  This work discloses some inconsistencies of quantum mechanics (QM) in two
different contexts. On the one hand the equivalence principle in his two
formulations is the starting point of all gravitational theories; yet there is
no way to make it consistent with the non-relativistic formulation of QM.
Moreover, the evaluation of the Schr\"odinger equation for a particle in an
accelerated reference frame sheds light on a trajectory-dependent phase term
subject to proper time. On the other hand, the Bargmann theorem unmistakably
proves that the Schr\"odinger equation does not admit any superposition of
different masses if the symmetry chosen for the system is the Galilean one.
However, this is incompatible with relativity, since mass and energy are
equivalent and one can certainly superimpose different energies. Both
inconsistencies support the hypothesis that mass and proper time could be
treated as variables. In the third chapter we will describe a theory in which
the Galilean symmetry is extended by assuming particle mass and proper time as
canonically conjugated classical variables, or conjugated observables in the
quantum theory.
  The first three chapters discuss subjects illustrated in several articles of
the professor Daniel M. Greenberger, which are listed in the bibliography.
Whereas in the last chapter the results of a recent theory on QW, say a
discrete model for quantum particle evolution, will be examined. In particular,
the striking result is that, in the most natural physical interpretation of QW
dynamics, it is not possible to have a constant and invariant mass in any
manner. The long term aim of this work is indeed to study the reasons and
consequences of a theory of particles with variable mass in the context of
wider studies still in progress on QW.
