  The Galilean symmetry and the Poincare symmetry are usually taken as the
fundamental (relativity) symmetries for `nonrelativistic' and `relativistic'
physics, respectively, quantum or classical. Our fully group theoretical
formulation approach to the theories, together with its natural companion of
mechanics from symplectic geometry, ask for different perspectives. We present
a sketch of the full picture here, emphasizing aspects which are different from
the more familiar picture. The letter summarizes our earlier presented
formulation while focusing on the part beyond, with an adjusted, or corrected,
identification of the basic representations having the (Newtonian) mass as a
Casimir invariant. Discussion on the limitations of the Poincare symmetry for
the purpose is particularly elaborated.
