Dirac equation based on the vector representation of the Lorentz group
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Dirac equation based on the vector representation of the Lorentz group Eckart Marsch1,a
, Yasuhito Narita2,b
1 Extraterrestrial Physics Group, Institute for Experimental and Applied Physics, Christian Albrechts
University at Kiel, Leibnizstraße 11, 24118 Kiel, Germany
2 Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria
Received: 22 July 2020 / Accepted: 22 September 2020 © The Author(s) 2020
Abstract In this paper, we derive an expanded Dirac equation for a massive fermion doublet, which has in addition to the particle/antiparticle and spin-up/spin-down degrees of freedom explicity an isospin-type degree of freedom. We begin with revisiting the four-vector Lorentz group generators, define the corresponding gamma matrices and then write a Dirac equation for the fermion doublet with eight spinor components. The appropriate Lagrangian density is established, and the related chiral and SU (2) symmetry is discussed in detail, as well as applications to an electroweak-style gauge theory. In “Appendix,” we present some of the relevant matrices.
1 Introduction The goal of this study is to reconsider an old subject, namely the Lorentz group and its representations [1–4], yet here with emphasis on the four-vector generators of the Lorentz group. The new result will be an extended Dirac equation [5] for a massive fermion doublet which has another intrinsic degree of freedom related to SU (2) symmetry. The long history of the Dirac and related equations has been reviewed some years ago by Esposito [6]. In the present context, we just mention that ideas similar than ours were discussed long time ago in the paper titled “Fermions without spinors” [7], whereby the authors used the classical Kähler tensor field equation [8], which was extended by them and then applied to fermions. In that paper, extensive use was made of differential forms and calculus and of the related advanced mathematics which even today is not a standard tool of theoretical physics or modern quantum field theory, as it is presented in text books [9–11] on the standard model (SM) of elementary particle physics. The attempts to establish extended Dirac equations continue until day, including recently also color [12] as an intrinsic degree of freedom of a fermion. The work by Kerner [13] provides a concise introduction to this subject. However, the algebraic effort in that theory is substantial, and the equation derived is mathematically more sophisticated than the original Dirac equation. In our paper, we remain modest and stay closer to the mathematical language used in the SM as described in the excellent textbook by Schwartz [11] published only a few years
a e-mail: [email protected] (corresponding author) b e-mail: [email protected]
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ago. When establishing here an extended Dirac wave equation, we aim thereby at least algebraic complexity and most physical transparency. Of course, any prediction made by this
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