Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1902.09805 · Scientific Reports

Singularity of a relativistic vortex beam and proper relativistic observables

Yeong Deok Han, Taeseung Choi, Sam Young Cho

Published May 4, 2020• 3 citations
arXivHTMLPDF

Abstract

We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. One set includes new spin and orbital angular momentum (OAM) operators, which are derived from the parity-extended Poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. On the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.

DOI
10.1038/s41598-020-64168-0

Related papers

Position, spin, and orbital angular momentum of a relativistic electron

Konstantin Yu Bliokh, Mark R. Dennis, Franco Nori

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)].

arXiv:1706.01658•Physical Review A•Aug 28, 2017•37 citations

Relativistic Electron Wave Packets Carrying Angular Momentum

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

There are important differences between the nonrelativistic and relativistic description of electron beams. In the relativistic case the orbital angular momentum quantum number cannot be used to specify the wave functions and the structure of vortex lines in these two descriptions is completely different. We introduce analytic solutions of the Dirac equation in the form of exponential wave packets and we argue that they properly describe relativistic electron beams carrying angular momentum.

arXiv:1611.04445•Physical Review Letters•Mar 13, 2017•79 citations

Vortex-Enhanced Zitterbewegung in Relativistic Electron Wave Packets

Zhongze Guo, Bei Xu, Qiang Gu

Zitterbewegung (ZBW), the trembling motion predicted by the Dirac equation, has long remained unobservable in free electrons due to its sub-Compton scale. We elaborately construct a relativistic vortex electron wave packet as a coherent superposition of both positive- and negative-energy Dirac states and derive their space-time dynamics. Our analysis demonstrates that introducing orbital angular momentum provides a mechanism for amplifying the ZBW amplitude far beyond that of conventional Gaussian packets, while maintaining coherence. The resulting relativistic vortex states unify Gaussian and Bessel-Gaussian models within a single framework and opens new possibilities for observing relativistic quantum dynamics in structured electron wave packets.

quant-pharXiv:2511.21142•Nov 27, 2025•0 citations

Spin Operators for Massive Particles

Taeseung Choi, Sam Young Cho

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.

arXiv:1410.0468•Oct 2, 2014•0 citations

Transfer of Information Using Helically Phased Modes

Miles J. Padgett, Graham M. Gibson, Johannes Courtial

In 1992, Les Allen and coworkers within Han Woerdman's group realized that light beams with an azimuthal phase dependence described by an additional exp(iℓθ) term, where ℓ can take any integer value and θ is the angular position within the beam, carry an orbital angular momentum. This orbital angular momentum is additional to the spin angular momentum associated with circular polarization. These azimuthally phased beams comprise ℓ intertwined helical phase fronts, skewed with respect to the beam axis, meaning that at all positions within the beam cross section, the wavefront normal and associated momentum density has a well-defined azimuthal component. The insight of Allen and coworkers was that for all beams described in this way, when integrated over the beam cross section, the orbital angular momentum (OAM) is ℓℏ per photon. Laguerre-Gaussian and high-order Bessel beams are both examples of mode families having these properties. A further feature of these beams is that the phase singularity on the beam axis precludes a nonzero on-axis intensity and therefore all such beams with ℓ ≠0 have an annular intensity cross section (see Fig. 4.1). The circulation of the optical momentum and energy around the singularity leads them also to be called optical vortices. The further recognition that beams carrying defined amounts of OAM could be created within the laboratory means that over the last 15 years they have formed the basis of many investigations, ranging from the transfer of OAM to microscopic particles (in optical spanners or optical wrenches) to studies of the fundamental properties of light in both the classical and quantum regimes. However, although OAM is an immensely useful concept, one must emphasize that all its properties are simply those predicted by standard electromagnetic theory. The purpose of this chapter is to consider how sets of helical laser modes carrying OAM can be used for information transfer.

SPIE eBooks•Mar 16, 2010•0 citations

Vortex-enhanced Zitterbewegung in relativistic Dirac wave packets

Zhongze Guo, Bei Xu, Qiang Gu

Zitterbewegung (ZBW), arising from the interference between positive- and negative-energy solutions of the Dirac equation, is a characteristic feature of relativistic single-particle dynamics, but its explicit manifestation depends on the choice of position operator. We formulate relativistic vortex electron wave packets as coherent superpositions of Dirac plane-wave states and derive their exact spacetime evolution. For the canonical Dirac position, we show that orbital angular momentum substantially enhances and spatially structures the relative-coordinate contribution conventionally associated with ZBW. We then analyze the operator dependence of this result by separating the canonical position into a Foldy–Wouthuysen/Newton–Wigner (FW/NW) mean position and an oscillatory relative coordinate. For the vortex packets considered here, the FW/NW mean position exhibits only the ordinary centroid drift and no transverse trembling. However, the same vortex-enhanced radial scale survives in the positive-energy sector as a Berry-type geometric dressing of the projected position operator. The present work therefore identifies not a representation-independent trembling of the one-particle centroid, but a vortex-amplified remnant of Dirac interband mixing in an operationally meaningful position observable.

Nuclear Physics B•Jun 2, 2026•0 citations
We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. One set includes new spin and orbital angular momentum (OAM) operators, which are derived from the parity-extended Poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. On the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.
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)].
Zitterbewegung (ZBW), the trembling motion predicted by the Dirac equation, has long remained unobservable in free electrons due to its sub-Compton scale. We elaborately construct a relativistic vortex electron wave packet as a coherent superposition of both positive- and negative-energy Dirac states and derive their space-time dynamics. Our analysis demonstrates that introducing orbital angular momentum provides a mechanism for amplifying the ZBW amplitude far beyond that of conventional Gaussian packets, while maintaining coherence. The resulting relativistic vortex states unify Gaussian and Bessel-Gaussian models within a single framework and opens new possibilities for observing relativistic quantum dynamics in structured electron wave packets.
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.
In 1992, Les Allen and coworkers within Han Woerdman's group realized that light beams with an azimuthal phase dependence described by an additional exp(iℓθ) term, where ℓ can take any integer value and θ is the angular position within the beam, carry an orbital angular momentum. This orbital angular momentum is additional to the spin angular momentum associated with circular polarization. These azimuthally phased beams comprise ℓ intertwined helical phase fronts, skewed with respect to the beam axis, meaning that at all positions within the beam cross section, the wavefront normal and associated momentum density has a well-defined azimuthal component. The insight of Allen and coworkers was that for all beams described in this way, when integrated over the beam cross section, the orbital angular momentum (OAM) is ℓℏ per photon. Laguerre-Gaussian and high-order Bessel beams are both examples of mode families having these properties. A further feature of these beams is that the phase singularity on the beam axis precludes a nonzero on-axis intensity and therefore all such beams with ℓ ≠0 have an annular intensity cross section (see Fig. 4.1). The circulation of the optical momentum and energy around the singularity leads them also to be called optical vortices. The further recognition that beams carrying defined amounts of OAM could be created within the laboratory means that over the last 15 years they have formed the basis of many investigations, ranging from the transfer of OAM to microscopic particles (in optical spanners or optical wrenches) to studies of the fundamental properties of light in both the classical and quantum regimes. However, although OAM is an immensely useful concept, one must emphasize that all its properties are simply those predicted by standard electromagnetic theory. The purpose of this chapter is to consider how sets of helical laser modes carrying OAM can be used for information transfer.
Zitterbewegung (ZBW), arising from the interference between positive- and negative-energy solutions of the Dirac equation, is a characteristic feature of relativistic single-particle dynamics, but its explicit manifestation depends on the choice of position operator. We formulate relativistic vortex electron wave packets as coherent superpositions of Dirac plane-wave states and derive their exact spacetime evolution. For the canonical Dirac position, we show that orbital angular momentum substantially enhances and spatially structures the relative-coordinate contribution conventionally associated with ZBW. We then analyze the operator dependence of this result by separating the canonical position into a Foldy–Wouthuysen/Newton–Wigner (FW/NW) mean position and an oscillatory relative coordinate. For the vortex packets considered here, the FW/NW mean position exhibits only the ordinary centroid drift and no transverse trembling. However, the same vortex-enhanced radial scale survives in the positive-energy sector as a Berry-type geometric dressing of the projected position operator. The present work therefore identifies not a representation-independent trembling of the one-particle centroid, but a vortex-amplified remnant of Dirac interband mixing in an operationally meaningful position observable.