This week I been following up the posts:

The starting point is to take the action of the area operator in the spin representation, and to consider the following normalized trace of the area operator:

If this give a well-defined operator on either the kinematical Hilbert space or on the Hilbert space of fully gauge-invariant wave functions, and also if we took the limit Λ_{fix} →∞ it is possible to read off the spectrum from the representation. There, the oscillatory behaviour of the sine function is suppressed by a factor of 1/d_{j }which leads to a discrete spectrum for sufficiently small spins j.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.1 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.3 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.05, 0.1 and 0.3 and j=1…100

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