Absence of submultiplicative norms for Wick-ordered Operator Products

TL;DR

This paper investigates the mathematical properties of norms used in the operator-theoretic renormalization group, demonstrating that none of the proposed norms satisfy all desired submultiplicative conditions related to Wick-ordered operator products.

Contribution

It introduces a family of norms for Hamiltonians and proves that none of these norms meet all the necessary submultiplicative properties for rigorous renormalization group analysis.

Findings

No introduced norm satisfies all submultiplicative conditions.

Highlights fundamental limitations in current norm formulations.

Provides insights into the mathematical structure of Hamiltonian norms.

Abstract

In this paper, a key problem of the rigorous formulation of the renormalization group as a continuous flow is identified. Some essential features of the operator-theoretic renormalization group are recalled, and a family of norms associated to 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.