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Journal of the Korean Physical Society

Anomalous Velocity Operator in a Multi-Particle Dirac System with Electron-Electron Interactions

Soo‐Yong Lee, Hyun‐Woo Lee

Published Dec 15, 2008• 1 citation
Publisher

Abstract

By examining the multi-particle Dirac system with the regularized Coulomb interaction including the Breit term, we found that there exist anomalous contributions to the velocity exist. Under the Foldy-Wouthuysen transformation, we showed that the effective meanings of the relativistic anomalous effect are spin-orbit coupling and the classical retardation effect in the non-relativistic limit. In this region, even though this result is intuitively appealing, the factor 2 anomaly appears in the term related to the spin-orbit coupling between relativistic and non-relativistic treatments. The reason for this discrepancy and the way to consistency via co-transformation of an operator corresponding to an observable are also discussed.

DOI
10.3938/jkps.53.3584

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By examining the multi-particle Dirac system with the regularized Coulomb interaction including the Breit term, we found that there exist anomalous contributions to the velocity exist. Under the Foldy-Wouthuysen transformation, we showed that the effective meanings of the relativistic anomalous effect are spin-orbit coupling and the classical retardation effect in the non-relativistic limit. In this region, even though this result is intuitively appealing, the factor 2 anomaly appears in the term related to the spin-orbit coupling between relativistic and non-relativistic treatments. The reason for this discrepancy and the way to consistency via co-transformation of an operator corresponding to an observable are also discussed.
Motivated by recent interest in relativistic electron vortex states, we revisit the spin and orbital angular momentum properties of Dirac electrons. These are uniquely determined by the choice of the position operator for a relativistic electron. We consider two main approaches discussed in the literature: (i) the projection of operators onto the positive-energy subspace, which removes the Zitterbewegung effects and correctly describes spin-orbit interaction effects, and (ii) the use of Newton-Wigner-Foldy-Wouthuysen operators based on the inverse Foldy-Wouthuysen transformation. We argue that the first approach [previously described in application to Dirac vortex beams in K. Y. Bliokh et al., Phys. Rev. Lett. 107, 174802 (2011)] has a more natural physical interpretation, including spin-orbit interactions and a nonsingular zero-mass limit, than the second one [S. M. Barnett, Phys. Rev. Lett. 118, 114802 (2017)].
It is natural to expect that the electromagnetic spin-spin interaction has roots in a more fundamental theory: quantum electrodynamics in this case. To show this, we take the non-relativistic limit of the Dirac-Maxwell Lagrangian, using the Foldy-Wouthuysen transformation and derive the Breit interaction by an alternative approach. By adding the Pauli term (i.e., a higher-order interaction between spin and the electromagnetic field), we calculate the correction terms to electromagnetic spin-spin interactions in the non-relativistic limit. Finally, we study some of the physical effects of these terms regarding spin-spin interactions and spin-spin entanglements. Accordingly, we provide theoretical estimates that are not yet accessible with current experiments.
The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.