On the group-theoretical approach to relativistic wave equations for arbitrary spin

L. Nanni

Teoretičeskaâ i matematičeskaâ fizika, Tome 209 (2021) no. 2, pp. 224-242

Résumé

Formulating a relativistic equation for particles with arbitrary spin remains an open challenge in theoretical physics. In this study, the main algebraic approaches used to generalize the Dirac and Kemmer–Duffin equations for arbitrary-spin particles are investigated. It is proved that an irreducible relativistic equation formulated using spin matrices satisfying the commutation relations of the anti-de Sitter group leads to inconsistent results, mainly as a consequence of the violation of unitarity and the appearance of a mass spectrum that does not reflect the physical reality of elementary particles. However, the introduction of subsidiary conditions resolves the problem of unitarity and restores the physical meaning of the mass spectrum. The equations obtained by these approaches are solved and the physical nature of the solutions is discussed.

Détail

Keywords: relativistic wave equation, higher spin, anti-de Sitter group, irreducible representations of Lorentz group.

Bibliographie

[1] W. Gordon, “Der Comptoneffekt nach der Schrödingerschen Theorie”, Z. Phys., 40:1–2 (1926), 117–133 | DOI

[2] P. A. M. Dirac, “The quantum theory of the electron”, Proc. Roy. Soc. London Ser. A, 117:778 (1928), 610–624 | DOI

[3] C. D. Anderson, “The positive electron”, Phys. Rev., 43:6 (1933), 491–498 | DOI

[4] J. Chadwick, “The existence of a neutron”, Proc. Roy. Soc. London Ser. A, 136:830 (1932), 692–708 | DOI

[5] E. Majorana, “Teoria relativistica di particelle con momentum internisico arbitrario”, Nuovo Cimento, 9:10 (1932), 335–344 | DOI

[6] E. Majorana, “Teoria simmetrica dell'elettrone e del positrone”, Nuovo Cimento, 14:4 (1937), 171–184 | DOI

[7] J. Iliopoulos, The Origin of Mass: Elementary Particles and Fundamental Symmetries, Oxford Univ. Press, Oxford, 2017 | Zbl

[8] F. Benatti, Dynamics, Information and Complexity in Quantum Systems, Theoretical and Mathematical Physics, Springer, Dordrecht, 2009 | DOI | MR

[9] Y.-P. Huo, C.-K. Ng, “Some algebraic structures related to a quantum system with infinite degrees of freedom”, Rep. Math. Phys., 67:1 (2011), 97–107 | DOI | MR

[10] G. Esposito, G. Marmo, G. Miele, G. Sudarshan, Advanced Concepts in Quantum Mechanics, Cambridge Univ. Press, Cambridge, 2015 | MR

[11] E. Wigner, “On unitary representations of the inhomogeneous Lorentz group”, Ann. Math. (2), 40:1 (1939), 149–204 | DOI | MR

[12] S. Esposito, The Physics of Ettore Majorana. Theoretical, Mathematical, and Phenomenological, Cambridge Univ. Press, Cambridge, 2015 | DOI | MR

[13] P. A. M. Dirac, “Relativistic wave equations”, Proc. Roy. Soc. London Ser. A, 155:886 (1936), 447–459 | DOI

[14] H. J. Bhabha, “Relativistic wave equations for elementary particles”, Rev. Modern Phys., 17:2–3 (1945), 200–216 | DOI | MR

[15] N. Kemmer, “The particle aspect of meson theory”, Proc. Roy. Soc. London Ser. A, 173:952 (1939), 91–116 | DOI | MR

[16] R. J. Duffin, “On the characteristic matrices of covariant systems”, Phys. Rev., 54:12 (1938), 1114–1114 | DOI

[17] W. Rarita, J. Schwinger, “On a theory of particles with half-integral spin”, Phys. Rev., 60:1 (1941), 61 | DOI

[18] V. Bargmann, E. P. Wigner, “Group theoretical discussion of relativistic wave equations”, Proc. Nat. Acad. Sci. USA, 34:5 (1948), 211–223 | DOI | MR

[19] R.-K. Loide, I. Ots, R. Saar, “Bhabha relativistic wave equations”, J. Phys. A: Math. Gen., 30:11 (1997), 4005–4017 | DOI | MR

[20] R. Aldrovandi, J. G. Pereira, An Introduction to Geometrical Physics, World Sci., Singapore, 1995 | DOI | MR

[21] R. Aldrovandi, J. P. Beltrán Almeida, J. G. Pereira, “de Sitter special relativity”, Class. Quantum Grav., 24:6 (2007), 1385–1404, arXiv: gr-qc/0606122 | DOI

[22] J. K. Lubański, “Sur la théorie des particules élémentaires de spin quelconque. I”, Physica, 9:3 (1942), 310–324 | DOI | MR

[23] W. Fulton, Young Tableaux, London Mathematical Society Student Texts, 35, Cambridge Univ. Press, Cambridge, 1997 | MR

[24] D. M. Fradkin, “Comments on a paper by Majorana concerning elementary particles”, Amer. J. Phys., 34:4 (1966), 314–318 | DOI

[25] R. Casalbuoni, “Majorana and the infinite component wave equations”, PoS (EMC2006), 37 (2006), 004, 15 pp., arXiv: hep-th/0610252

[26] E. Joung, J. Mourad, R. Parentani, “Group theoretical approach to quantum fields in de Sitter space. I The principal series”, JHEP, 08 (2006), 082, 36 pp. | DOI | MR

[27] B. Supriadi, T. Prihandono, V. Rizqiyah, Z. R. Ridlo, N. Faroh, S. Andika, “Angular momentum operator commutator against position and Hamiltonian of a free particle”, J. Phys.: Conf. Series, 1211 (2019), 012051, 8 pp. | DOI

[28] I. M. Gelfand, A. M. Yaglom, “Obschie relyativistski invariantnye uravneniya i beskonechnomernye predstavleniya gruppy Lorentsa”, ZhETF, 18:8 (1948), 703–733 | MR

[29] V. L. Ginzburg, “On relativistic wave equations with a mass spectrum”, Acta Phys. Polon., 15:1 (1956), 163–175 | MR

[30] I. M. Gelfand, R A. Minlos, Z. Ya. Shapiro, Predstavleniya gruppy vraschenii i gruppy Lorentsa, ikh primeneniya, Nauka, M., 1958 | MR

[31] M. A. Naimark, Lineinye predstavleniya gruppy Lorentsa, URSS, M., 2019 | MR | MR | Zbl

[32] R. G. Cooke, Infinite Matrices and Sequence Spaces, Dover, New York, 2014 | MR

[33] B. C. Chanyal, S. Karnatak, “A comparative study of quaternionic rotational Dirac equation and its interpretation”, Internat. J. Geom. Meth. Modern Phys., 17:2 (2020), 2050018, 21 pp., arXiv: 1909.02446 | DOI | MR

[34] E. H. Roffman, “Unitary and non-unitary representations of the complex inhomogeneous Lorentz group”, Commun. Math. Phys., 4:4 (1967), 237–257 | DOI | MR

[35] A. O. Barut, I. H. Duru, “Introduction of internal coordinates into the infinite-component Majorana equation”, Proc. Roy. Soc. London Ser. A, 333:1593 (1973), 217–224 | DOI | MR

[36] R. M. Mir-Kasimov, I. P. Volobujev, “Complex quaternions and spinor representations of de Sitter groups $SO(4,1)$ and $SO(3,2)$”, Acta Phys. Polon. B, 9:2 (1978), 91–105 | MR

[37] C. D. Greenman, “The generic spacing distribution of the two-dimensional harmonic oscillator”, J. Phys. A: Math. Gen., 29:14 (1996), 4065–4081 | DOI | MR

[38] V. L. Ginzburg, “Kakie problemy fiziki i astrofiziki predstavlyayutsya seichas osobenno vazhnymi i interesnymi (tridtsat let spustya, prichem uzhe na poroge KhKhI veka)?”, UFN, 169:4 (1999), 419–441 | DOI | DOI

[39] M. Gell-Mann, “Symmetries of baryons and mesons”, Phys. Rev., 125:3 (1962), 1067–1084 | DOI | MR

[40] V. V. Varlamov, “Spinor structure and internal symmetries”, Internat. J. Theor. Phys., 54:10 (2015), 3533–3576 | DOI | MR

[41] V. V. Varlamov, “Spinor structure and matter spectrum”, Internat. J. Theor. Phys., 55:11 (2015), 5008–5045 | DOI

[42] B. G. Sidharth, A formula for the mass spectrum of baryons and mesons, arXiv: physics/0306010

[43] C. Amsler, The Quark Structure of Hadrons. An Introduction to the Phenomenology and Spectroscopy, Lecture Notes in Physics, 949, Springer, Cham, 2018 | DOI

[44] I. L. Buchbinder, S. J. Gates, K. Koutrolikos, “Conserved higher spin supercurrents for arbitrary spin massless supermultiplets and higher spin superfield cubic interactions”, JHEP, 08 (2018), 055, 11 pp. | DOI | MR