The Dirac algebra and its physical interpretation

2015

The paper re-applies the 64-part algebra discussed by P. Rowlands in a series of (FERT and other) papers in the recent years. It demonstrates that the original introduction of the γ algebra by Dirac to ”the quantum theory of the electron” can be interpreted with the help of quaternions: both the α matrices and the Pauli (σ) matrices in Dirac’s original interpretation can be rewritten in quaternion forms. This allows to construct the Dirac γ matrices in two (quaternion-vector) matrix product representations in accordance with the double vector algebra introduced by P. Rowlands. The paper attempts to demonstrate that the Dirac equation in its form modified by P. Rowlands, essentially coincides with the original one. The paper shows that one of these representations affects the γ4 and γ5 matrices, but leaves the vector of the Pauli spinors intact; and the other representation leaves the γ matrices intact, while it transforms the spin vector into a quaternion pseudovector. So the paper ...

viXra, 2015

In its original form the Dirac equation for the free electron and the free positron is formulated by using complex number based spinors and matrices. That equation can be split into two equations, one for the electron and one for the positron. These equations appear to apply different parameter spaces. The equation for the electron and the equation for the positron differ in the symmetry flavor of their parameter spaces. This results in special considerations for the corresponding quaternionic second order partial differential equation.

2019

In this study, we develop the generalized Dirac like four-momentum equation for rotating spin-half particles in four-dimensional quaternionic division algebra. The generalized quaternionic Dirac equation consist the rotational energy and angular momentum of particle and anti-particle. Accordingly, we also discuss the quaternionic relativistic mass, moment of inertia and rotational energy-momentum four vector in Euclidean space-time. The quaternionic four angular momentum (i.e. the rotational analogy of four linear momentum) predicts the dual energy (rest mass energy and pure rotational energy) and dual momentum (linear like momentum and pure rotational momentum). Further, the solutions of quaternionic rotational Dirac energy-momentum are obtained by using one, two and four-component spinor forms of quaternionic wave function. We also demonstrate the solutions of quaternionic plane wave equation which analysis the rotational frequency and wave propagation vector of Dirac particles an...

International Journal of Geometric Methods in Modern Physics

In this study, we develop the generalized Dirac-like four-momentum equation for rotating spin-1/2 particles in four-dimensional quaternionic algebra. The generalized quaternionic Dirac equation consists of the rotational energy and angular momentum of particle and antiparticle. Accordingly, we also discuss the four-vector form of quaternionic relativistic mass, moment of inertia and rotational energy-momentum in Euclidean space-time. The quaternionic four-angular momentum, (i.e. the rotational analogy of four-linear momentum) predicts the dual energy (rest mass energy and pure rotational energy) and dual momentum (linear-like momentum and pure rotational momentum). Further, the solutions of quaternionic rotational Dirac energy-momentum are obtained by using one-, two- and four-component of quaternionic spinor. We also demonstrate the solutions of quaternionic plane wave equation which gives the rotational frequency and wave propagation vector of Dirac particles and antiparticles in ...

2010

The Dirac equation with Lorentz violation involves additional coefficients and yields a fourth-order polynomial that must be solved to yield the dispersion relation. The conventional method of taking the determinant of $4\times 4$ matrices of complex numbers often yields unwieldy dispersion relations. By using quaternions, the Dirac equation may be reduced to $2 \times 2$ form in which the structure

2001

The nilpotent version of the Dirac equation is applied to the baryon wavefunction, the strong interaction potential, electroweak mixing, and Dirac and Klein-Gordon propagators. The results are used to interpret a quaternion-vector model of particle structures.

European Journal of Physics, 2016

The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term by in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.

2003

The nilpotent Dirac formalism has been shown, in previous publications, to generate new physical explanations for aspects of particle physics, with the additional possibility of calculating some of the parameters involved in the Standard Model. The applications so far obtained are summarised, with an outline of some more recent developments.

2018

There are several 3 + 1 parameter quantities in physics (like vector + scalar potentials, four-currents, space-time, four-momentum, …). In most cases (but space-time), the three-and the one-parameter characterised elements of these quantities differ in the field-sources (e.g., inertial and gravitational masses, Lorentz-and Coulomb-type electric charges, …) associated with them. The members of the field-source pairs appear in the vector-and the scalar potentials, respectively. Sections 1 and 2 of this paper present an algebra what demonstrates that the members of the fieldsource siblings are subjects of an invariance group that can transform them into each other. (This includes, e.g., the conservation of the isotopic field-charge spin (IFCS), proven in previous publications by the author.) The paper identifies the algebra of that transformation and characterises the group of the invariance; it discusses the properties of this group, shows how they can be classified in the known nomenclature, and why is this pseudo-unitary group isomorphic with the SU(2) group. This algebra is denoted by tau (). The invariance group generated by the tau algebra is called hypersymmetry (HySy). The group of hypersymmetry had not been described. The defined symmetry group is able to make correspondence between scalars and vector components that appear often coupled in the characterisation of physical states. In accordance with conclusions in previous papers, the second part (Sections 3 and 4) shows that the equations describing the individual fundamental physical interacions are invariant under the combined application of the Lorentztransformation and the here explored invariance group at high energy approximation (while they are left intact at lower energies). As illustration, the paper presents a simple form for an extended Dirac equation and a set of matrices to describe the combined transformation in QED. The paper includes a short reference illustration (in Section 2.2) to another applicability of this algebra in the mathematical description of regularities for genetic matrices.

International Journal of Quantum Chemistry, 2012

The algebra of physical space (APS) is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example, the Dirac Hamiltonian can be presented as the scalar product of the electron's four-momentum and Dirac's fourvector of gamma matrices, ðc 0 ;cÞ, the latter of which is a Clifford algebra. We show here that a geometric spacetime or four-space solution of Dirac's equation conforms to the principles of APS, an early example of which is Schroedinger's solution of Dirac's equation for a free electron, which exhibits Zitterbewegung. In a four-space solution the spacetime coordinates,r and the scaled time ct, are treated on an equal footing as physical observables to avoid any suggestion of a preferred frame of reference. The geometric spacetime theory is studied here for the Coulomb problem. The positive-energy spectrum of states is found to be identical within numerical error to that of standard Dirac's theory, but the wave function exhibits Zitterbewegung. It is shown analytically how the geometric spacetime solution can be reduced to the standard solution of Dirac's equation, in which Zitterbewegung is absent. The rigor of APS and of its conforming geometric spacetime solution provide strong support for further investigation into the physical interpretation of the geometric spacetime Dirac's wave function and Zitterbewegung.