Noether's theorems and conserved currents in gauge theories in the presence of fixed fields

Abstract: We extend the standard construction of conserved currents for matter fields in general relativity to general gauge theories. In the original construction the conserved current associated with a spacetime symmetry generated by a Killing field $h^\mu$ is given by $\sqrt{-g}\,T^{\mu\nu}h_\nu$, where $T^{\mu\nu}$ is the energy-momentum tensor of the matter. We show that if in a Lagrangian field theory that has gauge symmetry in the general Noetherian sense some of the elementary fields are fixed and are invariant under a particular infinitesimal gauge transformation, then there is a current $\mathcal{B}^\mu$ that is analogous to $\sqrt{-g}\,T^{\mu\nu}h_\nu$ and is conserved if the non-fixed fields satisfy their Euler-Lagrange equations. The conservation of $\mathcal{B}^\mu$ can be seen as a consequence of an identity that is a generalization of $\nabla_\mu T^{\mu\nu}=0$ and is a consequence of the gauge symmetry of the Lagrangian. This identity holds in any configuration of the fixed fields if the non-fixed fields satisfy their Euler-Lagrange equations. We also show that $\mathcal{B}^\mu$ differs from the relevant canonical Noether current by the sum of an identically conserved current and a term that vanishes if the non-fixed fields are on-shell. As example we discuss the case of general, possibly fermionic, matter fields propagating in fixed gravitational and Yang-Mills background. We find that in this case the generalization of $\nabla_\mu T^{\mu\nu}=0$ is the Lorentz law $\nabla_\mu T^{\mu\nu} - F^{a\nu\lambda}\mathcal{J}_{a\lambda} = 0$, which holds as a consequence of the diffeomorphism, local Lorentz and Yang-Mills gauge symmetry of the matter Lagrangian. As a second simple example we consider the case of general fields propagating in a background that consists of a gravitational and a real scalar field.