Bohr-Sommerfeld conditions in geometric quantization

The corrected Bohr-Sommerfeld quantum conditions, ∫ pdq?d = integer, are studied in the framework of geometric quantization. It is shown, in the representation given by a polarization F, that a half-form corresponds to a wave function only if it vanishes on all closed curves with tangent vectors in F for which the quantum condition is not satisfied. The constant d is determined, for each closed curve y, by the element of the holonomy group of a bundle of metalinear frames for F induced by y. This result is applied to a one-dimensional harmonic oscillator and a two-dimensional relativistic Kepler problem. In the case of the one-dimensional harmonic oscillator there are two possibilities of choosing a metalinear frame bundle for F. One choice leads to the original Bohr-Sommerfeld condition while the other leads to the corrected version with d = $\text{1}\text{2}$. Similarly, choosing different metalinear frame bundles for F, we get for the relativistic Kepler problem the fine structures of the energy levels corresponding to spin 0 and spin $\text{1}\text{2}$.