Abstract
We study quantum mechanical systems of particles with Bose or Fermi statistics interacting via two-body potentials of positive type in thermal equilibrium. We rewrite partition functions, reduced density matrices (RDMs), and correlation functions in terms of Wiener and Gaussian functional integrals (sine-Gordon transformation). This permits us, e.g., to apply correlation inequalities. Our main results include an analysis of stability versus instability in the grand canonical ensemble and, for charge-conjugation-invariant systems, upper and lower bounds on RDMs, the existence of the thermodynamic limit of pressure, RDMs and correlation functions, an inequality comparing correlations with Fermi statistics to ones with Bose statistics, and inequalities which are important in the study of Bose-Einstein condensation and of superconductivity.
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References
S. Albeverio and R. Høegh-Krohn,Commun. Math. Phys. 30:171 (1973);J. Funct. Anal. 19:242 (1973).
D. Brydges,Commun. Math. Phys. 58:313 (1978); D. Brydges and P. Federbush, Debye screening, preprint.
D. Brydges and P. Federbush,Commun. Math. Phys. 49:233 (1976);53:19 (1977).
D. Brydges, J. Fröhlich, and E. Seiler,Ann. Phys. 121:227 (1979).
S. Edwards and A. Lenard,J. Math. Phys. 3:778 (1962).
C. Fortuin, P. Kasteleyn, and J. Ginibre,Commun. Math. Phys. 22:89 (1971).
J. Fröhlich,Commun. Math. Phys. 47:233 (1976).
J. Fröhlich, inProceedings of the International Conference on Mathematical Physics, Rome, 1977 (Springer Lecture Notes in Physics 80; Springer-Verlag, Berlin, 1978).
J. Fröhlich and Y. M. Park,Helv. Phys. Acta 50:315 (1977).
J. Fröhlich and Y. M. Park,Commun. Math. Phys. 59:235 (1978).
J. Fröhlich and E. Seiler,Helv. Phys. Acta 49:889 (1976).
J. Ginibre, inStatistical Mechanics and Quantum Field Theory, Les Houches, 1970, C. DeWitt and R. Stora, eds. (Gordon & Breach, New York, 1971).
J. Ginibre,Commun. Math. Phys. 16:310 (1970).
R. B. Griffiths,Phys. Rev. 176:655 (1968).
J. L. Lebowitz,Commun. Math. Phys. 28:313 (1972);35:87 (1977).
J. L. Lebowitz,J. Stat. Mech. 16:463 (1977).
E. H. Lieb and J. L. Lebowitz,Advan. Math. 9:316 (1972); E. H. Lieb,Rev. Mod. Phys. 48:553 (1976).
A. Messager, S. Miracle-Sole, and C. Pfister,Commun. Math. Phys. 58:19 (1978).
E. Nelson,J. Math. Phys. 5:332 (1964).
E. Nelson,Dynamical Theories of Brownian Motion (Princeton Univ. Press, Princeton, New Jersey, 1967).
Y. M. Park,J. Math. Phys. 18:12 (1977).
D. Ruelle,Statistical Mechanics (W. A. Benjamin, Reading, Mass., 1969).
D. Ruelle,J. Math. Phys. 12:901 (1971);Helv. Phys. Acta 45:215 (1972);36:183 (1963).
D. W. Robinson (Springer Lecture Notes in Physics 9; Springer-Verlag, Berlin, 1971).
A. J. F. Siegert,Physica 26:30 (1960).
J. Glimm, A. Jaffe, and T. Spencer, inConstructive Quantum Field Theory, G. Velo and A. S. Withtman, eds. (Springer Lecture Notes in Physics 25, Springer-Verlag, Berlin, 1973).
J. Glimm, A. Jaffe, and T. Spencer,Ann. Phys. 101:610, 631 (1975).
F. Berezin,The Method of Second Quantization (Academic Press, New York, 1966).
B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).
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This research was done in part during the author's stay at the Department of Physics of Princeton University and was partially supported by the NSF under grant NSF PHY 76-80958.
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Fröhlich, J., Park, Y.M. Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems II. Bose-Einstein and Fermi-Dirac statistics. J Stat Phys 23, 701–753 (1980). https://doi.org/10.1007/BF01008516
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DOI: https://doi.org/10.1007/BF01008516