dorsal/arxiv
View SchemaEffective Hamiltonians with Relativistic Corrections I: The Foldy--Wouthuysen transformation versus the direct Pauli reduction
| Authors | H. W. Fearing, G. I. Poulis, S. Scherer |
|---|---|
| Categories | |
| ArXiv ID | nucl-th/9302014 |
| URL | https://arxiv.org/abs/nucl-th/9302014 |
| DOI | 10.1016/0375-9474(94)90078-7 |
| Journal | Nucl.Phys. A570 (1994) 657-685 |
Abstract
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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"abstract": "Two different methods of obtaining ``effective $2\\times 2$ Hamiltonians\u0027\u0027\nwhich include relativistic corrections to nonrelativistic calculations are\ndiscussed, the standard Foldy--Wouthuysen transformation and what we call the\n``direct Pauli reduction\u0027\u0027. We wish to investigate under which circumstances\nthe two approaches yield the same result. Using a generic interaction with\nharmonic time dependence we show that differences in the corresponding\neffective S--matrices do arise beyond first--order perturbation theory. We\nattribute them to the fact that the use of the direct reduction effective\nHamiltonian involves the additional approximation of neglecting contributions\nfrom the negative--energy intermediate states, an approximation which is\nunnecessary in the Foldy--Wouthuysen case as there the $4\\times 4$ Hamiltonian\ndoes not connect positive-- and negative--energy states. We conclude that at\nleast in the cases where the relativistic Hamiltonian is known, using the\ndirect Pauli reduction effective Hamiltonian introduces spurious relativistic\neffects and therefore the Foldy--Wouthuysen reduction should be preferred.",
"arxiv_id": "nucl-th/9302014",
"authors": [
"H. W. Fearing",
"G. I. Poulis",
"S. Scherer"
],
"categories": [
"nucl-th",
"hep-ph"
],
"doi": "10.1016/0375-9474(94)90078-7",
"journal_ref": "Nucl.Phys. A570 (1994) 657-685",
"title": "Effective Hamiltonians with Relativistic Corrections I: The Foldy--Wouthuysen transformation versus the direct Pauli reduction",
"url": "https://arxiv.org/abs/nucl-th/9302014"
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