Higher-Order Foldy-Wouthuysen Transformations and Their Pathologies

Higher-Order Foldy-Wouthuysen Transformations and Their Pathologies 

Up to this point we have just been able to decouple a potential-free Dirac Hamiltonian exactly. In the presence of a scalar potential we are able to 

After having carried out an initial free-particle Foldy-Wouthuysen transformation Uo one could try to establish a sequence of further xmitary transformations Ui = ), i = 1,2,3.) written as 

The resulting expression for the expanded one-electron operator can then be formally written as 

Wouthuysen in 1950 [609] and has been presented in many textbooks dealing with relativistic quantum mechanics ever since [86,98,135,142], However, only the lowest-order corrections up to 0 c ) are discussed in these presentations without any warning not to proceed to higher orders in 1/c. The lowest-order term So is, of course, zeroth-order in V and minus-second-order in 1 /c due to the rest mass term — l)mgC, as can be verified by a formal Taylor series expansion of Eq. (11.37) in 1/c. 

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