Mathematical renormalization of Hamiltonian field theories

TL;DR

This paper provides a rigorous mathematical framework for renormalizing Hamiltonian field theories, ensuring the well-defined evolution operator and connecting to standard quantum field theory constructs.

Contribution

It introduces a rigorous definition of the renormalized evolution operator within the Weyl-Moyal algebra for arbitrary Hamiltonians and time intervals, linking to established QFT results.

Findings

Defines the renormalized evolution operator rigorously

Connects the construction to standard renormalized S-matrix and Green functions

Applicable to arbitrary Hamiltonians and full real-time intervals

Abstract

We rigorously define renormalized evolution operator of the Schr\"odinger equation in the infinite dimensional Weyl-Moyal algebra for any time interval for arbitrary Hamiltonian depending on time. We state that for renormalizable field theories, in the interaction representation, and for the time interval being the full real axis, our construction yields standard renormalized $S$-matrix and Green functions of perturbative quantum field theory.