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A length operator for canonical quantum gravity

T. Thiemann

TL;DR

This work defines a rigorous length operator for curves in four-dimensional Lorentzian quantum gravity within the real connection representation, constructing it from holonomies and the volume operator. The operator is densely defined, has self-adjoint extensions, and exhibits a purely discrete spectrum in units of the Planck length, mirroring the behavior of the area and volume operators. The spectrum is analyzed in simple graph settings (notably trivalent graphs), yielding explicit eigenvalues and confirming a quantum of length at the Planck scale, with large-spin regimes approaching semiclassical behavior. Additionally, the paper introduces a tube operator and weave-state framework to approximate classical 3-geometries, demonstrating how quantum geometry can reproduce classical metrics via discretized, lattice-like structures.

Abstract

We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which a $SU(2)$ connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which faciliates the construction of a new type of weave states which approximate a given classical 3-geometry.

A length operator for canonical quantum gravity

TL;DR

This work defines a rigorous length operator for curves in four-dimensional Lorentzian quantum gravity within the real connection representation, constructing it from holonomies and the volume operator. The operator is densely defined, has self-adjoint extensions, and exhibits a purely discrete spectrum in units of the Planck length, mirroring the behavior of the area and volume operators. The spectrum is analyzed in simple graph settings (notably trivalent graphs), yielding explicit eigenvalues and confirming a quantum of length at the Planck scale, with large-spin regimes approaching semiclassical behavior. Additionally, the paper introduces a tube operator and weave-state framework to approximate classical 3-geometries, demonstrating how quantum geometry can reproduce classical metrics via discretized, lattice-like structures.

Abstract

We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which a connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly is quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which faciliates the construction of a new type of weave states which approximate a given classical 3-geometry.

Paper Structure

This paper contains 17 sections, 6 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. The real connection representation
  3. The length operator
  4. Regularization of the length operator
  5. Finiteness and choice of the partition adapted to a graph
  6. Existence of the limit $n\to\infty$
  7. Cylindrical consistency
  8. Symmetry and positivity
  9. Self-adjointness
  10. Spectral analysis of the length operator
  11. Discreteness
  12. Eigenvalues
  13. Case A
  14. Case B
  15. Tube operator and weaves
  16. ...and 2 more sections

Key Result

Lemma 3.1

Every projection $\hat{L}_\gamma(c)$ defines a symmetric and positive semi-definite operator on $\cal H$ with domain ${\cal D}_\gamma:=\hbox{Cyl}_\gamma^3 ({\overline {{\cal A}/{\cal G}}})$.

Theorems & Definitions (8)

  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Theorem 4.1
  • Theorem 4.2
  • Definition 5.1
  • Definition 5.2