Complexity in Field Theory, Particle Physics and the Fractional Calculus
J. Weberszpil∗
UFRRJ- Inst. Multidisciplinar/DTL-Rio de Janeiro Abstract
Within the realm of complexity we analyze aspects of fractional eld theory and particle physics, justifying the uses of fractional calculus (FC) as fundamental tool. It is argued that adequate mod-eling of TeV physics demands the tools of fractal operators and fractional calculus. The non-local theories and memory eects are connected to complexity and the FC. Some realizations of theories with fractional dynamics or fractional geometric characteristics, like fractional electrodynamics, fractional quantum mechanics, fractional relativity, fractional dierential geometry, fractional grav-itational led theory and UN-particle physics are commented. The non dierentiable nature of the microscopic dynamics is evidenced and connected with time scales. A Modi ed Riemann-Liouvile approach are applied to describe a fractional wave equation with diferent fractional orders for space and time fractional partial derivatives.
I. INTRODUCTION
A complex system usually consists of a large number of simple members, elements or agents, which interact with one another and with the environment, and which have the potential to generate qualitatively NEW collective behav-ior, the manifestations of this behavior being the spontaneous creation of new spatial, temporal, or functional structures. A complex system is an `open' sys-tem involving `nonlinear interactions' among its subunits which can exhibit, under certain conditions, a marked degree of coherent or ordered behavior ex-tending well beyond the scale or range of the individual subunits. As said P. W. Anderson More Is Dierent [1] referring to the behavior of large and complex aggregates of elementary particles, that it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear, and the understand-ing of the new behaviors requires research which, in Andersons's think, is as fundamental in its nature as any other. the reductionist hypothesis does not by any means imply a "constructionist" one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe. The "reductionism" is the idea that macroscopic phenomena can be explained in terms of microscopic entities and/or events, but the speci c meaning of the term depends upon context and the conceptual identi cation within a particular science of levels of understanding. In physics and chemistry, the term "reductionism" may be applied to attempts to explain the macroscopic behavior of physical or chemical systems in terms of events at the level of atomic phenomena. Also in physics, the term "reductionism" may
be applied to attempts to explain both the macroscopic behavior of a physical system and/or the microscopic atomic behavior of the entities of the system in terms of events at the still more microscopic level of fundamental particles and fundamental forces.
Within the realm of complexity, new questions in fundamental physics have been raised, which cannot be formulated adequately using traditional methods. Consequently a new research area has emerged, which allows for new insights and intriguing results using new methods and approaches.
During the last decade the interest of physicists in non-local eld theories has been steadily increasing. The main reason for this development is the expectation, that the use of these eld theories will lead to a much more elegant and eective way of treating problems in particle and high-energy physics, as it is possible up to now with local eld theories. A particular subgroup of non-local eld theories is described with operators of fractional nature and is speci ed within the framework of fractional calculus.
Typically the complexity of systems is linked to the long term memory, long-range interactions, non markovianity of the kinetics, and particularly with the Levy-type processes (Levy ights). The microscopic and macroscopic levels of description of the process are separated in the time scale, and memory of the non dierentiable character peculiar to the microscopic dynamics is lost in the long time limit. Consequently, the results of observing the motion of an ensemble of trajectories can be predicted by means of theoretical prescriptions based on ordinary mathematical procedures proceeding from the dierentiabil-ity assumption. When the condition of time-scale separation is not available, the non dierentiable nature of the microscopic dynamics can be transmitted
to the macroscopic level. The key to this understanding is memory eects in stochastic systems. It is due to memory eects that the macroscopic behavior of stochastic systems contains a manifestation of microscopic dynamics. That is, It is due to memory eects that the macroscopic behavior of stochastic systems contains a manifestation of microscopic dynamics [2]. The memory eects is related to a macroscopic manifestation of randomness. The presence of complete memory in a system means that all its components have the same behavior. If the memory function has no characteristic time scales, the correct description of the macroscopic evolution of such systems has to be in terms of the fractional calculus[2]. The fractional calculus provides a bridge between purely deterministic processes and purely stochastic ones. The fact is of interest in its own right because chaos and order in Nature coexist.
Fractional calculus provides us with a set of axioms and methods to extend the concept of a derivative operator from integer order n to arbitrary order a, where a is a real value. Despite the fact, that this concept is discussed since the days of Leibniz and since then has occupied the great mathematicians of their times, no other research area has resisted a direct application for centuries. Consequently, Abel's treatment of the tautochrone problem from 1823 stood for a long time as a singular example for an application of fractional calculus.
In contrast to the low-energy regime of quantum theory, the high-energy dynamics of the vacuum is characterized in general by a large number of time-scales that are not reducible to a single average through coarse grained [3]. It follows that the ensemble system + vacuum evolves on multiple scales. This line of reasoning reproduces, in essence, the statistical physics argument for replacing ordinary derivatives and integrals with fractal operators [4].
The standard model embodies our current knowledge of elementary par-ticle physics and represents a well-tested framework for the study of non-gravitational phenomena at low energies. It is built on the foundations of relativistic quantum eld theory (QFT), which provides the correct description of electroweak and strong interactions involving leptons and quarks. It is gen-erally believed that, extending the validity of QFT to energies on or beyond the TeV range must include the unavoidable signature of vacuum uctuations and strong- eld gravity. Some authors [5] argue that an eective approach to the high-energy regime of QFT demands the tools of complex dynamics and fractal operators. Then, adequate modeling of TeV physics demands the tools of fractal operators and fractional calculus. Fractal operators are required when making the transition from a dynamical process characterized by smooth space-time or con guration paths to a process displaying irregular and non-dierentiable paths.
Goldfain arguments [9], given references that there are cosmic probes that can be aected in their propagation by the fundamental structure of our Uni-verse. These probes are either ultrahigh energy cosmic rays (UHECR) or gamma rays in the TeV range from very far and variable sources and that there is a growing body of theoretical arguments advocating that vacuum uctuations at high energy scales convert the smooth topology of spacetime continuum into an in nite dimensional hierarchical Cantor set, that is, a Cantorian geometry of spacetime at high-energy scales. In this context, it appears that the physics of these cosmic probes may hold valuable clues regarding the underlying geometry of spacetime in high-energy interactions.
of particles in a more stringent and accurate way, than actual theories do up to now, can be done by the help of fractional group theory, which allows express-ing and solvexpress-ing problems in a very elegant way, which cannot be su ciently described using traditional methods.
The interest in fractional wave equations aroused in the year 2000 with a publication of Raspini[7]. He demonstrated, that a 3-fold factorization of the Klein-Gordon equation leads to a fractional Dirac equation which contains fractional derivative operators of order a =2/3 and furthermore the resulting g - matrices obey an extended Cliord algebra [6].
In this panel we indicate some possible fractional generalization of standards equations that describes the dynamics associated with elds theory and parti-cles physics.
An approach based on Jumarie's Modi ed Riemann-Liouvile [8] is justi ed. Some possible realizations of fractional wave equations are given and the present state of our work is commented. Also, comments about Nottale's Scale Relativity and El-Naschie E-in nity Theory are made.
Citing Hawking: I think the next century will be the century of complexity. Stephen Hawking (Complexity Digest 2001/10, 5 March 2001.)
II. JUMARIE'S FRACTIONAL CALCULUS AND MODIFIED RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE
Recently, Guy Jumarie [8] proposed a simple alternative de nition to the RiemannLiouville derivative. His modi ed RiemannLiouville derivative has the advantages of both the standard RiemannLiouville and Caputo fractional derivatives: it is de ned for arbitrary continuous (non dierentiable) functions
and the fractional derivative of a constant is equal to zero.
Jumarie's calculus seams to give a mathematical framework for dealing with dynamical systems de ned in coarse-grained spaces and with coarse-grained time and, to this end, to use the fact that fractional calculus appears to be intimately related to fractal and self-similar functions.
As pointed out by Jumarie, non-dierentiability and randomness are mutu-ally related in their nature, in such a way that studies in fractals on the one hand and fractional Brownian motion on the other hand are often parallel in the same paper. A function which is continuous everywhere but is nowhere dif-ferentiable necessarily exhibits random-like or pseudo-random-features, in the sense that various samplings of this functions on the same given interval will be dierent. This may explain the huge amount of literature which extends the theory of stochastic dierential equation to stochastic dynamics driven by fractional Brownian motion.
Modi ed Riemann-Liouville (MRL) derivative:
Dαf (x) = 1 Γ(1 − α) d dx ˆ x 0 (x − t)−α(f (t) − f (0))dt, 0 < α < 1 (1) Advantages of MRL
Some advantages for the uses of MRL are: derivative of constant is zero, simple rules and not restrictive.
Simple rules
DαK = 0, Dαxγ = Γ(γ+1) Γ(γ+1−α)x
γ−α, γ > 0.
Simple Chain Rules dα dxαf [u(x)] = dα duαf d dxu !α (2) For non dierentiable functions
dα dxαf [u(x)] = d duf dα dxαu (3)
for coarse-grained space-time.
A. Fractal space-time in physics
The most natural and direct way to question the classical framework of physics is to remark that in the space of our real world, the generic point is not in nitely small (or thin) but rather has a thickness. A coarse-grained space is a space in which the generic point is not in nitely thin, but rather has a thickness; and here this feature is modeled as a space in which the generic increment is not dx, but rather (dx)αand likewise for coarse grained with respect to the time
variable t.
III. SOME POSSIBLE REALIZATIONS
Fractional Electrodynamics
Some attempts to implement an Electromagnetic Field Theory was taken by trying to construct fractional Maxwell equations, given in various senses of fractional derivatives. Authors like Engheta, Tarasov with a fractional vector calculus, Baleanu with Fractional Electromagnetic Equations Using Fractional
Forms, have been trying to develop a fractional electromagnetism.
Fractional Quantum Mechanics
Some author[10] have tried the approach of fractional quantum mechanics by de ning a fractional Schrodinger equation while others [11] developed a path integral approach.
Its important to note the dierence from a theory of fractional quantum me-chanics composed of fractional operators based on fractional derivatives de ni-tions or fractional power of operators [11] and a theory of quantum mechanical models in fractional dimensions [12].
Other approaches [13] includes obtainment of The fractional Schrï¾÷dinger equation using a proposed Lagrangian within an action with a fractional varia-tional principle and alternatively by a fracvaria-tional Klein-Gordon equation. A lo-cal version of fractional derivative to perform lolo-cal analysis of non-dierentiable functions was used with the called scale calculus to obtain a generalized Schrï¾÷dinger equation.
A fractional Schrï¾÷dinger's equation in fractal space
Fractional Relativity and Fractional Dierential Geometry
The fractional dynamics of relativistic particle is discussed in Tarasov article [15]. He formulate fractional dynamics of relativistic point particle as mechanics of the systems with nonholonomic constraint in the four-dimensional pseudo-Euclidean space-time.
Other approaches includes E-Naschie theory of Catorian space-time and Not-tale's scale relativity.
The main argument for dealing with fractal dierential operators concerns the fact that it may represent an appropriate structure for the description of physical phenomena that are expected to arise in the TeV realm of elementary particle physics. For example, it has been argued that the onset of large and persistent quantum vacuum uctuations, beside strong-gravity eects emerging from the short distance behavior of quantum eld theory, requires the use of fractional dierential operators. Moreover, the macroscopic description of phe-nomena in terms of standard dierential and integral operators breaks down due of dynamical instabilities developed on long time scales, i.e., unstable vacuum uctuations leading to self-organized criticality and consequently, this is one of the main arguments for using fractional dierential and integral operators within the context of eld theory
Nature appears not continuous, not periodic, but self-similar/fractal. Mo-hamed El-Naschie with e(∞) has introduced a mathematical formulation to describe the fractal phenomena that are resolution dependent [16]. The in nite dimensional e(∞)-space, when viewed at large scales, mimics the appearance of the four-dimensional complex space time.
The resolution-dependent character of physical laws as described by Nottale's scale relativity.
Fractional Gravitational Fields Theory
The cornerstone questions to be solved in constructing classical and quan-tum gravity theories [17] are positively connected to the meaning of space time dimension, extra dimension and/or fractional dimensions physical eects and possible contributions are from (non) holonomic and/or commutative/
non-commutative variables, of fractal dimension, nonlocal eld theories etc. With respect to dark energy and dark matter models, various attempts to construct the quantum gravity theory, to determine the status of singularities in funda-mental theories etc, it is thus natural to pursue alternative concepts of dier-ential and integral calculus and consider new ideas and models of space-time geometry. In this context El-Nabulsi [18] explored about the nature of the frac-tional gravitafrac-tional eld when usual integral operator is replaced by fracfrac-tional operator, in particular the RiemannLiouville fractional integral.
UN-particles, fractional spin particles and others
The possibility of a hidden sector of particle physics that lies beyond the energy range of the Standard Model has been recently advocated by many authors. A bizarre implication of this conjecture is the emergence of a contin-uous spectrum of mass-less elds with non-integral scaling dimensions called UN-particles
In 2006, Goldfain has formulated the concept of fractional number of eld quanta in connection with the development of quantum eld theory using com-plex dynamics [3] that he called Comcom-plex Field Theory-c-QFT. The mathe-matics of c-QFT is based on fractal dierential operators that generalize the momentum operators of conventional quantum eld theory (QFT), that is, fractional calculus.
Gaete, Helayel and Spallucci [19] obtained UN-gravity eects is the frac-tionation of the event horizon analyzing UN-graviton corrections to the Schwarzschild black hole.
IV. FRACTIONAL WAVE EQUATIONS
The present state of our work concerns the fractional wave equation in the MRL sense and its solutions as an initial value problem solved in a fractal light-cone space-time.
Fractional D'alembertian of same space-time partial derivative order
∂α ∂xα ∂α ∂xαφ (x, t) − 1 v2α ∂α ∂tα ∂α ∂tαφ (x, t) = 0 (4) Fractional D'alembertian of dierent space-time partial derivative orders
∂α ∂xα ∂α ∂xαφ (x, t) − 1 v2β ∂β ∂tβ ∂β ∂tβφ (x, t) = 0 (5)
The work is under development and will be published elsewhere.
V. COMMENTS AND CONCLUSIONS
There are many directions of investigation left to explore in fractional calcu-lus, fractional geometry and applications in various sciences. Here we outlined some of them. In this work, within the realm of complexity we connect complex systems, elds theory, particle physics, gravitational elds with the theory frac-tional order operators and fracfrac-tional calculus and aspects like coarse-grained space-time and fractional calculus were related. Concepts like fractals, com-plexity, fractal operators and fractional calculus were introduced as a tool to revisit the foundation of physics as natural science. Theories that includes
and associates concepts like dimension, resolution and scale in relativity are commented too.
In the Modi ed Riemann-Liouville sense, we proposed a fractional wave equation. The solution of that is under development.
[1] Science, New Series, Vol. 177, No. 4047 (Aug. 4, 1972), pp. 393-396
[2] A. A. Stanislavsky, Memory eects and macroscopic manifestation of randomness, Phys Rev. E 61, 5, 4752-4759 (2000)
[3] E. Goldfain, Complexity in quantum eld theory and physics beyond the standard model Chaos, Solitons and Fractals 28 (2006) 913922
[4] West JB, Bologna M, Grigolini P.Physics of fractal operators.New York: Springer-Verlag; 2003 [5] International Journal of Nonlinear Sciences and Numerical Simulation 6(3), 223-234, 2005 [6] Fractional Calculus - An Introduction for Physicists, World Scienti c, Singapore (2011), ISBN:
9789814340243.
[7] Andrea Raspini, DIRAC EQUATION WITH FRACTIONAL DERIVATIVES OF OR-DER 2/3, FIZIKA B (Zagreb) 9 (2000) 2, 4954.
[8] Guy Jumarie, Table of some basic fractional calculus formulae derived from a modi ed Rie-mannLiouville derivative for non-dierentiable functions, Applied Mathematics Letters 22 (2009) 378385
[9] E. Goldfain / Chaos, Solitons and Fractals 20 (2004) 427435
[10] A.M. Shahin, E.Ahmed and Yassmin A.Omar, On Fractional Order Quantum Mechanics, International Journal of Nonlinear Science,Vol.8(2009),No.4,pp. 469-472
[11] Nick Laskin, Fractional quantum mechanics, PRE 62, 3 (2000)
[12] M A Lohe and A Thilagam, Quantum mechanical models in fractional dimensions, J. Phys. A: Math. Gen. 37 (2004) 61816199
[13] Sami I. Muslih ï¾÷Om P. Agrawal ï¾÷ Dumitru Baleanu,A Fractional Schrï¾÷dinger Equation and Its Solution, Int J Theor Phys (2010) 49: 17461752
[14] Jacky Cresson ,Scale calculus and the Schrï¾÷dinger equation, J. Math. Phys., Vol. 44, No. 11, November 2003, 4907-4938.
[15] Vasily E., TarasovFractional Dynamics of Relativistic Particle, Int J Theor Phys (2010) 49: 293303
[16] M. S. El Naschie, Chaos, Solitons and Fractals 25(5) (2005) 955; 27(1) (2006) 9; 5(6) (1995) 1031; 8(11) (1997) 1865; 12(5) (2001) 851; 1(5) (1992) 485; 11 (2000) 1149; Int. J. Nonlin. Sci. Numer. Simul. 7(4) (2006) 477.
[17] Sergiu I. Vacaru, Fractional Dynamics from Einstein Gravity, General Solutions, and Black Holes, arXiv:1004.0628v1 [math-ph] 5 Apr 2010
[18] El-Nabulsi Ahmad Rami ,Fractional illusion theory of space: Fractional gravitational eld with fractional extra-dimensions,Chaos, Solitons and Fractals 42 (2009) 377384
[19] Patricio Gaete, Josï¾÷ A. Helaï¾÷el-Neto, Euro Spallucci,Un-graviton corrections to the Schwarzschild black hole, Physics Letters B 693 (2010) 155158
