[en] At first sight it is probably hard to believe that something new can be said about the harmonic oscillator (HO). But that is so indeed: Classically the Harmonic Oscillator (HO) is the generic example for the use of angle and action variables φ element of R mod 2π and I>0. However, the transformation q√(2I)cos φ, p=-√(2I)sin φ is only locally symplectic and singular for (q,p)=(0,0). Globally the phase space {(q,p)} has the topological structure of the plane R², whereas the phase space {(φ,I)} corresponds globally to the punctured plane R²-(0,0) or to a simple cone S¹ x R⁺ with the tip deleted. This makes a qualitative difference as to the quantum theory of the two phase spaces: The quantizing canonical group for the plane R² consists of the (centrally extended) translations generated by the Poisson Lie algebra basis {q,p,1}, whereas the corresponding canonical group of the phase space {(φ,I)} is the group SO↑(1,2)=Sp(2,R)/Z₂, where Sp(2,R) is the sympletic group of the plane, with the generating Poisson Lie algebra basis {h₀=I,h₁=Icosφ,h₂=-Isinφ} which provides also the basic ''observables'' on {(φ, I)}. In the quantum mechanics of the (φ,I)-model of the HO the three h_j correspond to self-adjoint generators K_j, j=0,1,2, of irreducible unitary representations from the positive discrete series of the group SO↑(1,2) or one of its infinitely many covering groups, the representations parametrized by the Bargmann index k>0. This index k determines the ground state energy E_k,n=0=ℎωk of the (φ,I)-Hamiltonian H(anti K)=ℎωK₀. For an m-fold covering the lowest possible value for k is k=1/m, which can be made arbitrarily small by choosing m accordingly. This is not in contraction to the usual approach in terms of the operators Q and P which are now expressed as functions of the K_j, but keep their usual properties. The richer structure of the K_j quantum model of the HO is ''erased'' when passing to the simpler Q,P model. This more refined approach to the quantum theory 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 strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant. (orig.)