Dirac particle with memory: Proper time non-locality

Abstract

A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described. PACS 45.10.Hj Perturbation and fractional calculus methods PACS 03.65.Yz Decoherence; open systems; quantum statistical methods PACS 11.90.+t Other topics in general theory of fields and particles

General Non-Markovian Quantum Dynamics

“…As proven in [58,59], operators (32) and (33) are connected (see Equation (47) in Definition 4 of [58] (p. 8)) by the equation…”

“…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].…”

Piecewise-Tunneled Captive Processes and Corridored Random Particle Systems

“…Note that although { t } t∈T captures glimpses of memory at discrete time points, it still does not account for the full historic path information. Systems with memory appear in numerous areas such as open-system oscillators [60], Dirac particles in electromagnetic fields [61], condensed matter dynamics in chemical physics [62], thermodynamics of materials [63,64], media propogation with viscoelasticity [65], particles in turbulent flow [59,66] and econophysics [67,68] to name a few. Below, we shall demonstrate how our framework can address enriched memory effects in confined dynamics by leveraging the deep structure of functional SDEs that communicate naturally with piecewise-tunneled captive processes.…”

Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus

“…For construction of a theory of non-Markovian quantum dynamics, it is important to consider time-dependent parameters [185,181] and the relativistic quantum systems with memory [186]. It is important for the non-Markovian quantum dynamics the generalization in the directions: the path integral for open quantum systems [187], [167, pp.…”