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.

 

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