Related papers: How `Complex' is the Dirac Equation?
We define a superalgebra S2(N/2) as a Z2 graded algebra of dimension 2N+3, where N is a positive, odd integer. The even component is a three-dimensional abelian subalgebra, while the odd component is made up of two N-dimensional, mutually…
This paper investigates the Lorentz invariance of the multidimensional Dirac-Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct…
Dirac matrices, also known as gamma matrices, are defined only up to a similarity transformation. Usually, some explicit representation of these matrices is assumed in order to deal with them. In this article, we show how it is possible to…
By exploring a spinor space whose elements carry a spin 1/2 representation of the Lorentz group and satisfy the the Fierz-Pauli-Kofink identities we show that certain symmetries operations form a Lie group. Moreover, we discuss the reflex…
We study pseudo-Riemanniasn manifolds $(M,g)$ with transitive group of conformal transformation which is essential, i.e. does not preserves any metric conformal to $g$. All such manifolds of Lorentz signature with non exact isotropy…
The Courant-Snyder theory for two-dimensional coupled linear optics is presented, based on the systematic use of the real representation of the Dirac matrices. Since any real $4\times 4$-matrix can be expressed as a linear combination of…
All possible permutations in the discrete $S_4$ group are classified by three rotation angles associated with the orthogonal group $O(3)$. We construct a spinor representation ${\bf 2}_D$ of $O(3)$, which is transformed by three 4$\times$4…
A generic-curved spacetime Dirac-like equation in 3D is constructed. It has, owing to the $\bar{SL}(n,R)$ group deunitarizing automorphism, a physically correct unitarity and flat spacetime particle properties. The construction is achieved…
We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a…
We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, $Q_8$, achieved by multiplying one of the gamma matrices by the imaginary number, $i$. The reason for doing this is to…
As discussed in a previous article, any (real) Lorentz algebra element possess a unique orthogonal decomposition as a sum of two mutually annihilating decomposable Lorentz algebra elements. In this article, this concept is extended to the…
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…
In this paper, starting from pure group-theoretical point of view, we develop a regular approach to describing particles with different spins in the framework of a theory of scalar fields on the Poincare group. Such fields can be considered…
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute…
There are several 3+1 parameter quantities in physics (like vector + scalar potentials, 4-currents, space-time, 4-momentum). In most cases (but space-time), the 3- and the 1-parameter characterised elements of these quantities differ in the…
An extended local Lorentz symmetry in four-dimensional (4D) theory is considered. A source of this symmetry is a group of general linear transformations of four-component Majorana spinors GL(4,M) which is isomorphic to GL(4,R) and is the…
This note supplements an earlier paper on conformal field theories. There it was shown how to construct tensor, spinor, and spinor-tensor primary fields in four dimensions from their counterparts in six dimensions, where conformal…
Properties of tensors equivalent to the direct product of two different 4-spinors are investigated. It is shown that the tensors obey additional 8 nonlinear restrictions, those are presented in Lorentz covariant form. In the context of the…
We present the rigorous derivation of covariant spin operators from a general linear combination of the components of the Pauli-Lubanski vector. It is shown that only two spin operators satisfy the spin algebra and transform properly under…
It is possible to construct representations of the Lorentz group using four-dimensional harmonic oscillators. This allows us to construct three-dimensional wave functions with the usual rotational symmetry for space-like coordinates and…