ua nt - p h / 98 05 02 5 v 1 9 M ay 1 99 8 Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding center motion

The knowledge about quantum effects on guiding center motion is important for low-temperature experiments with charged particles in external electromagnetic fields. In this paper we explicitly determine the quantum mechanical corrections to the classical guiding center Hamiltonian using a gauge invariant version of the Weyl calculus. 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 finally expand the guiding center Hamiltonian into an asymptotic power series with respect to both Planck's constant ¯ h and an adiabaticity parameter already present in the classical theory .