Intrinsic Spinor Techniques with Applications to the Lorentz Group and the Dirac Equation

AbstractIn 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.

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

Frontiers in Physics ◽

10.3389/fphy.2021.618392 ◽

2021 ◽

Vol 9 ◽

Author(s):

E. Marsch ◽

Y. Narita

Keyword(s):

Dirac Equation ◽

Lorentz Group ◽

Essential Role ◽

Fundamental Equation ◽

Extended Version ◽

Vector Representation ◽

Massive Fermion ◽

The Standard Model ◽

Cpt Theorem ◽

Extended Dirac Equation

We revisit the CPT theorem for the Dirac equation and its extended version based on the vector representation of the Lorentz group. Then it is proposed that CPTM may apply to this fundamental equation for a massive fermion a s a singlet or a doublet with isospin. The symbol M stands here for reversing the sign of the mass in the Dirac equation, which can be accomplished by operation on it with the so-called gamma-five matrix that plays an essential role for the chirality in the Standard Model. We define the CPTM symmetry for the standard and extended Dirac equation and discuss its physical implications and some possible consequences for general relativity.

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Fermions and the Dirac equation

10.1093/oso/9780198802877.003.0008 ◽

2018 ◽

Author(s):

Michael Kachelriess

Keyword(s):

Dirac Equation ◽

Lorentz Group ◽

Spinor Representation ◽

Majorana Fermions

Starting from the spinor representation of the Lorentz group,Weyl spinors and their transformation properties are derived. The Dirac equation and the properties of its solutions are discussed. Graßmann numbers and the gener-ating functional for fermions are introduced. Weyl and Majorana fermions are examined.

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The Dirac Equation and the Unitary Representations of the Inhomogeneous Lorentz Group

Journal of Mathematical Physics ◽

10.1063/1.1704283 ◽

1965 ◽

Vol 6 (2) ◽

pp. 322-326

Author(s):

L. B. Redei

Keyword(s):

Dirac Equation ◽

Lorentz Group ◽

Unitary Representations ◽

Inhomogeneous Lorentz Group

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Fermion Colour and Flavour Originating from Multiple Representations of the Lorentz Group and Clifford Algebra

Physical Science International Journal ◽

10.9734/psij/2019/v23i330158 ◽

2019 ◽

pp. 1-13

Author(s):

Eckart Marsch

Keyword(s):

Dirac Equation ◽

Clifford Algebra ◽

Lorentz Group ◽

Building Block ◽

Multiple Representations ◽

Degree Of Freedom ◽

Dirac Spinor ◽

Weyl Spinor ◽

Zero Mass ◽

Pauli Matrices

Where do such fermion properties as colour and flavour come from? We attempt to give a possible answer to this question in our paper. For that purpose we use the reducible (1/2,1/2) representation of the Lorentz group. Then the fermion corresponds to a doublet, each component of which can be described by the standard Dirac equation. In this way we conclude that quark and lepton, when being considered as doublets, originate from the discussed multiple representations of the Lorentz group (LG) and the related Clifford algebra. In particular the threefold colour degree of freedom emerges naturally, and similarly the threefold generation degree, both being enabled essentially by the fact that the SU(2) group has three generators given by the Pauli matrices. The Dirac spinor, or for zero mass the chiral Weyl spinor, remains the building block of that theory.

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