constants in the covariant canonical formalism.

For matter fields, the generator simplifies to zero, indicating a different transformation behavior.

Abstract

We investigate generators of local gauge transformations in the covariant canonical formalism (CCF) for matter fields, gauge fields and the second order formalism of gravity. 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 generator of the local gauge transformation in the CCF is given by $\varepsilon^r G_r + d\varepsilon^r \wedge F_r$ where $\varepsilon^r$ are infinitesimal parameters and $G_r$ are the Noether currents which are $(d-1)$-forms. $\{G_r , G_s \} = f^t_{\ rs}G_t$ holds where $\{\bullet, \bullet \}$ is the Poisson bracket of the CCF and $f^t_{\ rs}$ are the structure constants of the gauge group. For the gauge fields and the gravity, $G_r=-\{F_r, H \}$ holds. For the matter fields, $F_r=0$ holds.