Energy-momentum diffusion from spacetime discreteness

We study potentially observable consequences of spatiotemporal discreteness for the motion of massive and massless particles. First we describe some simple models for the motion of a massive point particle in a fixed causal set background. If the causal set is faithfully embeddable in Minkoswki spacetime, the models give rise to particle motion in the continuum spacetime. At large scales, the microscopic swerves induced by the underlying atomicity manifest themselves as a Lorentz invariant diffusion in energy-momentum governed by a single phenomenological parameter, and we derive in full the corresponding diffusion equation. Inspired by the simplicity of the result, we then derive the most general Lorentz invariant diffusion equation for a massless particle, which turns out to contain two phenomenological parameters describing, respectively, diffusion and drift in the particle’s energy. The particles do not leave the light cone however: their worldlines continue to be null geodesics. Finally, we deduce bounds on the drift and diffusion constants for photons from the blackbody nature of the spectrum of the cosmic microwave background radiation.

Figure 1

A trajectory constructed using (a) model 1 and (b) model 2.

Figure 2

Particle trajectories as flowlines in ${\mathcal{M}}^{\prime }={\mathbb{M}}^4\times {\mathbb{H}}^3\times \mathbb{R}$ (where we have suppressed two spatial dimensions).

Figure 3

The rms deviation between the evolved spectrum and a best-fit Planck spectrum (as a proportion of the spectrum peak): (a) varying $k_1$ with $k_2=0$, and (b) varying $k_2$ with $k_1=0$.

Figure 4

The rms deviation of the simulated spectrum from Planckian as a proportion of the spectrum peak, varying both $k_1$ and $k_2$. Values of $k_1$ and $k_2$ within the $5e-05$ contour give a spectrum that is Planckian to within 50 ppm of the peak.