In Mathematical Reviews, all of my papers so far have been reviewed. One of the reviews is more extensive and after reading that I felt it could be a good idea if I have the chance to review this review in Gauge Invariance blog. I thank Maria Cristina Abbati for writing the original review.

Let me emphasize this review by no mean is to criticize the original review as it is fair in its real meaning. Moreover, this paper under discussion has been already published in NPB a year ago and the review was written recently. The review written for Math Rev is only a way to explain, publicize and partially simplify the contents of the paper briefly. Mathematical Reviews is a large archive of Physics and Math papers where these papers are reviewed again to be resorted.

The original review is in green color. The red words in between are not so to criticize something. The are marked red only to make my review easy to read and conclude. The red words do not oppose anything in the green lines, as they are mostly after one another.

In a previous paper [M. Ansari, NPB 783 (2007), no. 3, 179--212; MR2356347], the author gave an accurate description of the spectrum of the area operator as it is defined in loop quantum gravity.

He improved on previous results in the area, gave a new minimum value of the area in Planck's area unit, a_{min} = (3^0.5) * 2 \pi \gamma (where \gamma is the Immirzi parameter), and found a classification of the spectrum as a union of equidistant subsets which he called generations.

[I should correct the first part of the 2nd paragraph. In that article, I have not tried to give any new minimum value to the area.

In fact, this minimum value of area was reported by Abhay Ashtekar and Jerzy Lewandowski. They reported a spectrum of area in 1996. A few months later, Carlo Rovelli and his colleague reported the same spectrum of area using a completely different technique. Later on, Thomas Thiemann also studied this spectrum in more details and reported the same spectrum for a quantum of area.

And Yes!, in that paper I reported a new classification of the specrum of area, the so-called ladder symmetry. In fact there exists a universal formula for area that is way more compact that the one reported originally.]

This paper is complementary to the above paper and contains a detailed analysis of the degeneracy of the area's eigenvalues. The numerical results for the first hundred levels of the spectrum are summarized in scatterplots. Taking into account the generating properties, the degeneracy for a large area is estimated and results in \Omega(A) = g(a_{min} ^N, where g(a_{min} is the degeneracy of the minimum value.

[Yes, I reported a hidden degeneracy as well as a ladder symmetry on area eigenvalues.

However, one should not mistake this "numerical result" mentioned here as an approximation to an analytical result. In fact, there is no numerical result in this work. Everything has been reported in terms of theorems and they are exact result. Recently, Takashi Tamaki extended this classification of area eigenvalues in a different way. I will give it a brief review in later posts in gauge invariance.]

A kinematical entropy S is assigned to each area value, defined as the log of its degeneration. Assuming that A=Na_{min} for a large area, the formula S=A log ( g(a_{min}) ) / a_{min} is obtained.

The formula is relevant when it is applied to the entropy of a non-rotating black hole and, compared with the Bekenstein-Hawking formula, determines the value of \gamma giving \gamma= log 3 / ( \pi 2^0.5 ) .

As the author remarks, he follows here the approach to black holes used by C. Rovelli (see the bibliography), where a surface in spacetime is initially considered and the expected properties of a black hole are imposed on its entropy at the semiclassical level. Another approach to black holes in loop quantum gravity was first studied by A. Ashtekar et al. [Phys. Rev. Lett. 80 (1998), no. 5, 904--907; MR1606489 (98k:83051)]. There, a boundary condition on spacetime was given at the classical level and the resulting Hamiltonian was quantized along the lines of loop quantization, giving a completely different kinematical space. In this approach the value for the Immirzi parameter is double Ansari's value.

[ The only notice to add here is that in the last sentence of above apargraph, one should note that this comparison is done in SO(3) group representations.]

A comparison of the two approaches is given by the author himself in the paper, but the question of the right black hole theory in loop quantum gravity and the value of the Immirzi parameter seems to be still open.

[ There is a main difference between the spin network black hole strategy, the one I studied in this paper, and the other opponent strategy that is called 'quantum isolated horizon picture.'

The difference is that in the latter picture one needs to *believe* that different parts of a black hole horizon are distinguishable from one another. If this is not assumed, one does not get the entropy proportional to area, instead it will be proportional to A^0.5. This is a strange fact that honestly is true! This distinguish-ability is an additional assumption and since 1996 till now nobody showed why should we assume this when the states associated with the Chern-Simons horizon does not show that as a quantum fact to us.

I neglectdiscussing the recent claims that the state of punctures are reported to be a sequence of punctures not a set of puncture. This assumption if is true generates an orderly difference between punctures, breaks a huge number of diffeomoerphism invariant class on the horizon and make each puncture distinguishable. However, there is no physical reason why should we prefer a sequence of punctures to a set of them. All of this seems to be passing a problem from distinguished states into an underlying diffeomorphism invariance breaking.

In the spin network black hole picture, which turns out to be a more quantum approach to understand a black hole, although the Thermodynamics of black hole and their evolutions have not been developed yet, but instead this picture does not suffer from the illness the other opponent suffers from. In other words, the entropy based on the degeneracy of spin network states is really proportional to the area A, and not A^0.5.

The reason is simple. The area operator when acts on the spin network states, it is in-sensitive to the completely tangential edges. The spin network states with different completely tangential edge states may belong to the same unification class of area eigenstates. However they may at the same time belong to completely different classes of other physical operators such as energy. As a consequence, any physical object that is described only by one parameter semi-classically, manifest area unification classes as a surface with distinguishable parts. In other words, two states with the same area that reside on a black hole horizon are distinguishable from one another only because their difference in their completely tangential edges, thus under the action of other operators they become recognizable. ]