# The Large-Volume Limit of a Quantum Tetrahedron is a Quantum Harmonic Oscillator by John Schliemann

In this paper the author shows that the volume operator of a quantum tetrahedron is for  large eigenvalues accurately described by a quantum harmonic oscillator. This result occurs because the volume operator couples only neighbouring states of its standard basis and  its matrix elements show a  maximum as a function of internal angular momentum quantum numbers. These quantum numbers are the coordinate of the oscillator describing its quadratic potential, while the corresponding derivative defines a momentum operator.

This paper has a great review of the basic mathematical structure of a quantum tetrahedron.

A quantum tetrahedron consists of four angular momenta Ji, where i = {1,2,3, 4} representing its faces and coupling to a vanishing total angular momentum – this means that the Hilbert space consists of all states ful filing:

j1 +j2 +j3 +j4 = 0

In the coupling scheme both pairs j1;j2 and j3;j4 couple frst to two irreducible SU(2)representations of dimension 2k+1 each, which are then added to give a singlet state.

The quantum number k ranges from kmin to kmax in integer steps with:
kmin = max(|j1-j2|,|j3 -j4|)
kmax = min(j1+j2,j3 +j4)

The total dimension of the Hilbert space is given by:
d = kmax -kmin + 1.

The volume operator can be formulated as:

where the operators

represent the faces of the tetrahedron with

and being the Immirzi parameter.

It is useful to use the operator

which, in the basis of the states can be represented as

with

and where

is the area of a triangle with edges a; b; c expressed via Herons formula,

These choices mean that the matrix representation of Q is real and antisymmetric and  the spectrum of Q consists for even d of pairs of eigenvalues q and -q differing in sign. This makes it much easier to find the eigenvalues as I found in  Numerical work with sage 7.

The large volume limt of Q is found to be given by:

The eigenstates of Q are just the well-known wave functions diagonalizing the harmonic oscillator in real-space representation:

This work leads to the result that the the classical volume Vcl of a general tetrahedron with face areas j1; j2; j3; j4 is, to leading order in all ji is given by:

I’ll be doing some sagemath work on this volume expression.