A FW transformation in two steps

Although a perturbation theory based on the FW transformation is obsolete - it is not only very laborious, but is also suffers from serious unphysical singularities - one cannot present the PT of relativistic effects, without demonstrating why the approach based on the FW transformation is bound to fail. 

There are various possible strategies for a formulation of this transformation [13], but for our purpose the FW transformation as a sequence of two transformations [49], is particularly recommended. Let us write the transformation operator Wpw as a product of two operators, such that the first one removes the non-diagonal blocks of D, while the second one reestablishes the normalization, such that the final transformation is unitary. 

Condition for vanishing of the off-diagonal blocks of the transformed matrix is that the operator X satisfies the equation [50, 12, 13] 

This equation plays a central role in relativistic quantum mechanics, far beyond the concept of a FW transformation. One can actually show that the upper and lower components (p and y of an exact 4-component spinor tp are related as [50, 12, 13] 

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