I am familiar with the probability current, and the closely related electric current, for an electron described by non-relativistic QM (the Pauli-Schrodinger equation). I also understand that this electric current can be obtained from a non-relativistic limit of the Dirac equation. In the context of the Dirac equation, there is an internal symmetry and Noether's theorem for this symmetry yields the appropriate current. What about the other, spacetime, symmetries of the Dirac equation? These have corresponding conserved currents, by Noether's theorem. What is the fate of these currents when one takes a non-relativistic limit? Do we get a 'momentum density/current', do we get an 'angular momentum density/current', etc, for the Schrodinger-Pauli equation?
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