[Paper Review] Configuration Spaces in Fundamental Physics

Edward Anderson|arXiv (Cornell University)|Mar 5, 2015

Black Holes and Theoretical Physics35 references17 citations

TL;DR

This paper argues that configuration spaces in fundamental physics—especially in N-body problems, gauge theories, and general relativity—are best modeled as stratified manifolds, requiring sheaf-theoretic methods rather than traditional fibre bundles. It advocates the 'accept' strategy for handling singularities in these spaces, demonstrating through 3-body triangle configurations that sheaves provide a natural framework for global consistency and obstruction theory.

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.

Motivation & Objective

• To analyze configuration spaces in fundamental physics as stratified manifolds arising from symmetry reductions, especially in N-body systems and general relativity.
• To evaluate and contrast three strategies—'excise', 'unfold', and 'accept'—for handling singularities in reduced configuration spaces.
• To argue that the 'accept' strategy, grounded in relational physics, is superior and necessitates sheaf-theoretic tools over fibre bundle methods.
• To demonstrate that pure-shape and relational configuration spaces (e.g., triangles in 3-body problems) serve as minimal models for all three strategies.
• To establish that sheaf cohomology provides a more general and computationally robust framework than Čech cohomology or fibre bundle constructions for topological obstructions in singular configuration spaces.

Proposed method

• Uses Jacobi and Lagrange coordinates to reduce N-body systems to relative configuration spaces, identifying the resulting spaces as stratified manifolds.
• Applies relational reduction by removing center-of-mass motion, rotations, and scale via constraints on momentum (total momentum, angular momentum, dilational momentum).
• Introduces the 'accept' strategy as the preferred method, preserving the full structure of the configuration space despite singularities.
• Employs sheaf theory—specifically sheaf axioms (locality and gluing)—to model global consistency of sections over open covers of configuration spaces.
• Contrasts sheaves with fibre bundles by emphasizing sheaves' ability to handle heterogeneous local data and global obstructions via sheaf cohomology.
• Uses stratifold theory (Kreck) as a dual structure pairing continuous functions with stratified manifolds, reinforcing the need for sheaf-based formalism.

Experimental results

Research questions

• RQ1 How do configuration spaces in fundamental physics—especially in N-body systems, gauge theories, and general relativity—fail to be smooth manifolds?
• RQ2 What are the relative merits and drawbacks of the 'excise', 'unfold', and 'accept' strategies for handling singularities in reduced configuration spaces?
• RQ3 Why is the 'accept' strategy—preserving the full singular structure—preferable from a relational physics perspective?
• RQ4 In what way do sheaf-theoretic methods generalize and surpass fibre bundle methods in describing configuration spaces with singularities?
• RQ5 How does sheaf cohomology provide a more comprehensive and computationally viable framework for obstruction theory in singular configuration spaces than Čech cohomology?

Key findings

• Configuration spaces in N-body problems, gauge theories, and general relativity are generically stratified manifolds, not smooth manifolds, due to symmetry reductions and singularities.
• The 3-body problem's shape space (triangles) serves as a minimal model for all three strategies—'excise', 'unfold', and 'accept'—demonstrating their distinct behaviors on a single geometric arena.
• The 'accept' strategy is favored on relational grounds: it preserves physical content and avoids artificial removal or extension of configuration space structure.
• Sheaf methods are necessary and sufficient for handling global consistency and obstruction theory in singular configuration spaces, particularly when fibre bundles fail.
• Sheaf cohomology generalizes Čech cohomology and provides a robust, computationally accessible framework for detecting obstructions to global sections in singular spaces.
• Stratifolds, as a pair of a stratified manifold and an algebra of continuous functions (interpreted as global sheaf sections), further validate the use of sheaves in modeling physical configuration spaces.