Élie Cartan developed modern differential geometry as theory of moving frames. Particles do not enter the equations of structure and thus play a less fundamental role. A Kaluza-Klein (KK) space without compactification brings particles into the core of geometry by making propertime (τ) the fifth dimension. (xi, τ) emerges as the subspace for the quantum sector. This KK space does not make sense in SR, by virtue of non-orthogonality of τ to 3-space (actually), which brings a preferred frame to the fore. In contrast, propertime is perpendicular to 3-space in the “para-Lorentzian structure” with absolute time dilation (PL). Its (xi,τ) subspace looks very much like (t,xi) in SR. And its (t,xi) sector is not made of orthogonal frames but does not cause contradictions with SR, which supporters of the thesis of conventionality of synchronizations have been claiming for many decades. In PL, the conjunction of the Clifford algebras of differential forms (Kaehler’s) and of their valuedness gives rise to a commutative algebra of primitive idempotents that embodies the U(1) × SU(2) symmetry. SU(3) also emerges in the process, but we do not deal with this issue beyond proposing the geometric palet of quarks.
- 11.30.Ly Other internal and higher symmetries 02.40.Hw Classical differential geometry
- Mathematical physics Gravitation and cosmology
- Issue 1 (2013)
Source: Email from Jose G. Vargas, 5 Dec. 2013, josegvargas_AT_earthlink.net