Prog. Theor. Phys. Vol. 103 No. 4 (2000) pp. 847-857
Bi-Local Field Equation Out of Bethe-Salpeter Equation
Department of Physics, College of Science and Technology
Nihon University, Tokyo 101-8308, Japan
(Received January 13, 2000)
The bi-local (BL) field equations offer a useful phenomenological
approach to two-body bound state problems by means of relativistic
potentials, although their field theoretical basis is obscure. On the
other hand, the Bethe-Salpeter (BS) equation for two-body bound states is
obtained under an approximation within the framework of field theory. In
some cases, the BS equations are known to be reduced to the BL field
equations, since the order of the BS equations is higher than that of the
BL field equations as differential equations. In this paper, we attempt
to find a systematic method of reduction by regarding those equations as
constraints in the homogeneous canonical formalism (HCF). It is shown
that if the interaction kernel contains a delta function representing an
instantaneous interaction, then reduction is possible even for the BS
equation for two-body scalar fields. Discussion is also given on the
relation between the normalization of the BS amplitude and that of the
reduced BL field.
DOI : 10.1143/PTP.103.847
E. E. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232[APS].
The following contains the bibliography of the recent articles:
N. Nakanishi (ed.), Prog. Theor. Phys. Suppl. No. 95 (1988).
- As for the review articles of bi-local models, see the following:
T. Takabayasi, Prog. Theor. Phys. Suppl. No. 67 (1979), 1[PTP].
T. Gotō, S. Naka and K. Kamimura, ibid, 69.
S. Ishida and M. Oda, ibid, 209.
T. Takabayasi, Nuovo Cim. 33 (1964), 668.
S. Ishida and M. Oda, Proc. of the International Symposium on Extended Objects and Bound Systems, Karuizawa, Japan, 19-21 March, 1992, ed. O Hara et al. (World Scientific Pub Co., 1992), p. 181.
- T. Gotō, S. Naka and K. Yamaguchi, Prog. Theor. Phys. 64 (1980), 1[PTP].
See also the following:
P. A. M. Dirac, Proc. Camb. Phil. Soc. 29 (1933), 389.
A. Mercier, Analytical and Canonical Formalism in Physics (North-Holland, Amsterdam, 1959), p. 131.
- H. Sazdjian, Phys. Lett. 27 (1985), 381;
J. Math. Phys. 28 (1987), 2618[CrossRef].
- J. Bijtebier and J. Broekaert, Nuovo Cim. 105A (1992), 351.
- L. D. Faddeev, Theor. Math. Phys. 1 (1970), 1.
S. Naka and T. Gotō, Phys. Lett. 92B (1980), 139.
- P. A. M. Dirac, R. Mod. Phys. 21 (1949), 392.
M. Ida, Nuovo Cim. 40A (1977), 354.
S. S. Schwever, Ann. of Phys. 20 (1962), 61[CrossRef].
E. E. Salpeter, Phys. Rev. 87 (1952), 328[APS].
- C. Alabiso and G. Schierholz, Nucl. Phys. B110 (1976), 81; B126 (1977), 461.
S. N. Biswas, K. Datta and A. Goyal, Phys. Rev. D25 (1982), 2199[APS].