Shape Theory. III. Comparative Theory of Backgound Independence

Abstract

Background Independence is the modern form of the relational side of the Absolute versus Relational Debate. Difficulties with its implementation form the Problem of Time. Its 9 facets - Isham and Kuchař's conceptual classification - correspond to 9 aspects of Background Independence as per the Author's classical-or-quantum theory-independent upgrade. 8 are local. The most significant arena for these is brackets algebra, the first 5 involving canonical constraints as follows. 1) Handling instantaneous gauge invariance. 2) Resolving its apparent timelessness. 3) Closure of 1-2)'s constraints. 4) Expression in terms of observables: commutants with constraints. 5) Reconstructing spacetime from constraint algebra rigidity. 6) is the spacetime counterpart of 2-4), and 7) is spacetime's foliation independence. 8) handles nonuniqueness. 9) renders 1-8) globally sound. We show how Shape(-and-Scale) Theory's mastery of 1) for N-point-particle models extends by placing a mechanics over shape(-and-scale) space to model 1-4). For flat-space Euclidean and similarity models, this gives a local resolution of the Problem of Time. This is moreover consistent within a global treatment if its reduced spaces are Hausdorff paracompact, which admit a Shrinking Lemma. GR's superspace is also Hausdorff paracompact. While 1-9) are poseable for all relativistic theories, and 1-4), 8), 9) for all theories - to all levels of mathematical structure: affine, projective, conformal, topological manifold, topological space... - resolution is on a case-by-case basis. Among N-point-particle theories, then, Article II's Hausdorff paracompact reduced space guarantee selects a very small subset. In particular, affine and projective shapes are precluded and conceiving in terms of shapes in space is preferred over doing so in spacetime. A substantial Selection Principle for Comparative Background Independence is thus born.

34 pages, including 6 figures. Minor adjustments

References in corpus (6)

• Some remarks on the notions of general covariance and background independence
• An effective approach to the problem of time
• Effective approach to the problem of time: general features and examples
• The Superspace of Geometrodynamics
• Nonparametric statistics on manifolds with applications to shape spaces
• Foundations of Relational Particle Dynamics

Cited by in corpus (14)

• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• 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
• 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
• A Local Resolution of the Problem of Time. VII. Constraint Closure
• Geometry from Brackets Consistency
• Problem of Time and Background Independence: classical version's higher Lie Theory
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• 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
• Comparative Theory of Background Independence
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables