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

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.

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 3+1DMassless Weyl Spinors from Bosonic Scalar-Tensor Duality

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Andrea Amoretti ◽  
Alessandro Braggio ◽  
Giacomo Caruso ◽  
Nicola Maggiore ◽  
Nicodemo Magnoli
Keyword(s):  
Dirac Equation ◽  
Charge Density ◽  
Dimensional Reduction ◽  
Degrees Of Freedom ◽  
Planar Boundary ◽  
Weyl Spinor ◽  
Bf Theory ◽  
Free Theory ◽  
Tomographic Representation

We consider the fermionization of a bosonic-free theory characterized by the3+1Dscalar-tensor duality. This duality can be interpreted as the dimensional reduction, via a planar boundary, of the4+1Dtopological BF theory. In this model, adopting the Sommerfield tomographic representation of quantized bosonic fields, we explicitly build a fermionic operator and its associated Klein factor such that it satisfies the correct anticommutation relations. Interestingly, we demonstrate that this operator satisfies the massless Dirac equation and that it can be identified with a3+1DWeyl spinor. Finally, as an explicit example, we write the integrated charge density in terms of the tomographic transformed bosonic degrees of freedom.

scholarly journals Trajectory construction of Dirac evolution

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190682
Author(s):  
Peter Holland
Keyword(s):  
Dirac Equation ◽  
Evolution Operator ◽  
Exact Formula ◽  
Dirac Spinor ◽  
Mean Flow ◽  
First Order ◽  
Continuity Equations ◽  
The Mean ◽  
Trajectory Models ◽  
Mass Dependent

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the ‘relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.

scholarly journals ELKO SPINOR FIELDS: LAGRANGIANS FOR GRAVITY DERIVED FROM SUPERGRAVITY

2009 ◽  
Vol 06 (03) ◽  
pp. 461-477 ◽  
Author(s):  
ROLDÃO DA ROCHA ◽  
J. M. HOFF DA SILVA
Keyword(s):  
Spinor Field ◽  
Sufficient Conditions ◽  
Dirac Spinor ◽  
Weyl Spinor ◽  
Conjugation Operator ◽  
Spinor Fields ◽  
Mass Dimension ◽  
Majorana Spinor ◽  
Dimension One ◽  
Elko Spinor Fields

Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong — together with Majorana spinor fields — to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor fields to the three classes Dirac spinor fields, it is shown that the Einstein–Hilbert, the Einstein–Palatini, and the Holst actions can be derived from the Quadratic Spinor Lagrangian (QSL), as the prime Lagrangian for supergravity. The Holst action is related to the Ashtekar's quantum gravity formulation. To each one of these classes, there corresponds a unique kind of action for a covariant gravity theory. Furthermore we consider the necessary and sufficient conditions to map Dirac spinor fields (DSFs) to ELKO, in order to naturally extend the Standard Model to spinor fields possessing mass dimension one. As ELKO is a prime candidate to describe dark matter and can be obtained from the DSFs, via a mapping explicitly constructed that does not preserve spinor field classes, we prove that — in particular — the Einstein–Hilbert, Einstein–Palatini, and Holst actions can be derived from the QSL, as a fundamental Lagrangian for supergravity, via ELKO spinor fields. The geometric meaning of the mass dimension-transmuting operator — leading ELKO Lagrangian into the Dirac Lagrangian — is also pointed out, together with its relationship to the instanton Hopf fibration.

Realizations of generators of complex angular momentum

1990 ◽  
Vol 68 (7-8) ◽  
pp. 599-603
Author(s):  
Shuchi Bora ◽  
H. C. Chandola ◽  
B. S. Rajput
Keyword(s):  
Angular Momentum ◽  
Lorentz Group ◽  
Continuous Spin ◽  
Zero Mass ◽  
Complex Angular Momentum ◽  
General Manner ◽  
Homogeneous Lorentz Group ◽  
Real Mass ◽  
Spin Representations ◽  
Spin Contribution

We use the generators of complex angular momentum in complex c3 space and derive the realizations of the homogeneous Lorentz group for nonzero real mass, zero mass, and imaginary mass systems. We use the appropriate little group for different systems to calculate the modifications in the spin contribution to angular momentum and the unphysical continuous spin representations are shown to be eliminated. We diagonalize the helicity operator in c3 space and obtain the generators of complex angular-momentum operators, which are shown to lead, in a general manner, to the standard helicity representations of the Poincare group for timelike and spacelike systems.

scholarly journals Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

Universe ◽  
2019 ◽  
Vol 5 (8) ◽  
pp. 184 ◽  
Author(s):  
Victor Miguel Banda Guzmán ◽  
Mariana Kirchbach
Keyword(s):  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Dirac Spinor ◽  
Product Spaces ◽  
Group Representations ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Irreducible Components ◽  
Carrier Space ◽  
Lorentz Group Representations

The momentum-independent Casimir operators of the homogeneous spin-Lorentz group are employed in the construction of covariant projector operators, which can decompose anyone of its reducible finite-dimensional representation spaces into irreducible components. One of the benefits from such operators is that any one of the finite-dimensional carrier spaces of the Lorentz group representations can be equipped with Lorentz vector indices because any such space can be embedded in a Lorentz tensor of a properly-designed rank and then be unambiguously found by a projector. In particular, all the carrier spaces of the single-spin-valued Lorentz group representations, which so far have been described as 2 ( 2 j + 1 ) column vectors, can now be described in terms of Lorentz tensors for bosons or Lorentz tensors with the Dirac spinor component, for fermions. This approach facilitates the construct of covariant interactions of high spins with external fields in so far as they can be obtained by simple contractions of the relevant S O ( 1 , 3 ) indices. Examples of Lorentz group projector operators for spins varying from 1 / 2 –2 and belonging to distinct product spaces are explicitly worked out. The decomposition of multiple-spin-valued product spaces into irreducible sectors suggests that not only the highest spin, but all the spins contained in an irreducible carrier space could correspond to physical degrees of freedom.

scholarly journals DIRAC SPINOR IN A NONSTATIONARY GÖDEL-TYPE COSMOLOGICAL UNIVERSE

1993 ◽  
Vol 08 (32) ◽  
pp. 3011-3015 ◽  
Author(s):  
VÍCTOR M. VILLALBA
Keyword(s):  
Dirac Equation ◽  
Frequency Spectrum ◽  
Rotational Speed ◽  
Separation Of Variables ◽  
Dirac Spinor ◽  
Dirac Particles

In this letter we solve, via separation of variables, the massless Dirac equation in a nonstationary rotating, causal Gödel-type cosmological universe, having a constant rotational speed in all the points of the space. We compute the frequency spectrum. We show that the spectrum of massless Dirac particles is discrete and unbounded.

The Dirac spinor in six dimensions

1968 ◽  
Vol 64 (3) ◽  
pp. 765-778 ◽  
Author(s):  
E. A. Lord
Keyword(s):  
Lorentz Group ◽  
Dimensional Space ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Indefinite Metric ◽  
Dirac Spinor ◽  
Six Dimensions ◽  
Spinor Representations ◽  
Dirac Spinors

AbstractThe spinor representations of the rotation group in a six-dimensional space with indefinite metric are shown to be four-component spinors, which become the usual Dirac spinors when the formalism is restricted to a four-dimensional subspace. Eriksson's work on the five-dimensional Lorentz group is found to result from a restriction of the six-dimensional treatment to a five-dimensional subspace, and the algebraic significance of Eriksson's work is thereby clarified.

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