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 is given by , where 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 that is analogous to and is conserved if the nonfixed fields satisfy their Euler-Lagrange equations. The conservation of can be seen as a consequence of an identity that is a generalization of and is a consequence of the gauge symmetry of the Lagrangian. This identity holds in any configuration of the fixed fields if the nonfixed fields satisfy their Euler-Lagrange equations. We also show that differs from the relevant canonical Noether current by the sum of an identically conserved current and a term that vanishes if the nonfixed fields are on shell. For an 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 is the Lorentz law , which holds as a consequence of the diffeomorphism, local Lorentz and Yang-Mills gauge symmetry of the matter Lagrangian. For 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.
- Received 2 November 2016
DOI:https://doi.org/10.1103/PhysRevD.96.025018
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