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PTP Online Top > Supplement (1995) > No.121

Prog. Theor. Phys. Supplement No.121 (1995) pp. 1-428

[ Full Text PDF : FREE ACCESS (35425K) ]

Novel Use of Legendre Transformation in Field Theory and Many Particle Systems

— On-Shell Expansion and Inversion Method —

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

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

(Received May 24, 1995)

Abstract:

The Legendre transformation is one of the ways to rewrite the theory by small number of selected variables. It leads naturally to the generalized action functional called the effective action and everything stars from this generating functional. It is applied to various problems of physics ranging from the relativistic field theory to the condensed matter physics. The main techniques used here are the on-shell expansion and the inversion method.
The scheme presented in this article is thus a generating functional formalism of the many particle systems.


URL : http://ptp.ipap.jp/link?PTPS/121/1/
DOI : 10.1143/PTPS.121.1

[ Full Text PDF : FREE ACCESS (35425K) ] Citation:

References:

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  12. [Chap. XII]
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    2) L. V. Keldysh, Zh. Eksp. Teor. Fiz. 39 (1960), 167 [Sov. Phys. -JETP 12 (1961), 142].
    3) R. P. Feynman and F. L. Vernon, Ann. of Phys. 24 (1963), 118[CrossRef].
    4) A. J. Niemi and G. W. Semenoff, Ann. of Phys. 152 (1984), 105[CrossRef]; Nucl. Phys. B 230 (1984), 181[CrossRef].
    5) K. Chou, Z. Su, B. Hao and L. Yu. Phys. Rep. 118 (1985), 1[CrossRef].
    6) R. Fukuda, Prog. Theor. Phys. 77 (1987), 825[PTP]; ibid. 77 (1987), 845[PTP].
    7) N. P. Landsman and Ch. G. van Weert, Phys. Rep. 145 (1987), 141[CrossRef].
    8) H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cim. 1 (1955), 205.
    9) As a textbook, see for example, C. Izykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
    10) See, for example, A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle System (McGraw-Hill, New York, 1971).
    11) Y. Suzuki and R. Fukuda, Phys. Rev. E 53 (1996), 4632[APS].
    12) M. Girardeau and R. Arnowitt, Phys. Rev. 106 (1957), 1117[APS]; ibid. 106 (1957), 1128[APS].
    D. N. Zubarev and Iu. A. Tserkov, Sov. Phys. -Dokl. 3 (1958), 603.
    G. Wentzel, Phys. Rev. 120 (1960), 1572[APS].
    M. Luban, Phys. Rev. 128 (1962), 965[APS].
    F. Takano, Phys. Rev. 123 (1961), 699[APS].
    A. Coniglio and M. Marinaro, Nuovo Cim. 48 (1967), 262.
    13) M. Sumino and R. Fukuda, J. Phys. Soc. Jpn. 59 (1990), 3553[JPSJ].
    14) M. Sumino, R. Fukuda and H. Higurashi, Prog. Theor. Phys. 86 (1991), 617[PTP].
    15) R. Fukuda, M. Sumino and K. Nomoto, Phys. Rev. A 45 (1992), 3559[APS].
    16) R. Fukuda and M. Sumino, Phys. Rev. A 44 (1991), 6252[APS].
    17) A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Method of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood, N. J., 1963).
    18) L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (W. A. Benjamin Publishing Company Inc., New York, 1962).
    19) Y. Takahashi and H. Umezawa, Collect. Phenom. 2 (1975), 55.
    H. Umezawa, H. Matsumoto and M. Tachiki, Thermo Field Dynamics and Condensed States (North Holland Publishing Company, Amsterdam, 1982).
    H. Matsumoto, J. Ojima and H. Umezawa, Ann. of Phys. 152 (1984), 348[CrossRef].
    20) M. Wagner, Phys. Rev. B 44 (1991), 6104[APS].
    21) R. Brout, Physica 22 (1956), 509[CrossRef].
    F. Mayne and I. Prigogine, Physica 22 (1956), 621[CrossRef].
    I. Prigogine, Non-Equilibrium Statistical Mechanics (Interscience Publishers Inc., New York, 1962).
    F. Mayne and I. Prigogine, Physica 63 (1973), 1[Elsevier].
    22) S. Fujita, J. Math. Phys. 6 (1965), 1877[CrossRef]; Non-Equilibrium Quantum Statistical Mechanics (Saunders Publishing Company, Philadelphia, 1966); J. Phys. Soc. Jpn. 27 (1969), 1096[JPSJ]; Phys. Rev. A 4 (1971), 1114[APS]; Physica 64 (1973), 1[Elsevier]; ibid. 64 (1973), 23[Elsevier].
    23) A. G. Hall, J. of Phys. A 8 (1975), 214[CrossRef]; Physica A 80 (1975), 369.
    Yu. A. Kukharenko and S. G. Tikhodeev, Zh. Eksp. Teor. Fiz. 83 (1982), 1444 [Sov. Phys. -JETP 56 (1982), 831].
    24) R. Fukuda, Prog. Theor. Phys. 85 (1991), 441[PTP].
    25) A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46 (1981), 211[APS]; Ann. of Phys. 149 (1983), 374 [CrossRef][ Errata; 153 (1984), 445[CrossRef]].
    See also, A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zweger, Rev. Mod. Phys. 59 (1987), 1[APS].
  13. [Chap. XIII]
    1) K. Shiraishi and R. Fukuda, Prog. Theor. Phys. 92 (1994), 863[PTP].
    2) E. E. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232[APS].
    3) For the review of field theoretic bound state problem,
    N. Nakanishi, Prog. Theor. Phys. Suppl. No. 43 (1969), 1[PTP]; ibid. No. 95 (1988), 1[PTP].
    4) G. C. Wick, Phys. Rev. 96 (1954), 1124[APS].
    R. E. Cutkosky, Phys. Rev. 96 (1954), 1135[APS].
    5) N. Mugibayashi, Prog. Theor. Phys. 25 (1961), 803[PTP].
    Y. Ohnuki and K. Watanabe, Prog. Theor. Phys. Suppl. Extra Number (1965), 416[PTP].
    6) N. Nakanishi, Phys. Rev. 138 (1965), B1182 [APS][Errata; 139 (1965), AB1]; Phys. Rev. 139 (1965), B1401[APS].
    7) K. Higashijima, KEK Preprint (1992).
    8) G. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Wiley, Interscience Pub. 1972); Mathematical and Conceptual Foundations of 20th-Century Physics (North-Holland Pub. 1984).
    D. Ruelle, Statistical Mechanics, Rigorous Results (W. A. Benjamin, Inc., 1969).
    9) R. Fukuda, Prog. Theor. Phys. 85 (1991), 441[PTP].
    10) M. Gell-Mann and F. E. Low, Phys. Rev. 84 (1951), 350[APS].
    11) H. Araki, Prog. Theor. Phys. 64 (1980), 719[PTP].
    12) R. Fukuda, in Proc. of 3rd Int. Symp. on Foundations of Quantum Mechanics in the Light of New Technology (ISQM) (Phys. Soc. Jpn. 1989); Proc. of Int. Colloq. on Quantum Theory Without State Vector Collapse (IOP Ltd.-Adam Higler, 1989).
    13) R. Fukuda, Phys. Rev. A 38 (1987), 8[APS]; ibid. 39 (1987), 3023[APS]; Prog. Theor. Phys. 81 (1989), 34[PTP].
    14) H. Everett III, Rev. Mod. Phys. 29 (1957), 454[APS].
  14. [Chap. XIV]
    1) J. Gavoret and P. Nosieres, Ann. of Phys. 28 (1964), 349[CrossRef].
    2) Y. A. Nepomnyashchii and A. A. Nepomnyashchii, Sov. Phys. -JETP 48 (1978), 493; JETP Lett. 21 (1975), 1.
    3) L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics (Pergamon Press, 1959), vol. 6.
    4) P. C. Hohenberg and P. C. Martin, Ann. of Phys. 34 (1965), 291[CrossRef].
    5) N. N. Bogolyubov, Jr., M. Yu. Kovalevskii, A. M. Kurbatov, S. V. Peletminskii and A. N. Tarasov, Usp. Fiz. Nauk. 159 (1989), 585.
    6) A. E. Glassgold, A. N. Kaufman and K. M. Watson, Phys. Rev. 120 (1960), 660[APS].
    7) A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle System (McGraw-Hill, New York, 1971).
    8) J. Schwinger, J. Math. Phys. 2 (1961), 407[CrossRef].
    9) L. V. Keldysh, Zh. Eksp. Teor. Fiz. 39 (1960), 167 [Sov. Phys. -JETP 12 (1961), 142].
    10) K. Chou, Z. Su, B. Hao and L. Yu, Phys. Rep. 118 (1985), 1[CrossRef].
    11) A. J. Niemi and G. W. Semenoff, Ann. of Phys. 152 (1984), 105[CrossRef]; Nucl. Phys. B 230 (1984), 181[CrossRef].
    12) M. Wagner, Phys. Rev. B 44 (1991), 6104[APS].
    13) R. Fukuda, M. Sumino and K. Nomoto, Phys. Rev. A 45 (1992), 3559[APS].
    R. Fukuda and M. Sumino, Phys. Rev. A 44 (1991), 6252[APS].
    14) C. De Dominicis and P. C. Martin, J. Math. Phys. 5 (1964), 14[CrossRef]; ibid. 5 (1964), 31[CrossRef].
    15) G. Jona-Lasinio, Nuovo Cim. 34 (1964), 1790.
    R. Jackiw, Phys. Rev. D 9 (1974), 1686[APS].
    M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D 10 (1974), 2428[APS].
    R. Fukuda and E. Kyriakopoulos, Nucl. Phys. B 85 (1975), 345.
    16) P. B. Weichman, Phys. Rev. B 38 (1988), 8739[APS].
    17) M. Girardeau and R. Arnowitt, Phys. Rev. 113 (1959), 755[APS].
    18) M. Girardeau and R. Arnowitt, Phys. Rev. 106 (1957), 1117[APS]; ibid. 106 (1957), 1128[APS].
    19) M. Girardeau, Phys. Rev. 115 (1959), 1090[APS]; J. Math. Phys. 3 (1962), 131[CrossRef].
    20) D. N. Zubarev and Iu. A. Tserkov, Sov. Phys. -Dokl. 3 (1958), 603.
    21) G. Wentzel, Phys. Rev. 120 (1960), 1572[APS].
    22) M. Luban, Phys. Rev. 128 (1962), 965[APS].
    23) F. Takano, Phys. Rev. 123 (1961), 699[APS].
    24) A. Coniglio and M. Marinaro, Nuovo Cim. 48 (1967), 262.
    25) Y. Suzuki and R. Fukuda, “Superfluid 4He through Legendre Transformation–Pairing Theory–”, Keio Univ. Preprint, 1995.
    26) N. M. Hugenholtz and D. Pines, Phys. Rev. 116 (1959), 489[APS].
    27) Y. Suzuki and R. Fukuda, “Microscopic Derivation of Two Fluid Model of 4He through Legendre Transformation”, Keio Univ. Preprint, 1995.
    28) Y. Suzuki and R. Fukuda, Phys. Rev. E 53 (1996), 4632[APS].
  15. [Appendix A]
    1) G. Jona-Lasinio, Nuovo Cim. 34 (1964), 1790.
    2) R. W. Haymaker, Riv. Nuovo Cim. 14 serie 3 (1991), No. 8.
  16. [Appendix B]
    1) T. Kanki, K. Murata and S. Sunakawa, Prog. Theor. Phys. 17 (1957), 7[PTP].
    2) H. Higgurashi and R. Fukuda, Prog. Theor. Phys. 83 (1990), 305[PTP].
    3) R. P. feynman and A. R. Hibbs, Quantum Mechanics and Path Integral (McGraw-Hill, Inc., New York, 1965).
  17. [Appendix C]
    1) C. De Dominicis and P. C. Martin, J. Math. Phys. 5 (1964), 14[CrossRef]; ibid. 5 (1964), 31[CrossRef].
    2) R. Jackiw, Phys. Rev. D 9 (1974), 1686[APS].
    3) M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D 10 (1974), 2428[APS].
    4) R. Fukuda and E. Kyriakopoulos, Nucl. Phys. B 85 (1975), 345.
    5) K. Okumura, Prog. Theor. Phys. 87 (1992), 703[PTP].
    6) See, for example, K. B. Efetov, Adv. Phys. 32 (1983), 53.
  18. [Appendix E]
    1) M. Sumino, R. Fukuda and H. Higurashi, Prog. Theor. Phys. 86 (1991), 617[PTP].
    2) R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, Inc., 1965).
  19. [Appendix F]
    1) K. Chou, Z. Su, B. Hao and L. Yu, Phys. Rep. 118 (1985), 1[CrossRef].
    2) R. Fukuda, Prog. Theor. Phys. 77 (1987), 825[PTP]; ibid. 77 (1987), 845[PTP].
    3) K. Nomoto and R. Fukuda, Phys. Rev. D 46 (1992), 1680[APS].
    4) M. Sumino, R. Fukuda and H. Higurashi, Prog. Theor. Phys. 86 (1991), 617[PTP].
  20. [Appendix G]
    1) R. Fukuda, M. Sumino and K. Nomoto, Phys. Rev. A 45 (1992), 3559[APS].
    2) T. Matsubara, Prog. Theor. Phys. 14 (1955), 351[PTP].
  21. [Appendix H]
    1) K. Okumura, Ph. D. Thesis, Keio Univ. 1994; Phys. Rev. B 52 (1995), 13358[APS].

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