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


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.


Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
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    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
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    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
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    Glimm, J., Jaffe, A.: Positivity and self-adjointness of theP(φ)2 Hamiltonian. Commun. math. Phys.22, 253 (1971)CrossRefGoogle Scholar
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    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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