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Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity

Johannes Brunnemann, Thomas Thiemann

TL;DR

This paper tackles the challenge of computing the Volume operator spectrum in Loop Quantum Gravity by deriving a compact, closed-form expression for its matrix elements using the Elliot–Biedenharn identity, thereby eliminating the heavy intertwiners and $6j$-symbol sums that impeded large-spin numerics. The authors reformulate the problem in gauge-invariant recoupling bases, express the matrix elements in terms of $3nj$ and eventually $6j$-symbols, and then obtain explicit formulas that reveal a banded Jacobi-like structure for the gauge-invariant 4-vertex. They provide both analytical results (zero-eigenvalue structure, monochromatic vertex cases) and comprehensive numerical studies up to $j_{ ext{max}}=50$, finding evidence for a volume gap, scaling laws, and a near-universal spectral density after appropriate rescaling. The work significantly improves computational feasibility for LQG dynamics, suggests applications to spin-foam analyses and many-particle spin systems, and lays groundwork for extending the approach to higher-valence vertices with high-performance computing.

Abstract

The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed. In this article we demonstrate that by means of the Elliot -- Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge invariant 4 -- vertex. The techniques derived in this paper could be of use also for the analysis of spin -- spin interaction Hamiltonians of many -- particle problems in atomic and nuclear physics.

Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity

TL;DR

This paper tackles the challenge of computing the Volume operator spectrum in Loop Quantum Gravity by deriving a compact, closed-form expression for its matrix elements using the Elliot–Biedenharn identity, thereby eliminating the heavy intertwiners and -symbol sums that impeded large-spin numerics. The authors reformulate the problem in gauge-invariant recoupling bases, express the matrix elements in terms of and eventually -symbols, and then obtain explicit formulas that reveal a banded Jacobi-like structure for the gauge-invariant 4-vertex. They provide both analytical results (zero-eigenvalue structure, monochromatic vertex cases) and comprehensive numerical studies up to , finding evidence for a volume gap, scaling laws, and a near-universal spectral density after appropriate rescaling. The work significantly improves computational feasibility for LQG dynamics, suggests applications to spin-foam analyses and many-particle spin systems, and lays groundwork for extending the approach to higher-valence vertices with high-performance computing.

Abstract

The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed. In this article we demonstrate that by means of the Elliot -- Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge invariant 4 -- vertex. The techniques derived in this paper could be of use also for the analysis of spin -- spin interaction Hamiltonians of many -- particle problems in atomic and nuclear physics.

Paper Structure

This paper contains 49 sections, 4 theorems, 139 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Revision of Known Results
  3. Matrix Elements in Terms of $3nj$-Symbols
  4. Closed Expression for the $3nj$-Symbols
  5. Preparations
  6. Closed Expresion for the $3nj$-Symbols
  7. Simplification of the Matrix Elements
  8. $3nj$-Symbols Expressed in Terms of $6j$-Symbols
  9. The Matrix Elements in Terms of $6j$-Symbols
  10. A Useful Identity
  11. Precalculation
  12. $I=1$, $J$ arbitrary
  13. $I=1$, $J=2$
  14. $I=2$, $J$ arbitrary
  15. $I=2$, $J=3$
  16. ...and 34 more sections

Key Result

Lemma 2.1

Contraction on Identical Coupling Order $<\vec{g}(IJ)|\vec{g}'>$$=<g_2(j_I,j_J),g_3(g_2,j_1),..,g_{I+1}(g_I,j_{I-1}), g_{I+2}(g_{I+1},j_{I+1}),..,g_J(g_{J-1},j_{J-1})|$$~~~~|g"_2(j_1,j_2),g"_3(g"_2,j_3),..,g"_{I+1}(g"_I,j_{I-1}), g"_{I+2}(g"_{I+1},j_{I+1}),...,g"_J(g"_{J-1},j_J)> \delta_{g_J,g"_J} .

Figures (13)

  • Figure 1: The configuration at the 4-vertex: 4 outgoing edges each carrying a representation of $SU(2)$ with a weight according to $j_1,j_2,j_3,j_4$
  • Figure 4: The logarithm of the number of eigenvalues in the intervals $I_n=[\lambda_n-\Delta\lambda,\lambda_N]$$\lambda_n=n\cdot\Delta\lambda$ as a function of $j_{max}$.
  • Figure 5: The evolution of the first positive 8 eigenvalues $\lambda_k(j):=V^2,\;k=1,..,8$ of $\hat{Q}_v$ in dependence of $j_4=j$. Note that each line represents eigenvalues with multiplicity given by the table in section \ref{['multiplicity tabular']} .
  • Figure 6: The quotient $\lambda_{max}^{(j)}/L_{max}$ as a function of $j$
  • Figure 7: The total number of 0-eigenvalues contained in all configurations allowed by (\ref{['order of angular momenta']}) depending on $j_{max}$
  • ...and 8 more figures

Theorems & Definitions (7)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 4.1
  • Theorem A.1
  • Definition A.1
  • Definition A.2
  • Definition A.3