  Andreka and her colleagues have described various geometrically inspired
first-order theories of special and general relativity, while Szekely's PhD
dissertation focuses on an intermediate logic of accelerated observers. In this
paper we will attempt to incorporate a model of quantum theoretical
trajectories that can reasonably claim to be physically meaningful into those
theories.
  We have recently shown that a model of quantum trajectories, based on
discrete finitary motion, is logically equivalent to Feynman's path-integral
formulation when spacetime is assumed to be Euclidean. In this paper we argue
that relativistic observers are subject to the same quantum illusions as in the
Euclidean case: even though motion is discrete and respects no built-in 'arrow
of time', observers have no choice but to perceive particle trajectories as
continuous paths in spacetime.
  Whereas the relativistic theories presuppose continuous paths as part of
their axioms, the illusion of continuity allows us to replace this axiom with a
lower-level quantum-inspired axiom concerning discrete jumps in spacetime. We
investigate the nature of these jumps, and how far they can be tied to the
underlying geometric structure of spacetime. In particular, we consider hops
which preserve features of the underlying number field, and investigate the
extent to which all hops can be restricted to be of this form.
  We conclude that hop-based motion can also be regarded as an illusion, one
caused by modelling physics in the 'wrong' number system.
