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We derive a product rule for gauge invariant Weyl symbols which provides ageneralization of the well-known Moyal formula to the case of non-vanishingelectromagnetic fields. Applying our result to the guiding center problem weexpand the guiding center Hamiltonian into an asymptotic power series withrespect to both Planck's constant $\hbar$ and an adiabaticity parameter alreadypresent in the classical theory. This expansion is used to determine theinfluence of quantum mechanical effects on guiding center motion.Comment: 24 pages, RevTeX, no figures; shortened version will be published in  J.Phys.

Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding centre motion