From a Relativistic Phenomenology of Anyons to a Model of Unification of Forces via the Spencer Theory of Lie Structures

TL;DR

This paper explores a theoretical framework unifying electromagnetism and gravitation using Spencer cohomology of Lie equations, proposing a geometric approach with potential extensions to other fundamental forces.

Contribution

It introduces a novel application of Spencer theory to model unification of fundamental forces through Lie pseudogroups and cohomology, extending geometric methods in physics.

Findings

Development of a non-linear Spencer complex for Poincaré and conformal groups

Derivation of a conformally equivariant Lagrangian density

Suggestions for extending the framework to weak and strong interactions

Abstract

Starting from a relativistic phenomenology of anyons in crystals, we discuss the concept of relativistic interaction and the need to unify electromagnetism and gravitation within the Spencer cohomology of Lie equations. Then, from the sophisticated non-linear Spencer complex of the Poincar\'e and conformal Lie pseudogroups, we build up a non-linear relative complex assigned to a gauge sequence for electromagnetic and gravitational potentials and fields. Then, using a conformally equivariant Lagrangian density, we deduce, first, the two first steps of its corresponding Janet complex and second, the dual relative linear complex. We conclude by giving suggestions for higher unification with the weak and strong interactions and interpretations of the Lagrangian density as a thermodynamical function and quantum wave-function.