Dirac equation based on the vector representation of the Lorentz group

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.

CPTM Symmetry for the Dirac Equation and Its Extended Version Based on the Vector Representation of the Lorentz Group

“…We have shown that the CPT theorem of the standard Dirac equation also holds for the extended version, which describes a massive fermion doublet obeying the isospin SU(2) symmetry [13]. This result was stated in Equation ( 16).…”

“…In this section we derive and discuss the basic symmetries of the standard and extended (including isospin) Dirac equation, which are charge conjugation, parity reflection and time inversion. We write the Dirac equation by using capital (lower) letters for the Gamma (gamma) matrices corresponding to the extended [13] and (standard) [17] Dirac equation. Lack of space forbids to explicitly quote here all related matrices in the Weyl representation.…”

“…with the well-known three-vector of the Pauli [23] matrices σ = (σ x , σ y , σ z ). According to the new derivations by Marsch and Narita [13], the Dirac matrices for the extended Dirac equation including isospin are based on the generalized 4×4 spin matrices, reading…”

“…According to our knowledge this idea in the present algebraic form has first been discussed by Marsch and Narita [12] before. We further define the CPTM symmetry for the new extended Dirac equation as proposed by Marsch and Narita [13,14] and provide here the related detailed algebra. There it was also shown that the extended Dirac equation can easily be expanded to include the SU(N) gauge fields of the standard model.…”

“…Changing the sign of the mass in the Dirac equation can be accomplished by operation of the so-called gamma-five matrix that plays an essential role for chirality in the SM. After revisiting the CPT theorem for the extended Dirac equation based on the vector representation of the Lorentz group [13], we newly define CPTM and discuss its consequences. A brief final section addresses some related theoretical aspects of general relativity.…”