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arXiv:1303.3862 · New Journal of Physics

What is the relativistic spin operator?

Heiko Bauke, Sven Ahrens, Christoph H. Keitel, R. Grobe

Published Apr 11, 2014• 51 citations
arXivHTMLPDF

Abstract

Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spin operator within the framework of relativistic quantum mechanics. We investigate the properties of different proposals for a relativistic spin operator. It is shown that most candidates are lacking essential features of proper angular momentum operators, leading to spurious zitterbewegung (quivering motion) or violation of the angular momentum algebra. Only the Foldy–Wouthuysen operator and the Pryce operator qualify as proper relativistic spin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally.

DOI
10.1088/1367-2630/16/4/043012

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Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spin operator within the framework of relativistic quantum mechanics. We investigate the properties of different proposals for a relativistic spin operator. It is shown that most candidates are lacking essential features of proper angular momentum operators, leading to spurious zitterbewegung (quivering motion) or violation of the angular momentum algebra. Only the Foldy–Wouthuysen operator and the Pryce operator qualify as proper relativistic spin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally.
Different operators have been suggested in the literature to describe the electron's spin degree of freedom within the relativistic Dirac theory. We compare concrete predictions of the various proposed relativistic spin operators in different physical situations. In particular, we investigate the so-called Pauli, Foldy-Wouthuysen, Czachor, Frenkel, Chakrabarti, Pryce, and Fradkin-Good spin operators. We demonstrate that when a quantum system interacts with electromagnetic potentials the various spin operators predict different expectation values. This is explicitly illustrated for the scattering dynamics at a potential step and in a standing laser field and also for energy eigenstates of hydrogenic ions. Therefore, one may distinguish between the proposed relativistic spin operators experimentally.
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)].
A relativistic spin operator is to be the difference between the total and orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all desirable commutation relations as a position operator, can give a proper spin operator. Historically important three spin operators respectively proposed by Bogolubov et al., Pryce, and Foldy-Woutheysen are investigated to manifest a corresponding spin operator to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics described by the Dirac Hamiltonian.
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.
A relativistic spin operator is the difference between the total and the orbital angular momentum. As the unique position operator for a localized state, the remarkable Newton-Wigner position operator, which has all the desirable commutation relations of a position operator, can give a proper spin operator. Historically, the three important spin operators proposed by Bogolubov et al. , Pryce, and Foldy-Woutheysen, respectively were investigated to manifest a spin operator corresponding to the Newton-Wigner position operator. We clarify a unique spin operator in relativistic quantum mechanics, which can be described by using the Dirac Hamiltonian.
Since the discovery a century ago, spin describing the intrinsic angular momentum of massive elementary particles has exposed its nature and significant roles in wide ranges of (relativistic) quantum phenomena and practical applications for future quantum technology. Emerging inconsistencies have also disclosed its telltale incomplete description. Finding relativistic spins (operators) of massive particles is a long-standing fundamental problem from the beginning of relativistic quantum mechanics. Here we present the rigorous derivation and the representation of spin operators from the spacetime symmetry. The covariant parity operation, defined by the spin operators, naturally leads to a fundamental equation equivalent to the covariant Dirac equation, which manifests existent relativistic spins. Proper understanding position operator in the Dirac theory on account of the spin operator through total angular momentum predicts no Zitterbewegung as well as conserving orbital and spin currents. The spin operators can be applicable for unraveling the inconsistencies and for exploring unveiled physics of massive particles.