Fractional Dynamics of Relativistic Particle

Tarasov, Vasily E.

doi:10.1007/s10773-009-0202-z

Cited by 27 publications

(8 citation statements)

References 30 publications

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“…We also note some other formulations of the non-Markovian quantum dynamics by using fractional calculus: the non-Markovian dynamics of OQS with time-dependent parameters [39,45], relativistic open classical systems [46], and quantum systems with memory [47].…”

Section: Introductionmentioning

confidence: 99%

General Non-Markovian Quantum Dynamics

Tarasov

2021

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A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.

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Section: Introductionmentioning

confidence: 99%

General Non-Markovian Quantum Dynamics

Tarasov

2021

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“…487–520, [ 9 ]). Relativistic open classical systems [ 58 , 59 ] and quantum systems with memory [ 60 ]. Classical system with memory as open system [ 61 ].…”

Section: Introductionmentioning

confidence: 99%

Quantum Maps with Memory from Generalized Lindblad Equation

Tarasov

2021

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In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.

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“…In [11], the mass spectroscopy of heavy mesons were investigated within the frame of conformable derivative searching for any ordering effect in their spectra that varies with the fractional order. In [12], the fractional dynamics of relativistic particles was studied, and it was found that fractional dynamics of such particles are described as non-Hamiltonian and dissipative. Possibility of being Hamiltonian system under some conditions was also presented.…”

Section: Introductionmentioning

confidence: 99%

The effect of deformation of special relativity by conformable derivative

Al-Jamel,

Al-Masaeed,

Rabei

et al. 2021

Preprint

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In the article, the deformation of special relativity within the frame of conformable derivative is formulated. Within this context, the two postulates of the theory were re-stated. And, the addition of velocity laws were derived and used to verify the constancy of the speed of light. The invariance principle of the laws of physics is demonstrated for some typical illustrative examples, namely, the conformable wave equation, the conformable Schrodinger equation, and the conformable Gordon-Klein equation. The current formalism may be applicable when using special relativity in a nonlinear or dispersive medium.

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Fractional Dynamics of Relativistic Particle

Cited by 27 publications

References 30 publications

General Non-Markovian Quantum Dynamics

General Non-Markovian Quantum Dynamics

Quantum Maps with Memory from Generalized Lindblad Equation

The effect of deformation of special relativity by conformable derivative

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