  This paper defines, on the Galilean space-time, the group of asymptotically
Euclidean transformations (AET), which are equivalent to Euclidean
transformations at space-time infinity, and proposes a formulation of
nonrelativistic quantum mechanics which is invariant under such
transformations. This formulation is based on the asymptotic quantum measure,
which is shown to be invariant under AET's. This invariance exposes an
important connection between AET's and Feynman path integrals, and reveals the
nonmetric character of the asymptotic quantum measure. The latter feature
becomes even clearer when the theory is formulated in terms of the
coordinate-free formalism of asymptotically Euclidean manifold, which do not
have a metric structure. This mathematical formalism suggests the following
physical interpretation: (i) Particles evolution is represented by trajectories
on an asymptotically Euclidean manifold; (ii) The metric and the law of motion
are not defined a priori as fundamental entities, but they are properties of a
particular class of reference frames; (iii) The universe is considered as a
probability space in which the asymptotic quantum measure plays the role of a
probability measure. Points (ii) and (iii) are used to build the asymptotic
measurement theory, which is shown to be consistent with traditional quantum
measurement theory. The most remarkable feature of this measurement theory is
the possibility of having a nonchaotic distribution of the initial conditions
(NCDIC), an extremely counterintuitive but not paradoxical phenomenon which
allows to interpret typical quantum phenomena, such as particle diffraction and
tunnel effect, while still providing a description of their motion in terms of
classical trajectories.
