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Communications in Mathematical Physics

, Volume 86, Issue 1, pp 1–48 | Cite as

Geometric analysis of φ4 fields and Ising models. Parts I and II

  • Michael Aizenman
Article

Abstract

We provide here the details of the proof, announced in [1], that ind>4 dimensions the (even) φ4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents — which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the φ4 field — and other aspects of the critical behavior in Ising models — are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimensiond=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensionsd>4, but not in the low dimensiond=2 — for which we establish the “universality” of hyperscaling.

Keywords

Brownian Motion Physical Interaction Ising Model Field Variable Continuum Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aizenman, M.: Phys. Rev. Lett.47, 1 (1981) [and a contribution in Mathematical problems in theoretical physics. (Proceedings, Berlin 1981). Berlin, Heidelberg, New York: Springer (to appear)Google Scholar
  2. 2.
    Symanzik, K.: Euclidean quantum field theory. In: Local quantum theory. Jost, R. (ed.). New York: Academic Press 1969Google Scholar
  3. 3.
    Fröhlich, J.: Nucl Phys.B200, [FS4] 281 (1982)Google Scholar
  4. 4.
    McBryan, O.A., Rosen, J.: Commun. Math. Phys.51, 97 (1976)Google Scholar
  5. 5.
    Simon, B.: Commun. Math. Phys.77, 111 (1980)Google Scholar
  6. 6.
    Griffiths, R., Hurst, C., Sherman, S.: J. Math. Phys.11, 790 (1970)Google Scholar
  7. 7.
    Newman, C., Schulman, L.: J. Stat. Phys.26, 613 (1981)Google Scholar
  8. 8.
    Sokal, A.D.: Phys. Lett.71A, 451 (1979)Google Scholar
  9. 9.
    Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976)Google Scholar
  10. 10.
    Kakutani, S.: Proc. Japan Acad.20, 648 (1944)Google Scholar
  11. 10a.
    Dvoretsky, A., Erdös, P., Kakutani, S.: Acta Sci. Math. (Szeged)12, 75 (1950)Google Scholar
  12. 11.
    Lebowitz, J.: Commun. Math. Phys.35, 87 (1974)Google Scholar
  13. 12.
    Glimm, J., Jaffe, A.: Ann. Inst. Henri Poincaré A22, 97 (1975)Google Scholar
  14. 13.
    Glimm, J., Jaffe, A.: Commun. Math. Phys.51, 1 (1976)Google Scholar
  15. 14.
    Sokal, A.D.: Ann. Inst. Henri Poincaré (to appear)Google Scholar
  16. 15.
    Glimm, J., Jaffe, A.: Phys. Rev. D10, 536 (1974)Google Scholar
  17. 16.
    Widom, R.: J. Chem. Phys.43, 3892 (1965)Google Scholar
  18. 16a.
    Kadanoff, L.P., et al.: Rev. Mod. Phys.39, 395 (1967)Google Scholar
  19. 16b.
    Fisher, M.: Rep. Prog. Phys.30, 615 (1967), and references thereinGoogle Scholar
  20. 17.
    Schrader, R.: Phys. Rev. B,14, 172 (1976)Google Scholar
  21. 18.
    Lieb, E.H., Sokal, A.D.: In preparationGoogle Scholar
  22. 19.
    Constructive quantum field theory. Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973Google Scholar
  23. 20.
    Simon, B.: TheP(φ)2 euclidean (quantum) field theory. Princeton, NS: Princeton University Press 1974Google Scholar
  24. 21.
    Glimm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981Google Scholar
  25. 22.
    Simon, B., Griffiths, R.: Commun. Math. Phys.33, 145 (1973)Google Scholar
  26. 23.
    Sokal, A.D.: Mean-field bounds and correlation inequalities. J. Stat. Phys. (to appear)Google Scholar
  27. 24.
    Newman, C.: Commun. Math. Phys.41, 1 (1975)Google Scholar
  28. 25.
    Newman, C.: Z. Wahrscheinlichkeitstheorie33, 75 (1975)Google Scholar
  29. 26.
    Schrader, R.: Phys. Rev. B15, 2798 (1977)Google Scholar
  30. 26a.
    Messager, A., Miracle-Sole, S.: J. Stat. Phys.17, 245 (1977)Google Scholar
  31. 27.
    Aizenman, M.: On brownian motion ind=4 dimensions (in preparation)Google Scholar
  32. 28.
    Graham, R.: Correlation inequalities for the truncated two-point function of an Ising ferromagnet (J. Stat. Phys., to appear) and: An improvement of the GHS inequality for the Ising ferromagnet (J. Stat. Phys., to appear)Google Scholar
  33. 29.
    Aizenman, M., Graham, R.: A bound on the renormalized coupling in the ϕd4 field theory ind=4 dimensions (in preparation)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Michael Aizenman
    • 1
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA

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