Related papers: Hyperbolic Numbers and the Dirac Spinor
4x4 Dirac (gamma) matrices (irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)) are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied…
The relativistic three-body problem is approached via the extension of the SL(2,C) group to the Sp(4,C) one. In terms of Sp(4,C) spinors, a Dirac-like equation with three-body kinematics is composed. After introducing the linear in…
This paper discusses a framework to parametrize and decompose operator matrix elements for particles with higher spin $(j > 1/2)$ using chiral representations of the Lorentz group, i.e. the $(j,0)$ and $(0,j)$ representations and their…
The goal of this paper is to develop a systematic method of locating the Mueller matrices within the class of the matrices of the real group SL(4, R). The main idea is to construct the general transformation of the group SL(4, R) (whose…
Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It…
The Cl(3,0) Clifford algebra is represented with the commutative ring of hyperbolic numbers H. The canonical form of the Poincare mass operator defined in this vector space corresponds to a sixteen-dimensional structure. This conflicts with…
We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann holomorphy conditions considered therein as fundamental field equations. The automorphism group SO(3,C) of the biquaternion…
We define a class of groups constructed from rings equipped with an involution. We show that under suitable conditions, these groups are either algebraic or arithmetic, including as special cases the orientation-preserving isometry group of…
We introduce spinors, at a level appropriate for an undergraduate or first year graduate course on relativity, astrophysics or particle physics. The treatment assumes very little mathematical knowledge (mainly just vector analysis and some…
Supergroups are defined in the framework of $\dZ_2$-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups…
Matrix elements of spinor and principal series representations of the Lorentz group are studied in the basis of complex angular momentum (helicity basis). It is shown that matrix elements are expressed via hyperspherical functions…
Part I: The geometric algebra of space is derived by extending the real number system to include three mutually anticommuting square roots of plus one. The resulting geometric algebra is isomorphic to the algebra of complex 2x2 matrices,…
A group theoretical description of basic discrete symmetries (space inversion P, time reversal T and charge conjugation C) is given. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex…
Using condition of relativistic invariance, group theory and Clifford algebra the component Lorentz invariance generalized Dirac equation for a particle with arbitrary mass and spin is suggested, where In the case of half-integral spin…
We describe explicitly the algebra of polynomial functions on the Hilbert space of four qubit states which are invariant under the SLOCC group $SL(2,{\mathbb C})^{4}$. From this description, we obtain a closed formula for the…
We relate the Lounesto classification of regular and singular spinors to the orbits of the $Spin(3,1)$ group in the space of Dirac spinors. We find that regular spinors are associated with the principal orbits of the spin group while…
We present a general model allowing "quantum simulation" of one-dimensional Dirac models with 2- and 4-component spinors using ultracold atoms in driven 1D tilted optical latices. The resulting Dirac physics is illustrated by one of its…
We summarize some previous work on SU(4) describing hadron representations and transformations as well as its noncompact 'counterpart' SU$*$(4) being the complex embedding of SL(2,$\mathbb{H}$). So after having related the 16-dim Dirac…
We show that the dynamical group of an electron in a constant magnetic field is the group of symplectomorphisms $Sp(4,\mathbb{R})$. It is generated by the spinorial realization of the conformal algebra $\mathfrak{so}(2,3)$ considered in…
In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3…