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149 2 2 2 2 , 22222 sin 1 sin sin 1 !! zyx LLLL (36) and commutes with all three components of L: 0, 2 j LL (37) These commutation rules tell us, that the magnitude of the angular momentum can always be exactly determined, but only one of its components can be measured at the same time with arbitrary accuracy. The other two remain indeterminable. Usually, L z is taken as the measurable component. For a stationary state, the part of the wave function L which describes the angular momentum has to be an eigenfunction of the angular momentum operators, thus it has to satisfy the two equations: LL L 22 L (38) and LzLz L L (39) where L 2 and L z represents the associated eigenvalues. The spherical harmonics l,m (): imP ml lml Y m l ml expcos )!(4 )12()!( ),( , (40) with the associated Legendre polynomials satisfy the relations: ),()1(),( , 2 , 2 mlml YllY !L (41) ),(),( ,,z mlml YmY !L (42) yielding the eigenvalues 22 )1( ! llL and Lm z ! (43) for L 2 and L z . The magnitude of the angular momentum )1( llL ! (44) as well as its z-component L z are quantised with l and m as quantum numbers. The latter can be changed applying the ladder operators SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS 150 yx iLLL (45) to create (+) or annihilate () a magnetic quantum according to LLz m LLL 1! (46) LL ll LLL 1 22 ! (47) Since L x and L y are both Hermitian (which implies ) and |L lm |2> 0, we have the well known restrictions l < m < l between the so-called angular l and magnetic m quantum numbers. Since the discovery of the electron spin, 1925, this quantum state keeps some mystery, since it cannot be described like an angular momentum, but shows experimental evidences to be something like an angular momentum due to the facts that spin acts experimentally like something rotating in space having an intrinsic couples with the normal orbital angular momentum to the resulting total Since all fundamental particles have spin 1/2, this quantum phenomenon is very important. The usual description of spin 1/2 is based on the Pauli matrices, avoiding any space-like imagination. People who are used to work with freedom. Indeed, this can easily be done. Let us look on the spherical harmonics Y l,m (,) given in (40) and the associated Legendre polynomials )( d d 1 2 2 xP x xxP l m m m m l (48) with 4.2 Spin-1/2-particles in fractional description angular momentum has the physical dimension of an angular momentum angular momentum like an angular momentum is conserved as part of angular momentum exhibits a magnetic moment expected from circulating currents and some other things remembering on angular momentum fractional order calculus will ask why physicists do not try to describe the spin similar to the well-known angular momentum, but only with a further degree of Krempl but do not change the magnitude: 151 l l l l l x xl xP 1 d d !2 1 2 (49) derivatives in the expressions: )( d d 1 2/1 2/1 2/1 4 1 22/1 2/1 xP x xxP (50) 1 d d2 2 2/1 2/1 2/1 x x xP (51) We can merge these two equations to: 2/1 2 2/12/1 2/12/1 4 1 22/1 2/1 1 d d 1 2 x x xxP (52) and need only to assume the existence of the fractional order derivatives, but the two independent solutions (+ or ): 2 expsincot 1 ),( ~ 2/1,2/12/1 iiY (53) satisfying the operator equations: 2/1 2 2/1, 2 2/1 2 ~ )1 2 1 ( 2 1~~ !!S (54) 2/12/12/1 ~ 2 ~~ ! ! i z S (55) for the spin operator S which is in complete analogy to the orbital angular momentum L. It might be noticed that usage of the generalised Legendre polynomials will also yield such solutions. The imaginary factor ( i) is due to the selected branches of the roots and can be omitted, because if is an eigenfunction, then c with arbitrary constant c is also an eigenfunction of the differential operator. However, the physical interpretation of the wave function asks for a normalisation. Thus, the integral of its norm |* | over the whole space has to to look on solution (53). Omitting for simplicity the unitary factor ( i) we have and extend them to fractional order l = 1/2, m = 1/2 i.e., to evaluate the semi- fortunately, do not need to evaluate them by fractional calculus. Thus we obtain be 1. But what is the “whole” space in our case ? To answer this question we have SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS 152 differ monumentally from their analogues in the orbital angular momentum functions. We immediately see, that there is a periodicity of 4 in the phase . This means that a spin-1/2-particle has to turn twice in the space to return in its initial state. This has intrigued physicists since the first experimental evidences of this phenomenon [12,13]. The interpretation of these experiments was taken into doubt by many physicists, but recently confirmed again with modified experiments avoiding the reasons for criticism [14]. Today, this 4-periodicity of spin-1/2-particles becomes generally accepted. If any theorist would have introduced the fractional description of the spin just after his discovery in the twenties of the last century, his model would have been made ridiculous due to wave function is the probability for the particle to occupy the volume element of the space, this space has to contain all possible configurations. This means that 3 integral for from 0 to 4. This means that our spin wave functions S for spin 1/2 have to look like: 2 expsincot 2 1 2/1 i (56) These functions form an orthonormal basis for all spin wave functions over the 3 2. The complete wave function of spin-1/2-particles is the product of the wave function ( r) describing the location of the particle in the 3 , and the spin wave function S over the spin space 3 2 S )(r (57) The ladder operators S (a) 4 Periodicity this 4-periodicity. Perhaps, this was one of the reasons, why one did not believe in a description similar to the orbital angular momentum. Nowadays, “dynamical picture. phases” or similar concepts [15] are proposed to save the “classical” spinor 2. We have to extend the Krempl Wave function of spin-1/2-particles Let us now return to our question about the “whole space”. Since the norm of the twice our civilian space, i.e., the “whole” space is (c) Ladder operators the phases /2, which do not affect the magnitude of these functions, but which (b) 153 yx iSSS (58) create (+) or annihilate () a magnetic spin quantum according to SsSz m SSS 1! (59) but maintains the magnitude s of the spin SS ss SS 1 22 !S (60) which can be proved analogue to the proof for L previously given. The nice aspect of the fractional description of spin-1/2-particles is beside the direct evidence of their 4-periodicity the possibility of an interpretation in the exponential term, which is also the sole complex part of the spin wave of a particle with the mass m P is proportional to the gradient of the phase of its wave function: P m ! v (61) }2/1,0,0{ sin P rm ! v (62) which tells us that we have a rotation around the chosen z-axes. Spin +1/2 LzLz L L (63) Since this fractional description of spin-1/2-particles allows its interpretation proven if a rigorous application of this fractional description of the spin can yield all the other observable results like the standard spinor description. 5 Conclusions (d) Interpretation of the spin space. In this picture, the spin-up and spin-down states differ only by the sign of function yielding the phase = /2. In quantum mechanics, the “mean velocity” Applying this to the spin wave functions (56) we get in polar coordinates: corresponds to a right-handed rotation, spin 1/2 to a left-handed rotation. in the real space, it is complementary to Pauli’s spinor picture. It has to be SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS These examples show that semi-integrals and semi-derivatives are appropriate to describe natural phenomena. Today, the application of semi-integrals in 154 connection with Abel-type integral equations pervades all natural and technical sciences, as well as modern medicine. The fractional description of spin-1/2- particles can perhaps contribute to enlighten the mystery of the spin. Krempl References 1. Oldham KB, Spanier J (1974) The Fractional Calculus. Academic Press, New York. 2. Abel NH (1823) Solution de quelques problèmes à l'aide d'intégrales définies. Mag. Naturvidenskaberne. 3. Abel NH (1826) Auflösung einer mechanischen Aufgabe. J. für die Reine Angew. Math, 1:153–157. 4. Laurent MH (1884) Sur le calcul des dérivées à indices quelconques. Nouv. Ann. Math. 3(3):240–252. 5. Krempl PW (1974) The Abel-type integral transformation with the kernel (t 2 - x 2 ) −1/2 and its application to density distributions of particle beams. CERN MPS/Int.BR/74-1, pp. 1–31. 6. Krempl PW (2005) Some Applications of Semi-Derivatives and Semi- Integrals in Physics. Proc. ENOC 05, Eindhoven, ID 11-363, 10 pp. 7. Deans SR (1996) Radon and Abel Transforms. in Poularikas AD (ed.), The Transforms and Applications Handbook. CRC Press, Boca Raton, pp. 631– 717. 8. Yuan Z-G (2003) The Filtered Abel Transform and its Application in Combustion Diagnostics, NASA/CR-2003-212121, pp 1–11. 9. Bruck H (1966) Accélérateurs circulaires de particules. Press Universitaires de France, Paris. 10. Krempl PW (1974) TMIBS–un programme pour le calcul de la densité projetée à partir des mesures effectuées avec les cibles. CERN MPS/BR Note 74-16, pp. 1–9. 11. Krempl PW (1975) Beamscope. CERN PSB/Machine Experiment News 126b. 12. Rauch H, Zeilinger A, Badurek G, Wilfing A (1975) Verification of coherent spinor rotation of fermions. Phys. Lett., 54A(6):425–427. 13. Werner SA, Colella R, Overhauser AW, Eagen CF (1975). Observation of the phase shift of a neutron due to precession in a magnetic field. Phys. Rev. Lett., 35(16):1053–1055. 14. Ioffe A, Mezei F (2001) 4π-symmetry of the neutron wave function under space rotation, Physica B, 297:303–306. 15. Hasegawa Y, Badurek G (1999) Noncommuting spinor rotation due to balaced geometrical and dynamical phases. Phys. Rev. A, 59(3):4614–4622. Part 2 Classical Mechanics and Particle Physics Raoul R. Nigmatullin 1 and Juan J. Trujillo 2 1 2 Abstract averaged collective motion in the mesoscale region. In other words, it means that after a proper statistical average the microscopic dynamics is converted into a relaxation that is widely used for description of relaxation phenomena in disordered media. It is shown that the generalized stretched-exponential function describes the integer integral and derivatives with real and complex exponents and their possible generalizations can be applicable for description of different relaxation or diffusion processes in the intermediate (mesoscale) region. Key words VERSUS A RIEMANN–LIOUVILLE INTEGRAL TYPE MESOSCOPIC FRACTIONAL KINETIC EQUATIONS Tenerife. Spain; E-mail: JTrujill@ull.es Kazan, Tatarstan, Russian Federation; E-mail: nigmat@knet.ru in the most cases the original of the memory function recovers the Riemann– fractal-branched processes one can derive the stretched exponential law of relaxation phenomena is also discussed. These kinetic equations containing non Generalized Riemann–Liouville fractional integral, universal decoupling procedure. Theoretical Physics Department, Kazan State University, Kremlevskaya 15, 420008, Departamento de Análisis Matemático, University of La Laguna, 38271, La Laguna. It is proved that kinetic equations containing noninteger integrals and deriva- tives are appeared in the result of reduction of a set of micromotions to some collective complex dynamics in the mesoscopic regime. A fractal medium con- taining strongly correlated relaxation units has been considered. It is shown that Liouville fractional integral. For a strongly correlated fractal medium a genera- lization of the Riemann–Liouville fractional integral is obtained. For the averaged collective motion in the fractal-branched complex systems. The appli- cation of the fractional kinetic equations for description of the dielectric © 2007 Springer. 155 in Physics and Engineering, 155 – 167. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications 156 1 Introduction integration/differentiation operators based on the given structure of a disordered mechanics is absent. So, there is a barest necessity to derive kinetic equations with the statistical mechanics, based on the consideration of an infinite chain of equations for a set of correlation functions. It becomes evident that equations with fractional derivatives can play a crucial role in description of kinetic and transfer phenomena in the mesoscale region. From our point of view this necessary fractional calculus. In present time the interest in application of the mathematical apparatus of the fractional calculus in different branches of techniques and natural sciences is considerably increased. Here one can remind the applications of the fractional calculus in constitutive relations and other properties of various engineering materials such as viscoelastic polymers, foam, gel, and animal tissues, and their engineering and Detailed references can be found in the recent review, in the proceedings of the The first attempt to understand the result of averaging of a smooth function over the given fractal (Cantor) set has been undertaken in [15]. In the note and later in paper some doubts were raised to the reliability of the previously obtained result this paper (RRN) to reconsider the former result, and the detailed study of this problem showed that the doubts had some grounds and were directly linked with the relatively delicate procedure of averaging a smooth function over fractal sets, in particular, on Cantor set and its generalizations. integer operators with real fractional exponent [1–7]. But in papers related to Recently much attention has been paid to existence of equations containing non integration or differentiation are realized on an “intuitive” level in the form of some medium with the usage of the modern methods of nonequilibrium statistical noninteger operators of differentiation and integration from the first principles of mathematical instrument should lie in deep understating of the “physics” of the 1. Fractional control of engineering systems. dynamic systems. 3. Analytical and numerical tools and techniques. scientific applications. measurements and verifications. 6. Bioengineering and biomedical applications conference and in papers [2, 4, 8–14]. [15–17]. The criticism expr essed in these publications forced one of the authors of consideration of the fractional equations containing noninteger operators of postulates/suppositions imposed on a structure or model considered. At the pre- sent time a systematic deduction of kinetic equations containing noninteger Nigmatullin and Trujillo 2. Advancement of calculus of variations and optimal control to fractional 4. Fundamental explorations of the mechanical, electrical, and thermal 5. Fundamental understanding of wave and diffusion phenomenon, their [...]... Nigmatullin RR (1984) Phys Stat Sol (b) 123:739 Nigmatullin RR, Tayurski DA (1991) Physica A, 1 75: 2 75 Nigmatullin RR (20 05) Physica B.: Phys Condens Matter, 358 :201 Nigmatullin RR, Osokin SI (2003) J Signal Proc., 83:2433 Nigmatullin RR, Osokin SI, Smith G (2003) J Phys D: Appl Phys, 36:2281 Nigmatullin RR, Osokin SI, Smith G (2003) Phys C.: Condens Matter 15: 3481 Oldham K, Spanier J (1974) The Fractional Calculus. .. obtain the macroscopic dynamics of the CTRW in an infinite medium, the method [9, 19] consists in transforming the Kolmogorov–Feller chain equation into the Fourier–Laplace domains and in taking the appropiate asymptotic limits In the latter case, the following space -fractional equation is obtained: t C(x t) K x C(x t) (8) ENHANCED TRACER DIFFUSION IN POROUS MEDIA In the latter equation, is the symmetric... imposed In particular, we study the case of a reflective barrier constraining the diffusing particles to a semi-infinite domain We obtain a modified kernel for the Riesz–Feller derivative with respect to the corresponding operator in an infinite medium Key words Superdiffusion, Space fractional equation, Reflective boundary, Lévy walks 1 Introduction Fick’s law is extensively used as a model for describing... conditions [22] From the geometrical point of view the temporal fractional integral is associated with Cantor set or its generalizations, occupying an intermediate position between the classical Euclidean point and continuous line But the meaning of fractional integral with real fractional exponent is not complete in the light of papers [8, 23–26], where the correct understanding of different self-similar objects... containing a memory function In section 3 we derive the memory function for a strongly correlated fractal medium In this section we show also how it is possible to generalize the Riemann– Liouville integral The general solutions containing log-periodic function help to understand the geometrical/physical meaning of noninteger operator containing an imaginary part of the complex fractional exponent 158 ... (t 1 t) (x x ) 2 (t t ) (5) Just as in case (i), we incorporate the boundary condition at x = 0 When advection is restricted to jumps, or more generally to time intervals of mean a containing all the jump durations, travelling from x to x implies jumping from x to x v a with respect to a frame moving only during the time intervals involving the jumps If we disregard possible interactions between the... enter into the corresponding kinetic equation merits a special examination 166 Nigmatullin and Trujillo References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Hilfer R (ed.) (2000) Applications of Fractional Calculus in Physics World Scientific, Singapore Zaslavsky GM (2002) Phys Rep., 371:461 Kupferman R (2002) J Stat Phys., 114:291 Nigmatullin RR, Le Mehaute A (20 05) ... equation, which involves a modified non-local Riesz–Feller derivative, whose kernel incorporates the boundary condition We further find a numerical solution of the obtained fractional model by modifying the numerical scheme that discretizes the spacefractional equation in an infinite domain [14] 2 A CTRW Model with an Impermeable Boundary We consider particles performing a CTRW with independent jump... effective all the time In this case, being in x without having performed any jump now means having been advected from x vt On the infinite line, particles which are in x at instant t either were subjected to jumps and advection, or were advected without any jump Hence in Fourier–Laplace coordinates [17] we have ˆ C (k u ) ˆ 2 (0 u ivk ) ivk )(1 ˆ 2 ( k u ivk )) 1 (u (26) which in physical variables... ( 25) Generally speaking, the influence of the reflective barrier is visible between the wall and the places, where solute was initially injected, as shown in Figs 4 and 5 When v is increased, the influence becomes smaller Fig 4 Solutions to the advective fractional equation ( 1 .5, v 1 ) with a reflective barrier at x = 0 (left) and without a border (right) Initial condition: a Dirac pulse at x = 5 . 2007 Springer. 155 in Physics and Engineering, 155 – 167. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications 156 1 Introduction integration/differentiation. generalizations, occupying an intermediate position between the classical Euclidean point and continuous line. But the meaning of fractional integral with real fractional exponent is not complete in the light. facts that spin acts experimentally like something rotating in space having an intrinsic couples with the normal orbital angular momentum to the resulting total Since all fundamental
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