Lie Theory suffices to understand, and Locally Resolve, the Problem of Time

TL;DR

This paper demonstrates that Lie theory provides a comprehensive framework to understand and locally resolve the Problem of Time in physics, emphasizing the role of Lie brackets, derivatives, and automorphisms.

Contribution

It introduces a Lie claw digraph framework that unifies background independence, the Problem of Time, and fundamental physical laws through Lie algebraic structures.

Findings

Resolved the facet ordering problem of the Problem of Time.

Established the sufficiency of Lie brackets and derivatives in understanding background independence.

Extended the Dirac Algorithm using Lie algebraic structures.

Abstract

The Lie claw digraph controls Background Independence and thus the Problem of Time and indeed the Fundamental Nature of Physical Law. This has been established in the realms of Flat and Differential Geometry with varying amounts of extra mathematical structure. This Lie claw digraph has Generator Closure at its centre (Lie brackets), Relationalism at its root (implemented by Lie derivatives), and, as its leaves, Assignment of Observables (zero commutants under Lie brackets) and Constructability from Less Structure Assumed (working if generator Deformation leads to Lie brackets algebraic Rigidity). This centre is enabled by automorphisms and powered by the Generalized Lie Algorithm extension of the Dirac Algorithm (itself sufficing for the canonical subcase, for which generators are constraints). The Problem of Time's facet ordering problem is resolved.