Quantum origin of (Newtonian) mass and Galilean relativity symmetry

The Galilei group has been taken as the fundamental symmetry for ‘nonrelativistic’ physics, quantum or classical. Our fully group theoretical formulation approach to the quantum theory asks for some adjustments. We present a sketch of the full picture here, emphasizing aspects that are different from the more familiar picture. The analysis involves a more careful treatment of the relation between the exact mathematics and its physical application in the dynamical theories, and a more serious full implementation of the mathematical logic than what is usually available in the physics literature. The article 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 and the notion of center of mass as dictated by the symmetry, that is particularly also to be seen as the Heisenberg–Weyl symmetry inside it. Another result is the necessary exclusion of the time translational symmetry. The time translational symmetry in the Galilei group plays no role in the formulation of the dynamical theory and does not correspond to the physical time in any nontrivial setting.