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From Invariant Decomposition to Spinors

Martin Roelfs, David Eelbode, Steven De Keninck

Abstract

Plane-based Geometric Algebra (PGA) has revealed points in a $d$-dimensional pseudo-Euclidean space $\mathbb{R}_{p,q,1}$ to be represented by $d$-blades rather than vectors. This discovery allows points to be factored into $d$ orthogonal hyperplanes, establishing points as pseudoscalars of a local geometric algebra $\mathbb{R}_{pq}$. Astonishingly, the non-uniqueness of this factorization reveals the existence of a local $\text{Spin}(p,q)$ geometric gauge group at each point. Moreover, a point can alternatively be factored into a product of the elements of the Cartan subalgebra of $\mathfrak{spin}(p,q)$, which are traditionally used to label spinor representations. Therefore, points reveal previously hidden geometric foundations for some of quantum field theory's mysteries. This work outlines the impact of PGA on the study of spinor representations in any number of dimensions, and is the first in a research programme exploring the consequences of this insight.

From Invariant Decomposition to Spinors

Abstract

Plane-based Geometric Algebra (PGA) has revealed points in a -dimensional pseudo-Euclidean space to be represented by -blades rather than vectors. This discovery allows points to be factored into orthogonal hyperplanes, establishing points as pseudoscalars of a local geometric algebra . Astonishingly, the non-uniqueness of this factorization reveals the existence of a local geometric gauge group at each point. Moreover, a point can alternatively be factored into a product of the elements of the Cartan subalgebra of , which are traditionally used to label spinor representations. Therefore, points reveal previously hidden geometric foundations for some of quantum field theory's mysteries. This work outlines the impact of PGA on the study of spinor representations in any number of dimensions, and is the first in a research programme exploring the consequences of this insight.
Paper Structure (13 sections, 3 theorems, 34 equations, 6 figures)

This paper contains 13 sections, 3 theorems, 34 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Geometry 101
  3. First Postulate Violation and the Principle of Covariance
  4. Geometric Gauges
  5. Double cover on the flip side
  6. Invariant Decomposition
  7. What is the point?
  8. Spinors
  9. Algebraic Spinors
  10. Back to Reality: Pointors
  11. A Problem in Accounting?
  12. Hestenes Spinors
  13. Conclusion & Future Research

Key Result

Theorem 1

A product $U=u_1 u_2 \cdots u_\ell$ of $\ell$ reflections $u_i$ can be decomposed into exactly $\lceil\frac{\ell}{2}\rceil$ simple commuting factors. These are $\lfloor\frac{\ell}{2}\rfloor$ products of two reflections, and, for odd $\ell$, one extra reflection.

Figures (6)

  • Figure 1: All products of reflections in 3DPGA ($\mathbb{R}^{{{}}}_{3,0,1}$). From left to right: plane(reflection), represented by a vector $u \in \mathbb{R}^{{(1)}}_{3,0,1}$, line(reflection), represented by two orthogonal plane-reflections $u \wedge v \in \mathbb{R}^{{(2)}}_{3,0,1}$, point(reflection), represented by three orthogonal plane-reflections $u \wedge v \wedge w \in \mathbb{R}^{{(3)}}_{3,0,1}$. More generally, plane-reflections are the atoms of all the orthogonal transformations in a space.
  • Figure 2: The three fundamental types of bireflection. Left) two reflection in line-time make a boost. Middle) two reflections in parallel lines make a translation. Right) Two reflections in intersecting lines in the plane make a rotation.
  • Figure 3: Reflections in 1, 2 and 3 lines (green), and their main invariants (orange).
  • Figure 4: Four possible factorisations of a bireflection. Notice that the initial and final states are the same, and hence there is a gauge degree of freedom that can be chosen freely.
  • Figure 5: From left to center: the gauge at $P_1$ is used to make $v' \perp w$. From center to right: the gauge at $P_2$ is used to make $w'\perp u'$. This reveals any composition of three reflections in the plane to always be a transflection, with invariant line $w'$.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1: Invariant decomposition
  • proof
  • Theorem 2
  • proof
  • Definition 3.1: Pointor
  • Theorem 3
  • proof