scholarly journals Biquaternionic Dirac Equation Predicts Zero Mass for Majorana Fermions

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1144
Author(s):  
Avraham Nofech
Keyword(s):  
Dirac Equation ◽  
Outer Automorphism ◽  
Charge Conjugation ◽  
Majorana Fermions ◽  
Algebraic Proof ◽  
Conjugation Operator ◽  
Equal Zero ◽  
Zero Mass ◽  
Mass Coefficient ◽  
Pauli Algebra

A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism of the Pauli algebra. The charge conjugation operator takes a particulary simple form in this formulation and switches the sign of the mass coefficient, so that for a solution invariant under charge conjugation the mass has to equal zero. The multiple of the charge conjugation operator by the imaginary unit turns out to be a complex Lorentz transformation. It commutes with the outer automorphism, while the charge conjugation operator itself anticommutes with it, providing a second more algebraic proof of the main theorem. Considering the Majorana equation, it is shown that non-zero mass of its solution is imaginary.

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 Nonstandard Dirac equations for nonstandard spinors

2014 ◽  
Vol 23 (14) ◽  
pp. 1444007 ◽  
Author(s):  
A. G. Nikitin
Keyword(s):  
Dirac Equation ◽  
Mass Term ◽  
Charge Conjugation ◽  
Poincaré Group ◽  
Poincare Group ◽  
Dirac Equations ◽  
Generalized Dirac Equation ◽  
Carrier Space ◽  
New Look

Generalized Dirac equation with operator mass term is presented. Its solutions are nonstandard spinors (NSS) which, like eigenspinoren des Ladungskonjugationsoperators (ELKO), are eigenvectors of the charge conjugation and dual-helicity operators. It is demonstrated that in spite of their noncovariant nature the NSS can serve as a carrier space of a representation of Poincaré group. However, the corresponding boost generators are not manifestly covariant and generate nonlocal momentum dependent transformations, which are presented explicitly. These results can present a new look on group-theoretical grounds of ELKO theories.

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.

scholarly journals Antimatter Gravity: Second Quantization and Lagrangian Formalism

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 397-411
Author(s):  
Ulrich D. Jentschura
Keyword(s):  
Dirac Equation ◽  
Flat Space ◽  
Space Time ◽  
Charge Conjugation ◽  
Lagrangian Formalism ◽  
Curved Space ◽  
The Earth ◽  
Antimatter Gravity ◽  
Quantized Field ◽  
Charge Conjugation Parity

The application of the CPT (charge-conjugation, parity, and time reversal) theorem to an apple falling on Earth leads to the description of an anti-apple falling on anti–Earth (not on Earth). On the microscopic level, the Dirac equation in curved space-time simultaneously describes spin-1/2 particles and their antiparticles coupled to the same curved space-time metric (e.g., the metric describing the gravitational field of the Earth). On the macroscopic level, the electromagnetically and gravitationally coupled Dirac equation therefore describes apples and anti-apples, falling on Earth, simultaneously. A particle-to-antiparticle transformation of the gravitationally coupled Dirac equation therefore yields information on the behavior of “anti-apples on Earth”. However, the problem is exacerbated by the fact that the operation of charge conjugation is much more complicated in curved, as opposed to flat, space-time. Our treatment is based on second-quantized field operators and uses the Lagrangian formalism. As an additional helpful result, prerequisite to our calculations, we establish the general form of the Dirac adjoint in curved space-time. On the basis of a theorem, we refute the existence of tiny, but potentially important, particle-antiparticle symmetry breaking terms in which possible existence has been investigated in the literature. Consequences for antimatter gravity experiments are discussed.

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 Global Evolution of the U(1) Higgs Boson: Nonlinear Stability and Uniform Energy Bounds

2020 ◽  
Author(s):  
Shijie Dong ◽  
Philippe G. LeFloch ◽  
Zoe Wyatt
Keyword(s):  
Dirac Equation ◽  
Nonlinear Stability ◽  
Minkowski Spacetime ◽  
Energy Functional ◽  
Initial Value ◽  
Energy Bounds ◽  
New Energy ◽  
Mass Coefficient ◽  
Klein Gordon ◽  
Higgs Fields

Abstract Relying on the hyperboloidal foliation method, we establish the nonlinear stability of the ground state of the U(1) standard model of electroweak interactions. This amounts to establishing a global-in-time theory for the initial value problem for a nonlinear wave–Klein–Gordon system that couples (Dirac, scalar, gauge) massive equations together. In particular, we investigate here the Dirac equation and study a new energy functional defined with respect to the hyperboloidal foliation of Minkowski spacetime. We provide a decay result for the Dirac equation which is uniform in the mass coefficient and thus allows for the Dirac mass coefficient to be arbitrarily small. Furthermore, we establish energy bounds for the Higgs fields and gauge bosons that are uniform with respect to the hyperboloidal time variable.

scholarly journals On the Majorana Equation: Relations between Its Complex Two-Component and Real Four-Component Eigenfunctions

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Eckart Marsch
Keyword(s):  
Dispersion Relation ◽  
Dirac Equation ◽  
Massive Particle ◽  
Mass Term ◽  
Complex Conjugation ◽  
Conjugation Operator ◽  
Direct Linearization ◽  
Two Component ◽  
Relativistic Dispersion ◽  
Component Spinor

We first derive without recourse to the Dirac equation the two-component Majorana equation with a mass term by a direct linearization of the relativistic dispersion relation of a massive particle. Thereby, we make only use of the complex conjugation operator and the Pauli spin matrices, corresponding to the irreducible representation of the Lorentz group. Then we derive the complex two-component eigenfunctions of the Majorana equation and the related quantum fields in a concise way, by exploiting the so-called chirality conjugation operator that involves the spin-flip operator. Subsequently, the four-component spinor solutions of the real Majorana equation are derived, and their intrinsic relations with the spinors of the complex two-component version of the Majorana equation are revealed and discussed extensively.

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