Noether Currents and Generators of Local Gauge Transformations in the Covariant Canonical Formalism

Abstract

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