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Beyond Expectation Values: Generalized Semiclassical Expansions for Matrix Elements of Gauge Coherent States

Haida Li, Hongguang Liu

Abstract

We derive an asymptotic expansion for off-diagonal coherent-state matrix elements of non-polynomial operators in gauge theories admitting holomorphic coherent-state representations. The derivation combines stationary-phase analysis with an operator-level treatment of the Taylor remainder, and yields explicit semiclassical error control under stated assumptions. As a primary application, we formulate the expansion for volume and flux related operators in Loop Quantum Gravity and compare it with the standard diagonal expansion proposed in arXiv:gr-qc/0607101. By organizing the expansion around the genuine off-diagonal Berezin symbol rather than a diagonal expectation value, the resulting formula preserves the full holomorphic structure of the geometric phase and reproduces benchmark matrix elements accurately in the numerical regimes tested here, particularly when the coherent-state labels are well separated.

Beyond Expectation Values: Generalized Semiclassical Expansions for Matrix Elements of Gauge Coherent States

Abstract

We derive an asymptotic expansion for off-diagonal coherent-state matrix elements of non-polynomial operators in gauge theories admitting holomorphic coherent-state representations. The derivation combines stationary-phase analysis with an operator-level treatment of the Taylor remainder, and yields explicit semiclassical error control under stated assumptions. As a primary application, we formulate the expansion for volume and flux related operators in Loop Quantum Gravity and compare it with the standard diagonal expansion proposed in arXiv:gr-qc/0607101. By organizing the expansion around the genuine off-diagonal Berezin symbol rather than a diagonal expectation value, the resulting formula preserves the full holomorphic structure of the geometric phase and reproduces benchmark matrix elements accurately in the numerical regimes tested here, particularly when the coherent-state labels are well separated.

Paper Structure

This paper contains 12 sections, 3 theorems, 132 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. General Expansion Formula
  3. Berezin symbol and general coherent states
  4. Asymptotic estimate for shifted operator powers
  5. The main expansion fomula
  6. Application in LQG
  7. LQG coherent states
  8. Application of the expansion formula in LQG
  9. Numerical Comparison
  10. Setup
  11. Comparison of the two expansions
  12. Discussion

Key Result

Theorem 2.1

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 (3)

  • Figure 3.1: Spin-network structure of a gauge-variant 3-bridges graph.
  • Figure 3.2: Berezin Symbol of $\widehat{V}_{V_1}$ for $\theta=120^{\circ}$.
  • Figure 3.3: Relative discrepancy between the numerical data and the old/new expansion formulas for $\theta=120^\circ$.

Theorems & Definitions (6)

  • Theorem 2.1: Hörmander’s theorem 7.7.5
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof