Research Article | Open Access
L. Marek-Crnjac, "The Hausdorff Dimension of the Penrose Universe", Physics Research International, vol. 2011, Article ID 874302, 4 pages, 2011. https://doi.org/10.1155/2011/874302
The Hausdorff Dimension of the Penrose Universe
Penrose fractal tiling is one of the simplest generic examples for a noncommutative space. In the present work, we determine the Hausdorff dimension corresponding to a four-dimensional analogue of the so-calledPenrose Universe and show how it could be used in resolving various fundamental problems in high energy physics and cosmology.
As explained in detail in Connes’ , Penrose fractal tiling constitutes mathematically a quotient space . Using this fact A. Connes following earlier work due to von Neumann deduced a dimensional function which we generalize to a simple formula function linking both the Menger-Urysohn topological dimension and the corresponding Hausdorff dimension.
The present work is subdivided into three main parts. First, we show an explicit application and generalization of the Connes’ dimensional function. Second, we derive the Hausdorff dimension of the Hilbert space which X represents. Finally, we show the relevance of these results in high energy physics and cosmology.
2. The Dimensional Function and the Hilbert Space
Let us start from the Connes’ dimensional function for the Penrose universe :
Writing and using the Fibonacci sequence, it is easy to see that, starting from the seed and , we obtain the following dimensional hierarchy: By complete induction, one finds We obtain an exceptional Fibonacci sequence : The classical Fibonacci sequence is defined by the recurrence relation where , , and . The first few Fibonacci numbers of the classical Fibonacci sequence are given .
The th Fibonacci number is given by the formula which is called the Binet form, named after Jaques Binet, where and are the solutions of the quadratic equation : The Binet form of the th Fibonacci number of the sequence can be expressed similar to the classical Fibonacci sequence: The Fibonacci sequence can be presented as an infinite geometric sequence: The Golden Section principle that connects the adjacent powers of the golden mean is seen from the infinite geometric sequence.
The formula for the th Fibonacci number is clearly identical to the bijection formula of E-infinity algebra and rings, namely [2, 3], Here, is the Menger-Urysohn topological dimension which should not be confused with the embedding dimension and is the Hausdorff dimension whose topological dimension is . To see that this extends in a simple fashion to negative dimensions , we set and find that the empty set is structured and possesses a finite Hausdorff dimension equal to because Now, we claim that is effectively a random Hilbert space and is four dimensional topologically speaking while the Hausdorff dimension is given by [5–7] The zero set is given by One finds In an analogous manner, we see that the topological dimension is given by In other words, although has a finite average topological dimension equal to 4, its formal topological dimension is infinity so that it is a real infinite-dimensional Hilbert space. Thus, the fact that does not contradict the fact that the formal dimension is because 4 is an expectation value of the topological dimension in exactly the same manner that is the expectation value of the Hausdorff dimension. This fact is easily grasped when remembering that a random Cantor set has, according to Mauldin-Williams theorem, a Hausdorff dimension equal to . Consequently, building a space from the union and the intersection of infinitely many such random Cantor sets gives us a hierarchal infinite-dimensional Cantor set with an average Hausdorff dimension given by the centre of gravity theorem [8–10]: exactly as anticipated.
In fact it is well known from the work of Jones and Sunder’s  and Connes’ that the Hilbert space is enclosed in which has a dimension related to Jones’ index of the subfactor by [1, 12] It was shown in the E-infinity theory that [1, 12] as well as This clearly shows that is a Hilbert space as claimed. This very special Hilbert space is fixed by three and not only one dimension. It is formally infinite dimensional. Secondly, it has a finite average topological dimension equal to 4. Thirdly, it has a finite average Hausdorff dimension equal to . The infinite hierarchal nature of is reflected in the continued fraction expansion :
3. Relevance in High Energy Physics
With at hand we can obtain a plethora of results in high energy physics by applying a simple intersection theory. As an example, we consider determining the value of the inverse electromagnetic fine structure constant . To do that, we generalize the two-dimensional Penrose tiling with to a holographic boundary for a compact E-infinity manifold obtained from the fuzzy version of E8E8 exceptional Lie symmetry groups for which we have  where .
Thus, intersecting with the Narain lattice of superstring compactification with its well-known 80 dimensions gives where 336 are the dimensions of Lie group and the comes from the compactification. Noting that the number of symmetries of pure Einsteinium gravity in eight dimensions is exactly equal to 20, then we can write the fundamental equation of the E-infinity theory as follows: That means Inserting and solving for , one finds which is the integer approximation of the well-known experimental value. It should be noted that the 339 is the number of gluons-like states at ultra-high energy.
The vital role of the most irrational number, namely, the golden mean in the KAM theorem, is well understood and used extensively in the theory of quantum chaos. A similar role is played by the golden mean in the VAK  as well as the theory of vague torus  which links classical chaos and quantum mechanics. In some sense, the irrationality of is a substitute for the lack of friction in quantum Hamiltonian dynamics and gives elementary particles their stability. Researchers in cosmology have suspected for a long time that the Kirkwood gaps [14, 15] in the asteroids belt are due to the resonance instability connected to the lack of irrationality in the corresponding winding numbers and thus the lack of stability of the concerned orbits. We can transfer the same cosmological argument to high energy physics and claim that the hierarchal problem is due to similar stability considerations. It is reasoned in the E-infinity theory that the and quantization observed in the mass spectrum of elementary and composite particles is due to stability of KAM [14, 15].
Penrose tiling is an incredibly rich but basically simple structure which is quantum mechanical in its very essence. It is a generic example of noncommutative geometry and the E-infinity theory. Penrose tiling may be generalized to 4 dimensions. The Hausdorff dimension in this case is and when intersected with the 80-dimensional Narain lattice one obtains the compactified Klein modular space. In all these circumstances, the golden mean which was recently discovered experimentally in a quantum system plays a central role . The golden mean is the organizing centre for the Connes’ dimensional function as well as the Jones’ index. The irrationality of this most irrational number could be drawn into explaining the stability of elementary particles and the hierarchal problem (the desert hypothesis) as well as in explaining the Kirkwood gaps in the asteroid belt.
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Copyright © 2011 L. Marek-Crnjac. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.