The Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator

Abstract

It is shown that the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator. This result relies on the fact that (i) the volume operator couples only neighboring states of its standard basis, and (ii) its matrix elements show a unique maximum as a function of internal angular momentum quantum numbers. These quantum numbers, considered as a continuous variable, are the coordinate of the oscillator describing its quadratic potential, while the corresponding derivative defines a momentum operator. We also analyze the scaling properties of the oscillator parameters as a function of the size of the tetrahedron, and the role of different angular momentum coupling schemes.

11 pages, 3 figures; a few remarks (and pertaining references) added, version to appear in Class. Quant. Grav

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Cited by in corpus (3)

• Coherent Quantum Dynamics: What Fluctuations Can Tell
• Classical and Quantum Polyhedra
• Semiclassical Length Measure from a Quantum-Gravity Wave Function