No-Counterterm approach to quantum field theory

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We give a conjectural way for computing the S-matrix and the correlation functions in quantum field theory beyond perturbation theory. The basic idea seems universal and naively simple: to compute the physical quantities one should consider the functional differential Schrodinger equation (without normal orderings), regularize it, consider the regularized evolution operator in the Fock space from t=T₁ to t=T₂, where the interval (T₁,T₂) contains the support of the interaction cutoff function, remove regularization (without adding counterterms), and tend the interaction cutoff function to a constant. We call this approach to QFT the No-Counterterm approach. We show how to compute the No-Counterterm perturbation series for the ϕ⁴ model in Rᵈ⁺¹. We give rough estimates which show that some summands of this perturbation series are finite without renormalization (in particular, one-loop integrals for d=3 and all integrals for d≥ 6).