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That section does not make much sense. There is something crucial missing from the formulas, but I suspect that it masks a conceptual misapprehension. Is this saying more than "any integer <math>a</math> can be uniquely represented as <math>a=n^2\zeta,</math> where <math>\zeta</math> is square-free"? What is the mathematical statement there, and what is result of some experimental spetroscopy? Unless someone comes up with a really compelling reason, I would propose to remove (or at least move) this section from the article. [[User:Arcfrk|Arcfrk]] 07:32, 10 March 2007 (UTC)
I have moved the whacky section from the main text to here. [[User:Arcfrk|Arcfrk]] 22:28, 23 March 2007 (UTC)
:'''Application in Loop Quantum Gravity'''
In the theory of [[loop quantum gravity]] area is an observable operator. As a consequence, the area of a quantum surface is quantized. [[Abhay Ashtekar]] and his colleagues in 1996 found that three incident edges of spins ''j<sub>1</sub>'', ''j<sub>2</sub>'', and ''j<sub>3</sub>'' at a trivalent vertex generate the patch of area:
<math>a = \ell_P^2 \sqrt{2j_1(j_1+1)+2j_2(j_2+1) - j_3(j_3+1)},</math>
where <math>\ell_P</math> is the [[Planck length]].
The spectroscopy of a canonically quantized black hole showed that the area eigenvalue formula fits into the following reduced formula
<math> \forall n \in N, a = \ell_P^2 n \sqrt{2\zeta} </math>
(subject to the identification of repeated numbers) where <math>\zeta</math> is a square-free number and <math>\{ \zeta \} = </math> the set of all square-free numbers.
This helps to expect that black hole Hawking radiation is concentraited on a few lines whose energy is proportional to the square root of square-free numbers.
:'''References'''
* [http://arxiv.org/abs/hep-th/0607081 Spectroscopy of a Canonically Quantized Black Hole]
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