A mathematical framework for a standard theory using extended representations of paths and world lines

TL;DR

This paper develops a mathematical framework that unifies path representations with local and global space-time interactions, aligning with standard particle physics theories like Dirac, Klein-Gordon, and electro-weak models.

Contribution

It introduces a novel extended path representation framework that integrates local and global space-time interactions consistent with established particle physics theories.

Findings

Compatibility with Dirac and Klein-Gordon equations

Incorporates standard charge assignments and SU(3) confinement

Provides a unified mathematical structure for particle interactions

Abstract

An analysis using a composition of currently-accepted theories is given. Starting with a synthesis of what may be generically termed ``paths'', analysis of representations for these ``paths'' is developed. Foreground and background interactions are explicitly treated by using a local representation that treats the two representations equally and symmetrically. A restriction coupling from the global space-time representation to local interaction source terms is treated in terms of mass and charge couplings. Rewriting the connection in terms of the global manifold and the coupled terms yields compatibility with Dirac and Klein-Gordan equations for electro-weak coupled particles and fields. Compatibility with currently-accepted theories that includes standard charge assignments, SU(3) confinement, and a definition for particle flavor generations is used to constrain and validate the composition and the analysis.