Effective hamiltonians with relativistic corrections: The Foldy-Wouthuysen transformation versus the direct Pauli reduction

Two different methods of obtaining “effective 2 × 2 hamiltonians” which include relativistic corrections to nonrelativistic calculations are discussed. The standard Foldy-Wouthuysen transformation generates hamiltonians which order by order in 1/M decouple the upper from the lower components. The upper left-hand block then defines an effective 2 × 2 Foldy-Wouthuysen hamiltonian. In the second method the matrix element of the interaction hamiltonian of the Dirac representation is evaluated between free positive-energy states and reduced to two-component form. The resulting expression (possibly expanded in 1/M) then defines what we call the “direct Pauli reduction” effective 2 × 2 hamiltonian. We wish to investigate under which circumstances the two approaches yield the same result. Using a generic interaction with harmonic time dependence we show that differences in the corresponding effective S-matrices do arise beyond first-order perturbation theory. We attribute them to the fact that the use of the direct reduction effective hamiltonian involves the additional approximation of neglecting contributions from the negative-energy intermediate states, an approximation which is unnecessary in the Foldy-Wouthuysen case as there the 4 × 4 hamiltonian does not connect positive- and negative-energy states. We conclude that at least in the cases where the relativistic hamiltonian is known, using the direct Pauli reduction effective hamiltonian introduces spurious relativistic effects and therefore the Foldy-Wouthuysen reduction should be preferred.