  The algebra of linear and quadratic functions of basic observables on the
phase space of either the free particle or the harmonic oscillator possesses a
finite-dimensional anomaly. The quantization of these systems outside the
critical values of the anomaly leads to a new degree of freedom which shares
its internal character with spin, but nevertheless features an infinite number
of different states. Both are associated with the transformation properties of
wave functions under the Weyl-symplectic group $WSp(6,\Re)$. The physical
meaning of this new degree of freedom can be established, with a major scope,
only by analysing the quantization of an infinite-dimensional algebra of
diffeomorphisms generalizing string symmetry and leading to more general
extended objects.
