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).

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