An Alternative Approach to Unification of Gauge and Geometric Interactions

TL;DR

This paper presents a novel algebraic framework where the Minkowski metric emerges from the internal group space of causal spinor fields, unifying gauge and geometric interactions through group representations.

Contribution

It introduces a new approach to unifying gauge and geometric interactions by constructing internal symmetry groups within the algebra of causal spinor fields, highlighting the role of CKM mixing.

Findings

Minkowski metric arises from internal group space algebra

Construction of internal SU(3) and SU(2) groups involves CKM mixing

Representation is a subgroup of GL(4) generators

Abstract

The algebra of the generators for infinitesimal transformations of the $\Gamma={1 \over 2}$ representation of causal spinor fields (Dirac fields) \emph{explicitly} constructs the Minkowski metric \emph{within} the internal group space as a consequence of non-vanishing commutation relations between generators that carry a single space-time index. This representation is a subgroup of the set of all of the generators that transform under the group GL(4). The sixteen hermitian generators of GL(4) include the three angular momentum spin matrices, a matrix proportional to the Dirac matrix $\gamma^0$, and 12 additional matrices that have the same number of degrees of freedom as SU(3)$\times$SU(2)$\times$U(1). In this paper, the construction of linearly independent internal SU(3) and SU(2) local symmetry groups for the causal spinor fields is demonstrated to necessarily involve the CKM mixing of three sets of the SU(3) eigenstates as related to the SU(2) eigenstates.