Emergence of special and doubly special relativity

Abstract

Building on our previous work [Phys. Rev. D82, 085016 (2010)], we show in this paper how a Brownian motion on a short scale can originate a relativistic motion on scales that are larger than particle's Compton wavelength. This can be described in terms of polycrystalline vacuum. Viewed in this way, special relativity is not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength is taken. By analyzing the robustness of such a special relativity under small variations in the polycrystalline grain-size distribution we naturally arrive at the notion of doubly-special relativistic dynamics. In this way, a previously unsuspected, common statistical origin of the two frameworks is brought to light. Salient issues such as the rôle of gauge fixing in emergent relativity, generalized commutation relations, Hausdorff dimensions of representative path-integral trajectories and a connection with Feynman chessboard model are also discussed.

Granular universe: statistical origin of special and doubly special relativity

“…The particle then acquires a sharp mass equal to Einstein's mass, and the process (not being hindered by fluctuating masses) is purely Brownian. This conclusion is in line with the well-known Feynman checkerboard picture [14,17] to which it reduces in the case of (1 + 1) dimensional relativistic Dirac particle.…”

“…Further generalization to external electromagnetic potential has been reported in Refs. [10,14]. We further demonstrate that the above formulation can be looked at as if the particle would randomly propagate (in the sense of Brownian motion) through inhomogeneous or granular medium ("vacuum") [14].…”

“…Our argument is based upon a recent observation [9,10,14] that the Euclidean path integral (PI) for relativistic particles may be interpreted as describing a doubly-stochastic process that operates at two separate spatio-temporal scales. The short spatial scale, which is much smaller than particle's Compton length λ C = 1/mc ( = 1 is assumed throughout this paper), describes a Wiener (i.e., Galilean relativity) process with a sharp Newtonian mass.…”

Symmetries of relativistic world lines

“…It has also been argued that it is simply impossible to formulate a probability conserving relativistic quantum mechanics and one has to go to quantum field theory right away. Taking the problem more seriously it has also been argued that the usual notion of probability has to be changed [15][16][17].…”

Relational causality and classical probability: Grounding quantum phenomenology in a superclassical theory

“…within the framework of superclassical physics) makes possible new types of phenomena due to emergent features unexpected from either the very small or the large scale physics ( Fig. 2.1; compare also the related concept of "emergent relativity" as discussed, e.g., by Jizba and Scardigli, this volume, [18], and [19]). In our development of an emergent quantum mechanics, we shall thus further on describe it as a superclassical approach, in order to avoid confusions when using "classical" explanations.…”