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Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context

A. Ashtekar, D. Marolf, J. Mourão, T. Thiemann

TL;DR

The paper generalizes the Osterwalder-Schrader reconstruction to diffeomorphism-invariant, background-free field theories by replacing Euclidean invariance with diffeomorphism covariance and by formulating a set of generalized axioms. It demonstrates that, for each foliation of spacetime, a physical Hilbert space and a Hamiltonian can be recovered from a measure on quantum histories, with unitary equivalences ensuring consistency across foliation choices. The authors illustrate the framework with explicit examples, including a uniform measure for gauge theories, 2D Yang-Mills, and 2+1 gravity/BF theories, showing how the canonical and path-integral pictures align without relying on Wick rotation. This diffeomorphism-invariant reconstruction provides a robust route to quantizing gravity and topological gauge theories, contingent on constructing appropriate measures for given classical theories.

Abstract

Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian) Hamiltonian quantum theory starting from a measure on the space of (Euclidean) histories of a scalar quantum field. In this paper, we extend that construction to more general theories which do not refer to any background, space-time metric (and in which the space of histories does not admit a natural linear structure). Examples include certain gauge theories, topological field theories and relativistic gravitational theories. The treatment is self-contained in the sense that an a priori knowledge of the Osterwalder-Schrader theorem is not assumed.

Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context

TL;DR

The paper generalizes the Osterwalder-Schrader reconstruction to diffeomorphism-invariant, background-free field theories by replacing Euclidean invariance with diffeomorphism covariance and by formulating a set of generalized axioms. It demonstrates that, for each foliation of spacetime, a physical Hilbert space and a Hamiltonian can be recovered from a measure on quantum histories, with unitary equivalences ensuring consistency across foliation choices. The authors illustrate the framework with explicit examples, including a uniform measure for gauge theories, 2D Yang-Mills, and 2+1 gravity/BF theories, showing how the canonical and path-integral pictures align without relying on Wick rotation. This diffeomorphism-invariant reconstruction provides a robust route to quantizing gravity and topological gauge theories, contingent on constructing appropriate measures for given classical theories.

Abstract

Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian) Hamiltonian quantum theory starting from a measure on the space of (Euclidean) histories of a scalar quantum field. In this paper, we extend that construction to more general theories which do not refer to any background, space-time metric (and in which the space of histories does not admit a natural linear structure). Examples include certain gauge theories, topological field theories and relativistic gravitational theories. The treatment is self-contained in the sense that an a priori knowledge of the Osterwalder-Schrader theorem is not assumed.

Paper Structure

This paper contains 19 sections, 13 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The general setting
  3. Basic framework
  4. Subtleties
  5. Generalized Osterwalder-Schrader Axioms
  6. Recovery of the Hilbert Space Theory
  7. Examples
  8. The Uniform Measure for Gauge Theories
  9. Two-dimensional Yang-Mills Theory
  10. 2+1 gravity and BF-Theories
  11. Discussion
  12. A Universality Class of Generally Covariant Quantum Gauge Field Theories in Two Spacetime Dimensions
  13. Introduction
  14. Review of the classical Osterwalder-Schrader framework
  15. Reconstruction Theorem
  16. ...and 4 more sections

Key Result

Theorem 1

i) For each $E\in {\rm Fol}(\sigma,(M,s))$, axioms (II) and (III) imply the existence of a Hilbert space ${\cal H}^E_D$ of physical states. There is a natural class of unitary equivalences between ${\cal H}^E_D$ and ${\cal H}^{\tilde{E}}_D$ for all $E$ and $\tilde{E}$ in the (weak) equivalence class

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Definition 3.1
  • Theorem 3.1
  • ...and 6 more