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:

large vol equ 1

where the operators

large vol equ 2

represent the faces of the tetrahedron with

large vol equ 3and being the Immirzi parameter.

It is useful to use the operator

large vol equ 4

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

large vol equ 5

with

large vol equ 6

and where

large vol equ 7

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:

large vol equ 8

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

large vol equ 9

large vol fig 1

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:

large vol equ 10

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

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