On Fermions in Compact momentum Spaces Bilinearly Constructed with Pure Spinors

TL;DR

This paper extends Cartan's conjecture to quantum fermions in compact momentum spaces constructed with pure spinors, linking internal symmetries to Clifford algebras and suggesting a natural emergence of four-dimensional spacetime.

Contribution

It introduces a novel spinor-based framework for fermions in compact momentum spaces, connecting internal symmetries to Clifford algebras and proposing a natural derivation of 4D spacetime without extra configuration dimensions.

Findings

Fermions in compact momentum spaces can be described using pure spinors and Clifford algebras.

Internal symmetries like SU(3) and SU(2)×U(1) emerge from the algebraic structure.

Four-dimensional spacetime naturally arises from the momentum space construction.

Abstract

It is shown how the old Cartan's conjecture on the fundamental role of the geometry of simple (or pure) spinors, as bilinearly underlying euclidean geometry, may be extended also to quantum mechanics of fermions (in first quantization), however in compact momentum spaces, bilinearly constructed with spinors, with signatures unambiguously resulting from the construction, up to sixteen component Majorana-Weyl spinors associated with the real Clifford algebra $\Cl(1,9)$, where, because of the known periodicity theorem, the construction naturally ends. $\Cl(1,9)$ may be formulated in terms of the octonion division algebra, at the origin of SU(3) internal symmetry. In this approach the extra dimensions beyond 4 appear as interaction terms in the equations of motion of the fermion multiplet; more precisely the directions from 5$^{th}$ to 8$^{th}$ correspond to electric, weak and isospin interactions $(SU(2) \otimes U(1))$, while those from 8$^{th}$ to 10$^{th}$ to strong ones SU(3). There seems to be no need of extra dimension in configuration-space. Only four dimensional space-time is needed - for the equations of motion and for the local fields - and also naturally generated by four-momenta as Poincar\'e translations. This spinor approach could be compatible with string theories and even explain their origin, since also strings may be bilinearly obtained from simple (or pure) spinors through sums; that is integrals of null vectors.