Can the zero-point energy of the quantized harmonic oscillator be lower? Possible implications for the physics of "dark energy" and "dark matter"
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.