Origin of Picture Change Errors

The trace of this one-electron density matrix D gives the one-electron (probability) density p, 

This representation is called the Dirac picture. Alternatively, this expectation value may be reformulated within the two-component DKH framework employing the large component of the transformed spinor ip only, 

It must be noted that the unitary transformation U and its adjoint do not commute with the property operator X even if it was a multiplicative electric-field-like operator in position space. The reason for this is the fact that the transformation U always contains momentum operators. 

Solely due to the vanishing small component of the transformed (decoupled) spinor ip only the upper left block X  

From these considerations it should be unambiguously clear that the change of picture is an artificial effect resulting from basic properties of the theory of unitary linear transformations rather than a complicated feature of two-component theories. It has already been discussed by Newton and Wigner [863] and Foldy and Wouthuysen [609]. 

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