Constructive Quantum Field Theories via Lattice Regularizations and Critical Phenomena in Statistical Mechanics : Part I

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  • Constructive Quantum Field Theories via Lattice Regularizations and Critical Phenomena in Statistical Mechanics
  • Constructive Quantum Field Theories via

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We discuss construction of the continuum scalar (in particular, φ^4) quantum field theory as subsequence limits of lattice theories and triviality or nontriviality of the limiting theory. Close connection between taking the continuum limit of lattice field theories and approaching the critical point of the corresonding statistical ? mechanical models is exploited to obtain the bound on the renormalized coupling constant of the φ^4 theory. We show that the continuum limits are trivial (i.e. Gaussian) for the (φ^4)_d or (Ising)_d models in dimensions d > 4 and nontrivial (i.e. non-Gaussian) for weakly coupled (φ^4)_d models in dimensions d < 4. Important ingredients in the proofs are the random walk representation and correlation inequalities in rigorous statistical mechanics. Moreover, in the case d = 4, any deviation from the mean field theory at the critical point implies triviality of the continuum limit.

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