Emergence of special and doubly special relativity

Building on our previous work [Phys. Rev. D 82, 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.

Figure 1

The roughness of the representative trajectories in the relativistic path integral (11) depends on a spatial/temporal scale. On a fine scale (A), where $t\ll t_C$ (or $\ell \ll {\lambda }_C$), a particle can be considered as propagating with a sharp Newtonian mass $\tilde{m}$ in a single spatial grain with a Brownian motion controlled by the Hamiltonian ${\mathit{p}}^2/2\tilde{m}$. On the intermediate scale of order ${\lambda }_C$ the particle propagates with an average velocity equal to the speed of light $c$. On a coarser scale (B), the particle appears to follow a Brownian process with a sharp Lorentz invariant mass $m$, and the particle’s net velocity is then less than $c$—as it should be for a massive relativistic particle governed by the relativistic Hamiltonian $c({\mathit{p}}^2+m^2{c}^2{)}^{1/2}$.