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Communications in Mathematical Physics

, Volume 61, Issue 3, pp 275–284 | Cite as

The ϕ 2 4 quantum field as a limit of Sine-Gordon fields

  • Oliver A. McBryan
Article

Abstract

We exhibit the λϕ 2 4 quantum field theory as the limit of Sine-Gordon fields as suggested by the identity
$$\varphi ^4 /4! = \mathop {\lim }\limits_{\varepsilon \to 0} (\varepsilon ^{ - 4} \cos \varepsilon \varphi - \varepsilon ^{ - 4} + \tfrac{1}{2}\varepsilon ^{ - 2} \varphi ^2 ).$$
The proofs of finite volume stability for the two models, due to Nelson and Fröhlich respectively, are unrelated. We find a generalized stability argument that incorporates ideas from both of the simpler cases. The above limit, for the Schwinger functions, then proceeds uniformly in ɛ.
As a by-product, let (ϕ,dμ) be a Gaussian random field, ϕ K (1≦κ<∞) a regularization of ϕ, andV a function satisfying:
  1. (i)

    V K )≧−ak α

     
  2. (ii)

    V(ϕ) −V K )∥pbpβk−γ, 2≦p < ∞

     
TheneV(ϕ)L1(dμ) provided α(β−1)<γ.

Keywords

Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
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.
    Nelson, E.: Probability theory and Euclidean field theory. In: Constructive quantum field theory (ed. G. Velo, A. S. Wightman). Berlin-Heidelberg-New York: Springer 1973Google Scholar
  2. 2.
    Fröhlich, J.: Classical and quantum statistical mechanics in one and two dimensions: Two-component Yukawa and Coulomb systems. Commun. math. Phys.47, 233 (1976)CrossRefGoogle Scholar
  3. 3.
    Glimm, J., Jaffe, A.: Positivity and self-adjointness of theP(φ)2 Hamiltonian. Commun. math. Phys.22, 253 (1971)CrossRefGoogle Scholar
  4. 4.
    Simon, B.: TheP(φ)2 Euclidean quantum field theory. Princeton: Princeton University Press 1974Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Oliver A. McBryan
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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