The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
The largest group of coordinate transformations leaving invariant the Schroedinger equation of the n-dimensional harmonic oscillator is determined and shown to be isomorphic to the corresponding group of the free-particle equation. It can be described as a Galilei group in which the time translations have been replaced by the group SL(2,R) of projective transformations. The relation between the oscillator group and the spectrumgenenating algebra of the harmonic oscillator is investigated. The relevance of the oscillator group and the group SL(2,R) for general quantum systems is discussed. (auth)