Twisted geometry coherent states in all dimensional loop quantum gravity: I. Construction and Peakedness properties

TL;DR

This paper introduces a new family of coherent states for all dimensional loop quantum gravity based on twisted geometry parametrization, demonstrating their peakedness, classical correspondence, and suitability for semi-classical analysis.

Contribution

It constructs and analyzes a novel family of coherent states for all dimensional LQG that solve edge simplicity constraints and exhibit classical-like peakedness properties.

Findings

Coherent states form an over-complete basis of the Hilbert space.

Expectation values of operators match classical values up to gauge.

Peakedness and uncertainties are well-controlled by the semi-classical parameter t.

Abstract

A new family of coherent states for all dimensional loop quantum gravity are proposed, which is based on the generalized twisted geometry parametrization of the phase space of $SO(D+1)$ connection theory. We prove that this family of coherent states provide an over-complete basis of the Hilbert space in which edge simplicity constraint is solved. Moreover, according to our explicit calculation, the expectation values of holonomy and flux operators with respect to this family of coherent states coincide with the corresponding classical values given by the labels of the coherent states, up to some gauge degrees of freedom. Besides, we study the peakedness properties of this family of coherent states, including the peakedness of the wave functions of this family of coherent states in holonomy, momentum and phase space representations. It turns out that the peakedness in these various representations and the (relative) uncertainty of the expectation values of the operators are well controlled by the semi-classical parameter $t$. Therefore, this family of coherent states provide a candidate for the semi-classical analysis of all dimensional loop quantum gravity.