  We derive a product rule for gauge invariant Weyl symbols which provides a
generalization of the well-known Moyal formula to the case of non-vanishing
electromagnetic fields. Applying our result to the guiding center problem we
expand the guiding center Hamiltonian into an asymptotic power series with
respect to both Planck's constant $\hbar$ and an adiabaticity parameter already
present in the classical theory. This expansion is used to determine the
influence of quantum mechanical effects on guiding center motion.
