The Dirac algebra and grand unification

2000

A version of the Dirac equation is derived from first principles using a combination of quaternions and multivariate 4-vectors. The nilpotent form of the operators used allows us to derive explicit expressions for the wavefunctions of free fermions, vector bosons, scalar bosons; Bose-Einstein condensates, and baryons;annihilation, creation and vacuum operators; the quantum field integrals; and C, P and T transformations;

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 ...

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.

The Mathematical Gazette, 1991

Bantam books, 1988) says that he was told that if a popular science book contains one formula, the sale of the book becomes halved. I shall try to use no formulae at all (except in the footnotes)-apart from Einstein's notorious equation, "E = me 1 ", and powers of 10-which are inevitable in this subject. f I owe this remark to Victor Weisskopf.

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.

Nuclear Physics B - Proceedings Supplements, 1997

A review is given of the status of the program of classical reduction to Dirac's observables of the four interactions (standard SU(3)xSU(2)xU(1) particle model and tetrad gravity) with the matter described either by Grassmannvalued fermion fields or by particles with Grassmann charges.

Progress of Theoretical Physics, 1987

A new algebraic theory is developed to describe the characteristic features of leptons and quarks as a whole. A pair of master fields with up and down 'weak-isospin is introduced and postulated to obey the generalized Dirac equations with coefficient matrices which belong to an algebra, a triplet algebra, consisting of triple-direct-products of Dirac's I-matrices. The triplet algebra is decomposed into three subalgebras, in a non-intersecting manner, which describe respectively the external Lorentz symmetry, the internal colour symmetry and the degrees of freedom for fourfold-family-replication of fundamental fermionic particle modes. The master fields belonging to a 64 dimensional multi-spinor space form non-irreducible representations of the Lorentz group and represent fourfold-replications of families of spin 1/2 particles, each one of which accomodates triply-degenerate quark modes and singlet leptonic modes. Canonical quantization of master fields leads naturally to the renormalizable unified field theories of fundamental fermions with universal gauge interactions of local symmetries having the route of descent from SUc(4) x SUL(2) x SUR(2) to SUc(3) x SUL(2) x Uy(l).

2009

The purpose of this contribution is to provide an introduction for a general physics audience to the recent results of Emile Grgin that unifies quantum mechanics and relativity into the same mathematical structure. This structure is the algebra of quantions, a non-division algebra that is the natural framework for electroweak theory on curved space-time. Similar with quaternions, quantions preserve the

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.

A Clifford Cl(5, C) Unified Gauge Field Theory formulation of Conformal Gravity and U (4) × U (4) × U (4) Yang-Mills in 4D, is reviewed, along with its implications for the Pati-Salam group SU (4) × SU (2)L × SU (2)R, and T rinif ication GUT models of 3 fermion generations based on the group SU (3)C × SU (3)L × SU (3)R. We proceed with a brief review of a unification program of 4D Gravity and SU (3) × SU (2) × U (1) Yang-Mills emerging from 8D pure Quaternionic Gravity. A realization of E8 in terms of the Cl(16) = Cl(8) ⊗ Cl(8) generators follows, as a preamble to Tony Smith's E8 and Cl(16) = Cl(8) ⊗ Cl(8) unification model in 8D. The study of Chiral Fermions and Instanton Backgrounds in CP 2 , CP 3 related to the problem of obtaining 3 fermion generations is thoroughly studied. We continue with the evaluation of the coupling constants and particle masses based on the geometry of bounded complex homogeneous domains and geometric probability theory. An analysis of neutrino masses, Cabbibo-Kobayashi-Maskawa quark-mixing matrix parameters and neutrino-mixing matrix parameters follows. We finalize with some concluding remarks about other proposals for the unification of Gravity and the Standard Model, like string, M, F theory and Noncommutative and Nonassociative Geometry.