The Scattering Algebra of Physical Space: Squared Massive Constructive Amplitudes

TL;DR

This paper introduces the Scattering Algebra, a novel framework connecting the Algebra of Physical Space to the Constructive Standard Model, enabling simplified, geometric calculations of massive particle amplitudes and revealing new insights into spinor formalism.

Contribution

It presents the Scattering Algebra formalism that unifies APS and CSM, providing a new geometric approach to particle amplitudes with confirmed consistency and potential for extending to massless cases.

Findings

Confirmed equivalence between APS-CSM and traditional CSM formalisms

Identified spinors with Lorentz rotors in APS

Demonstrated the approach's effectiveness for massive 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.