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Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff Dimension, Percolation, Clusters

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Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff Dimension, Percolation, Clusters Applied Mathematics 2018, 8(1): 5-8 DOI: 10.5923/j.am.20180801.02 Why Fermions and Bosons are Observable as Single Particles while Quarks are not? Sencer Taneri Private Researcher, Turkey Abstract Bosons and Fermions are observable in nature while Quarks appear only in triplets for matter particles. We find a theoretical proof for this statement in this paper by investigating 2-dim model. The occupation numbers q are calculated by a power law dependence of occupation probability and utilizing Hausdorff dimension for the infinitely small mesh in the phase space. The occupation number for Quarks are manipulated and found to be equal to approximately three as they are Parafermions. Keywords Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff dimension, Percolation, Clusters There may be particles that obey some kind of statistics, 1. Introduction generally called parastatistics. The Parastatistics proposed in 1952 by H. Green was deduced using a quantum field The essential difference in classical and quantum theory (QFT) [2, 3]. Whenever we discover a new particle, descriptions of N identical particles is in their individuality, it is almost certain that we attribute its behavior to the rather than in their indistinguishability [1]. Spin of the property that it obeys some form of parastatistics and the particle is one of its intrinsic physical quantity that is unique maximum occupation number q of a given quantum state to its individuality. The basis for how the quantum states for would be a finite number that could assume any integer N identical particles will be occupied may be taken as value as 1<q <∞ (see Table 1). But, the evident experimental, as hypothetical, or as quantum field theoretical, fundamental result is that all particles are either Fermions or but most probably suggesting a combination of the three is Bosons. It is not so clear why we call all particles as either best. Fermions or Bosons, and this situation may be discovered in this article by consideration of the quantum mechanical Table 1. The Character of Various Particles quantity spin. Occupation Particle Generic Name Spin(s) Number(q) Electron Fermion 1/2 1 2. Theory Positron Fermion 1/2 1 Particles which are described by anti-symmetric Proton Fermion 1/2 1 eigenfunctions are called Fermions, and particles which are Neutron Fermion 1/2 1 described by symmetric eigenfunctions are called Bosons in Muon Fermion 1/2 1 quantum statistics [4]. That is, the eigenfunctions for a system of several identical Fermions switch sign if the Quark Parafermion 1/2 3 labels of any two of them are exchanged, while the α particle Boson 0 ∞ eigenfunction for a system of several identical Bosons does He atom (ground state) Boson 0 ∞ not switch sign in such a particle exchange. The symmetry π meson Boson 0 ∞ and antisymmetry under the interchange of two particles is Photon Boson 1 ∞ a characteristics of the particles, and not something that can be arranged in preparation of the initial state [5-11]. The Deutron Boson 1 ∞ law, which was discovered by Pauli states that 1. Systems consisting of identical particles for * Corresponding author: half-odd-integral spin (i.e., spin 1/2, 3/2..) are [email protected] (Sencer Taneri) described by antisymmetric wave functions. Such Published online at http://journal.sapub.org/am kind of particles are called Fermions and are said to Copyright © 2018 Scientific & Academic Publishing. All Rights Reserved obey Fermi-Dirac statistics. 6 Sencer Taneri: Why Fermions and Bosons are Observable as Single Particles while Quarks are not? 2. Systems consisting of identical spin (spin 0,1,2,..) are The mesh can still be divided into infinitesimally smaller described by symmetric wave functions. Such kind of segments where matter percolates in 2-dim till it reaches the particles are called Bosons, and are said to obey exterior edges of Lx and Ly where it is called a cluster Bose-Einstein statistics. now. Let P the probability that spin S cluster with In 1964, Gell-Mann and Feynmann's PhD student George Zweig, who was working at CERN, proposed that baryons mass M occupies an area A and traces an arc of phase φ and mesons are bound states of hypothetical triplet particles. angle and radius while being at the boundaries of a Gell-Mann called the triplet particles "quarks", using a word drum to which twice infinitesimally small meshes with that had been introduced by James Joyce in his novel edges 2′ belong in space. If M <<1 and Finnegans Wake [12]. 1 Px(′′ , y )= δδ ( x ′ ) ( y ′ )~ , one can write for the In order to understand the statistics of a particle, it is M useful to assemble a collection of them. Then, one can find probability P that the smaller meshes at the origin are the maximum number that can occur in completely occupied for isotropic space, symmetric or antisymmetric states. It is known that when no SS more than q particles can exist in any completely Px(,′′ y )=zz = = Px()(′′ Py ) (1) antisymmetric (symmetric) state, the particles are called 22πMN πα Parafermions (Parabosons) of order q. Ohnuki e Kamefuchi where Sz is the magnitude of z component of spin analyzed the possibility of quarks be Parafermions, as proposed by Greenberg in 1964 [13, 14]. The conclusion vector S to take spin degeneracy into account and N is was that the quarks cinematic properties obeying the number of occupied sites. Further by mass action law in parafermionic statistics can not explain the quark d percolation theory, MN=αα(2 ) = is valid for big confinement by themselves and that the para-quark model is compatible with the SU(3)c color gauge. Here, we set up a circle composed of smaller sites in 2-dim. Here, α is theoretical construction in the next section to observe the some proportionality constant and d is dimension of the parastatistics of matter particles. cluster [15]. Percolation theory forms the basis of the liquid dynamics of the lepton liquid problem for electron. Returning to equation (1) for ()xy′′= , we set the 3. Model probabilities, S Px()′′= Py () = z . (2) (2πα )1/2 N Then, one can write the probability for the small mesh, SS P= P()( x′ P y ′′ ) dN = zzdN′= . ∫∫(2πα )1/2 N (2 πα )1/2 (3) 1/2 1 If P =1 then Sz = (2πα ) where α = is 22π the concentration density of the lepton liquid in the electron with Sz =1/2. Conclusively, equation (2) becomes − ′ SNq 1 S 1 d Px()′′= Py () = zz= Nd′ (4) Figure 1. Infinitesimally small mesh in phase space with edges ∆x (2πα )1/2 N (2 πα )1/2 and ∆y yet divided into twice infinitesimally smaller meshes. and = ′ ′ are radii of the circles while φ phase angle is same for both radii. where qd2/ is the occupation number equal to 1 for the electron and d' is dimension equal to 2 as well. Further, S whose z component S is perpendicular to both of ∆x and z mass M equals N / (2 2π ) , number of occupied sites ∆y , is the spin vector of the cluster N ()M = . The spectral probability P' (spectral density, We see infinitesimally small mesh in space with edges 22π having uncertainty in position ∆=xLx and position with exponential time dependence (traveling wave solution)) is the Gamma function integral [16] which yields following ∆=yLy in Fig. 1. Let this small mesh contain an electron with convolution of Faltung theorem [17] for very large N, cluster rotating with spin S and occupation number q . N' ( NN, ′ →∞), Applied Mathematics 2018, 8(1): 5-8 7 P′= ∫∫( eiM ′τ P ()( x ′′ P y ) dτ dM ′ ) as the Hausdorff dimension for clusters. Note that we have two states of elements, P+ and P− . If this dimension is −−′′ S 11dd (5) = ∫∫( z Ndd′′ N′′ d(αα Nd ) ( N )) equal to the dimension d in equation (7), one can obtain, 2πα P logP+−(1 P ) log (1 − P ) d = − ++ + + (9) log2 where 0<<NN ,′′ 22ππ M ,22 M . This is nothing but, where, 2 ssφφ′′1/dd1/ Sz 2 d′ P′′= dN P+ = − int( ). (10) 4(2πα )3/2 (1− d′ ) 2 ππ (6) 4SS11 So, given the spin of the cluster s, we have two = zzdN2 dd= φ′ unknowns, the phase φ′ and dimension d. 4(2πα )3/2 (1− 2d ) 2 π ′ ′ where dd= ′ /2 and N = φ where φ is the phase 4. Results and Discussion angle, are purely empirical in Fig. 1. Here, electrons are Fermions with d′ = 2 and d =1 , and thus with d 2 =1 Our calculation here reminds techniques of self consistent solution of the method above. Equation set (9-10) and occupation number q =1. This yields a number for P' have self consistent trivial solutions; which may be larger than one as the numerator is some number on the Riemann surface. We make the φ′ =0,d = 0, P+ = 0, sn = (11) transformation P′′= P − int() P ′ with int() P′ , integer ′ 11 ′ φπ=,d = 1, P+ = , sn = + (12) value of P' in order to have 01<<P and to have the 22 numerator equals some number between 0 and 2π with for spin s and integer n = 0,1,2,... The occupation branch cut at 2π . So, number mentioned in Theory section may be taken as SSφφ′′1/dd1/ qd=1/ and the results may be summarized as, P′ =zz − int() (7) ππ φ′ =0,q =∞= , s n ( Boson ) (13) is the final form of the formula for the spectral probability 1 for the infinitesimally small mesh with sides ∆x and ∆y , φπ′ =,q = 1, s = n + ( Fermion ). (14) see Fig.
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