Properties of the Salpeter-Bethe Two-Nucleon Equation

Goldstein, Jack

doi:10.1103/physrev.91.1516

Cited by 132 publications

(48 citation statements)

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“…This is a special example of the radial wave equation (4a) for a hypercentral inverse cube law of 4-force. Putting"Y = _KjS2 in equation (4a) we find (105) which is identical with Goldstein's (1953) equation provided R = QS and 4'1') = A2+K = n2+K-1. In the non-relativistic limit, this force gives the solutions appropriate to the r-2 potential.…”

Section: (C) Inverse Oube Law Oj Forcementioning

confidence: 64%

“…When Goldstein (1953) formulated the Bethe-Salpeter (1951) wavefunctions for two spinors interacting via the ladder exchange of neutral bosons, he obtained a radial wave equation from the quantum field theory of the form (104) in the case of equal masses. This is a special example of the radial wave equation (4a) for a hypercentral inverse cube law of 4-force.…”

Section: (C) Inverse Oube Law Oj Forcementioning

confidence: 99%

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Solutions of the Relativistic Two-Body Problem. II. Quantum Mechanics

Cook¹

1972

Aust. J. Phys.

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This paper discusses the formulation of a quantum mechanical equivalent of the relative time classical theory proposed in Part 1. The relativistic wavefunction is derived and a covariant addition theorem is put forward which allows a covariant scattering theory to be established. The free particle eigenfunctions that are given are found not to be plane waves. A covariant partial wave analysis is also given. A means is described of converting wavefunctions that yield probability densities in 4-space to ones that yield the 3·space equivalents. Bound states are considered and covariant analogues of the Coulomb potential, harmonic oscillator potential, inverse cube law of force, square well potential, and two-body fermion interactions are discussed.

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Section: (C) Inverse Oube Law Oj Forcementioning

confidence: 64%

Section: (C) Inverse Oube Law Oj Forcementioning

confidence: 99%

Solutions of the Relativistic Two-Body Problem. II. Quantum Mechanics

Cook¹

1972

Aust. J. Phys.

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“…First, we consider the spectrum for constituents of equal mass in the limit of massless bound states. Goldstein [26] was the first to consider this limiting case. He found massless solutions for all values of the coupling constant.…”

Section: Bound States In Qedmentioning

confidence: 98%

(In-) consistencies in the Relativistic Description of Excited States in the Bethe–Salpeter Equation

Ahlig

Alkofer

1999

Annals of Physics

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The Bethe Salpeter equation provides the most widely used technique to extract bound states and resonances in a relativistic quantum field theory. Nevertheless a thorough discussion of how to identify its solutions with physical states is still missing. The occurrence of complex eigenvalues of the homogeneous Bethe Salpeter equation complicates this issue further. Using a perturbative expansion in the mass difference of the constituents we demonstrate for scalar fields bound by a scalar exchange that the underlying mechanism which results in complex eigenvalues is the crossing of a normal (or abnormal) with an abnormal state. Based on an investigation of the renormalization of one-particle properties we argue that these crossings happen beyond the applicability region of the ladder Bethe Salpeter equation. The implications for a fermion antifermion bound state in QED are discussed, and a consistent interpretation of the bound state spectrum of QED is proposed. Academic Press

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“…( -i/7 +M) +Om, (l)m=s-112 fmxCfm+1 (z) +2~m(r4 -x4 xjs) C2;,. (z), Om=rrmC.Jm(z) +2ir 1 rm[x, 74]C2~(z),where z=x4 s-112 and according to(4)(5) rrm is a solution of{J0-4m(m+1) +sg(s)}rrm=O and fm, ~m' and (m=rm+~m are coupled by {p-2~4 (m+ 1) 2 -sg(s) }fm+2{4 (m+ 1) 2 -1}(m=O, {p +2-4 (m+ 1) 2 +sg(s)} (m+2fm=2s{g(s) +sg 1 (s)} ~m> P -4s· ojos;. :__4 (in+1} 2 -sg(s)} ~m+2fm+2 (2s· ojos+ 1}(m=O.…”

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confidence: 99%

Covariant Solutions of the Bethe-Salpeter Equation

Green

Biswas

1957

Prog. Theor. Phys.

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A fully covariant investigation is made of the Bethe-Salpeter equation for a pair of nucleons, with pseudo-scalar interaction. The " ladder " approximation is adopted, but pair creation and nucleonic recoil are accounted for exactly. Matrix solutions are obtained, with varying degrees of explicitness, for instantaneous and delayed interaction, vanishing and non-vanishing meson mass, and for vanishing and non-vanishing total energy. Important properties are disclosed which are either obscured or do not appear at all in non-relativistic approximation.First the Bethe-Salpeter equatiort is reduced to a pair of coupled differential equations in which the Dirac matrices appear only in the coupling. 1n the instantaneous interaction approximation, these can be reduced to a single covariant equation in a single variable, showing the radical influence of nucleon recoil on pair effects. When the instantaneous. interaction approximation is discarded, new features appear. There is a discrete infinity of stable states corresponding to each one of the nonrelativistic theories, requiring a new quantum number for their enumeration. Jastrow's hypothesis of a repulsive "core" interaction is rigorously f'Stablished, and the singularity is isolated. It is shown how to obtain solutions corresponding to states of higher angular momentum from those with ]=0, making use of the relativistic quantum enumeration. The conclusion is drawn that the relativistic quantum number is a property of the state of any pair of interacting particles, and its possible connection with the " strangeness " number is discussed. § 1. IntroductionIt has been suspected for some time that the covariant wave equations, which have been derived on the basis of quantized field theory/> have certain features differing essentially from those of corresponding SchrOdinger equations. It is true that by making non-covariant approximations it has been possible to obtain analogues of the Schrodinger equations, even to the extent of predicting effective potentials. Among others Salpeter and Bethe, 2 ' Uvy 31 and Klein 4 > have treated the inter-nucleon interaction in this way. But, on removal of the non-covariant approximations difficulties arise. It was first noticed by Goldstein, 5 > in a special (unrealistic) case, that the Bethe-Salpeter (B-S) equation appeared not to possess solutions of the type expected for the prediction of bound states. A variety of diagnoses of this trouble have been made, blaming sometimes the extreme value of the binding energy assumed by Goldstein and sometimes the highly singular nature of the field-theoretical potential. The remedy suggested by Goldstein himself was to introduce a less singular potential which by a limiting process might be made to coincide with the exact theoretical potential. Though such, a procedure does not work 6 > in the extreme case studied by Goldstein, possibly it would do for less extreme binding at

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Properties of the Salpeter-Bethe Two-Nucleon Equation

Cited by 132 publications

References 9 publications

Solutions of the Relativistic Two-Body Problem. II. Quantum Mechanics

Solutions of the Relativistic Two-Body Problem. II. Quantum Mechanics

(In-) consistencies in the Relativistic Description of Excited States in the Bethe–Salpeter Equation

Covariant Solutions of the Bethe-Salpeter Equation

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