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Bridging Quantum and Semiclassical Volume: A Numerical Study of Coherent State Matrix Elements in Loop Quantum Gravity

Haida Li, Hongguang Liu

Abstract

In Loop Quantum Gravity, the quantum action of the volume operator is crucial in understanding quantum dynamics. In this work, we implement a generalized numerical algorithm that can compute the quantum action of the volume operator on a broad class of gauge-variant and gauge-invariant spin-network states. This algorithm is later used to calculate the coherent state expectation value and coherent state matrix elements of the volume operator. By comparing the results generated by our numerical model with the analytical results in various scenarios at the near-semiclassical region, not only is our numerical model validated with high accuracy, but it also provides a complete picture of how the full quantum action of the volume operator connects with its semiclassical approximations. We further find that the maximal eigenvalue approaches the classical polyhedral volume in the semiclassical regime. For irregular geometries, we also observe that the relative volume magnitudes can change in the deep quantum regime.

Bridging Quantum and Semiclassical Volume: A Numerical Study of Coherent State Matrix Elements in Loop Quantum Gravity

Abstract

In Loop Quantum Gravity, the quantum action of the volume operator is crucial in understanding quantum dynamics. In this work, we implement a generalized numerical algorithm that can compute the quantum action of the volume operator on a broad class of gauge-variant and gauge-invariant spin-network states. This algorithm is later used to calculate the coherent state expectation value and coherent state matrix elements of the volume operator. By comparing the results generated by our numerical model with the analytical results in various scenarios at the near-semiclassical region, not only is our numerical model validated with high accuracy, but it also provides a complete picture of how the full quantum action of the volume operator connects with its semiclassical approximations. We further find that the maximal eigenvalue approaches the classical polyhedral volume in the semiclassical regime. For irregular geometries, we also observe that the relative volume magnitudes can change in the deep quantum regime.

Paper Structure

This paper contains 27 sections, 1 theorem, 47 equations, 22 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Volume Operator in Canonical LQG
  3. Volume Operator acting on gauge-variant and gauge-invariant spin-network states
  4. Coherent States in LQG
  5. Semiclassical expansion of the volume operator matrix elements
  6. Computational Framework and Algorithm
  7. Calculation of the Transformation Matrix
  8. Computing the Action of the Volume Operator
  9. The Final $j$-Summation and Parallelization
  10. Analytical Computation of the $(\widehat{Q}_v)^q$ Operators
  11. Main Results
  12. Expectation value of the volume operator on the gauge-variant 3-bridges
  13. Numerical Accuracy and $n$-Winding Corrections
  14. Matrix Elements of Higher-Order $\widehat{Q}^q_{V_1}$ Operators
  15. Convergence of the Volume Operator Expansion
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let K be a compact subset in $\mathbb{R}^n$, X an open neighborhood of K, and k a positive integer. If: (1) the complex functions $u \in C_0^{2 k}(K)$, $f \in C^{3 k+1}(X)$ and $\operatorname{Im} f(x) \geq 0 \quad\forall x\in X$, (2) there is a unique point $x_0 \in K$ satisfying $\operatorname{Im}\ Here the constant $C$ is bounded when $f$ stays in a bounded set in $C^{3 k+1}(X)$. We use the stan

Figures (22)

  • Figure 4.1: Graphical illustration of the spin-networks of: (a) Gauge variant 3-bridges, (b) gauge-invariant 4-bridges (each vertex corresponds to a tetrahedron), and (c) Gauge variant 3-flower (the vertex corresponds to a parallelepiped).
  • Figure 4.2: Consistency check of the normalization factor. (a) The normalization factor is obtained numerically (circles, using spin-network representation) and analytically (blue line, computed purely in coherent state representation) for different $j_{\mathrm{cap}}$. The parameters are set to be $t=2.5$, $\vec{z}_I=\vec{0}$ and $\vec{p}_I=20\vec{n}_I$. (b) The relative difference between analytical results and numerical results (computed as a percentage over the classical result). High accuracy can be achieved by raising $j_{\mathrm{cap}}$ higher. (c) Peakedness along $\vec{j}=(j_0,j_0,j_0)$ direction of the intermediate value before the final $j$-summation (where $\iota$ and $M$ indices are already summed). For $j_0\leq 5$ and $j_0\geq 10$, the contribution to the final result is negligible, which coincides with the results shown in (a) and (b). (d) The computation time of our numerical algorithm corresponds to each circle on the left. An exponential increase can be observed as $j_{cap}$ (the limit of $j$ summation) increases.
  • Figure 4.3: (a) Relative difference between the normalization factor of $n_{\mathrm{cap}}=0,1,2$ and $n_{\mathrm{cap}}=3$. (b) Difference on the normalization factor between the numerical result for $\vec{p}_a=6\vec{n}_a$ and the analytical result on the same setting for $n_{\mathrm{cap}}=3$.
  • Figure 4.4: Comparison between the analytical results ($n_{\mathrm{cap}}=0$) and numerically computed results of the expectation value of (a) $\widehat{Q}_{V_1}$, (b) $\widehat{Q}^2_{V_1}$.
  • Figure 4.5: the $n_{\mathrm{cap}}$ difference between $n_{\mathrm{cap}}=0$, $n _{\mathrm{cap}}=1$, $n_{\mathrm{cap}}=2$ cases and $n_{\mathrm{cap}}=3$ for the numerically computed (normalized) expectation value of (a) $\widehat{Q}_{V_1}$, (b) $\widehat{Q}^2_{V_1}$ operators for $\vec{p}_I=6\vec{n}_I$.
  • ...and 17 more figures

Theorems & Definitions (1)

  • Theorem 3.1: Hörmander’s theorem 7.7.5