Exactly Solvable Quantum Mechanical Models with Infinite Renormalization of the Wave Function

TL;DR

This paper introduces an exactly solvable quantum mechanical model that addresses infinite wave function renormalization, providing insights beyond perturbative approaches in quantum field theory.

Contribution

It constructs a simplified, exactly solvable model with infinite wave function renormalization, utilizing Pontriagin space theory and indefinite inner products.

Findings

Model is exactly solvable with infinite wave function renormalization.

Both field and conjugate momentum become well-defined operators after renormalization.

Uses Pontriagin space theory to handle indefinite inner products.

Abstract

The main difficulty of quantum field theory is the problem of divergences and renormalization. However, realistic models of quantum field theory are renormalized within the perturbative framework only. It is important to investigate renormalization beyond perturbation theory. However, known models of constructive field theory do not contain such difficulties as infinite renormalization of the wave function. In this paper an exactly solvable quantum mechanical model with such a difficulty is constructed. This model is a simplified analog of the large-N approximation to the $\Phi\phi^a\phi^a$-model in 6-dimensional space-time. It is necessary to introduce an indefinite inner product to renormalize the theory. The mathematical results of the theory of Pontriagin spaces are essentially used. It is remarkable that not only the field but also the canonically conjugated momentum become well-defined operators after adding counterterms.