Modified Riemann-Liouville Approach to Field Theory and Particle Physics

Weberszpil, José; Godinho, Cresus F. L.; Helayël-Neto, J. A.

doi:10.5540/03.2015.003.01.0209

Cited by 4 publications

(6 citation statements)

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“…Weberszpil, Lazo and Helayël-Neto [42] have shown that the linear ole-derivative (59a) is the first order expansion of the Hausdorff derivative. Whether the other generalized derivatives are also connected to fractal derivatives and fractal metrics remains to be investigated.…”

Section: Final Remarksmentioning

confidence: 99%

Deformed mathematical objects stemming from the $q$-logarithm function

Borges¹,

Costa²

2021

Preprint

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Generalized numbers, arithmetic operators and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined.Commutativity, associativity and distributivity, and also a pair of linear/nonlinear derivatives are observed within each class. Two entropic functionals emerge from the formalism, one of them is the nonadditive Tsallis entropy.

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Section: Final Remarksmentioning

confidence: 99%

Deformed mathematical objects stemming from the $q$-logarithm function

Borges¹,

Costa²

2021

Preprint

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“…We could revisit the ZB phenomena by applying the Balankin's [50,51] approach based on mapping to a fractional continuum [52] and by adopting a local version of fractional called Hausdorff derivative. Doing so, an expansion of the fractional Newton binomial, that appears as a pre-factor in the derivative of [50,51], can be shown to lead to a q-derivative [53] as a lowerorder term [54]. The resulting equations would be non-linear due to a local factor, with integer-order derivatives acting on a continuum Euclidean space of fractional metric as a result of the mapping [50].…”

Section: Some Remarks On the Fractional Derivativementioning

confidence: 99%

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

Weberszpil

Helayël-Neto

2015

JAP

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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 = −

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“…These concepts were used to identify ultraviolet cosmological solutions and their effects in the early universe. Fractional calculus has also been applied in particle physics 38,39 where the fractional D'Alembertian operator was introduced and the fractional Lorentz transforms and fractional Klein-Gordon equations were developed. In addition this version of the Klein-Gordon equation has been employed for modeling linear dispersive phenomena 40 .…”

Section: Introductionmentioning

confidence: 99%

Investigation of dephasing in an open quantum system under chaotic influence via a fractional Kohn–Sham scheme

Ganesan

2014

Int J of Quantum Chemistry

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In this work, the dynamics of dephasing (without relaxation) in the presence of a chaotic oscillator is theoretically investigated. The time‐dependent density functional theory framework was used in tandem with the Lindblad master equation approach for modeling the dissipative dynamics. Using the Kohn–Sham (K–S) scheme under certain approximations, the exact model for the potentials was acquired. In addition, a space‐fractional K–S scheme was developed (using the modified Riemann–Liouville operator) for modeling the dephasing phenomenon. Extensive analyses and comparative studies were then done on the results obtained using the space‐fractional K–S system and the conventional K–S system. © 2014 Wiley Periodicals, Inc.

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Modified Riemann-Liouville Approach to Field Theory and Particle Physics

Cited by 4 publications

References 1 publication

Deformed mathematical objects stemming from the $q$-logarithm function

Deformed mathematical objects stemming from the $q$-logarithm function

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

Investigation of dephasing in an open quantum system under chaotic influence via a fractional Kohn–Sham scheme

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