Operators for quantized directions

TL;DR

This paper introduces two quantum operators inspired by spin geometry that measure angles between surfaces and edge bundles at spin network vertices, analyzing their properties and spectra to connect quantum geometry with classical angles.

Contribution

It defines novel operators for quantized angles in quantum geometry, extending previous geometric operators and exploring their spectral and semiclassical properties.

Findings

Operators effectively measure angles in quantum geometry

Spectral analysis of the operators reveals their properties

Conditions identified for semiclassical states to replicate classical angles

Abstract

Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two ``bundles'' of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles.