Quantum Harmonic Oscillator as Zariski Geometry

We describe the structure QHO = QHON (dependent on the positive integer number N) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that the structure is not interpretable in an algebraically closed field and, for the case F = C, is not a structure on a complex manifold. There are several reasons that motivate our interest in this particular example. First, this Zariski geometry differs considerably from the series of examples in [HZ] which all are based on the actions of certain kinds of noncommutative groups as the groups of Zariski automorphisms of the structures constructed. In the present case we represent the well-known noncommutative algebra with generators P and Q satisfying the relation QP− PQ = i, (1)