Adiabatic Limit in Perturbation Theory

Abstract

We show that, with correct mass and wave function renormalization, the time-ordered products for Wick polynomials T(ℒ(y₁)…ℒ(y_n)) constructed by a method outlined in a previous paper[1] are such that the vectors of the form

$$ \int {T\left( {\mathcal{L}\left( {{{y}_{1}}} \right) \ldots \mathcal{L}\left( {{{y}_{n}}} \right)} \right)g\left( {{{y}_{1}}} \right) \ldots g\left( {{{y}_{1}}} \right) \ldots g\left( {{{y}_{n}}} \right)\psi d{{y}_{1}} \ldots d{{y}_{n}}} $$

have limits when g tends to a constant, provided ¥ is chosen in a suitable dense domain. It follows that the S matrix has unitary adiabatic limit as an operator-valued formal power series in Fock space.