The Dirac Equation in Six-dimensional SO(3,3) Symmetry Group and a Non-chiral “Electroweak” Theory

Abstract

We propose a model of electroweak interactions without chirality in a sixdimensional spacetime with 3 time-like and 3 space-like coordinates, which allows a geometrical meaning for gauge symmetries. The spacetime interval ds 2 = dx μ dx μ is left invariant under the symmetry group SO(3, 3). We obtain the six-dimensional version of the Dirac gamma matrices, μ , and write down a Dirac-like Lagrangian density, L = iψ μ ∇ μ ψ. The spinor ψ can be decomposed into two Dirac spinors, ψ 1 and ψ 2 , interpreted as the electron and neutrino fields, respectively. In six-dimensional spacetime the electron and neutrino fields appear as parts of the same entity in a natural manner. The SO(3, 3) Lorentz symmetry group is locally broken to the observable SO(1, 3) Lorentz group, with only one observable time component, t z . The t z -axis may not be the same at all points of the spacetime, and the effect of breaking the SO(3, 3) spacetime symmetry group locally to an SO(1, 3) Lorentz group, is perceived by the observers as the existence of the gauge fields. We interpret the origin of mass and gauge interactions as a consequence of extra time dimensions, without the need of introducing the so-called Higgs mechanism for the generation of mass. Further, in our 'toy' model, we are able to give a geometric meaning to the electromagnetic and non-Abelian gauge symmetries.

An $E_8 \otimes E_8$ Unification of the Standard Model with Pre-Gravitation, on an Exceptional Lie algebra - Valued Space

“…Furthermore, we have argued earlier that an elementary particle such as the electron does not experience spacetime as classical, or real. Therefore, the appearance of the imaginary unit i in the spacetime vector (52) and in the operator (54) should not be a surprise. We do get a classical spacetime description after squaring, as in (53) and in (55), and this classical spacetime is what is experienced by bosons and by classical objects.…”

“…Very relevant for us is also Kritov (2021) [53] who shows that the Clifford algebra Cl(3, 0) can be used to make two copies of 4D spacetime with relatively flipped signatures. Dartora and Cabrera (2009) [54] have studied 'The Dirac equation in six-dimensional SO (3,3) symmetry group and a non-chiral 'electroweak' theory'. An old (1950) paper by Podolanski [55] studies unified field theory in six dimensions, and in fact the abstract starts by saying 'The geometry of the Dirac equation is actually six-dimensional'.…”

Quantum number conservation: a tool in the design and analysis of high energy experiments

“…Furey [6] showed that some features of unified theories can be reproduced by the structure of complex Clifford algebras. Dartora and Cabrera [7] showed that the main features of electro-weak theory can be explained in terms of the Cl 3,3 algebra if chirality is omitted. Stoica [8] has shown that the results in [4][5][6] can also be expressed using the complex Clifford algebra * Cl 6 , and has investigated how this algebra might incorporate chiral symmetry breaking.…”

Energy Vector and Time Vector in the Dirac Theory

“…Numerous authors [2][3][4][5][6] have already discussed this. From our studies, we found the topic equally interesting when we encountered the energy vector.…”