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) for a gauge field 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 π A are defined as derivatives of the Lagrangian d-form L(ψ A , dψ A ) with respect to dψ A , namely π A := ∂L/∂dψ A , where ψ A are p-form dynamical fields. The formcanonical equations are derived from the form-Legendre transformation of the Lagrangian form H := dψ A ∧ π A − 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 are described by only ψ 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.