Transformed Operators for Electric and Magnetic Properties

In this appendix, we provide the tools for deriving transformed property operators in which the momentum appears in complicated functions, and separating the terms of these operators (Dyall 2000b). The reason that we cannot simply differentiate the functions with respect to the property operators is that the property operators do not commute with the momentum operators, and therefore the commutation must be considered explicitly. 

For this purpose, we consider the functions of momentum to be defined by their power series expansion. The expansion is not convergent for p me, but provided we do not truncate the expansion we may make transformations on the series outside its radius of convergence as a representation of the operator and re-sum the series in the 

The scalar functions of momentum that appear in the transformation operators are all functions of which we will denote x. The terms involving ( r p) 

The expansion of the commutator becomes essentially a power series in x, which means that it is an expansion whose terms can be defined by a leading power of The 

The magnetic terms for the free-particle Foldy-Wouthuysen transformation are more complicated because the operator is not linear in the momentum. To do so, we start with the field-dependent operator corresponding to Sp, which we will call p, 

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