scholarly journals Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of SU(N) Gauge Symmetry

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 475
Author(s):  
Eckart Marsch ◽  
Yasuhito Narita
Keyword(s):  
Gauge Symmetry ◽  
Dirac Equation ◽  
Clifford Algebra ◽  
Lie Algebra ◽  
Matrix Algebra ◽  
Lorentz Invariance ◽  
Mathematical Connections ◽  
Time Symmetry ◽  
Massive Spin ◽  
Tensor Multiplication

In this paper, we study possible mathematical connections of the Clifford algebra with the su(N)-Lie algebra, or in more physical terms the links between space-time symmetry (Lorentz invariance) and internal SU(N) gauge-symmetry for a massive spin one-half fermion described by the Dirac equation. The related matrix algebra is worked out in particular for the SU(2) symmetry and outlined as well for the color gauge group SU(3). Possible perspectives of this approach to unification of symmetries are briefly discussed. The calculations make extensive use of tensor multiplication of the matrices involved, whereby our focus is on revisiting the Coleman–Mandula theorem. This permits us to construct unified symmetries between Lorentz invariance and gauge symmetry in a direct product sense.

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 Matrix 3-Lie superalgebras and BRST supersymmetry

2017 ◽  
Vol 14 (11) ◽  
pp. 1750160 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Clifford Algebra ◽  
Lie Algebra ◽  
Vector Fields ◽  
Jacobi Identity ◽  
Lie Superalgebras ◽  
Infinite Dimensional ◽  
Algebra Approach ◽  
The Matrix ◽  
Infinite Dimensional Lie Algebra

Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

scholarly journals DIMENSIONAL REDUCTION

1999 ◽  
Vol 14 (02) ◽  
pp. 99-103 ◽  
Author(s):  
CORINNE A. MANOGUE ◽  
TEVIAN DRAY
Keyword(s):  
Dirac Equation ◽  
Space Time ◽  
Particle Spectrum ◽  
Spin States ◽  
Helicity State ◽  
Curved Space ◽  
Massless Spin ◽  
Generation Structure ◽  
Massive Spin ◽  
Opposite Helicity

Using an octonionic formalism, we introduce a new mechanism for reducing ten space–time dimensions to four without compactification. Applying this mechanism to the free, ten-dimensional, massless (momentum space) Dirac equation results in a particle spectrum consisting of exactly three generations. Each generation contains one massive spin-1/2 particle with two spin states, one massless spin-1/2 particle with only one helicity state, and their antiparticles — precisely one generation of leptons. There is also a single massless spin-1/2 particle/antiparticle pair with the opposite helicity and no generation structure. We conclude with a discussion of some further consequences of this approach, including those which could arise when using the formalism on a curved space–time background, as well as the implications for the nature of space–time itself.

scholarly journals Equations for massless and massive spin-1/2 particles with varying speed and neutrino in matter

2014 ◽  
Vol 29 (06) ◽  
pp. 1450031 ◽  
Author(s):  
S. I. Kruglov
Keyword(s):  
Energy Spectrum ◽  
Density Matrix ◽  
Lorentz Invariance ◽  
First Order ◽  
Dirac Equations ◽  
Order Form ◽  
Invariance Violation ◽  
Massive Spin ◽  
Lorentz Invariance Violation ◽  
Massless Fermions

The modified Dirac equations describing massless and massive spin-1/2 particles violating the Lorentz invariance are considered. The equation for massless fermions with varying speed is formulated in the 16-component first-order form. The projection matrix, which is the density matrix, extracting solutions to the equation has been obtained. Exact solutions to the equation and energy spectrum for a massive neutrino are obtained in the presence of background matter. We have considered as subluminal as well as superluminal propagations of neutrinos. The bounds on the Lorentz invariance violation parameter from astrophysical observations by IceCube, OPERA, MINOS collaborations and by the SN1987A supernova are obtained for superluminal neutrinos.

scholarly journals THE STABILIZED POINCARE–HEISENBERG ALGEBRA: A CLIFFORD ALGEBRA VIEWPOINT

2007 ◽  
Vol 16 (09) ◽  
pp. 1519-1529 ◽  
Author(s):  
N. G. GRESNIGT ◽  
P. F. RENAUD ◽  
P. H. BUTLER
Keyword(s):  
Quantum Mechanics ◽  
Clifford Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Clifford Algebras ◽  
Heisenberg Algebra ◽  
Stability Conditions ◽  
Relativistic Kinematics ◽  
Evolutionary Step

The stabilized Poincare–Heisenberg algebra (SPHA) is a Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after combining the Lie algebras of quantum mechanics and relativity. In this paper, we show how the sixteen-dimensional real Clifford algebras Cℓ(1,3) and Cℓ(3,1) can both be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations. It is conceptually easier and more straightforward to work with a Clifford algebra. The Clifford algebra path suggests that the next evolutionary step toward a theory of physics at the interface of GR and QM might be to depart from working in spacetime and instead to work in spacetime–momentum.

scholarly journals FIELD THEORETIC REALIZATIONS FOR CUBIC SUPERSYMMETRY

2004 ◽  
Vol 19 (32) ◽  
pp. 5585-5608 ◽  
Author(s):  
N. MOHAMMEDI ◽  
G. MOULTAKA ◽  
M. RAUSCH DE TRAUBENBERG
Keyword(s):  
Gauge Symmetry ◽  
Dimensional Space ◽  
A Priori ◽  
Space Time ◽  
Time Symmetry ◽  
Poincaré Algebra ◽  
Dimensional Space Time

We consider a four-dimensional space–time symmetry which is a nontrivial extension of the Poincaré algebra, different from supersymmetry and not contradicting a priori the well-known no-go theorems. We investigate some field theoretical aspects of this new symmetry and construct invariant actions for noninteracting fermion and noninteracting boson multiplets. In the case of the bosonic multiplet, where two-form fields appear naturally, we find that this symmetry is compatible with a local U(1) gauge symmetry, only when the latter is gauge fixed by a 't Hooft–Feynman term.

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