Algebraic structures, physics and geometry from a Unified Field Theoretical framework

TL;DR

This paper explores a Unified Field Theory revealing that the fundamental geometric structure naturally incorporates biquaternionic algebra, providing new insights into fermionic representations, symmetries, and potential supersymmetric extensions.

Contribution

It demonstrates that the underlying geometry of the UFT inherently possesses a biquaternionic structure, offering a novel algebraic and geometric interpretation of fermionic fields and symmetries.

Findings

Biquaternionic structure replaces complex structure in the principal fiber bundle

Analysis of Majorana and Dirac representations within this framework

Identification of hidden symmetries and supersymmetric possibilities

Abstract

Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in this UFT general context that the underlying basis of the single geometrical structure P (G,M) (the principal fiber bundle over the real spacetime manifold M with structural group G) reflecting the symmetries of the different fields carry naturally a biquaternionic structure instead of a complex one. This fact allows us to analyze algebraically and to interpret physically in a straighforward way the Majorana and Dirac representations and the relation of such structures with the spacetime signature and non-hermitian (CP) dynamic operators. Also, from the underlying structure of the tangent space, the existence of hidden (super) symmetries and the possibility of supersymmetric extensions of these UFT models are given showing that Rothstein's theorem is incomplete for that description. The importance of the Clifford algebras in the description of all symmetries, mainly the interaction of gravity with the other fields, is briefly discussed.