Structure of the Spinor Rotation Operator

In the QED approach k also contains terms of the form a/5, ap aq and a ap, which were not covered by the discussion of the previous section. We therefore need to investigate how these terms affect the first-order correction to the energy, (01, K j 10). However, we must apply the normal ordering after we have performed 

To illustrate this approach, it is again sufficient to consider the one-electron part of the Hamiltonian, and the relevant commutator in the Dirac approach (filled or empty) becomes 

Collecting all the terms arising from evaluation of the commutator, we get from the Dirac approach (empty or filled) 

If we now take the reference expectation of this operator in the QED approach, 

Thus we find that in both the QED approach and the empty Dirac approach all terms except the first vanish, because the reference contains only electrons in positive-energy states. But in the filled Dirac approach all terms in principle survive. Thus the properly normal-ordered QED approach gives the same result for this term in the energy expansion as the empty Dirac approach, and not the same as the filled Dirac approach. The parameter space for the QED approach is therefore identical with that of the empty Dirac approach. 

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