Prog. Theor. Phys. Vol. 87 No. 3 (1992) pp. 703-726

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Derivation of the 4-Body Bound-State Equation from the Effective Action

— Diagrammatical Legendre Transformation —

Ko Okumura

Department of Physics, Faculty of Science and Technology, Keio University, Yokoyama 223

(Received October 30, 1991)

Abstract:

The 4-body bound-state equation is derived by use of on-shell expansion of the effective action. Derivation is based on the second derivative of the effective action which makes the graphical rule much easier. The essential tool is the sum-up rule of the graphs which is equivalent to the Legendre transformation. This rule can be widely used for any operator that has external point(s). Applied to the 1, 2, 3-body channels our method gives another way of deriving the known bound-state equations. The obtained equations hold whether the vacuum state is condensed or not.

URL : http://ptp.ipap.jp/link?PTP/87/703/
DOI : 10.1143/PTP.87.703

References:

Y. Nambu, Prog. Theor. Phys. 5 (1950), 614[PTP].
E. E. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232[APS].
N. Nakanishi, Prog. Theor. Phys. Suppl. No. 43 (1969); Prog. Theor. Phys. Suppl. No. 95 (1988), ed. N. Nakanishi.
S. Machida and H. Nakagawa, Prog. Theor. Phys. 54 (1975), 243[PTP]; ibid. 54 (1975), 510[PTP].
K. Tesima, Prog. Theor. Phys. Suppl. No. 77 (1983), 285[PTP].
M. Komachiya, M. Ukita and R. Fukuda, Phys. Rev. D40 (1990), 2654[APS].
J. Glimm and A. Jaffe, Lecture Notes in Physics (Springer, Berlin, 1975), 39, p. 118.
M. Combescure and F. Dunlop, Ann. of Phys. 122 (1979), 102[CrossRef].
A. Cooper, J. Feldman and L. Rosen, Ann. of Phys. 137 (1981), 146[CrossRef]; ibid. 137 (1981), 213[CrossRef]; J. Math. Phys. 23 (1982), 846[CrossRef]; Phys. Rev. D25 (1982), 1565[APS].
Y. Yokojima, M. Komachiya and R. Fukuda, Keio Univ. Preprint (1991).
R. Fukuda, Prog. Theor. Phys. 78 (1987), 1487[PTP].
C. De Dominicis and P. C. Martin, J. Math. Phys. 5 (1964), 14[CrossRef]; ibid. 5 (1964), 31[CrossRef].
A. N. Vasil'ev and A. K. Kazanskii, Theor. Math. Phys. 12 (1972), 875; ibid. 14 (1973), 215.
Yu. M. Pis'mak, Theor. Math. Phys. 18 (1974), 211.
A. N. Vasil'ev, A. K. Kazanskii and M. Pis'mak, Theor. Math. Phys. 19 (1974), 443; ibid. 20 (1974), 754.
T. Spencer, Commun. Math. Phys. 44 (1975), 144.
G. Jona-Lasinio, Nuovo Cim. 34 (1964), 1790.
For example,
S. Coleman and E. Weinberg, Phys. Rev. D7 (1973), 1888[APS].
R. Jackiw, Phys. Rev. D9 (1974), 1686[APS].
John M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974), 2428[APS].
D. J. Gross and A. Noveu, Phys. Rev. D10 (1974), 3235[APS].
R. Fukuda, M. Komachiya and M. Ukita, Phys. Rev. D38 (1988), 3747[APS].
M. Komachiya, M. Ukita and R. Fukuda, Phys. Rev. D42 (1990), 2792[APS].

Citing Article(s) :

Progress of Theoretical Physics Supplement No.121 (1995) pp. 1-428 :
Novel Use of Legendre Transformation in Field Theory and Many Particle Systems

Reijiro Fukuda, Masahiro Komachiya, Satoshi Yokojima, Yoko Suzuki, Ko Okumura and Takeshi Inagaki