Configuration Spaces in Fundamental Physics

Abstract

I consider configuration spaces for $N$-body problems, gauge theories and for GR in both geometrodynamical and Ashtekar variables forms, including minisuperspace and inhomogeneous perturbations thereabout in the former case. These examples include many interesting spaces of shapes (with and without whichever of local or global notions of scale). In considering reduced configuration spaces, stratified manifolds arise. Three strategies to deal with these are `excise', `unfold' and `accept'. I show that spaces of triangles arising from various interpretations of 3-body problems already serve as model arena for all three. I furthermore argue in favour of the `accept' strategy on relational grounds. This approach requires sheaf methods (which go beyond fibre bundles and general bundles, which I contrast with sheaves and presheaves in some appendices). Sheaf methods are also required for the stratifold construct that pairs some well-behaved stratified manifolds with sheaves. I apply arguing against `excise' and `unfold' to GR's superspace and thin sandwich, and to the removal of collinear configurations in mechanics. Non-redundant configurations are also useful in providing more accurate names for various spaces and theories.

35 pages including 16 Figures. Several new figures and Appendices added

References in corpus (11)

• The Superspace of Geometrodynamics
• Gribov no-pole condition, Zwanziger horizon function, Kugo-Ojima confinement criterion, boundary conditions, BRST breaking and all that
• Foundations of Relational Particle Dynamics
• Problem of Time and Background Independence: the Individual Facets
• Conformal Superspace: the configuration space of general relativity
• Triangleland. I. Classical dynamics with exchange of relative angular momentum
• On the Configuration Spaces of Homogeneous Loop Quantum Cosmology and Loop Quantum Gravity
• Consciousness and the Quantum
• Relational mechanics of shape and scale
• Origin of Structure in the Universe: Quantum Cosmology Reconsidered
• A New Approach to Quantising Space-Time: I. Quantising on a General Category

Cited by in corpus (14)

• Explicit partial and functional differential equations for beables or observables
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• On Types of Observables in Constrained Theories
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity
• Origin of Structure in the Universe: Quantum Cosmology Reconsidered
• On the Conceptual Issues Surrounding the Notion of Relational Bohmian Dynamics
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Pure shape dynamics: General framework
• A Proposal for a Metaphysics of Self-Subsisting Structures. I. Classical Physics
• Problem of Time: Temporal Relationalism Compatibility of Other Local Classical Facets Completed
• Background Independence: ${\mathbb{S}}^1$ and $\mathbb{R}$ absolute spaces differ greatly in Shape-and-Scale Theory
• Topological Shape Theory
• 'Shape Dynamics': Foundations Reassessed