Abstract
In the present article we investigate the spin-1/2 and spin-1 cases in different bases. Next, we look for relations with the Majorana-like field operator. We show explicitly incompatibility of the Majorana anzatzen with the Dirac-like field operators in both the original Majorana theory and its generalizations. Several explicit examples are presented for higher spins too. It seems that the calculations in the helicity basis only give mathematically and physically reasonable results.
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Notes
See Ref. [6] for discussion.
Of course, there are no any mathematical difficulties to change it to the normalization to ± m, which may be more convenient for the study of the massless limit.
The only possible changes may be related to different forms of normalization of 4-spinors, which would have influence on the factor before δ-function.
We should have the same contradiction even if \(\varphi \rightarrow \alpha \).
Please note that the phase factors may have physical significance in quantum field theories as opposed to the textbook nonrelativistic quantum mechanics, as was discussed recently by several authors.
Such definitions of 4-spinors differ, of course, from the original Majorana definition in x-representation:
$$ \nu (x) = \frac{1}{\sqrt{2}} ({\Psi}_{D} (x) + {{\Psi}_{D}^{c}} (x)) , $$(49)Cν(x) = ν(x) that represents the positive real C − parity field operator. However, the momentum-space Majorana-like spinors open various possibilities for description of neutral particles (with experimental consequences, see [27]). For instance, “for imaginary C parities, the neutrino mass can drop out from the single β decay trace and reappear in 0νββ, a curious and in principle experimentally testable signature for a non-trivial impact of Majorana framework in experiments with polarized sources.”
The change of the mass dimension of the field operator [28] has no sufficient foundations because the Lagrangian can be constructed on using the coupled Dirac equations, see Ref. [29]. After that one can play with \(\sqrt {m}\) to reproduce all possible mathematical results, which may (or may not) answer to the physical reality.
However, remember, that we have the p0 = 0 solution of the Maxwell equations. It has the experimental confirmation (for instance, the stationary magnetic field curlB = 0).
00 in other notation.
References
Dvoeglazov, V.V., Hadronic, J.: Suppl. 18, 239 (2003)
Dvoeglazov, V.V.: Int. J. Mod. Phys. B 20, 1317 (2006)
Dvoeglazov, V.V. In: Dvoeglazov, V.V. (ed.) Einstein and Others: Unification, p 211. Nova Science Publishers, Hauppauge (2015)
Dvoeglazov, V.V., Naturforsch, Z.: A 71, 345 (2016)
Dvoeglazov, V.V.: SFIN 22(A1), 157 (2009)
Dvoeglazov, V.V.: J. Phys. Conf. Ser. 284, 012024 (2011)
Dvoeglazov, V.V.: Bled Workshops 10(2), 52 (2009)
Dvoeglazov, V.V.: Bled Workshops 11(2), 9 (2010)
Dvoeglazov, V.V.. In: Beyond the Standard Model. Proceedings of the Vigier Symposium, p 388. Noetic Press, Orinda (2005)
Dvoeglazov, V.V.: Bled Workshops 14(2), 199 (2013)
Dirac, P.A.M.: Proc. Roy. Soc. Lond. A 117, 610 (1928)
Sakurai, J.J.: Advanced Quantum Mechanics. Addison-Wesley, Reading (1967)
Ryder, L.H.: Quantum Field Theory. Cambridge University Press, Cambridge (1985)
Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill Book Co., New York (1980)
Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, 2nd edn. Nauka, Moscow (1973)
Varshalovich, D.A., Moskalev, A.N., Khersonskiĭ, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988). §6.2.5
Dvoeglazov, V.V.: Fizika B 6, 111 (1997)
Dvoeglazov, V.V.: Int. J. Theor. Phys. 43, 1287 (2004)
Berestetskiĭ, V.B., Lifshitz, E.M., Pitaevskiĭ, L.P.: Quantum Electrodynamics. Pergamon, New York (1982). translated from the Russian, §16
Dvoeglazov, V.V.: Nuovo Cim. B 111, 483 (1996)
Ahluwalia, D.V.: Int. J. Mod. Phys. A 11, 1855 (1996)
Weinberg, S.: The Quantum Theory of Fields. Vol I. Foundations. Cambridge University Press, Cambridge (1995)
Greiner, W.: Field Quantization. Springer, Berlin (1996)
Tokuoka, Z.: Prog. Theor. Phys. 37, 603 (1967)
Rück, H.M., Greiner, W.: J. Phys. G: Nucl. Phys. 3, 657 (1977)
Majorana, E.: Nuovo Cim. 14, 171 (1937)
Kirchbach, M., Compean, C., Noriega, L.: Eur. Phys. J. A 22, 149 (2004)
Ahluwalia, D.V., Grumiller, D.: JCAP 0507, 012 (2005)
Dvoeglazov, V.V.: Nuovo Cim. A 108, 1467 (1995)
Dvoeglazov, V.V.: Mod. Phys. Lett. A 12, 2741 (1997)
Dvoeglazov, V.V.: J. Phys. Conf. Ser. 1010, 012011 (2018)
Novozhilov, Y.V.: Introduction to Elementary Particle Physics. Pergamon, New York (1975)
Dvoeglazov, V.V. In: Dvoeglazov, V.V. (ed.) Photon: Old Problems in Light of New Ideas. Nova Science Publishers, Huntington (2000)
Ahluwalia, D.V., Sawicki, M.: Phys. Rev. D 47, 5161 (1993)
Ahluwalia, D.V., Sawicki, M.: Phys. Lett. B 335, 24 (1994)
Sankaranarayanan, A., Good Jr., R.H.: Nuovo Cim. 36, 1303 (1965)
Ahluwalia, D.V., Johnson, M.B., Goldman, T.: Phys. Lett. B 316, 102 (1993)
Dvoeglazov, V.V.: Int. J. Theor. Phys. 37, 1915 (1998)
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I acknowledge discussions with colleagues at recent conferences. I am grateful to the Zacatecas University for professorship.
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Dvoeglazov, V.V. Incompatibility of the Dirac-like Field Operators with the Majorana Anzatzen. Int J Theor Phys 58, 1369–1383 (2019). https://doi.org/10.1007/s10773-019-04008-8
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DOI: https://doi.org/10.1007/s10773-019-04008-8