Random Motion of Light-Speed Particles
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Random Motion of Light-Speed Particles Maurizio Serva1 Received: 16 July 2020 / Accepted: 12 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In 1956 Mark Kac proposed a process related to the telegrapher equation where the particle travels at constant speed (say the speed of light c) and randomly inverts its velocity. This process had important applications concerning the path-integral solution and the probabilistic interpretation of the 1+1 dimensions Dirac equation. The extension to 3+1 dimensions requires that the particle only moves at light-speed, which implies that velocity can be represented as a point on the surface of a sphere of radius c. The realizations of the process for the velocity only may connect these points, and, by strict analogy with the Kac model, it can be assumed that the velocity jumps from one value to another. In this paper we follow a new and different strategy assuming that the velocity performs continuous trajectories (velocity changes direction in a continuous way) which are the realization of a Wiener process on the surface. The processes which emerge transform one in the other by Lorentz boost. The associate Forward Kolmogorov Equation for the joint probability density of position and velocity, which is the (3+1) dimensional analogous of the telegrapher equation, is examined and a simplification is performed by means of variables separation. Keywords Brownian motion · Wiener process · Relativity · Lorentz boost · Ito calculus
1 Introduction In 1956 the Polish-American physicist and mathematician Mark Kac proposed a process where the particle travels in one space dimension at constant speed (left or right) and randomly inverts its velocity, and he proved that the associated probability density satisfies the telegrapher equation [1]. About 30 years after the Kac pioneering work, Gaveau et al. noticed that this process could be easily associated to the Dirac equation in 1+1 dimensions (one space dimension + time) by an analytical continuation of time. Then, using this equivalence, they were able to provide a probabilistic solution of the Dirac equation in terms of the Kac light-speed trajectories [2].
Communicated by Michael Kiessling.
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Maurizio Serva [email protected] Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università dell’Aquila, L’Aquila, Italy
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M. Serva
Indeed, the process considered by Gaveau et al. is part of a larger class, in fact, by Lorentz boosts new processes can be obtained. The particles still move the speed of light c (a simple consequence of the fact that a light-speed particle in an inertial frame is also light-speed in any other inertial frame) but velocity inversions from right to left and from left to right occur at a different probability rate. The class of these (1+1)-dimensional light-speed process can be further extended by considering inversion rates which not only depend on the sign of the velocity but also on position and time. This extension gave the possibility to reformu
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