Einstein-Dirac theory on gauge-natural bundles (Q1422466)

The author presents a clear-cut example of the importance of the functional approach of gauge-natural bundles and the general theory of Lie derivatives for classical field theory, where the sole correct geometrical formulation of Einstein(-Cartan) gravity coupled with Dirac fields gives rise to an unexpected indeterminacy in the concept of conserved quantities. A version of Noether's theorem suitable for gauge-natural bundles is given. The concepts of a spin-tetrad and a spin-connection are defined. Finally, the author briefly recalls the Lagrangian formulation of the Einstein(-Cartan)-Dirac theory and, on applying the theory of conserved quantities, finds a general superpotential, which essentially proves the aforementioned indeterminacy of any conserved charge associated with the gravitational field.