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Prog. Theor. Phys. Vol. 18 No. 2 (1957) pp. 121-138

Covariant Solutions of the Bethe-Salpeter Equation

H. S. Green and S. N. Biswas

University of Adelaide, S. Australia

(Received April 22, 1957)

Abstract:

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 equation is reduced to a pair of coupled differential equations in which the Dirac matrices appear only in the coupling. In 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 non-relativistic theories, requiring a new quantum number for their enumeration. Jastrow's hypothesis of a repulsive “core” interaction is rigorously established, and the singularity is isolated.
It is shown how to obtain solutions corresponding to states of higher angular momentum from those with J = 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.

URL : http://ptp.ipap.jp/link?PTP/18/121/
DOI : 10.1143/PTP.18.121

[ Full Text PDF : FREE ACCESS (1375K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

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Citing Article(s) :

1. Progress of Theoretical Physics Vol. 19 No. 6 (1958) pp. 725-739 :

   General Solution of the Bethe-Salpeter Equation in Instantaneous Interaction Approximation

   S. N. Biswas

2. Progress of Theoretical Physics Vol. 23 No. 2 (1960) pp. 273-283 :

   On the Redundant Solutions of the Bethe-Salpeter Equation

   Yoshio Ohnuki, Yasutaro Takao and Hiroomi Umezawa

3. Progress of Theoretical Physics Vol. 25 No. 2 (1961) pp. 178-188 :

   Approximate Solution of the Bethe-Salpeter Equation for Two Fermions

   Masaaki Kawaguchi

4. Progress of Theoretical Physics Vol. 25 No. 5 (1961) pp. 803-821 :

   Exact Treatment of the Bound State Problems in the Non-Relativistic Quantum Field Theory

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5. Progress of Theoretical Physics Vol. 27 No. 6 (1962) pp. 1165-1187 :

   A Derivation of the Probability Density for the Bethe-Salpeter Equation

   J. Reinfelds

6. Progress of Theoretical Physics Vol. 47 No. 2 (1972) pp. 626-647 :

   Dual Realistic Quark Model

   Masako Bando, Shigeru Machida, Hisao Nakkagawa and Koichi Yamawaki

7. Progress of Theoretical Physics Vol. 51 No. 4 (1974) pp. 1159-1171 :

   Relativistic Composite Model Consistent with the Bethe-Salpeter Equation, Approximate SU(6) Symmetry and Duality. I

   Shigeru Machida

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