Numerical work with sagemath 9 – Area eigenvalues in LQG

In this work I am studying the  area eigenvalues in LQG. To study the spectrum numerically I have to specify the triple (j1, j2, j3) for each vertex, the only restriction being that:

|j1 − j2| < j3 < j1 + j2

Hence, j1 and j2 run over all integer and half-integer numbers, whilst j3 runs from |j1− j2| to j1 + j2 in unit steps.

The formula A = sqrt(j(j+1) gives the main sequence area eigenvlaues . The formula A = sqrt(2*j1*(j1+1)+ 2*j2*(j2+1)-j3*(j3+1)) gives  all Area eigenvlaues . The degenerate sector area eigenvalues are the A = sqrt(2*j1*(j1+1)+ 2*j2*(j2+1)-j3*(j3+1)) eigenvalues minus the main sequence A = sqrt(j(j+1)) eigenvalues;

Numerical work 9

This work is based on the paper A numerical study of spectral properties of the area operator in loop quantum gravity by Helesfai and Bene


A numerical study of spectral properties of the area operator in loop quantum gravity by Helesfai and Bene

In this paper the authors investigate the lowest 37000 eigenvalues of the area operator in loop quantum gravity. They obtain an asymptotic formula for the eigenvalues as a function of their sequential number. The multiplicity of the lowest few hundred eigenvalues is also determined and the smoothed spectral density is calculated. The authors present the spectral density for various number of vertices, edges and SU(2) representations.

In LQG during  quantization, geometrical quantities like area become operators acting on the Hilbert space. Area operators are especially interesting, since their spectral properties enable LQG to be tested via the Bekenstein-Hawking entropy of Black Holes in the quasiclassical limit.

Eigenvalues of the area operator

A lot is known about its eigenvalues and eigenstates analytically. Each spin network state is an eigenstate of the area operator. The eigenvalues are explicitly expressed as:

helesfai equ1

where lp stands for the Planck length and b for the Immirzi parameter whose value is argued to be:helesfai immirz

To study the spectrum numerically – then if only the eigenvalues are of interest, it is enough to specify the triple (j1, j2, j2) for each vertex, the only restriction being that |j1 − j2| < j3 < j1 + j2

Hence, j1 and j2 run over all integers and half-integer numbers, while j3 runs from |j1− j2| to j1 + j2 in unit steps.

helesfai lowest eigenvalues

helesfai log fit

I have coded these ideas into sagemath and have prepared a sagemath notebook and interactive: Area eigenvalues in LQG.


Angle and Volume Studies in Quantized Space by Michael Seifert

In this paper seifert reviews the angle and volume operators in Loop Quantum Gravity. I’m particularly interested at the moment in his treatment of the angle operator.

In LQG, quantum states are represented by spin networks –  graphs with weighted edges. Spatial observables such as area, volume, and angle are given by the eigenvalues of Hermitian operators on the spin network states. The author presents results on  the angle and volume operators which act on the vertices of spin networks. He finds that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, with a fairly slow decrease to  zero. He also presents numerical results indicating that the angle operator can reproduce the classical angle distribution. Noting that the volume operator is significantly harder to investigate analytically the author presents analytical and  numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, a result which corresponds to the classical model of space.

Angular Momentum Operators

One of the aims of this paper is to describe the angles associated with a given vertex.  To define angles, need to be able to associate a direction with an edge or a group of edges. In Penrose’s original formulation of spin networks, the labels on the edges refer to angular momenta carried by the edges  so one of the natural directions to associate with an edge is its angular momentum vector. The angular momenta can be measured by the angular momentum operators, which act upon the edges. Angular momentum operators are expressed in terms of the Pauli spin matrices σi:

seirfeit pauli matrices

Using this, we can define angular momentum operators Ji = 1/2hσi which act on the edges. Can also construct the Jˆ2 operator this is equal to the sum of the squares of the three Ji operators.

The eigenvalues simplify to the  form:

Jˆ2 = 1/2hj(j +1)

The Area Operatorseirfeit areas

The total area of  is then given by

Discreteness of Area and Volume fig 3

where the summation of i is over all edges that intersect , and Ji is the edge label of the edge I.

The Angle Operator

seirfeit angle partition

The angle operator will measure the angle between the edges that traverse S1 and those that traverse S2. The  direction that is associated with the edges traversing S1, S2, and Sr is the angular momentum vectors associated with their internal edges J1,J2, and Jr.

Classically it is expected that

cos θ = J1.J2 / |J1||J2|

The angle operator is then defined as

seirfeit angle op

The eigenvalues of the angle operator:

seirfeit angle eigenvalue

where ji = ni/2.

From the discreteness of the angle operator Seifert derives three  results: will be referred to as

Small angle  property for angle e

seirfeit small angle

where the total value of core spin nT = n1 + n2 + nr

Angular resolution property

seirfeit angle resolution

where n is valence of the vertex.

Angular distribution property

Shape in an atom of space: exploring quantum geometry phenomenology by Seth A. Major

This week I have been reading about and working on operators other than the volume operator. One paper I particularly enjoyed was by Seth. A. Major who is an interesting physicist. Not only does he do a lot of work in the area of quantum gravity but he is one of the leading physicists exploring the idea of the universe as a computer. In this paper he develops a phenomenology for the deep spatial geometry of loop quantum gravity. In the context of a simple model – an atom of space, he is shows how combinatorial structures can affect physical observations. The angle operator is used to develop a model of angular corrections to local, continuum flat-space 3-geometries. This is similar to the work of Roger Penrose described in an earlier post: Angular momentum: An approach to combinatorial space-time by Roger Penrose. The physical effects reply on the combinatorics of SU(2) recoupling. Bhabha scattering, that is, electron-positron scattering  is used as an example of how these effects might be observed.

The Angle operator

The angle operator acts on nodes and the spectrum may be expressed in terms of the SU(2) representations of the intertwiner at a single node. equ1 atom of space

fig 1 atom of spaceThe angle operator may also be defined in terms of the electric flux variables and the usual embedded graphs of LQG. I have also been working on a sagemath program to calculate the angle operator eigenvalues for given J1, J2, J3 and J4.

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


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.

Loop Quantum Gravity –

This week as I have been getting deeper into the sagemath community. I’ve setup a Loop Quantum Gravity page at So far I have posted an intereactive version of the Quantum Tetrahedron Volume Eigenvalue calculator which I developed a few weeks ago. At present I am  working on the graphical representation, area operator and  angle operator aspects of the quantum tetrahedron.

sage interactions Loop quantum Gravity


Quantum tetrahedron volume eigenvalues in loop quantum gravity by Meissner

In this paper the author presents a simple method to calculate certain sums of the eigenvalues of the volume operator in loop quantum gravity. He derives the asymptotic distribution of the eigenvalues in the classical limit of very large spins which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity. Meissner shows that relatively simple formulae can be obtained for the sums of squares (or higher even powers) of the eigenvalues, with the most important quantum numbers in the vertex fixed: j, j1, j2, j3, where j is the total spin and j1, j2, j3 are spins of the incoming legs.

Volume operator
The volume operator V is defined as:

meissner fig2

where q is:

meissner fig 3Where, I, J,K run through 1, 2, …,N (N ≥ 3) and i, j, k are SU(2) indices. There is an unshown overall constant proportional to the Planck volume. Eigenvalues of V describe a contribution to the volume from a given vertex with N + 1 legs with spin conservation. The most important
case is  N = 3 the so called 4-valent vertex.

The operator q has several important properties:

  • It commutes with all Ji^2:

[q, Ji^2] = 0             i= 1, 2, 3

  • It commutes with all components of the total spin J

[q, J] = 0                       J = J1 + J2 + J3

  • q treats all four spins J1, J2, J3, J4 satisfying J1+J2+J3+J4 = 0  in the same way.
  • q is self adjoint and the trace of any odd power of q vanishes, therefore q has only real eigenvalues that are either 0 or come in pairs (λi,−λi).