Absence of submultiplicative norms for Wick-ordered operator products

Abstract

In this paper, a key problem in the rigorous formulation of the renormalization group as a continuous flow is identified. Some essential features of the operator-theoretic renormalization group of Bach, Chen, Fröhlich, and Sigal are recalled, and a family of norms associated with different Banach spaces of Hamiltonians is introduced. From this, three different properties, which should be satisfied by an adequate norm, are derived, e.g., a submultiplicativity with respect to Wick-ordering. A proof is given that none of the norms introduced in this paper satisfy all of these conditions.