There is a large body of literature on the application of fractals and deterministic chaos in quantum physics [1-69]. In particular, the present authors integrated Cantor sets and fractals as well as number theory into the foundation of quantum mechanics [5-22]. Modern transfinite set theory [6] has unraveled many undreamed of logical possibilities such as hierarchies of things and “numbers” larger than infinity [25] and smaller than zero in addition to non integer and even negative dimensions [1,2,4,5,7,8,10]. Some of these novel and particularly revolutionary mathematical notions have sometimes ago found their way into theoretical physics [1-16]. This is definitely the case with fractal-Hausdorff dimensions [11, 14,15] which led to the theory of fractal space time pioneered by G. Ord [7]. Other even more demanding concepts such as the empty set and negative topological dimensions form the basis of El Naschie’s Cantorian E-Infinity theory [5,10,31,32]. The present note continues this line of thinking by pointing out the urgent need for a new theory of general fractal logic in high energy quantum physics when counting and discovering new particles which is an essential part of the trade. The idea is related but not identical to Zadeh’s theory of fuzzy logic [33-38]. Our present main thrust is to reason that although counting which is at the basis of statistical mechanics may seem like the simplest thing in the universe, high energy physics calls for another form of fractal transfinite counting which is anything but straightforward in the traditional sense [5-10]. As a clear cut example of profound consequence we show that while the number of gauge bosons of the standard model is indeed 12 in the ordinary sense of counting using integer numbers [55-57], these 12 corresponding to only 11.7082394 bosons which have in turn the fractal weight of 14 rather than 12 gauge bosons [38-54]. In this sense we are asserting what on its face value is an absurdity namely the equality of 12 to 14 via 11.708294. This indicates that 2 more elementary particles are hidden allowing 14 particles to appear as if they were 12. We conjecture that these two extra particles may be the Higgs and the graviton. It is further concluded that this fractal nature which is inert even to the mere number of elementary quantum particles is in one way or another behind the enormous difficulties of discovering the Higgs and the graviton experimentally [55-57]. This is more than sufficient reason to call for the introduction of a new theory for fractal logic.

In order to be able to analytically observe the transfinite irrational fingerprint of the golden mean entanglement [65-67], we must fulfill the following conditions:

a) First we must have an accurate mathematical model which can capture a substantial and relevant part of Reality.

b) Second we must solve the model exactly.

If we do not observe both of the above conditions, the golden mean fine structure, i.e. the fractal Cantorian structure disappears theoretically and would unlikely be observed experimentally [52].

Adding or subtracting small quantities with the objecttive of simplifying the corresponding field equation is a method which goes back to the work of F. John on the derivation of the equations of thin elastic shells using Taylor expansion from the 3D field equations of the classical theory of Elasticity [63]. Interestingly John based his method on the work of Radon and his theory of operators. In turn Radon’s work was extended by Von Neumann and later on by A. Connes in his theory of non-commutative geometry [10-63]. Finally all of this is strongly related to El Naschie’s E-Infinity theory [10,63]. It is thus understandable that E-Infinity extended and made extensive use of F. John’s method to make it applicable to quantum high energy physics [52,59]. The astonishing point is that the so-called transfinite correction corresponding to F. John’s method is almost entirely reducible to Hardy’s generic value f⁵ = 0.09016994393 which is the quantum probability of Hardy’s entanglement [30] where      

is the inverse of the golden mean. For instance is simply the prime number 137 plus a function of f⁵ namely k₀ = f⁵(1 – f⁵). We may thus write that [10,31,32]

where is the inverse electromagnetic fine structure constant [56,58].

Now Measure Theory [29] gives us the tools to estimate the quantities needed to make things and equations “fit” harmonically together. In a manner of speaking transfinite corrections a la F. John and El Naschie is developed to make the rough or sharp edges of the equation and their geometrical shapes fit together like in Penrose’s golden mean fractal tiling [1] or a bathroom floor where transfinite “Mortel” fill the gaps between the edges the tiling to make for a smooth tiling similar to what we have in “Al Hambra” Islamic wallpaper groups, the 2 and 3 Stein spaces and the compactified Kleinian curve SL(2, 7) used for the Holographic boundary of E-Infinity [1-73]. Meantime it is well understood that this Holographic boundary manifold [2] is related to the bulk of E₈E₈ exceptional symmetry group via the fundamental equation B = H + G + E where B is the Bulk, H is the Holographic boundary, G is Einstein gravity and E is Electromagnetism. Since H = 339, G = 20 and E = =137 one finds the integer value of B to be B = 496 = E₈E₈ exactly as should be [20,63]. However the exact transfinite-corrected value of B is found directly from the corresponding transfinite value of = 137 + k₀ by the exact golden mean scaling which replaces the classical approximate logarithmic scaling of conventional quantum field theory [62]. The golden mean scaling exponent in this case is λ = 3 + f = 3.618033989 so that the exact B is found to be [10,63].

where k = 2f⁵ = f³(1 – f³) and f⁵ is the probability of Hardy’s quantum entanglement [30].

The preceding result may be interpreted in terms of quasi particles as follows: We have an expectation average KAON [10] with mass equal 496 GeV meeting a virtual particle with negative mass equal –k² = –0.032522475 GeV and producing 496 – k² which in this unit-less form is the quasi dimension of the bulk. A similar interpretation may be given to = 137 + k₀ [10,63].

We may also note a very nice symmetry or rather anti-symmetry between the transfinite correction of E₈E₈ and that of and.

This is obvious from contrasting k and k₀ and remembering that, we subtract k² from the exceptional 496 symmetry group dimension while we add k₀ to the 137 of alfa:

and

We recall that f³ is the contra factual or global probability of quantum entanglement [30-32] which together with the local two-particle quantum entanglement f² gives us the Hardy generic value of the probability of quantum entanglement namely f⁵ = (f³)(f²). For readings on the method of adding and subtracting small quantities to smooth the given equation of classical field theories, we refer to the work of F. John and M. S. El Naschie [10,60-63].

The connection between “fractal” sphere packing and E-Infinity theory was considered in detail by L. MarekCrnjac and El Naschie [2,10]. Here we recall some remarkable observation which serves to deepen our understanding of transfinite golden mean corrections [30-32].

a)    First we note that the density of sphere packing in four dimensions is Δ = 0.61685. This is very close to the golden mean f = 0.618033989.

b)    In the case of ten dimensions i.e. the dimensionality of superstring theory, the lattice sphere packing has a density of Δ = 0.09202 ≈ 9%.

This is rather close to Hardy’s quantum entanglement probability.

c)    Curiously, at this ten (10) dimensionality the central density is 0.03608. This is very close indeed to k₀(exp) = 0.0359 which appears in the experimental determination of

(4)

We note that the theoretical value of k₀ is k₀ = f⁵(1 – f⁵) where f⁵ is Hardy’s quantum entanglement probability [30].

The classical standard model which is at the core of conventional quantum field theory relies and stems from 12 massless gauge bosons described by a 12 dimensional combined Lie symmetry group SU(3)SU(2)U(1). Apart from superstring theory which proceeds in an entirely different way with 496 massless bosons corresponding to the generator of E₈E₈ exceptional Lie group, there is a more conventional approach which is based on an eleven dimensional space-time similar to the acclaimed eleven dimensional M-theory of E. Witten [49,50]. This theory is known as super gravity and relies neither on E₈E₈ nor on SU(3)SU(2)SU(1) but rather on a less known group with 14 dimensions [55]. This is the ortho-symplectic group OSP(m/n) where [55,56]

where

exactly as anticipated [55].

In what follows we will attempt to reason that the 12 bosons of the standard model are in reality 14 by virtue of a fractal number

which give a real fractal weight to 12 bosons and make them equivalent to 14 in this fractal counting of fractal elementary particles using what we may call fractal logic. This is a notion related but not identical to fuzzy logic as mentioned earlier on [33,38].

Our next and most important novel task is to devise a fractal number to massless gauge bosons of the standard model. There is no automatic mechanistic way to do that at present but we are guided by intelligent guess work and the overriding principle of a harmonic number theoretical symphony which fits everything with everything else seamlessly [9,65]. This principle served us well in the past and as we will see shortly will lead us to also here astonishing undreamed of possibilities coupled with the utmost simplicity [36-53].

The first and may be the most important messenger particle of them all, the illusive photon (γ) will be given the fractal number weight not 1 but f. This f we recall is at the same time the Hausdorff dimension of a quantum particle whose cobordism [20,30] namely the quantum wave was given the Hausdorff dimension f² [20]. We also recall that f corresponds to the zero set while f² corresponds to the classical empty set with the MengerUhryson topological dimensions zero and minus 1 respectively [10,61].

Second the three bosons of the electro weak are given a priori an easily justifiable weighted number namely 16k = 32f⁵. Third the 8 gluons are given the fractal weight number 8 + () = 8 + f⁵. The total is now 11.5936411 weighed fractal particles corresponding to 12 particles. To this we add two new particles which we conjecture that they could be one Higgs and a graviton. The fractal number of the Higgs is k₀ = f⁵(1 – f⁵) while that of the graviton is k² = [f³(1 – f³)]². The total is now 11.70820393 which happen to be where = 137 + k₀ is the E-Infinity value of the inverse electromagnetic fine structure constant [10,63].

The massless bosons correspond therefore to 14 particles when counting in the ordinary way using integers. Consequently they could be represented by the OSP(1/4) symmetry group of super gravity [55] and it is granted that this astonishing unexpected fact gives a feel similar to that of Alice in the Fractal Land of high-energy quantum physics [10,30-32]. The preceding result may be summarized in the following table:

The discovery of the logarithmic scaling of high energy physics was a major step forward for establishing quantum field theory [56,57]. Nonetheless logarithmic scaling as will be shown here is merely an excellent approximation of the exact golden mean scaling of E-Infinity golden field theory. As excellent as it is, this approximation masks Hardy’s golden mean quantum entanglement and makes algebraic manipulation of the governing equations unnecessarily heavy if not clumsy and sometime interactable and messy [55-57]. As a simple demonstration testifying for the above, let us consider the logarithmic scaling of the inverse fine structure constant of a Cooper Pair. This is clearly

[58-64]. By comparison the golden mean scaling for namely leads to

which is equal to the exact Hausdorff dimension of the core of E-Infinity Cantorian space-time.

The above result must make us wonder why not reconsider the famous renormalization equation of quantum gravity unification using golden mean scaling. Let us write the concerned renormalization equation in the logarithmic scaling form first. This is [61,62]

Let us replace in this equation the logarithmic scaling term by the golden mean scaling as follows [60-62]

Then insert all the exact E-Infinity value of = 9 and                 = 1,

and = 137.082039 in the renormalization equation of ordinary quantum field theory. That way we find [60-64]

where k = f³(1 – f³).

This is the well known exact E-Infinity result [65,67].

To ascertain the high accuracy of the classical theory based on logarithmic scaling compared to our exact solution, we repeat the analysis and replace by what we have derived at the very beginning namely

and find that [61-63]

This is pretty close to the exact result of E-Infinity theory namely 26.18033989 [10,43,63]. Of course the above equation could have been performed without using nor f². We could have used the conventional purely logarithmic running of coupling by taking for instance M_u to be that of grand unification i.e. M_u(10)¹⁶ GeV while taking M_x = M_z = 91 GeV. This leads to [61-63]

For non-super symmetric and super symmetric unification respectively, we found the values = 42.33050198 and = 26.16525099. Both values are very close to the exact results [48].

Even today some would dispute that Fuzzy logic is a new theory [35]. We beg to differ and insist that Fuzzy logic [33-38] was and is still a radically novel theory with far reaching consequences which has played a significant role not only in running production lines of modern care industries but also in furthering and deepening understanding of fundamental theories such as quantum physics and deterministic chaos [61,62]. In what follows we list a large number of points and observations which when taken together indicate in our opinion that we are on the verge of discovering even more deep logical theories than Fuzzy Logic [33-38] which we have dubbed for the time being Fractal logic for want of a better name. Without much ado here are the points which we gathered for the purpose of the present article.

The key to understanding the superficial inconsistency coupled with undreamed of possibilities which fractal logic and high energy particle physics offers is to understand the dual nature of the geometry and topology of random Cantor sets and its golden mean Hausdorff dimension [63].

Let us start by asking why random Cantor sets? The answer may be found in the following:

1)    It has a zero measure i.e. a zero length as well as a zero topological dimension and yet it has a non-zero Hausdorff-fractal dimension. In a sense it is both there and not there at the same time [10,63].

2)    A Cantor set has the cardinality of the continuum yet it is a totally disjointed discretuum [10,63].

3)    It possesses the golden mean Hausdorff dimension which is the epitome of regularity and harmonic order, yet it is random [10,63].

4)    Although its Lebesque measure is zero, its Hausdorff measure is 1 so that with certain fine print one could write for it an absurd looking equation namely that zero = one [31,32].

5)    The geometry of a Cantor set is the reason behind quantum entanglement [30].

The next question to ask when seeking a deeper understanding of fractal logic is why the golden mean? This may be considered as follows:

1)    It is the real solution for the simplest quadratic equation.

2)    It is the most irrational number and thus belongs to most stable periodic orbit as shown in KAM theorems of non linear dynamics [45,62].

3)    It preserves the simplicity of fractal tiling and symmetry groups such Penrose tiling [1].

4)    It is the Eigen frequency of the simplest two degrees of freedom mechanical oscillators [10,18].

5)    It has the simplest continued fraction and continued square root expansion representation [10,62,63].

6)    It is found abundantly in philo texts of plants as well as in art, paintings and music being a bridge between art and science [9,10].

7)    It is the raison d’etre for Hardy’s quantum entanglement.

8)    Golden mean arithmetic and number system are a naturally quantized calculus with transition to zero at infinity [31,32].

For all of the above reasons and the other fundamental findings presented in the main body of this paper, it seems that fractal logic is unavoidable and in time it will become the standard tool of theoretical scientists who work in high energy particle physics [65]. We have reached a sophisticated level of our mathematical civilization moving from performing computations and counting using numbers to performing computations and counting using fractals and Cantor sets [65-69].