Effective Hamiltonians with Relativistic Corrections I: The Foldy--Wouthuysen transformation versus the direct Pauli reduction

Details Effective Hamiltonians with Relativistic Corrections I: The Foldy--Wouthuysen transformation versus the direct Pauli reduction Effective Hamiltonians with Relativistic Corrections I: The Foldy--Wouthuysen transformation versus the direct Pauli reduction

1993-02-25

arxiv.org/abs/nucl-th/9302014v1

Nucl.Phys. A570 (1994) 657-685

Physics

Nuclear Physics

Nuclear Theory

TRIUMF preprint TRI-PP-93-12

Scientific paper

10.1016/0375-9474(94)90078-7

Two different methods of obtaining ``effective $2\times 2$ Hamiltonians'' which include relativistic corrections to nonrelativistic calculations are discussed, the standard Foldy--Wouthuysen transformation and what we call the ``direct Pauli reduction''. 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 $4\times 4$ 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.

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