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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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The pure phases (harmonic functions) of generalized processes or: Mathematical physics of phase transitions and symmetry breaking
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by J. Fröhlich PDF
Bull. Amer. Math. Soc. 84 (1978), 165-193
  • J. Fröhlich, B. Simon, and Thomas Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Comm. Math. Phys. 50 (1976), no. 1, 79–95. MR 421531, DOI 10.1007/BF01608557
  • Freeman J. Dyson, Elliott H. Lieb, and Barry Simon, Phase transitions in the quantum Heisenberg model, Phys. Rev. Lett. 37 (1976), no. 3, 120–123. MR 432128, DOI 10.1103/PhysRevLett.37.120
  • 3. J. Fröhlich and E. H. Lieb, Existence of phase transitions for anisotropic Heisenberg models, (to appear). See also Phys. Rev. Lett. 38 (1977), 440. (Note: The proofs of the results on quantum mechanical ferromagnets announced in references 2 and 3 contain a gap, as they were based on an incorrect lemma of 2.) 4. J. Fröhlich, R. Israel, E. H. Lieb and B. Simon, papers concerning phase transitions in lattice systems with long range interactions (to appear).
  • J. Fröhlich, Phase transitions, Goldstone bosons and topological superselection rules, Current problems in elementary particle and mathematical physics (Proc. XV. Internat. Universitätswochen Kernphysik, Univ. Graz, Schladming, 1976) Acta Phys. Austriaca, Suppl. XV, Springer, Vienna, 1976, pp. 133–269. MR 0523547
  • J. Fröhlich and T. Spencer, Phase transitions in statistical mechanics and quantum field theory, New developments in quantum field theory and statistical mechanics (Proc. Cargèse Summer Inst., Cargèse, 1976) NATO Adv. Study Inst. Ser. B: Physics, vol. 26, Plenum, New York-London, 1977, pp. 79–130. MR 508189
  • James Glimm, Arthur Jaffe, and Thomas Spencer, Phase transitions for $\phi _{2}^{4}$ quantum fields, Comm. Math. Phys. 45 (1975), no. 3, 203–216. MR 391797, DOI 10.1007/BF01608328
  • 8. J. Glimm, Ann. Physics 101 (1976), 610; 101 (1976), 631. 9. D. Ruelle, Statistical mechanics, Math. Phys. Monograph Ser., Benjamin, London and Amsterdam, 1969. 10. D. Ruelle, in Méchanique Statistique et Théorie Quantique des Champs, Les Houches 1970, C. DeWitt and R. Stora, editors, Gordon and Breach, New York, 1971.
  • A. Lenard (ed.), Statistical mechanics and mathematical problems, Lecture Notes in Physics, Vol. 20, Springer-Verlag, Berlin-New York, 1973. Battelle Rencontres, Seattle, Wash., 1971. MR 0395622
  • Huzihiro Araki and P. D. F. Ion, On the equivalence of $KMS$ and Gibbs conditions for states of quantum lattice systems, Comm. Math. Phys. 35 (1974), 1–12. MR 376049, DOI 10.1007/BF01646450
  • Huzihiro Araki, On the equivalence of the KMS condition and the variational principal for quantum lattice systems, Comm. Math. Phys. 38 (1974), 1–10. MR 475531, DOI 10.1007/BF01651545
  • 14. R. Israel, Tangents to the pressure as invariant equilibrium states in statistical mechanics of lattice systems, Thesis, Princeton Univ., 1975; Princeton Ser. in Physics, Princeton Univ. Press, Princeton, N.J. (to appear). See also Comm. Math. Phys. 43 (1975), 59, and D. Ruelle, On manifolds of phase coexistence, I.H.E.S. Preprint, 1975. 15. R. L. Dobrushin, Functional Anal. Appl. 2 (1968, 302; see also Thcor. Probability Appl. 13 (1968), 197.
  • Edward Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis 12 (1973), 97–112. MR 0343815, DOI 10.1016/0022-1236(73)90091-8
  • Frank Spitzer, Markov random fields and Gibbs ensembles, Amer. Math. Monthly 78 (1971), 142–154. MR 277036, DOI 10.2307/2317621
  • Robert B. Israel, High-temperature analyticity in classical lattice systems, Comm. Math. Phys. 50 (1976), no. 3, 245–257. MR 446250, DOI 10.1007/BF01609405
  • D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys. 9 (1968), 267–278. MR 234697, DOI 10.1007/BF01654281
  • 19. J. Fröhlich, Lectures on equilibrium statistical mechanics, Princeton Univ., 1976/77 and paper in preparation.
  • Huzihiro Araki, On representations of the canonical commutation relations, Comm. Math. Phys. 20 (1971), 9–25. MR 290709, DOI 10.1007/BF01646731
  • Hermann Weyl, Symmetry, Princeton University Press, Princeton, N. J., 1952. MR 0048449, DOI 10.1515/9781400874347
  • 22. S. Coleman, Secret symmetry: An introduction to spontaneous symmetry breakdown and gauge fields, Proc. of the 1973 Internat. Summer School of Physics, Ettore Majorana, A. Zicchichi, editors.
  • Jürg Fröhlich, Schwinger functions and their generating functionals. I, Helv. Phys. Acta 47 (1974), 265–306. MR 436830
  • 24. D. Ruelle, J. Math. Phys. 12 (1971), 901; Helv. Phys. Acta 45 (1972), 215.
  • Jürg Fröhlich, The reconstruction of quantum fields from Euclidean Green’s functions at arbitrary temperatures, Helv. Phys. Acta 48 (1975), no. 3, 355–363. MR 406205
  • 25. M. Takesaki, in Statistical Mechanics and Mathematical Problems, see reference 11, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Math., vol. 128, Springer-Verlag, Berlin and New York, 1970. See also: R. Haag, N. Hugenholtz and M. Winnink, Comm. Math. Phys. 5 (1967), 215.
  • William Greenberg, Correlation functionals of infinite volume quantum spin systems, Comm. Math. Phys. 11 (1968/69), 314–320. MR 242452, DOI 10.1007/BF01645852
  • 27. O. E. Lanford III, Cargèse Lectures, Gordon and Breach, New York, 1969.
  • R. L. Dobrushin and S. B. Shlosman, Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics, Comm. Math. Phys. 42 (1975), 31–40. MR 424106, DOI 10.1007/BF01609432
  • J. Slawny, A family of equilibrium states relevant to low temperature behavior of spin $1/2$ classical ferromagnets. Breaking of translation symmetry, Comm. Math. Phys. 35 (1974), 297–305. MR 339739, DOI 10.1007/BF01646351
  • Joel L. Lebowitz and Anders Martin-Löf, On the uniqueness of the equilibrium state for Ising spin systems, Comm. Math. Phys. 25 (1972), 276–282. MR 312854, DOI 10.1007/BF01877686
  • Joel L. Lebowitz, Coexistence of phases in Ising ferromagnets, J. Statist. Phys. 16 (1977), no. 6, 463–476. MR 456178, DOI 10.1007/BF01152284
  • 31. O. E. Lanford III, in Méchanique Statistique et Théorie Quantique des Champs, see reference 10. 32. J. Fröhlich, unpublished notes, 1976. 33. G. Hegerfeldt and C. Nappi, ZiF-Univ. of Bielefeld, preprint, 1976, Comm. Math. Phys. (to appear).
  • E. Seiler and B. Simon, Nelson’s symmetry and all that in the Yukawa2 and $(\phi ^{4})_{3}$ field theories, Ann. Physics 97 (1976), no. 2, 470–518. MR 438960, DOI 10.1016/0003-4916(76)90044-0
  • Y. Choquet-Bruhat, Construction de solutions radiatives approchées des équations d’Einstein, Comm. Math. Phys. 12 (1969), 16–35 (French, with English summary). MR 241087, DOI 10.1007/BF01646432
  • 36. N. D. Mermin, J. Math. Phys. 8 (1967), 1061. See also M. Kac, Quart. Appl. Math. 30 (1972), 17; O. McBryan and T. Spencer, Comm. Math. Phys. 53 (1977), 299. 37. J. Bricmont, J. R. Fontaine and L. J. Landau, On the uniqueness of the equilibrium state in plane rotators, Univ. Louvain, preprint UCL-IPT-77/03.
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Additional Information
  • Journal: Bull. Amer. Math. Soc. 84 (1978), 165-193
  • MSC (1970): Primary 60G20, 81A18
  • DOI:
  • MathSciNet review: 475121