From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras

TL;DR

This paper revisits special relativity to characterize 4D fundamental structures using quantized tangent spaces, Lie algebra roots, and Cartan subalgebras, linking particles, velocities, and masses within a novel algebraic framework.

Contribution

It introduces a new algebraic model connecting special relativity, quantized tangent spaces, and Lie algebra structures to describe particles and their properties.

Findings

Quantized tangent space characterized by zero-momentum particles.

Particles split from roots in a rank-4 Lie algebra with Cartan generators.

Domains for particle evolution in spacetime are explicitly calculated.

Abstract

Starting by revisiting Special Relativity, here we provide a reliable characterization of the entire 4-dimensional fundamental structures in our reality where the frame of discrete tangent space of $F^{1,3}$ is quantized to massless, zero-momentum particles distributing on a 4-dimensional regular base $\{\mathbb{N}\cdot c_\alpha\}$ with metric $B(c_\alpha,c_\beta)=\delta_{\alpha\beta}=\text{diag}(+,+,+,+)$, determining the constant $c$ locally, as well as instant characterizations on all particles moving along the proper time of $\tau\in\mathbb{N}$. Together with $\phi_\text{IV}$ on 1-dimensional space $\{\mathbb{N}\cdot c_4\}$ of $B(c_\alpha,c_4)=0$, the quantized particles of tangent frame are split anti-symmetrically from roots $\gamma_{\alpha4}$ in a rank-4 Lie algebra with exactly $c_\alpha$ the generators of its Cartan subalgebra. As with the combined frame-Higgs valued in $\gamma_{\alpha4}$, every massive particle is related to some split frame and Higgs to obtain its unique quantized relative 3-velocity, 4-velocity, and rest mass under the restriction of Pauli Exclusion Principle at each $\tau$. We calculated the domains $S_i\subset F$ and $S_0\subset F_\geq$ in $F_\geq\oplus F^3$ the spacetime in which a massive particle $\mathtt{p}$ evolves with $\{x^\alpha\}$ of $x^\alpha\in S_\alpha$ the coordinate its Center-of-Mass under an eligible observation that $\vert F\vert=\vert F_\geq\vert=\aleph_0<\vert\mathbb{R}\vert$, available for post-Newtonian approaches.