Communications in Mathematical Physics

, Volume 64, Issue 1, pp 49–72 | Cite as

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

  • J. Bricmont
  • J. R. Fontaine
  • L. Landau


Correlation inequalities are used to show that the two component λ(φ2)2 model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant.

For then-componentP(|φ|2)2+μφ1 model, absence of continuous symmetry breaking, as μ goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingN loc τ and higher order estimates.


Neural Network Nonlinear Dynamics Symmetry Breaking Finite Volume Quantum Computing 
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  1. 1.
    Bricmont, J.: Correlation inequalities for two component fields. Ann. Soc. Sc. Brux.90, 245–252 (1976)Google Scholar
  2. 2.
    Bricmont, J., Fontaine, J.-R., Landau, L.J.: On the uniqueness of the equilibrium state for plane rotators. Commun. math. Phys.56, 281–296 (1977)Google Scholar
  3. 3.
    Coleman, S.: There are no Goldstone bosons in two dimensions. Commun. math. Phys.31, 259–264 (1973)Google Scholar
  4. 4.
    Dobrushin, R.L., Shlosman, S.B.: Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics. Commun. math. Phys.42, 31–40 (1975)Google Scholar
  5. 5.
    Dunlop, F., Newman, C.: Multicomponent field theories and classical rotators. Commun. math. Phys.44, 223–235 (1975)Google Scholar
  6. 6.
    Dunlop, F.: Correlation inequalities for multicomponent rotators. Commun. math. Phys.49, 247–256 (1976)Google Scholar
  7. 7.
    Fröhlich, J.: Marseille conference: Poetic phenomena in (2-dim) quantum field theory: non uniqueness of the vacuum, the solitons and all that, pp. 112–130 (1975)Google Scholar
  8. 8.
    Fröhlich, J.: New super-selection sectors “soliton-state” in two dimensional bose quantum field models. Commun. math. Phys.47, 269–310 (1976)Google Scholar
  9. 9.
    Fröhlich, J.: Private communicationGoogle Scholar
  10. 10.
    Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. math. Phys.50, 78–85 (1976)Google Scholar
  11. 11.
    Gal-Ezer, E.: Spontaneous breakdown in two dimensional space-time. Commun. math. Phys.44, 191–195 (1975)Google Scholar
  12. 12.
    Ginibre, J.: General formulation of Griffith's inequalities. Commun. math. Phys.16, 310–328 (1970)Google Scholar
  13. 13.
    Glimm, J., Jaffe, A.: A λφ4 quantum field theory without cutoff. I. Phys. Rev.176, 1945–1951 (1968)Google Scholar
  14. 14.
    Glimm, J., Jaffe, A.: The λφ4 quantum field theory without cutoffs. II. The field operators and the approximate vacuum. Ann. Math.91, 362–401 (1970)Google Scholar
  15. 15.
    Glimm, J., Jaffe, A.: The λφ4 quantum field theory without cutoffs. III. The physical vacuum. Acta Math.125, 203–261 (1970)Google Scholar
  16. 16.
    Glimm, J., Jaffe, A.: The λ(φ4)2 quantum field theory without cutoffs. IV. Perturbations of the hamiltonian. J. Math. Phys.13, 1568–1584 (1972)Google Scholar
  17. 17.
    Glimm, J., Jaffe, A.: The energy momentum spectrum and vacuum expectation values in quantum field theories. II. Commun. math. Phys.22, 1–22 (1971)Google Scholar
  18. 18.
    Glimm, J., Jaffe, A.: Quantum field theory models in statistical mechanics and quantum field theory. Dewitt, C., Stora, R. (eds.). New York: Gordon and Breach 1971Google Scholar
  19. 19.
    Glimm, J., Spencer, T.: The Wightman axioms and the mass gap for theP(φ)2 quantum field theory (preprint)Google Scholar
  20. 19a.
    Glimm, J., Jaffe, A.: Ann. Inst. H. Poincaré Sect. A22, 109 (1975)Google Scholar
  21. 20.
    Guerra, F., Rosen, L., Simon, B.: TheP(φ)2 euclidean quantum field theory as classical statistical mechanics. Ann. Math.101, 111–259 (1975)Google Scholar
  22. 21.
    Guerra, F., Rosen, L., Simon, B.: Boundary conditions for theP(φ)2 euclidean field theory. Ann. inst. H. Poincaré Sect. A25, 231–334 (1976)Google Scholar
  23. 22.
    Heifets, E.P., Osipov, E.P.: The energy momentum spectrum in theP(φ)2 quantum field theory. Commun. math. Phys.56, 161–172 (1977)Google Scholar
  24. 22a.
    Kunz, H., Pfister, Ch.Ed, Vuillermot, P.A.: Inequalities for some classical spin vector models. J. Ph. A Math. and Gen.9, 1673 (1976)Google Scholar
  25. 23.
    Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York, London: Academic Press 1972Google Scholar
  26. 24.
    Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis self adjointness. New York, London: Academic Press 1975Google Scholar
  27. 25.
    Reeh, H.: Symmetry operations and spontaneously broken symmetries in relativistic quantum field. Fortschr. Physik16, 687–706 (1968)Google Scholar
  28. 26.
    Rosen, L.: A λφ2n field theory: Higher order estimates. Com. Pure Appl. Math.24, 417–457 (1971)Google Scholar
  29. 27.
    Segal, I.: Notes torward the construction of non linear relativistic quantum fields. Proc. Nat. Acad. Sci. USA37, 1178 (1967)Google Scholar
  30. 28.
    Simon, B.: TheP(φ)2 euclidean (quantum) field theory. Princeton: Princeton University Press 1974Google Scholar
  31. 29.
    Spencer, T.: Perturbations of theP(φ)2 quantum field Hamiltonian. J. Math. Phys.14, 823–828 (1973)Google Scholar
  32. 30.
    Spencer, T.: The mass gap for theP(φ)2 quantum field model with a strong external field. Commun. math. Phys.39, 63–76 (1974)Google Scholar
  33. 31.
    Vilenkin, N.Ja: Fonctions spéciales et théorie de la représentation des groupes. Paris: Dunod 1969Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • J. Bricmont
    • 1
  • J. R. Fontaine
    • 1
  • L. Landau
    • 2
  1. 1.Institut de Physique ThéoriqueUniversité de LouvainLouvain-La-NeuveBelgium
  2. 2.Mathematics Department, Bedford CollegeUniversity of LondonLondonEngland

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