Regular Hamiltonians for non-relativistic interacting quantum field theories

TL;DR

This paper introduces a method to rigorously define interaction Hamiltonians in non-relativistic quantum field theories by modifying the configuration space measure to handle singularities at coincident particle positions.

Contribution

It proposes a novel approach to regularize Hamiltonians in non-relativistic quantum field theories through measure modification to manage singular operator products.

Findings

Provides a rigorous framework for operator multiplication at coincident points.

Enables well-defined interaction terms in non-relativistic quantum fields.

Lays groundwork for further mathematical analysis of quantum field interactions.

Abstract

In the context of non-relativistic quantum field theory, a method is proposed for multiplying field operators at the same spatial point and obtaining regular (i.e. rigorously defined) interaction terms for the Hamiltonian. The basic idea is to modify the Lebesgue measure of configuration space of many particles by adding singular measures over the subspaces of configuration space in which the positions of two or more particles coincide.