  Spinor fields are written in polar form so as to compute their tensorial
connection, an object that contains the same information of the connection but
which is also proven to be a real tensor. From this, one can still compute the
Riemann curvature, encoding the information about gravity. But even in absence
of gravity, when the Riemann curvature vanishes, it may still be possible that
the tensorial connection remains different from zero, and this can have effects
on matter. This is shown with examples in the two known integrable cases: the
hydrogen atom and the harmonic oscillator. The fact that a spinor can feel
effects due to sourceless actions is already known in electrodynamics as the
Aharonov-Bohm phenomenon. A parallel between the electrodynamics case and the
situation encountered here will be drawn. Some ideas about relativistic effects
and their role for general treatments of quantum field theories are also
underlined.
