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A general method to derive the diagonal representation for a generic matrixvalued quantum Hamiltonian is proposed. In this approach new mathematicalobjects like non-commuting operators evolving with the Planck constant promotedas a running variable are introduced. This method leads to a formal compactexpression for the diagonal Hamiltonian which can be expanded in a power seriesof the Planck constant. In particular, we provide an explicit expression forthe diagonal representation of a generic Hamiltonian to the second order in thePlanck constant. This last result is applied, as a physical illustration, toDirac electrons and neutrinos in external fields.

Diagonal Representation for a Generic Matrix Valued Quantum Hamiltonian