  In a one-dimensional lattice, the induced metric (from a noncommutative
geometry calculation) breaks translation invariance. This leads to some
inconsistencies among different spectator frames, in the observation of the
hoppings of a test particle between lattice sites. To resolve the
inconsistencies between the different spectator frames, we replace the test
particle's bare mass by an effective locally dependent mass. This effective
mass also depends on the lattice constant - i.e. it is a scale dependent
variable (a "running" mass). We also develop an alternative approach based on a
compensating potential. The induced potential between a spectator frame and the
test particle is attractive on the average. We then show that the entire
formalism holds for a quantum particle represented by a wave function, just as
it applies to the mechanics of a classical point particle.
