A New Look at the Quantum Mechanics of the Harmonic Oscillator
  Classically the Harmonic Oscillator (HO) is the generic example for the use of angle and action variables phi in R mod 2 pi and I > 0. But the symplectic transformation (\phi,I) to (q,p) is singular for (q,p) = (0,0). Globally {(q,p)} has the structure of the plane R^2, but {(phi,I)} that of the punctured plane R^2 -(0,0). This implies qualitative differences for the QM of the two phase spaces: The quantizing group for the plane R^2 consists of the (centrally extended) translations generated by {q,p,1}, but the corresponding group for {(phi,I)} is SO(1,2) = Sp(2,R)/Z_2, (Sp(2,R): symplectic group of the plane), with Lie algebra basis {h_0 = I, h_1 = I cos phi, h_2 = - I sin phi}. In the QM for the (phi,I)-model the three h_j correspond to self-adjoint generators K_j, j=0,1,2, of irreducible unitary representations (positive discrete series) for SO(1,2) or one of its infinitely many covering groups, the Bargmann index k > 0 of which determines the ground state energy E (k, n=0) = hbar omega k of the (phi,I)-Hamiltonian H(K). For an m-fold covering the lowest possible value is k=1/m, which can be made arbitrarily small! The operators Q and P, now expressed as functions of the K_j, keep their usual properties, but the richer structure of the K_j quantum model of the HO is ``erased'' when passing to the simpler Q,P model! The (phi,I)-variant of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with the (Landau) levels of charged particles in magnetic fields, with the propagation of light in vacuum, passing through electric or magnetic fields. Finally it leads to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant. 
H.A. Kastrup