  A "minimal" generalization of Quantum Mechanics is proposed, where the
Lagrangian or the action functional is a mapping from the (classical) states of
a system to the Lie algebra of a general compact Lie group, and the wave
function takes values in the corresponding group algebra. This formalism admits
a probability interpretation and a suitable dynamics, but has no obvious
classical correspondence. Allowing the Lagrangian or the action functional to
take values in a general Lie algebra instead of only the real number field
(actually the u(1) algebra) enlarges the extent of possible physical laws that
can describe the real world. The generalized quantum dynamics of a point
particle in a background gauge field is given as an example, which realizes the
gauge invariance by a Wilson line structure and shows that some
Schrodinger-like equation can be deduced within this formalism. Some possible
developments of this formalism are also discussed.
