Helicity is a topological invariant of massless particles: C = − 2 h

Palmerduca, Eric; Qin, Hong

doi:10.1103/PhysRevResearch.7.L022001

Helicity is a topological invariant of massless particles: $C = - 2 h$

Journal Article · Wed Apr 02 04:00:00 EDT 2025 · Physical Review Research

DOI:https://doi.org/10.1103/PhysRevResearch.7.L022001· OSTI ID:2545136

;

There is an elementary but indispensable relationship between the topology and geometry of massive particles. The geometric spin $s$ is related to the topological dimension of the internal space $V$ by $dim V = 2 s + 1$ . This breaks down for massless particles, which are geometrically characterized by their helicity $h$ , but all have 1D internal spaces. We show that a subtler relation exists between the topology and geometry of massless particles. Wave functions of massless particles are sections of nontrivial line bundles over the light cone whose topology is completely characterized by their first Chern number $C$ . We prove that in general $C = - 2 h$ . In doing so, we also exhibit a method of generating all massless bundle representations via an Abelian group structure of massless particles.

Published by the American Physical Society 2025

View Journal Article

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC02-09CH11466

OSTI ID:: 2545136

Alternate ID(s):: OSTI ID: 2558892

Journal Information:: Physical Review Research, Journal Name: Physical Review Research Journal Issue: 2 Vol. 7; ISSN PPRHAI; ISSN 2643-1564

Publisher:: American Physical SocietyCopyright Statement

Country of Publication:: United States

Language:: English

References (43)

Lie Groups, Lie Algebras, and Representations Hall, Brian C. Graduate Texts in Mathematics https://doi.org/10.1007/978-3-319-13467-3	book	January 2015
Differential Geometry: Connections, Curvature, and Characteristic Classes Tu, Loring W. Graduate Texts in Mathematics https://doi.org/10.1007/978-3-319-55084-8	book	June 2017
Difficulties with massless particles? Flato, M.; Sternheimer, D.; Fronsdal, C. Communications in Mathematical Physics, Vol. 90, Issue 4 https://doi.org/10.1007/BF01216186	journal	December 1983
Lie Groups and Quantum Mechanics Simms, D. J. Lecture Notes in Mathematics, Vol. 52 https://doi.org/10.1007/BFb0069914	book	January 1968
Graviton topology Palmerduca, E.; Qin, H. Journal of High Energy Physics, Vol. 2024, Issue 11 https://doi.org/10.1007/JHEP11(2024)150	journal	November 2024
Bundle Structure of Massless Unitary Representations of the Poincaré Group Dragon, Norbert International Journal of Theoretical Physics, Vol. 63, Issue 6 https://doi.org/10.1007/s10773-024-05612-z	journal	June 2024
Topological invariant and the quantization of the Hall conductance Kohmoto, Mahito Annals of Physics, Vol. 160, Issue 2 https://doi.org/10.1016/0003-4916(85)90148-4	journal	April 1985
Covariant representations in a fibre bundle framework Asorey, Manuel; Boya, Luis J.; Cariñena, JoséF. Reports on Mathematical Physics, Vol. 21, Issue 3 https://doi.org/10.1016/0034-4877(85)90040-0	journal	June 1985
IV The Orbital Angular Momentum of Light Allen, L.; Padgett, M. J.; Babiker, M. Progress in Optics, Vol. 39, p. 291-372 https://doi.org/10.1016/S0079-6638(08)70391-3	book	May 1999
The angular momentum controversy: What’s it all about and does it matter? Leader, Elliot; Lorcé, Cédric Physics Reports, Vol. 541, Issue 3 https://doi.org/10.1016/j.physrep.2014.02.010	journal	August 2014
A First Course in General Relativity Schutz, Bernard https://doi.org/10.1017/CBO9780511984181	journal	June 2009
The Quantum Theory of Fields Weinberg, Steven https://doi.org/10.1017/CBO9781139644167	book	May 2013
The dispersion and propagation of topological Langmuir-cyclotron waves in cold magnetized plasmas Fu, Yichen; Qin, Hong Journal of Plasma Physics, Vol. 88, Issue 4 https://doi.org/10.1017/S0022377822000629	journal	July 2022
A bulk-interface correspondence for equatorial waves Tauber, C.; Delplace, P.; Venaille, A. Journal of Fluid Mechanics, Vol. 868 https://doi.org/10.1017/jfm.2019.233	journal	April 2019
Photonic Weyl degeneracies in magnetized plasma Gao, Wenlong; Yang, Biao; Lawrence, Mark Nature Communications, Vol. 7, Issue 1 https://doi.org/10.1038/ncomms12435	journal	August 2016
Topological sound in active-liquid metamaterials Souslov, Anton; van Zuiden, Benjamin C.; Bartolo, Denis Nature Physics, Vol. 13, Issue 11 https://doi.org/10.1038/nphys4193	journal	July 2017
Topological phases and bulk-edge correspondence of magnetized cold plasmas Fu, Yichen; Qin, Hong Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-021-24189-3	journal	June 2021
Topological transition in stratified fluids Perrot, Manolis; Delplace, Pierre; Venaille, Antoine Nature Physics, Vol. 15, Issue 8 https://doi.org/10.1038/s41567-019-0561-1	journal	June 2019
One-way helical electromagnetic wave propagation supported by magnetized plasma Yang, Biao; Lawrence, Mark; Gao, Wenlong Scientific Reports, Vol. 6, Issue 1 https://doi.org/10.1038/srep21461	journal	February 2016
Note on the Bondi-Metzner-Sachs Group Newman, E. T.; Penrose, R. Journal of Mathematical Physics, Vol. 7, Issue 5 https://doi.org/10.1063/1.1931221	journal	May 1966
Gravitational waves carrying orbital angular momentum Bialynicki-Birula, Iwo; Bialynicka-Birula, Zofia New Journal of Physics, Vol. 18, Issue 2 https://doi.org/10.1088/1367-2630/18/2/023022	journal	February 2016
Canonical separation of angular momentum of light into its orbital and spin parts Bialynicki-Birula, Iwo; Bialynicka-Birula, Zofia Journal of Optics, Vol. 13, Issue 6 https://doi.org/10.1088/2040-8978/13/6/064014	journal	April 2011
Quantum Optics Garrison, John; Chiao, Raymond https://doi.org/10.1093/acprof:oso/9780198508861.001.0001	journal	September 2008
Introduction to Symplectic Topology McDuff, Dusa; Salamon, Dietmar Oxford Scholarship Online https://doi.org/10.1093/oso/9780198794899.001.0001	book	June 2017
Quantal Phase Factors Accompanying Adiabatic Changes Berry, M. V. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 392, Issue 1802 https://doi.org/10.1098/rspa.1984.0023	journal	March 1984
Significance of Electromagnetic Potentials in the Quantum Theory Aharonov, Y.; Bohm, D. Physical Review, Vol. 115, Issue 3 https://doi.org/10.1103/PhysRev.115.485	journal	August 1959
Twisting and entangling gravitons in high-dimensional orbital angular momentum states via photon-graviton conversion Wu, Haorong; Chen, Lixiang Physical Review D, Vol. 107, Issue 12 https://doi.org/10.1103/PhysRevD.107.125027	journal	June 2023
Photon topology Palmerduca, Eric; Qin, Hong Physical Review D, Vol. 109, Issue 8 https://doi.org/10.1103/PhysRevD.109.085005	journal	April 2024
Topological Waves in Fluids with Odd Viscosity Souslov, Anton; Dasbiswas, Kinjal; Fruchart, Michel Physical Review Letters, Vol. 122, Issue 12 https://doi.org/10.1103/PhysRevLett.122.128001	journal	March 2019
Topological Gaseous Plasmon Polariton in Realistic Plasma Parker, Jeffrey B.; Marston, J. B.; Tobias, Steven M. Physical Review Letters, Vol. 124, Issue 19 https://doi.org/10.1103/PhysRevLett.124.195001	journal	May 2020
Quantized Hall Conductance in a Two-Dimensional Periodic Potential Thouless, D. J.; Kohmoto, M.; Nightingale, M. P. Physical Review Letters, Vol. 49, Issue 6 https://doi.org/10.1103/PhysRevLett.49.405	journal	August 1982
Homotopy and Quantization in Condensed Matter Physics Avron, J. E.; Seiler, R.; Simon, B. Physical Review Letters, Vol. 51, Issue 1 https://doi.org/10.1103/PhysRevLett.51.51	journal	July 1983
Quantum field theory for spin operator of the photon Yang, Li-Ping; Khosravi, Farhad; Jacob, Zubin Physical Review Research, Vol. 4, Issue 2 https://doi.org/10.1103/PhysRevResearch.4.023165	journal	May 2022
Colloquium: Topological insulators Hasan, M. Z.; Kane, C. L. Reviews of Modern Physics, Vol. 82, Issue 4, p. 3045-3067 https://doi.org/10.1103/RevModPhys.82.3045	journal	November 2010
Topological Langmuir-cyclotron wave Qin, Hong; Fu, Yichen Science Advances, Vol. 9, Issue 13 https://doi.org/10.1126/sciadv.add8041	journal	March 2023
Quantum spin Hall effect of light Bliokh, K. Y.; Smirnova, D.; Nori, F. Science, Vol. 348, Issue 6242 https://doi.org/10.1126/science.aaa9519	journal	June 2015
Topological origin of equatorial waves Delplace, Pierre; Marston, J. B.; Venaille, Antoine Science, Vol. 358, Issue 6366 https://doi.org/10.1126/science.aan8819	journal	October 2017
Gravitational waves with orbital angular momentum Baral, Pratyusava; Ray, Anarya; Koley, Ratna The European Physical Journal C, Vol. 80, Issue 4 https://doi.org/10.1140/epjc/s10052-020-7881-2	journal	April 2020
Spin and Orbital Angular Momentum of Photons Enk, S. J. van; Nienhuis, G. Europhysics Letters (EPL), Vol. 25, Issue 7 https://doi.org/10.1209/0295-5075/25/7/004	journal	March 1994
On the transformations generated by the electromagnetic spin and orbital angular momentum operators Fernandez-Corbaton, Ivan; Zambrana-Puyalto, Xavier; Molina-Terriza, Gabriel Journal of the Optical Society of America B, Vol. 31, Issue 9 https://doi.org/10.1364/JOSAB.31.002136	journal	August 2014
What are the quantum commutation relations for the total angular momentum of light? tutorial Das, Pronoy; Yang, Li-Ping; Jacob, Zubin Journal of the Optical Society of America B, Vol. 41, Issue 8 https://doi.org/10.1364/JOSAB.524752	journal	July 2024
On Unitary Representations of the Inhomogeneous Lorentz Group Wigner, E. The Annals of Mathematics, Vol. 40, Issue 1 https://doi.org/10.2307/1968551	journal	January 1939
Manifestation of the topological index formula in quantum waves and geophysical waves Faure, Frédéric Annales Henri Lebesgue, Vol. 6 https://doi.org/10.5802/ahl.169	journal	July 2023