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A Groupoid Construction of Functional Integrals: Brownian Motion and Some TQFTs

Joshua Lackman

Abstract

We formalize Feynman's construction of the quantum mechanical path integral. To do this, we shift the emphasis in differential geometry from the tangent bundle onto the pair groupoid. This allows us to use the van Est map and the piecewise linear structure of manifolds to develop a coordinate-free, partition of unity-free approach to integration of differential forms, etc. This framework makes sense for any field theory valued in a Lie algebroid. We apply it to define the Wiener measure, stochastic integrals and other observables in a coordinate-free way. We use it to reconstruct Chern-Simons with finite gauge group and to obtain some non-perturbative deformation quantizations via the Poisson sigma model on a disk.

A Groupoid Construction of Functional Integrals: Brownian Motion and Some TQFTs

Abstract

We formalize Feynman's construction of the quantum mechanical path integral. To do this, we shift the emphasis in differential geometry from the tangent bundle onto the pair groupoid. This allows us to use the van Est map and the piecewise linear structure of manifolds to develop a coordinate-free, partition of unity-free approach to integration of differential forms, etc. This framework makes sense for any field theory valued in a Lie algebroid. We apply it to define the Wiener measure, stochastic integrals and other observables in a coordinate-free way. We use it to reconstruct Chern-Simons with finite gauge group and to obtain some non-perturbative deformation quantizations via the Poisson sigma model on a disk.
Paper Structure (46 sections, 19 theorems, 210 equations)

This paper contains 46 sections, 19 theorems, 210 equations.

Table of Contents

  1. Introduction and Overview
  2. Feynman's Construction of Path Integrals
  3. The Idea
  4. The Pair Groupoid's Role in Integration
  5. The Fundamental Theorem of Calculus on Manifolds
  6. Graded van Est Map
  7. Approximating Functional Integrals and Examples
  8. Outline of Paper
  9. Convention for Wedge Products
  10. Main Result
  11. Statement and Proof
  12. *Sigma Algebras and Approximations
  13. *Connection With Feynman's Construction
  14. *The Wiener Measure as a Direct Limit of Approximations
  15. *The Lebesgue Measure as a Direct Limit of Finite Sigma Algebras
  16. ...and 31 more sections

Key Result

Theorem 1

Let $M$ be an $n$-dimensional manifold and let $(\Omega_M,\Omega_{\partial M})\in H^{n}\textup{Pair}\,(M,\partial M)_{\textup{loc}}\,.$ Then

Theorems & Definitions (119)

  • Theorem
  • Theorem 1.1.1
  • Remark 1.1.2
  • proof
  • Proposition 1.2.1
  • Definition 2.1.1
  • Definition 2.1.2
  • Example 2.1.3
  • Definition 2.1.4
  • Definition 2.1.5
  • ...and 109 more