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On the structure of the energy-momentum and the spin currents in Dirac's electron theory

F. W. Hehl·A. Macías·E. W. Mielke·Yu. N. Obukhov
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Abstract

We consider a classical Dirac field in flat Minkowski spacetime. We perform a Gordon decomposition of its canonical energy-momentum and spin currents, respectively. Thereby we find for each of these currents a convective and a polarization piece. The polarization pieces can be expressed as exterior covariant derivatives of the two-forms $\check M_α$ and $M_{αβ}=-M_{βα}$, respectively. In analogy to the magnetic moment in electrodynamics, we identify these two-forms as gravitational moments connected with the translation group and the Lorentz group, respectively. We point out the relation between the Gordon decomposition of the energy-momentum current and its Belinfante-Rosenfeld symmetrization. In the non-relativistic limit, the translational gravitational moment of the Dirac field is found to be proportional to the spin covector of the electron.

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F. W. Hehl, A. Macías, E. W. Mielke, Yu. N. Obukhov. 1997-06-03. On the structure of the energy-momentum and the spin currents in Dirac's electron theory. https://arxiv.org/abs/gr-qc/9706009

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