The quantum tetrahedron as a quantum harmonic oscillator

It is known that the large-volume limit of a quantum tetrahedron is a quantum  harmonic oscillator. In particular the volume operator of a quantum tetrahedron is, in the sector of large eigenvalues, accurately described by a quantum harmonic oscillator.

Using  Vpython, it is possible to visualise the quantum tetrahedron as a  semi classical quantum simple harmonic oscillator.  The implementation of this is shown below;

Quantum simple harmonic oscillator

The quantum tetrahedron as a  Quantum  harmonic oscillator .

quantum tetrahedron as quantum SHM

Click to view animated gif

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


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

For a monochromatic quantum tetrahedron we can obtain closed analytical expressions:


Below is shown a sagemath program which displays the wavefunction  of a Quantum simple harmonic oscillator using Hermite polynomials Hn:


This an really important step in the development of Loop Quantum Gravity, because now we have  a  quantum field theory of geometry. This leads from combinatorial or simplicial description of geometry, through spin foams and the  quantum tetrahedron, to many particle states of quantum tetrahedra and the emergence of spacetime as a Bose-Einstein condensate.

quantum tetrahedron concepts

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