Ultraviolet Stability in Field Theory. The ϕ 3 4 Model

Abstract

Renormalization group ideas appeared in the early papers on constructive quantum field theory [5, 6, 7], although they were never stated explicitly. The model (ϕ4)d was considered in these papers and in the Euclidean approach a generating functional for Schwinger functions in this model is given formally by the expression $$Z(f) = {Z^{ - 1}}\int {{\pi _x}} d\phi (x)\;\exp \{ - \int {dx[\frac{1}{2}\phi (x)( - \Delta \phi )(x) + \frac{1}{2}m_0^2{\phi ^2}(x) + \lambda {\phi ^4}(x) + \int {dxf(x)\phi (x)\} } } $$ (1) where Z is defined by the integral above with f = 0, x ∈ Rd and ϕ is a function from Rd to R. To give a sense to this expression it is necessary to introduce space and ultraviolet cut-offs, e.g. in the papers mentioned above the expression (1) was replaced by $${Z_k}( \wedge ,f) = {Z_k}{( \wedge )^{ - 1}}\int {d{\mu _{{c_{{m^2}}}}}} (\phi )\;\exp \{ - \int\limits_ \wedge {dx[\frac{1}{2}\delta {m^2}(\kappa ,\lambda )\phi _\kappa ^2(x) + \lambda \phi _\kappa ^4(x)] + (f,\phi )\} } $$ (2) where \(d{\mu _{{c_{{m^2}}}}}\) is a Gaussian measure defined on S′(Rd), with covariance \({C_{{m^2}}} = {( - \Delta + {m^2})^{ - 1}},{m^2} > 0, \wedge \) is a compact subset of Rd, δm2(κ,λ) is some constant (mass renormalization counterterm) and ϕ͂κ(p) = ζκ(p)ϕ͂(p), where ζκ(p) is a regular function with support in |p| ≤ κ, ζκ(p) → 1 as κ → ∞. An essential point of constructions in these papers was a division of the set |p| ≤ κ into a sequence of subsets κr-1 ≤ |p| ≤ κr, r = 1,2,...,n, 0(1) = κ0 < κ1 < ... < κn−1 < κn = κ, and calculations of integrals with field ϕ͂κ restricted to each shell separately (e.g. integrating with respect to the restricted fields).