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The Scattering Algebra of Physical Space: Squared Massive Constructive Amplitudes

Moab Croft, Neil Christensen

Abstract

The Algebra of Physical Space (APS) is used to explore the Constructive Standard Model (CSM) of particle physics. Namely, this paper connects the spinor formalism of the APS to massive amplitudes in the CSM. A novel equivalency between traditional CSM and APS-CSM formalisms is introduced, called the Scattering Algebra (SA), with example calculations confirming the consistency of results between both frameworks. Through this all, two significant insights are revealed: The identification of traditional CSM spin spinors with Lorentz rotors in the APS, and the connection of the CSM to various formalisms through ray spinor structure. The CSM's results are replicated in massive cases, showcasing the power of the index-free, matrix-free, coordinate-free, geometric approach and paving the way for future research into massless cases, amplitude-construction, and Wigner little group methods within the APS.

The Scattering Algebra of Physical Space: Squared Massive Constructive Amplitudes

Abstract

The Algebra of Physical Space (APS) is used to explore the Constructive Standard Model (CSM) of particle physics. Namely, this paper connects the spinor formalism of the APS to massive amplitudes in the CSM. A novel equivalency between traditional CSM and APS-CSM formalisms is introduced, called the Scattering Algebra (SA), with example calculations confirming the consistency of results between both frameworks. Through this all, two significant insights are revealed: The identification of traditional CSM spin spinors with Lorentz rotors in the APS, and the connection of the CSM to various formalisms through ray spinor structure. The CSM's results are replicated in massive cases, showcasing the power of the index-free, matrix-free, coordinate-free, geometric approach and paving the way for future research into massless cases, amplitude-construction, and Wigner little group methods within the APS.
Paper Structure (37 sections, 118 equations, 2 figures)

This paper contains 37 sections, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Essential Conventions
  3. $\mathrm{Spin}(3)$ in $\mathbb{G}_3$
  4. Spacetime in $\mathbb{G}_3$
  5. $\mathrm{Spin}(1,3)$ in $\mathbb{G}_3$
  6. Spinor Formalism
  7. Lorentz Spinors
  8. Ray Spinor Structure
  9. Scattering Algebra
  10. Initial Dictionary between Traditional and Geometric Formalism
  11. Example Calculations
  12. $hf\overline{f}$ with Indices
  13. $hf\overline{f}$ without Indices
  14. $W q\overline{q}$ with Indices
  15. $W q\overline{q}$ without Indices
  16. ...and 22 more sections

Figures (2)

  • Figure 1: The lightrays $\ell_\pm=(1\pm\hat{a})/2$ shown on the lightcone, partitioning time.
  • Figure 2: A directed-graph summary of the (light)ray spinor structure in the Algebra of Physical Space.