Renormalized Scattering Operator in Closed Form. I.φ3Model

Abstract

We present a method of obtaining the results of renormalization which makes no explicit reference either to perturbation theory or to the removal of infinities, but instead is based directly on physical requirements. Applied to scalar ${\ensuremath{\varphi}}^{3}$ theory, the method yields a closed expression for the renormalized scattering operator $S$ in the Dyson form as an implicitly time-ordered exponential of an interaction Hamiltonian plus quasilocal counterterms. Except for the over-all phase of $S$, these counterterms are given as explicit functionals of the vacuum expectation value of the bilinear product of operator derivatives of $S$ with respect to the asymptotic in-field $a(x)$, which enables them to be calculated recursively to any given order of perturbation theory from lower orders. $S$ is calculated in a straightforward manner up to third order of perturbation theory, and the two-point function to fourth order, and all are shown to be finite, the infinities canceling automatically. The second- and third-order results are identical with those of conventional renormalized perturbation theory. No comparable calculation of the fourth-order result seems to be available.