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.

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.

scholarly journals THE MASS OF THE NEUTRINOS

2010 ◽  
Vol 19 (11) ◽  
pp. 2285-2292
Author(s):  
B. G. SIDHARTH
Keyword(s):  
Dirac Equation ◽  
Standard Model ◽  
Special Relativity ◽  
Lorentz Invariance ◽  
The Standard Model

In the theory of the Dirac equation and in the standard model, the neutrino is massless. Both these theories use Lorentz invariance. In modern approaches however, spacetime is no longer smooth, and this modifies special relativity. We show how such a modification throws up the mass of the neutrino.

scholarly journals An extended Dirac equation in noncommutative spacetime

2016 ◽  
Vol 31 (15) ◽  
pp. 1650089 ◽  
Author(s):  
R. Vilela Mendes
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Quantum ◽  
Large Mass ◽  
Exterior Algebra ◽  
Relativistic Quantum Mechanics ◽  
Spacetime Geometry ◽  
Noncommutative Spacetime ◽  
Extended Dirac Equation

Stabilizing, by deformation, the algebra of relativistic quantum mechanics a noncommutative spacetime geometry is obtained. The exterior algebra of this geometry leads to an extended massless Dirac equation which has both a massless and a large mass solution. The nature of the solutions is discussed as well as the effects of coupling the two solutions.

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 Rigorous Derivation of the Pauli Equation With Time-dependent Electromagnetic Field

VLSI Design ◽  
1999 ◽  
Vol 9 (4) ◽  
pp. 415-426 ◽  
Author(s):  
Norbert J. Mauser
Keyword(s):  
Dirac Equation ◽  
Order Approximation ◽  
Fundamental Equation ◽  
Time Dependent ◽  
Projection Operators ◽  
Rigorous Derivation ◽  
Solid State Physics ◽  
Pauli Equation ◽  
Rigorous Results ◽  
Small Component

In this work we discuss relativistic corrections for the description of charge carriers in a quantum mechanical framework. The fundamental equation is the Dirac equation which takes into account also the electron's spin. However, this equation intrinsically also incorporates positrons which play no role in applications in solid state physics. We give a rigorous derivation of the Pauli equation describing electrons in a first order approximation of the Dirac equation in the limit of infinite velocity of light. We deal with time-dependent electromagnetic potentials where no rigorous results have been given before. Our approach is based on the use of appropriate projection operators for the electron and the positron component of the spinor which are better suited than the widely used simple splitting into ‘upper (large)’ and ‘lower (small) component’. We also systematically derive corrections at second order in 1/c where we essentially recover the results of the Foldy-Wouthuysen approach. However, due to the non-static problem, differences occur in the term which couples the electric field with the spin.

scholarly journals Lorentz invariant CPT breaking in the Dirac equation

2017 ◽  
Vol 32 (09) ◽  
pp. 1741014 ◽  
Author(s):  
Kazuo Fujikawa ◽  
Anca Tureanu
Keyword(s):  
Dirac Equation ◽  
Explicit Construction ◽  
Mass Splitting ◽  
Negative Energy ◽  
Coordinate Space ◽  
Step Functions ◽  
The Standard Model ◽  
Breaking Mechanism ◽  
Mass Degeneracy ◽  
Breaking Term

If one modifies the Dirac equation in momentum space to [Formula: see text], the symmetry of positive and negative energy eigenvalues is lifted by [Formula: see text] for a small [Formula: see text]. The mass degeneracy of the particle and antiparticle is thus lifted in a Lorentz invariant manner since the combinations [Formula: see text] with step functions are manifestly Lorentz invariant. We explain an explicit construction of this CPT breaking term in coordinate space, which is Lorentz invariant but nonlocal at the distance scale of the Planck length. The application of this Lorentz invariant CPT breaking mechanism to the possible mass splitting of the neutrino and antineutrino in the Standard Model is briefly discussed.

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