Transformation of Magnetic Property Operators

If applying a transformation derived from the decoupling of the unperturbed Dirac Hamiltonian was potentially unreliable for electric properties, for magnetic properties it simply will not work. The reason is that magnetic perturbations enter through the vector potential A and thus are odd operators. 

None of the transformations we have considered will transform this operator completely into an even operator. If we use the exponential ansatz of (16.23), the first term in the 

Only the second term of this operator is even the other two are odd. 

We must therefore derive a transformation that includes the vector potential. In whichever way we define the sequence of transformations, we are using a momentum-dependent operator in the first step—and indeed in subsequent steps as well. The vector potential A enters the Dirac equation in the same way as the momentum p. Consequently, any transformation that contains the momentum must also contain the vector potential. The correct procedure is to replace p with jr = p+eA in the transformation, and transform the Hamiltonian with this new transformation operator. 

From our experience to date with the transformations, we can immediately foresee a problem. If the transformation is some complicated function of the momentum, we might not be able to separate out the perturbation from the zeroth-order Hamiltonian. This would be unfortunate, because magnetic operators break Kramers symmetry and we would be forced to perform calculations without spin (or time-reversal) symmetry. We might also be forced to perform finite-field calculations. We will address this problem as it arises. 

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