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Quantum Spin Dynamics (QSD) II

T. Thiemann

TL;DR

Thieme-Thiemann develop a rigorous non-perturbative quantization of the Lorentzian Wheeler-DeWitt constraint for four-dimensional vacuum gravity in the continuum, constructing both non-symmetric and symmetric operator realizations. They analyze the complete physical Hilbert space using group-averaged, diffeomorphism-invariant distributions built from spin-network states, and show how the Euclidean and Lorentzian constraints act by finite, controlled graph changes, enabling a tractable spectral analysis. A key advance is the introduction of a symmetric operator via a modified loop assignment with smooth exceptional edges, yielding anomaly-freeness, cylindrical consistency, and provable self-adjoint extensions; they also outline a path to a Wick-rotation-based connection to Euclidean gravity and discuss the potential for a group-averaging physical inner product. Collectively, the work lays a foundation for explicit construction of physical states and observables in Quantum Spin Dynamics (QSD) and points toward applications in black-hole physics. The framework highlights the role of the volume operator and spin-½ edge insertions in governing dynamics and points to future work on spectral analysis, regularization choices, and the practical extraction of observables.

Abstract

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results.

Quantum Spin Dynamics (QSD) II

TL;DR

Thieme-Thiemann develop a rigorous non-perturbative quantization of the Lorentzian Wheeler-DeWitt constraint for four-dimensional vacuum gravity in the continuum, constructing both non-symmetric and symmetric operator realizations. They analyze the complete physical Hilbert space using group-averaged, diffeomorphism-invariant distributions built from spin-network states, and show how the Euclidean and Lorentzian constraints act by finite, controlled graph changes, enabling a tractable spectral analysis. A key advance is the introduction of a symmetric operator via a modified loop assignment with smooth exceptional edges, yielding anomaly-freeness, cylindrical consistency, and provable self-adjoint extensions; they also outline a path to a Wick-rotation-based connection to Euclidean gravity and discuss the potential for a group-averaging physical inner product. Collectively, the work lays a foundation for explicit construction of physical states and observables in Quantum Spin Dynamics (QSD) and points toward applications in black-hole physics. The framework highlights the role of the volume operator and spin-½ edge insertions in governing dynamics and points to future work on spectral analysis, regularization choices, and the practical extraction of observables.

Abstract

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well as for the generator of the Wick transform from the Euclidean to the Lorentzian regime we prove existence of self-adjoint extensions and based on these we present a method of proof of self-adjoint extensions for the Lorentzian operator. Finally we comment on the status of the Wick rotation transform in the light of the present results.

Paper Structure

This paper contains 13 sections, 10 theorems, 25 equations.

Table of Contents

  1. Complete physical Hilbert space and observables
  2. Method to compute $a^{(n)}_{[T']}([w_0])$
  3. The symmetric operator
  4. Motivation
  5. The symmetric Euclidean Operator
  6. Symmetry
  7. Self-adjointness
  8. The symmetric Lorentzian operator
  9. Symmetry and cylindrical consistency
  10. Self-Adjointness
  11. Solutions
  12. Wick rotation transform
  13. Conclusions

Key Result

Theorem 1.1

Each distributional solution to all constraints of Lorentzian quantum gravity is a finite linear combination of states $\Psi$ of the following two types : Type I) $\Psi=[f]$ where $f$ is an arbitrary linear combination of spin-network states based on spin-nets $w_0\in W_0$. Type II) $\Psi=[f]$ where

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.1
  • Corollary 1.1
  • Definition 1.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 10 more