Infinite Infrared Regularization and a State Space for the Heisenberg Algebra

TL;DR

This paper introduces a novel regularization technique for infinite infrared singularities in quantum field theory, constructing a Krein space completion for test functions with an indefinite inner product, and applies it to the Heisenberg algebra.

Contribution

It generalizes existing infrared regularization methods to handle singularities of infinite order and provides a framework for constructing positive definite state spaces in such cases.

Findings

Constructed a Krein space completion for highly singular kernels.

Provided conditions for the regularization procedure using differential operators.

Built a maximally positive state space for the Heisenberg algebra with infinite infrared singularity.

Abstract

We present a method for the construction of a Krein space completion for spaces of test functions, equipped with an indefinite inner product induced by a kernel which is more singular than a distribution of finite order. This generalizes a regularization method for infrared singularities in quantum field theory, introduced by G. Morchio and F. Strocchi, to the case of singularites of infinite order. We give conditions for the possibility of this procedure in terms of local differential operators and the Gelfand- Shilov test function spaces, as well as an abstract sufficient condition. As a model case we construct a maximally positive definite state space for the Heisenberg algebra in the presence of an infinite infrared singularity.