Planar spin network coherent states II. Small corrections

TL;DR

This paper analyzes small correction states in planar spin network coherent states, proving their completeness and using them to develop a perturbation expansion of the inverse volume operator, with detailed variance and spreading estimates.

Contribution

It demonstrates the completeness of small correction states in planar spin network coherent states and applies them to expand the inverse volume operator perturbatively.

Findings

Small correction states form a complete subset of coherent states.

Standard deviations of angles are given by matrix elements of SC states.

Estimated spreading rate of coherent state wave packets.

Abstract

This paper is the second of two which construct coherent states for spin networks with planar symmetry. Paper 1 constructs set of coherent states peaked at specific values of holonomy and triad. These operators acting on the coherent state give back the coherent state plus small correction (SC) states. The present paper proves that these SC states form a complete subset of the overcomplete set of coherent states. The subset is used to construct a perturbation expansion of the inverse of the volume operator. Appendices calculate the standard deviations of the angles occurring in the holonomies, demonstrate that standard deviations are given by matrix elements of the SC states, and estimate the rate of spreading of a coherent state wave packet.