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My question is why Dirac's Hamiltonian isn't diagonal? As much as I understand, the momentum of the particle and it's spin belong to the complete set of commuting variables, which define the state of the particle, and their eigenstates must also be energy eigenstates. But because Hamiltonian is non-diagonal (because gamma matrices are not diagonal), it follows that if we have a particle with definite momentum and definite spin, with time it is going to change it's state to become a particle with no definite spin.

Where am I wrong?