dorsal/arxiv
View SchemaRelativistic bound-state equations in three dimensions
| Authors | D. R. Phillips, S. J. Wallace |
|---|---|
| Categories | |
| ArXiv ID | nucl-th/9603008 |
| URL | https://arxiv.org/abs/nucl-th/9603008 |
| DOI | 10.1103/PhysRevC.54.507 |
| Journal | Phys.Rev.C54:507-522,1996 |
Abstract
Firstly, a systematic procedure is derived for obtaining three-dimensional bound-state equations from four-dimensional ones. Unlike ``quasi-potential approaches'' this procedure does not involve the use of delta-function constraints on the relative four-momentum. In the absence of negative-energy states, the kernels of the three-dimensional equations derived by this technique may be represented as sums of time-ordered perturbation theory diagrams. Consequently, such equations have two major advantages over quasi-potential equations: they may easily be written down in any Lorentz frame, and they include the meson-retardation effects present in the original four-dimensional equation. Secondly, a simple four-dimensional equation with the correct one-body limit is obtained by a reorganization of the generalized ladder Bethe-Salpeter kernel. Thirdly, our approach to deriving three-dimensional equations is applied to this four-dimensional equation, thus yielding a retarded interaction for use in the three-dimensional bound-state equation of Wallace and Mandelzweig. The resulting three-dimensional equation has the correct one-body limit and may be systematically improved upon. The quality of the three-dimensional equation, and our general technique for deriving such equations, is then tested by calculating bound-state properties in a scalar field theory using six different bound-state equations. It is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four-dimensional equations significantly better than the corresponding quasi-potential equations do.
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"abstract": "Firstly, a systematic procedure is derived for obtaining three-dimensional\nbound-state equations from four-dimensional ones. Unlike ``quasi-potential\napproaches\u0027\u0027 this procedure does not involve the use of delta-function\nconstraints on the relative four-momentum. In the absence of negative-energy\nstates, the kernels of the three-dimensional equations derived by this\ntechnique may be represented as sums of time-ordered perturbation theory\ndiagrams. Consequently, such equations have two major advantages over\nquasi-potential equations: they may easily be written down in any Lorentz\nframe, and they include the meson-retardation effects present in the original\nfour-dimensional equation. Secondly, a simple four-dimensional equation with\nthe correct one-body limit is obtained by a reorganization of the generalized\nladder Bethe-Salpeter kernel. Thirdly, our approach to deriving\nthree-dimensional equations is applied to this four-dimensional equation, thus\nyielding a retarded interaction for use in the three-dimensional bound-state\nequation of Wallace and Mandelzweig. The resulting three-dimensional equation\nhas the correct one-body limit and may be systematically improved upon. The\nquality of the three-dimensional equation, and our general technique for\nderiving such equations, is then tested by calculating bound-state properties\nin a scalar field theory using six different bound-state equations. It is found\nthat equations obtained using the method espoused here approximate the wave\nfunctions obtained from their parent four-dimensional equations significantly\nbetter than the corresponding quasi-potential equations do.",
"arxiv_id": "nucl-th/9603008",
"authors": [
"D. R. Phillips",
"S. J. Wallace"
],
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"doi": "10.1103/PhysRevC.54.507",
"journal_ref": "Phys.Rev.C54:507-522,1996",
"title": "Relativistic bound-state equations in three dimensions",
"url": "https://arxiv.org/abs/nucl-th/9603008"
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