Semiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent States

TL;DR

This paper investigates the semiclassical limit of the LQG volume operator using flux coherent states, finding that only cubic graphs with six-valent vertices reproduce classical values, impacting spin foam models.

Contribution

It introduces a generalized flux coherent state framework for LQG and demonstrates that only cubic graphs with six-valent vertices yield correct semiclassical volume expectations.

Findings

Expectation value matches classical volume only for six-valent cubic graphs.

Coherent states are non-normalizable but define finite cut-off states.

Results influence the understanding of spin foam models based on four-valent networks.

Abstract

The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG), both in the full theory and in truncated models adapted to cosmological situations coined Loop Quantum Cosmology (LQC). It is therefore crucial to check whether its semiclassical limit coincides with the classical volume operator plus quantum corrections. In the present article we investigate this question by generalizing and employing previously defined coherent states for LQG which derive from a cylindrically consistently defined complexifier operator which is the quantization of a known classical function. These coherent states are not normalizable due to the non separability of the LQG Hilbert space but they define uniquely define cut off states depending on a finite graph. The result of our analysis is that the expectation value of the volume operator with respect to coherent states depending on a graph with only n valent verticies reproduces its classical value at the phase space point at which the coherent state is peaked only if n = 6. In other words, the semiclassical sector of LQG defined by those states is described by graphs with cubic topology! This has some bearing on current spin foam models which are all based on four valent boundary spin networks.