  Information measures for relativistic quantum spinors are constructed to
satisfy various postulated properties such as normalisation invariance and
positivity. Those measures are then used to motivate generalised Lagrangians
meant to probe shorter distance physics within the maximum uncertainty
framework. The modified evolution equations that follow are necessarily
nonlinear and simultaneously violate Lorentz invariance, supporting previous
heuristic arguments linking quantum nonlinearity with Lorentz violation. The
nonlinear equations also break discrete symmetries. We discuss the implications
of our results for physics in the neutrino sector and cosmology.
