Abstract
We construct a consistent theory of a quantum massive Weyl field. We start with the formulation of the classical field theory approach for the description of massive Weyl fields. It is demonstrated that the standard Lagrange formalism cannot be applied for the studies of massive first-quantized Weyl spinors. Nevertheless we show that the classical field theory description of massive Weyl fields can be implemented in frames of the Hamilton formalism or using the extended Lagrange formalism. Then we carry out a canonical quantization of the system. The independent ways for the quantization of a massive Weyl field are discussed. We also compare our results with the previous approaches for the treatment of massive Weyl spinors. Finally the new interpretation of the Majorana condition is proposed.
Similar content being viewed by others
References
Kobzarev, I.Y., Martem’yanov, B.V., Okun’, L.B., Shchepkin, M.G.: The phenomenology of neutrino oscillations. Sov. J. Nucl. Phys. 32, 823–828 (1980)
Schechter, J., Valle, J.W.F.: Neutrino masses in SU(2)⊗U(1) theories. Phys. Rev. D 22, 2227–2235 (1980)
Auger, M., et al. (EXO Collaboration): Search for neutrinoless double-beta decay in 136Xe with EXO-200. Phys. Rev. Lett. 109, 032505 (2012). arXiv:1205.5608 [hep-ex]
Andreotti, E., et al. (CUORICINO Collaboration): 130Te neutrinoless double-beta decay with CUORICINO. Astropart. Phys. 34, 822–831 (2011). arXiv:1012.3266 [nucl-ex]
Chamon, C., Jackiw, R., Nishida, Y., Pi, S.-Y., Santos, L.: Quantizing Majorana fermions in a superconductor. Phys. Rev. B 81, 224515 (2010). arXiv:1001.2760 [cond-mat.str-el]
Itzykson, C., Zuber, J.-B.: Quantum Field Theory, p. 694. McGraw-Hill, New York (1980)
Goldhaber, M., Grodzins, L., Sunyar, A.W.: Helicity of neutrinos. Phys. Rev. 109, 1015–1017 (1958)
Fukugita, M., Yanagida, T.: Physics of Neutrinos and Applications to Astrophysics, pp. 289–319. Springer, Berlin (2003)
Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, 3rd edn. pp. 10–89. Wiley, New York (1980)
Schechter, J., Valle, J.W.F.: Majorana neutrinos and magnetic fields. Phys. Rev. D 24, 1883–1889 (1981)
Weinberg, S.: The Quantum Theory of Fields: Foundations, 2nd edn. pp. 292–338. Cambridge University Press, Cambridge (1996)
Ahluwalia, D.V., Lee, C.-Y., Schritt, D.: Self-interacting Elko dark matter with an axis of locality. Phys. Rev. D 83, 065017 (2011). arXiv:0911.2947 [hep-ph]
An, F.P., et al. (Daya Bay Collaboration): Observation of electron-antineutrino disappearance at Daya Bay. Phys. Rev. Lett. 108, 171803 (2012). arXiv:1203.1669 [hep-ex]
Abe, Y., et al. (Double Chooz Collaboration): Indication of reactor \(\bar{\nu}_{e}\) disappearance in the double Chooz experiment. Phys. Rev. Lett. 108, 131801 (2012). arXiv:1112.6353 [hep-ex]
Dvornikov, M.: Field theory description of neutrino oscillations. In: Greene, J.P. (ed.) Neutrinos: Properties, Sources and Detection, pp. 23–90. NOVA Science Publishers, New York (2011). arXiv:1011.4300 [hep-ph]
Faddeev, L., Jackiw, R.: Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett. 60, 1692–1694 (1988)
Gantmacher, F.: Lectures in Analytical Mechanics, pp. 71–80. Mir Publishers, Moscow (1975)
Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints, pp. 13–21. Springer, Berlin (1990)
Berestetskiĭ, V.B., Lifshitz, E.M., Pitaevskiĭ, L.P.: Quantum Electrodynamics, 2nd edn. p. 86. Pergamon, Oxford (1980)
Case, K.M.: Reformulation of the Majorana theory of the neutrino. Phys. Rev. 107, 307–316 (1957)
Gitman, D.M., Gonçalves, A.E., Tyutin, I.V.: New pseudoclassical model for Weyl particles. Phys. Rev. D 50, 5439–5442 (1994)
Barut, A.O., Zanghi, N.: Classical model of the Dirac electron. Phys. Rev. Lett. 52, 2009–2012 (1984)
Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, 4th edn. pp. 140–143. Butterworth-Heinemann, Amsterdam (1994)
Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian formalism for nonlinear waves. Phys. Usp. 40, 1087–1116 (1997)
Dvornikov, M., Maalampi, J.: Oscillations of Dirac and Majorana neutrinos in matter and a magnetic field. Phys. Rev. D 79, 113015 (2009). arXiv:0809.0963 [hep-ph]
Joos, E., Zeh, H.D., Kiefer, C., Giulini, D.J.W., Kupsch, J., Stamatescu, I.-O.: Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)
Acknowledgements
I am very thankful to D.M. Gitman, J. Lukierski, and J. Maalampi for helpful discussions, to S. Forte for bringing Ref. [16] to my attention, as well as to FAPESP (Brazil) for a grant.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dvornikov, M. Canonical Quantization of a Massive Weyl Field. Found Phys 42, 1469–1479 (2012). https://doi.org/10.1007/s10701-012-9679-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-012-9679-z