Spaces of Spaces

TL;DR

This paper explores generalized configuration spaces in theoretical physics, examining their classical and quantum properties, and introduces probabilistic and topological tools to better understand quantum gravity and records theory.

Contribution

It extends the study of configuration spaces to deeper levels of structure, providing classical preliminaries, criteria for relational and background independence, and probabilistic approaches including new examples.

Findings

Relational and Background Independence criteria applied to all levels of structure.

Probability theory developed for configuration spaces, aiding records theory.

New examples of records theories including Cech cohomology, random sets, and sheaves.

Abstract

Wheeler emphasized the study of Superspace - the space of 3-geometries on a spatial manifold of fixed topology. This is a configuration space for GR; knowledge of configuration spaces is useful as regards dynamics and QM.In this Article I consider furthmore generalized configuration spaces to all levels within the conventional `equipped sets' paradigm of mathematical structure used in fundamental Theoretical Physics. This covers A) the more familiar issue of topology change in the sense of topological manifolds (tied to cobordisms), including via pinched manifolds. B) The less familiar issue of not regarding as fixed the yet deeper levels of structure: topological spaces themselves (and their metric space subcase), collections of subsets and sets. Isham has previously presented quantization schemes for a number of these. I consider some classical preliminaries for this program, aside from the most obvious (classical dynamics for each). Rather, I provide I) to all levels Relational and Background Independence criteria, which have Problem of Time facets as consequences. I demonstrate that many of these issues descend all the way down, whilst also documenting at which level the others cease to apply. II) Probability theory on configuration spaces. In fact such a stochastic treatment is how to further mathematize the hitherto fairly formal and sketchy subject of records theory (a type of formultion of quantum gravity). Along these lines I provide a number of further examples of records theories. This is in addition to Kendall's shape statistics being the example corresponding to relational mechanics models. To this example I now add 1) Cech cohomology, 2) Kendall's random sets, 3) the lattice of topologies on a fixed set. I finally consider 4) sheaves, both as a generalization of Cech cohomology and in connection to the study of stratified manifolds such as Superspace itself.