2024/03/02 by James Lindesay, Lindesay, James
Mathematics · Physics and Astronomy · #FOS: Physical sciences #General Physics (physics.gen-ph) #Mathematical and Theoretical Analysis #Quantum Mechanics and Applications
paper · pdf · doi:10.48550/arxiv.2403.07930
openalex publication_date 2024/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An extension of the Lorentz group to include generators Γμ carrying a space-time index is demonstrated to explicitly construct the Minkowski metric within the internal group space as a consequence of the non-vanishing commutation relations between those generators. For fundamental representation states, this group forms a subgroup of GL(4) consistent with Dirac fermions. Since the representations inherently pair particles with anti-particles, the generators define a minimal group for transforming fields that fundamentally satisfy microscopic causality, thus treating causality as a crucial physical property. For fermions transforming under GL(4), beyond three spin matrices and Γ0 there are 12 additional hermitian generators. It is demonstrated that a closed invariance subgroup of SU(2)×U(1) for fermions of defined mass can be constructed. However, any closed subgroup of SU(3) transformations necessarily mix the mass-SU(2) eigenstates. The first part of exploration of group representations presented in this paper examines associating global invariance to these additional generators. The extended group requires that any charges arising due to establishment of internal local symmetries necessarily exhibit geometric equivalence under curvilinear transformations on the parameterization of space-time translations. Furthermore, a formulation for mixing the degenerate massless fermions (which explicitly cannot manifest SU(3) symmetry or charges) while maintaining Lorentz invariance, is presented.