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Closed formula for the matrix elements of the volume operator in canonical quantum gravity

T. Thiemann

TL;DR

The paper derives a closed-form expression for the matrix elements of the volume operator in loop quantum gravity, using a state-dependent regularization that preserves diffeomorphism covariance and reproduces the Ashtekar–Lewandowski operator up to a constant. It shows the spectrum is discrete and computable via finite-dimensional matrices, with a rigorous treatment of cylindrical consistency, symmetry, and anomaly-freeness. By recasting matrix elements as cubic polynomials in angular momentum operators and employing recoupling theory and 6j-symbols, the work provides a practical framework for both analytical evaluation and numerical spectroscopy of the volume operator across spin-network states. The results pave the way for detailed spectral analysis and improved numerical studies of quantum geometric observables in four-dimensional canonical quantum gravity.

Abstract

We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four spacetime dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state independent factor.

Closed formula for the matrix elements of the volume operator in canonical quantum gravity

TL;DR

The paper derives a closed-form expression for the matrix elements of the volume operator in loop quantum gravity, using a state-dependent regularization that preserves diffeomorphism covariance and reproduces the Ashtekar–Lewandowski operator up to a constant. It shows the spectrum is discrete and computable via finite-dimensional matrices, with a rigorous treatment of cylindrical consistency, symmetry, and anomaly-freeness. By recasting matrix elements as cubic polynomials in angular momentum operators and employing recoupling theory and 6j-symbols, the work provides a practical framework for both analytical evaluation and numerical spectroscopy of the volume operator across spin-network states. The results pave the way for detailed spectral analysis and improved numerical studies of quantum geometric observables in four-dimensional canonical quantum gravity.

Abstract

We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four spacetime dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state independent factor.

Paper Structure

This paper contains 11 sections, 3 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Regularization of the volume operator
  4. Spectral Analysis
  5. Cylindrical Consistency
  6. Symmetry, Positivity and Self-Adjointness
  7. Discreteness and Anomaly-freeness
  8. The complete set of matrix elements
  9. The left regular representation and the spin-network representation
  10. Recoupling Theory for $n$ angular momenta
  11. Examples

Key Result

Lemma 3.1

For each triple of edges $e,e',e^{\prime\prime}$ there exists a choice of vectors $\vec{n}_i,\vec{n}_i',\vec{n}_i^{\prime\prime}$ and a way to guide the limit $\Delta_i,\Delta'_i,\Delta^{\prime\prime}_i\to 0$ such that vanishes a) if $e,e',e^{\prime\prime}$ do not all intersect $p$ or b) $\det(\frac{\partial(x^a_{e e' e^{\prime\prime}})}{\partial(t,t',t^{\prime\prime})})_p=0$ (which is a diffeomo

Theorems & Definitions (5)

  • Lemma 3.1
  • Definition 5.1
  • Lemma 5.1
  • Lemma 5.2
  • Definition 5.2