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;
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