Derivation of relativistic wave equation from the Poisson process

Abstract

A Poisson process is one of the fundamental descriptions for relativistic particles: both fermions and bosons. A generalized linear photon wave equation in dispersive and homogeneous medium with dissipation is derived using the formulation of the Poisson process. This formulation provides a possible interpretation of the passage time of a photon moving in the medium, which never exceeds the speed of light in vacuum. The motion of relativistic particles is represented by the Poisson process. Kac proposed the continuous version of Poisson process in 1956 [1]. Gaveauet al derived the Dirac equation in 1984 using the formulation of Poisson process [2]. McKeon and Old constructed a modified version of the Poisson process in 1992 [3], where the relativistic particles can move both forward and backward in time. These Poisson processes were extended in 2001 [4] to more general frameworks of three-dimensional Dirac equation including an external field. Bialynicki-Birula invented a linear photon wave equation in 1994 [5] as a linearized version of the d'Alembert wave equation. Sipe verified in 1995 [6] that this wave equation identifies the ordinary canonical quantization.We treat the case that a time-dependent external potential V´tµ exists [4]. Let an absorption probability in a time interval δ be I´tµδ , where the path can be absorbed at most once in a time interval δ by the influence of the potential at time t. The conditional expectation is represented aswhere 1 I δ corresponds to a time shift generator in a time interval δ , and the onedimensional space-shift generator is L c∂ ∂ x. The one-dimensional space-shift generator has to be generalized to a three-dimensional generator with minimal coupling, L c σ σ ¡´∇ ∇ ie A´xµ µ in the case that vector field A exists, where σ σ gives the Pauli 413

Some stochastic aspects of quantization

“…This method has been extended to the more general case of a three-dimensional Dirac equation in an external field [8]. Recently we have derived a linearized photon wave equation in a dispersive medium [9].…”

The Central Limit Theorem for Random Dynamical Systems

“…A large class of applications of such models, both in physics and biology, is worth mentioning here: the shot noise, the photo conductive detectors, the growth of the size of structural populations, the motion of relativistic particles, both fermions and bosons (see [10,23,26]), the generalized stochastic process introduced in the recent model of gene expression by Lipniacki et al [30] see also [3,14,18]. The results bring some information important from biological point of view.…”

Dimension of invariant measures for continuous random dynamical systems

“…A large class of applications of such models, both in physics and biology, is worth mentioning here: The shot noise, the photo conductive detectors, the growth of the size structured populations of cells, the motion of relativistic particles, both fermions and bosons generalized stochastic process introduced in the recent model of gene expression by Lipniacki et al . .…”

“…Continuous random dynamical systems take into consideration some very important and widely studied cases, namely, dynamical systems generated by learning systems [8][9][10][11], Poisson-driven stochastic differential equations [12][13][14][15], iterated function systems with an infinite family of transformations [16][17][18], random evolutions [19,20], randomly controlled dynamical systems, and irreducible Markov systems [21]. Our model is similar to the so-called piecewise-deterministic Markov process introduced by Davis [22,23].A large class of applications of such models, both in physics and biology, is worth mentioning here: The shot noise, the photo conductive detectors, the growth of the size structured populations of cells, the motion of relativistic particles, both fermions and bosons [24][25][26][27] generalized stochastic process introduced in the recent model of gene expression by Lipniacki et al [2,28].After time t 1 , we restart the whole procedure with .x, i/ replaced by the new initial condition X.t 1 /, so that the process moves along the integral curves of one of the system (5.3) and (5.4) until the time t 2 of the second jump, and so on.Hence, the solution of (5.1) and (5.2) is now given by X.t/ D … t tn n .X.t n // for t n Ä t < t nC1 .We will consider process f.X.t/, .t//g t 0 , .X.t/, .t// : ! R 2 f0, 1g, where the norm in R 2 is defined by kxk D maxfjx 1 j, jx 2 jg for x D .x 1 , x 2 / 2 R 2 .We are going to show that for i 2 f0, 1g dynamical systems .… t i / t 0 satisfy assumptions of Theorem 3.1.…”