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Prog. Theor. Phys. Supplement No.92 (1987) pp. 1-13
Scalar Field Theory from Statistical Mechanics
Takashi Hara,
Tetsuya Hattori,*
Kei-ichi Kondo** and
Hal Tasaki***
Courant Institute of Mathematical Sciences, New York University, New York, N. Y. 10012, U.S.A.
*Department of Physics, Gakushuin University, Tokyo 171
**Department of Physics, Nagoya University, Nagoya 464
***Physics Department, Princeton University, Princeton, N. J. 08544, U.S.A.
(Received August 12, 1987)
Abstract:
We present an introduction to the basic ideas of rigorous approaches to the quantum scalar field theories which are based on the statistical mechanical point of view. We also discuss several aspects of the rigorous studies in critical phenomena and phase transitions, and refer to some related open questions.
URL :
http://ptp.ipap.jp/link?PTPS/92/1/
DOI : 10.1143/PTPS.92.1
References:
- J. Glimm and A. Jaffe, Quantum Physics, 2nd ed. (Springer-Verlag, New York, 1987).
- J. Fröhlich, in Progress in Gauge Field Theory (Cargese Lectures, 1983), ed. G. 't Hooft et al. (Plenum Press, New York, 1984).
- G. A. Baker, Jr., in Phase Transitions and Critical Phenomena, ed. C. Domb and M. Green (Academic Press, London, 1984), vol. 9.
- K.-I. Kondo, Soryusiron Kenkyu (Kyoto) 70 (1984), 11.
- H. Tasaki, private communication, Bussei Kenkyu 42 (1984), 739.
-
D. C. Brydges, J. Fröhlich and A. D. Sokal, Commun. Math. Phys. 91 (1983), 141[CrossRef].
See also A. D. Sokal, Ann. Inst. Henri Poincaré 37 (1982), 317.
-
K. Osterwalder an R. Schrader, Commun. Math. Phys. 31 (1973), 83[CrossRef];
ibid. 42 (1975), 281[CrossRef].
See also K.-I. Kondo, Butsuri 42 (1987) (in Japanese).
- R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (Benjamin, New York, 1964).
-
D. C. Brydges, J. Fröhlich and T. Spencer, Commun. Math. Phys. 83 (1982), 123[CrossRef].
D. C. Brydges, J. Fröhlich and A. D. Sokal, Commun. Math. Phys. 91 (1983), 117[CrossRef].
See T. Hattori, Suuri Kagaku 272 (1986), 27, for a guide for beginners (in Japanese).
- For a systematic derivation of related inequalities, see
G. S. Sylvester, J. Stat. Phys. 15 (1976), 327.
See S. B. Schlosman, Commun. Math. Phys. 102 (1986), 679, for some of the recent results.
Also see G. S. Sylvester, Commun. Math. Phys. 73 (1980), 105,
for a difficulty that arises when one tries to extend the Griffiths inequalities to multi-component systems.
- D. Ruelle, Commun. Math. Phys. 18 (1970), 127; ibid. 50 (1976), 189.
See T. Hattori, Wakate Kainhou 40 (1986), 35; ibid. 41 (1987), 76 for a guide for beginners (in Japanese).
For the particular model in question, we have a simple alternate proof by T. Hara (unpublished), see Ref. 5), p. 759.
- M. Aizenman, D. J. Barsky and R. Fernandez, J. Stat. Phys. 47 (1987), 343.
- M. Aizenman and D. J. Barsky, Commun. Math. Phys. 108 (1987), 489.
-
M. Aizenman, Phys. Rev. Lett. 47 (1981), 1[APS];
Commun. Math. Phys. 86 (1982), 1[CrossRef].
- M. Aizenman and R. Fernandez, J. Stat. Phys. 44 (1986), 393.
-
T. Hara, T. Hattori and H. Tasaki, J. Math. Phys. 26 (1985), 2922[CrossRef].
-
K.-I. Kondo and T. Hara, J. Math. Phys. 28 (1987), 1206[CrossRef].
- H. Tasaki, in Fractal Kagaku, ed. H. Takayasu et al. (Asakura, Tokyo, 1987) (in Japanese).
- H. Tasaki, Commun. Math. Phys. 112 (1987), to appear.
-
J. Fröhlich, Nucl. Phys. B 200 (1982), 281[CrossRef].
- K. Symanzik, in Local Quantum Theory, ed. R. Jost (Academic Press, New York, 1969).
- K.-I. Kondo, Nagoya Univ. Preprint (1987); J. Stat. Phys. 46 (1987), 35.
K.-I. Kondo, T. Otofuji and Y. Sugiyama, J. Stat. Phys. 40 (1985), 563.
- G. A. Baker, Jr. and A. S. Wightman, in Progress in Quantum Field Theory, ed. H. Ezawa and S. Kamefuchi (Elsevier Science Publ., 1986).
- G. Gallavotti and V. Rivasseau, Ann. Inst. Henri Poincaré 40 (1984), 185.
- K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 99 (1985), 197.
See also T. Hara and H. Tasaki, Prog. Theor. Phys. Suppl. No. 92 (1987), 26[PTP].
-
M. Aizenman and R. Graham, Nucl. Phys. B 225 (1983), 261[CrossRef].
-
C. Aragao de Carvalho, S. Caracciolo and J. Fröhlich, Nucl. Phys. B 215 (1983), 209[CrossRef].
- K.-I. Kondo, Prog. Theor. Phys. 73 (1985), 504[PTP].
-
K. G. Wilson and J. Kogut, Phys. Rep. 12 (1974), 75[CrossRef].
- T. Hara, Prog. Theor. Phys. 76 (1986), 318[PTP].
-
T. Hattori, J. Math. Phys. 24 (1983), 2200[CrossRef].
- T. Hara, unpublished;
see K.-I. Kondo, Prog. Theor. Phys. 77 (1987), 473[PTP].