Fermions and the Dirac equation

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.

scholarly journals Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1373
Author(s):  
Louis H. Kauffman
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Clifford Algebras ◽  
Discrete Systems ◽  
Complex Numbers ◽  
Majorana Fermions ◽  
Matrix Algebras ◽  
Dirac Equations ◽  
Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

scholarly journals Dirac equation based on the vector representation of the Lorentz group

2020 ◽  
Vol 135 (10) ◽  
Author(s):  
Eckart Marsch ◽  
Yasuhito Narita
Keyword(s):  
Gauge Theory ◽  
Dirac Equation ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Lagrangian Density ◽  
Degree Of Freedom ◽  
Vector Representation ◽  
Massive Fermion

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.

Fermionic Formulation of the Ising Model

2020 ◽  
pp. 319-340
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Ising Model ◽  
Dirac Equation ◽  
Quadratic Form ◽  
Continuum Limit ◽  
Majorana Fermions ◽  
Bogoliubov Transformation ◽  
Crucial Aspect ◽  
Creation And Annihilation Operators ◽  
Order And Disorder ◽  
The Continuum

A crucial aspect of the Ising model is its fermionic nature and this chapter is devoted to this property of the model. In the continuum limit, a Dirac equation for neutral Majorana fermions emerges. The details of the derivation are much less important than understanding why it is possible. The chapter emphasizes the simplicity and the exactness of the result, and covers the so-called Wigner-Jordan transformation, which brings the original Hamiltonian to a quadratic form in the creation and annihilation operators of the fermions. It covers the role played by the Bogoliubov transformation and the importance of the order and disorder operators.

Unified Space-Time Trigonometry and its Applications to Relativistic Kinematics

1973 ◽  
Vol 51 (12) ◽  
pp. 1304-1312 ◽  
Author(s):  
Antoine Jaccarini
Keyword(s):  
Lorentz Group ◽  
Space Time ◽  
Spinor Representation ◽  
Lorentz Transformations ◽  
Geometrical Approach ◽  
Collision Process ◽  
Spherical Trigonometry ◽  
Relativistic Kinematics

This paper presents a geometrical approach to relativistic kinematics. Owing to a unified space-time trigonometry, the spherical trigonometry formalism may be used to describe and study the kinematics of any collision process. Lorentz transformations may thus be treated as purely geometrical problems.A different way to define a unified trigonometry is also proposed, which is based on the spinor representation of the Lorentz group. This leads to a different and more general formalism than the former one.

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

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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