Prog. Theor. Phys. Supplement Extra Number (1965) pp. 416-435
On the Abnormal Solution of Bethe-Salpeter Equation
— A Relation between Static and Non-Static Solutions
Department of Physics, Nagoya University, Nagoya
(Received June 29, 1965)
It is shown that the Wick-Cutkosky solution of the covariant Bethe-Salpeter equation has a close connection with that of the static model, that is, the former is a bound state within a potential obtained from the Bethe-Salpeter equation in the static model. Hence the abnormal solution can be understood as a bound state in the physically meaningless potential. Further it is proved that in the case of non-vanishing intermediate boson mass, the covariant equation has no abnormal solution if λ≦1/4. This corresponds to the fact that the redundant potential, the abnormal energy eigenvalue of the fixed source Bethe-Salpeter equation, does not appear when λ≦1/4. The correspondence between the static and the non-static models exists also in the norm of the state vector, which is found to be (-1)κ for both cases.
DOI : 10.1143/PTPS.E65.416
G. C. Wick, Phys. Rev. 96 (1954), 1124[APS].
R. E. Cutkosky, Phys. Rev. 96 (1954), 1135[APS].
- Y. Ohnuki, Y. Takao and H. Umezawa, Prog. Theor. Phys. 23 (1960), 273, [PTP]which is referred to as I.
N. Nakanishi, Phys. Rev. 138 (1965), B1182[APS].
- N. Nakanishi, preprint.
In references 4) and 5), by using the Wick-Cutkosky solution with n = l + 1, the norm (-1)κ is proved in the cases: (i) κ arbitrary, Pµ = 0, (ii) κ arbitrary, s infinitesimal, (iii) κ= 0, 0 < s ≦2, (iv) κ= 1, 0 ≦s ≦2 + (n + 2)-1, (v) κ= 0, 4 - s infinitesimal where Pµ is the total energy-momentum, and s1/2 denotes the bound-state mass in units of constituent-particle mass. The authors are grateful to Prof. Yukawa for informing them the recent works by Nakanishi before publication.
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis 4th edition (The University Press, Cambridge, 1927).