Gauge Gravity and Electroweak Theory

Abstract

Reformulation of the Dirac equation in terms of the real Spacetime Algebra (STA) reveals hidden geometric structure, including a geometric role for the unit imaginary as generator of rotations in a spacelike plane. The STA and the real Dirac equation play essential roles in a new Gauge Theory Gravity (GTG) version of General Relativity (GR). Besides clarifying the conceptual foundations of GR and facilitating complex computations, GTG opens up new possibilities for a unified gauge theory of gravity and quantum mechanics, including spacetime geometry of electroweak interactions. The Weinberg-Salam model fits perfectly into this geometric framework, and a promising variant that replaces chiral states with Majorana states is formulated to incorporate zitterbewegung in electron states.

Zitterbewegung in Quantum Mechanics

“…To see where (35) comes from, we introduce the orbital angular momentum p ∧ z so the total angular momentum is…”

A demonstration that electroweak theory can violate parity automatically (leptonic case)

“…In short, the model yields quite efficiently the basic electroweak behaviour for one generation of standard model leptons, together with a right-handed neutrino. This result may be useful for authors with an interest in left-right asymmetry models, 6-21 existing Clifford algebraic particle models, [23][24][25][26][27][28][29][30][31][32][33][34] beyond-the-standardmodel proposals, [35][36][37] and noncommutative geometry.…”

On the Unification of Interactions by Clifford Algebra

“…The role of the latter transformations is then just like that of the gauge transformations acting in an "internal" space. Such idea has been proposed by Hestenes [5,6], with the aim to explain the Weinberg-Salam model of electro-weak interactions, based on the group U(1)×SU(2) in geometric terms. Hestenes also argued that the color group SU(3) might be explained in such terms.…”

“…On the other hand, many authors have employed the power of Clifford algebra in describing geometry, physics, and 782 M. Pavšič A A C A their generalization [3], [4]. Hestenes showed how the Weingerg-Salam electroweak model fits perfectly into the geometric framework based on Clifford algebra [5,6]. The idea that Clifford algebra provides a framework for the unification of interactions has been investigated in a number of illuminating and penetrating works [7]- [9].…”

“…The space of minimal left ideals can thus be considered as an 'internal' space that forms a representation for the gauge groups associated with the fundamental interactions [18,19]. Such idea has also been considered by Hestenes [5,6]. Here we explore it further, within a somewhat different framework, since we build on the concept of Clifford space C. Moreover, the imaginary unit of quantum theory in our approach comes from phase space, and not from the Clifford algebra of spacetime.…”