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Gauge Invariance for Classical Massless Particles with Spin

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Abstract

Wigner’s quantum-mechanical classification of particle-types in terms of irreducible representations of the Poincaré group has a classical analogue, which we extend in this paper. We study the compactness properties of the resulting phase spaces at fixed energy, and show that in order for a classical massless particle to be physically sensible, its phase space must feature a classical-particle counterpart of electromagnetic gauge invariance. By examining the connection between massless and massive particles in the massless limit, we also derive a classical-particle version of the Higgs mechanism.

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Notes

  1. For a more comprehensive treatment of the results in this paper, see [3].

  2. For an early example of this technique, see [5]. For a more modern, pedagogical treatment, see [4].

  3. See [11] for a pedagogical review.

  4. See, for example, [1, 13], but also [7] for a more optimistic take.

  5. For a derivation, see, for example, [3, 11].

  6. Note that if we permit parity transformations, which map \(\sigma \mapsto -\sigma\), then we must require that the equivalence relation (49) hold only for states that share the same helicity \(\sigma\).

References

  1. Abbott, L.F.: Massless particles with continuous spin indices. Phys. Rev. D 13(8), 2291–2294 (1976). https://doi.org/10.1103/PhysRevD.13.2291

    Article  ADS  MathSciNet  Google Scholar 

  2. Balachandran, A.P., Marmo, G., Skagerstam, B.S., Stern, A.: Gauge Symmetries and Fibre Bundles—Applications to Particle Dynamics, 1st edn. Springer, Berlin, Heidelberg (1983). https://doi.org/10.1007/3-540-12724-0. https://www.springer.com/gp/book/9783540127246

  3. Barandes, J.A.: Manifestly Covariant Lagrangians, Classical Particles with Spin, and the Origins of Gauge Invariance (2019). https://arxiv.org/abs/1911.08892

  4. Deriglazov, A., Rizzuti, B.: Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation. Am. J. Phys. 79(8), 882–885 (2011). https://doi.org/10.1119/1.3593270

    Article  ADS  Google Scholar 

  5. Dirac, P.A.M.: Relativity quantum mechanics with an application to compton scattering. Proc. R. Soc. A 111(758), 405–423 (1926). https://doi.org/10.1098/rspa.1926.0074

    Article  ADS  MATH  Google Scholar 

  6. Rivas, M.: Kinematical Theory of Spinning Particles. Springer, Berlin (2002)

    Book  Google Scholar 

  7. Schuster, P., Toro, N.: On the theory of continuous-spin particles: wavefunctions and soft-factor scattering amplitudes. J. High Energy Phys. 20, 104 (2013). https://doi.org/10.1007/JHEP09(2013)104

    Article  MathSciNet  MATH  Google Scholar 

  8. Skagerstam, B.S., Stern, A.: Lagrangian descriptions of classical charged particles with spin. Phys. Scr. 24, 493–497 (1981). https://doi.org/10.1088/0031-8949/24/3/002

    Article  ADS  Google Scholar 

  9. Souriau, J.M.: Structure of Dynamical Systems, 1st edn. Birkhäuser, Boston (1997)

    Book  Google Scholar 

  10. Stueckelberg, E.C.G.: “Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kernkraäfte”. Helvetica Physica Acta 11(3), 225–244, 299–328 (1938). https://doi.org/10.5169/seals-110852. https://www.e-periodica.ch/digbib/view?pid=hpa-001:1938:11::636#227

  11. Weinberg, S.: The Quantum Theory of Fields, 1st edn. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  12. Wigner, E.P.: On unitary representations of the inhomogeneous Lorentz group. Ann. Math. 40(1), 149–204 (1939). https://doi.org/10.2307/1968551

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Wigner, E.P.: Invariant Quantum Mechanical Equations of Motion, pp. 161–184. International Atomic Energy Agency, Vienna (1963). https://doi.org/10.1007/978-3-662-09203-3_18

    Book  Google Scholar 

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Acknowledgements

J. A. B. has benefited from personal communications with Howard Georgi, Andrew Strominger, David Griffiths, David Kagan, David Morin, Logan McCarty, Monica Pate, and Alex Lupsasca.

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  1. Jefferson Physical Laboratory, Harvard University, 17 Oxford Street, Cambridge, MA, 02138, USA

    Jacob A. Barandes

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Correspondence to Jacob A. Barandes.

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Barandes, J.A. Gauge Invariance for Classical Massless Particles with Spin. Found Phys 51, 7 (2021). https://doi.org/10.1007/s10701-021-00415-2

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