Dispersion of Ultra-Relativistic Tardyonic and Tachyonic Wave Packets on Cosmic Scales

We investigate the time propagation of tachyonic (superluminal) and tardyonic (subluminal, ordinary) massive wave packets on cosmic scales. A normalizable wave packet cannot be monochromatic in momentum space and thus acquires a positional uncertainty (or packet width) that increases with travel distance. We investigate the question of how this positional uncertainty affects the uncertainty in the detection time for cosmic radiation on Earth. In the ultrarelativistic limit, we find a unified result, $\delta x(t)\mathrm{/}{c}^3=m^2\delta pt\mathrm{/}{p}_0^3$, where $\delta x(t)$ is the positional uncertainty, $m$ is the mass parameter, $\delta p$ is the initial momentum spread of the wave function, and $p_0$ is the central momentum of the wave packet, which, in the ultrarelativistic limit, is equal to its energy. This result is valid for tachyons and tardyons; its interpretation is being discussed.