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

TL;DR

This paper develops a generalized semiclassical expansion for off-diagonal matrix elements of gauge theory operators in coherent states, improving accuracy and preserving geometric phases, with applications to Loop Quantum Gravity.

Contribution

It introduces a novel off-diagonal expansion method that maintains the full holomorphic structure and provides explicit error control, advancing semiclassical analysis in gauge theories.

Findings

The expansion accurately reproduces benchmark matrix elements in numerical tests.

It preserves the full geometric phase structure of the operators.

The method offers explicit error bounds under certain assumptions.

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.