  Replacing the canonical pair q and p of the harmonic oscillator (HO) by the
locally and symplectically equivalent pair angle phi and action variable I
implies a qualitative change of the global topological structure of the
associated phase spaces: the pair (q,p) is an element of a topologically
trivial plane R^2 whereas the pair (phi,I>0) is an element of a topologically
non-trivial, infinitely connected, punctured plane R^2-{0}, which has the group
SO(1,2) as its "canonical" group. Due to its infinitely many covering groups
the resulting ("symplectic") spectrum of the associated quantum Hamiltonian is
given by {hbar omega (n+b), n =0,1,...; b in (0,1]}, in contrast to the
"orthodox" spectrum {hbar omega (n+1/2)}.
  The potentially most important implications concern the vibrations of
diatomic molecules in the infrared, e.g. those of molecular hydrogen H_2. Those
symplectic spectra of the HO may provide a simultaneous key to two outstanding
astrophysical puzzles, namely the nature of dark (vacuum) energy and that of
dark matter: To the former because the zero-point energy b hbar omega of free
electromagnetic wave oscillator modes can be extremely small > 0 (b ca.
exp(-35) for the measured dark energy density). And a key to the dark matter
problem because the quantum zero-point energies of the Born-Oppenheimer
potentials in which the two nuclei of H_2 or the nuclei of other primordial
diatomic molecules vibrate can be lower, too, and, therefore, may lead to
spectrally detuned "dark" H_2 molecules during the "Dark Ages" of the universe
and forming WIMPs in the hypothesized sense. All results appear to be in
surprisingly good agreement with the LambdaCDM model of the universe. Besides
laboratory experiments the search for 21-cm radio signals from the Dark Ages of
the universe and other astrophysical observations can help to explore those
hypothetical implications.
