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.