Classical and Quantum Polyhedra

TL;DR

This paper extends the semiclassical analysis of quantum polyhedra, comparing different quantization methods, and introduces a new framework for understanding their volume operators in quantum gravity.

Contribution

It presents a detailed comparison of canonical and wave function approaches for quantum polyhedra, and introduces a new formulation of quantum operators with dimensioned position variables.

Findings

Both methods agree at leading order with harmonic oscillator approximation.

Perturbative corrections improve eigenstate approximations.

Full wave function description for large negative volume eigenvalues.

Abstract

Quantum polyhedra constructed from angular momentum operators are the building blocks of space in its quantum description as advocated by Loop Quantum Gravity. Here we extend previous results on the semiclassical properties of quantum polyhedra. Regarding tetrahedra, we compare the results from a canonical quantization of the classical system with a recent wave function based approach to the large-volume sector of the quantum system. Both methods agree in the leading order of the resulting effective operator (given by an harmonic oscillator), while minor differences occur in higher corrections. Perturbative inclusion of such corrections improves the approximation to the eigenstates. Moreover, the comparison of both methods leads also to a full wave function description of the eigenstates of the (square of the) volume operator at negative eigenvalues of large modulus. For the case of general quantum polyhedra described by discrete angular momentum quantum numbers we formulate a set of quantum operators fulfilling in the semiclassical regime the standard commutation relations between momentum and position. Differently from previous formulations, the position variable here is chosen to have dimension of (Planck) length squared which facilitates the identification of quantum corrections. Finally, we provide expressions for the pentahedral volume in terms of Kapovich-Millson variables.