EFFECTIVE-HAMILTONIANS WITH RELATIVISTIC CORRECTIONS - THE FOLDY-WOUTHUYSEN TRANSFORMATION VERSUS THE DIRECT PAULI REDUCTION

Two different methods of obtaining ''effective 2 x 2 hamiltonians'' wh ich include relativistic corrections to nonrelativistic calculations a re discussed. The standard Foldy-Wouthuysen transformation generates h amiltonians which order by order in 1/M decouple the upper from the lo wer components. The upper left-hand block then defines an effective 2 x 2 Foldy-Wouthuysen hamiltonian. In the second method the matrix elem ent of the interaction hamiltonian of the Dirac representation is eval uated between free positive-energy states and reduced to two-component form. The resulting expression (possibly expanded in 1/M) then define s what we call the ''direct Pauli reduction'' effective 2 x 2 hamilton ian. We wish to investigate under which circumstances the two approach es yield the same result. Using a generic interaction with harmomic ti me dependence we show that differences in the corresponding effective S-matrices do arise beyond first-order perturbation theory. We attribu te them to the fact that the use of the direct reduction effective ham iltonian involves the additional approximation of neglecting contribut ions from the negative-energy intermediate states, an approximation wh ich is unnecessary in the Foldy-Wouthuysen case as there the 4 x 4 ham iltonian does not connect positive- and negative-energy states. We con clude that at least in the cases where the relativistic hamiltonian is known, using the direct Pauli reduction effective hamiltonian introdu ces spurious relativistic effects and therefore the Foldy-Wouthuysen r eduction should be preferred.