Common aspects of -deformed Lie algebras and fractional calculus

Lazo²,

Physica A: Statistical Mechanics and its Applications

2015

Over the recent decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the q-calculus in Tsallis' version of Non-Extensive Statistics with its undeniable success whenever applied to a wide class of different systems; Kaniadakis' approach, based on the compatibility between relativity and thermodynamics;Fractional Calculus (FC), that deals with the dynamics of anomalous transport and other natural phenomena, and also some local versions of FC that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations.The question we might ask is whether or not there are common aspects that connect these alternative approaches.In this short communication, we discuss a possible relationship between q−deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis' framework and the Kaniadakis' scenario, with local form of fractional-derivative operators defined in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. This connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics, such as porous or granular media. Possible connections with other alternative definitions of FC are also contempled. Insights on complexity connected to concepts like coarse-grained space-time and physics in general are pointed out.

Section: Conclusion and Outtlookmentioning

confidence: 99%

On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric

Lazo²,

Physica A: Statistical Mechanics and its Applications

2015

“…For some alternative definitions of fractional derivatives, see [44] and the interesting work [45]. Some relevant comments and remarks on the similarities between different approaches or even on fractional q-calculus may be found in [12,17,43,46,47,48].…”

Section: Some Remarks On the Fractional Derivativementioning

confidence: 99%

On the Zitterbewegung Transient Regime in a Coarse-Grained Space-Time

2015

JAP

In the present contribution, by studying a fractional version of Dirac's equation for the electron, we show that the phenomenon of Zitterbewegung in a coarse-grained medium exhibits a transient oscillatory behavior, rather than a purely oscillatory regime, as it occurs in the integer case, α = 1. Our result suggests that, in such systems, the Zitterbewegung-type term related to a trembling motion of a quasiparticle is tamed by its complex interactions with other particles and the medium. This can justify the difficulties in the observation of this interesting phenomenon. The possibility that the Zitterbewegung be accompanied by a damping factor supports the viewpoint of particle substructures in Quantum Mechanics. introductionDirac's equation unifies both Quantum Mechanics and Special Relativity by providing a relativistic description of the electron's spin; it predicts the existence of antimatter and is able to reproduce accurately the spectrum of the hydrogen atom. It also embodies the 'Zitterbewegung' (ZB) effect as an unexpected quivering motion of a free relativistic quantum particle, like the electron, for instance. This name was coined by Schrödinger, who first observed that, in describing relativistic electrons by the Dirac's equation, the components of the relativistic quadri-velocity do not commute with the free-electron Hamiltonian, with the consequence that the electron's velocity is not a constant of the motion even in the absence of external fields. Such an effect must be of a quantum nature, as it does not obey Newton's laws of classical dynamics. Schrödinger calculated the resulting time dependence of the electron's velocity and position, concluding that, in addition to its classical motion, the electron experiences very fast periodic oscillations [1].One of the motivations for analyzing ZB-models is to describe the spin of the electron, S, and its magnetic moment, μ, as generated by a local circulation of mass and charge. Experiments indicate the possibility for an internal structure for the electron, considering it as an extended object (wavelength 10 −13 m < δ < 10 −16 m, where δ is the supposed dimension for the where A S α is the operator A α in the Schrödinger representation. In order to set up a fractional version of Heisenberg evolution equation, we can use the fractional Leibniz rule in the MRL approach. The result isWe can now calculate the fractional evolution of the position asThe commutation relations yield:so that the fractional evolution equation turns out to the be the fractional Breit equation:As a next step, we proceed to calculate J 0 D α t α. Using that the Poisson bracket,and the fractional Heisenberg equation, we get thatorwhere η α is given byNow, assuming that the conservation relations holdand by verifying that {H α , η α } = 0, then, we can write the fractional differential equation for η α as J 0 D α t η α = −

“…[44,45,46] or the approaches with Hausdorff derivative, also called fractal derivative [47,48], that can be applied to power-law phenomena and the recently developed α − derivative [49]. The MRL approach seems to us to be an integral version of the calculus mentioned above and all of them deserve to be more deeply investigated, under a mathematical point of view, in order to give exact differences and similarities respect to the traditional fractional calculus with Riemann-Liouville or Caputo definition and with local fractional calculus and even fractional q-calculus [50,15,51], as well as in the comparative point of view of physics [48,51,52], for the scope of applicability. Now that we have set up these fundamental expressions, we are ready to carry out the calculations of main interest.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Anomalousg-Factors for Charged Leptons in a Fractional Coarse-Grained Approach

Advances in High Energy Physics

2014

In this work, we investigate aspects of the electron, muon and tau gyromagnetic ratios (g − f actor) in a fractional coarse-grained scenario, by adopting a Modified Riemann-Liouville (MRL) fractional calculus. We point out the possibility of mapping the experimental values of the specie's g − f actors into a theoretical parameter which accounts for fractionality, without computing higher-order QED calculations. We wish to understand whether the value of (g − 2) may be traced back to a fractionality of space-time.The justification for the difference between the experimental and the theoretical value g = 2 stemming from the Dirac equation is given in the terms of the complexity of the interactions of the charged leptons, considered as pseudo-particles and "dressed" by the interactions and the medium. Stepwise, we build up a fractional Dirac equation from the fractional Weyl equation that, on the other hand, was formulated exclusively in terms of the helicity operator. From the fractional angular momentum algebra, in a coarse-grained scenario, we work out the eigenvalues of the spin operator. Based on the standard electromagnetic current, as an analogy case, we write down a fractional Lagrangian density, with the electromagnetic field minimally coupled to the particular charged lepton. We then study a fractional gauge-like invariance symmetry, formulate the covariant fractional derivative and propose the spinor field transformation. Finally, by taking the non-relativistic regime of the fractional Dirac equation, the fractional Pauli equation is obtained and, from that, an explicit expression for the fractional g − f actor comes out that is compared with the experimental CODATA value. Our claim is that the different lepton species must probe space-time by experiencing different fractionalities, once the latter may be associated to the dressing of the particles and, then, to the effective interactions of the different families with the medium.Some authors argue that the anomalous magnetic moment (AMM) of the muon can be considered one of the most promising observables that can give rises to a rich new physics, specially in particle physics [1]. Other authors claim that the muons AMM is one of the most precisely measured quantities in particle physics, reaching a precision of 0.54ppm [2]. This argumentation is also reinforced because of the relatively large value of the muons mass [3].The tau anomaly is not very well known experimentally and, by this reason do not provide a good test of the StandardThe non-relativistic quantum mechanics predict by the reduction of the Dirac equation to the Pauli equation that g e = 2. Experimentally, the electron g − f actor has been measured with high precision, and the "2010 CODATA recommended 1