A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics

Abstract

In a major advance and simplification of this field, we show that A Local Resolution of the Problem of Time - also viewable as A Local Theory of Background Independence - can at the classical level be described solely by of Lie's Mathematics. This comprises i) Lie derivatives to encode Relationalism, including via solving the generalized Killing equation. ii) Lie brackets to formulate Closure, via Lie's Algorithm suitably extended to accommodate Dirac and topological insights, producing generator Lie algebraic structures: Lie algebras or algebroids. iii) Observables defined by Lie brackets relations, recastable as explicit PDE systems to be solved using the Flow Method, and constitute observables Lie algebras. iv) The `passing families of theories through the Dirac Algorithm' approach to Spacetime Construction from Space, and to obtaining more structure from less internally to each of space and spacetime separately, are identified as deformations that work selectively when Lie Algebraic Rigidity is encountered. v) Reallocation of Intermediary-Object (RIO) Invariance: the general Lie Theory's commuting-pentagon analogue of posing Refoliation Invariance for GR. i) to v) cover respectively the Relationalism, Closure, Observables, Deformations and RIO super-aspects of Background Independence, Lie Theory moreover already collates i) to iii) and the internal case of iv) as multiple interacting aspects. The Problem of Time's multiple interacting facets are then explained as, firstly, having 2 copies of this Lie collation, 1 for each of spacetime and space primalities. Secondly, a Wheelerian two-way route between these two primalities, comprising v) and iv)'s `spacetime from space' version. We further develop the Comparative Theory of Background Independence thus. We can even give a `pillars of the Foundations of Geometry' parallel of our Background Independence super-aspects.

arXiv comment44 pages including 12 figures

References in corpus (38)

• Deformation quantization of Poisson manifolds, I
• The Dynamics of General Relativity
• Problem of Time in Quantum Gravity
• Canonical Quantum Gravity
• The Problem of Time in Quantum Gravity
• The Superspace of Geometrodynamics
• The Problem of Time and Quantum Cosmology in the Relational Particle Mechanics Arena
• Lie algebroid structures on double vector bundles and representation theory of Lie algebroids
• Proof of the Thin Sandwich Conjecture
• Beables/Observables in Classical and Quantum Gravity
• Hypersurface-deformation algebroids and effective space-time models
• Problem of Time and Background Independence: the Individual Facets
• New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split
• Classical Machian Resolution of the Spacetime Reconstruction Problem
• Explicit partial and functional differential equations for beables or observables
• A Local Resolution of the Problem of Time
• On Types of Observables in Constrained Theories
• Theory of Transformation Groups, by S. Lie and F. Engel (Vol. I, 1888). Modern Presentation and English Translation
• Shape Theory. III. Comparative Theory of Backgound Independence
• A Local Resolution of the Problem of Time. II. Configurational Relationalism via a generalization of Group Averaging
• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• A Local Resolution of the Problem of Time. IV. Quantum outline and piecemeal Conclusion
• A loop quantum multiverse?
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• A Local Resolution of the Problem of Time. V. Combining Temporal and Configurational Relationalism for Finite Theories
• A Local Resolution of the Problem of Time. IX. Spacetime Constructability
• Geometry from Brackets Consistency
• A Local Resolution of the Problem of Time. VII. Constraint Closure
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Shape Theories. I. Their Diversity is Killing-Based and thus Nongeneric
• Monopoles of Twelve Types in 3-Body Problems
• Shape Theories. II. Compactness Selection Principles
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
• Problem of Time and Background Independence: classical version's higher Lie Theory
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• Lie algebras : Classification, Deformations and Rigidity

Cited by in corpus (7)

• A Local Resolution of the Problem of Time. IV. Quantum outline and piecemeal Conclusion
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• Nijenhuis-type variants of Local Theory of Background Independence
• Nambu variant of Local Resolution of Problem of Time and Background Independence
• Lie Theory suffices to understand, and Locally Resolve, the Problem of Time
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables
• Comparative Theory of Background Independence