Translation invariant models in QFT without ultraviolet cutoffs

TL;DR

This paper proves ultraviolet renormalization for translation invariant quantum field models like the Nelson and polaron models using functional integration, establishing ground states in certain cases and advancing the mathematical understanding of these models.

Contribution

It introduces a method for ultraviolet renormalization of translation invariant QFT models via functional integrations, including new results for the Nelson and polaron models.

Findings

Ultraviolet renormalization of the Nelson model with fixed total momentum is established.

The Nelson Hamiltonian with zero total momentum has a ground state for all coupling constants in two dimensions.

Ultraviolet renormalization of the polaron model is also achieved.

Abstract

The translation invariant model in quantum field theory is considered by functional integrations. Ultraviolet renormalization of the translation invariant Nelson model with a fixed total momentum is proven by functional integrations. As a corollary it can be shown that the Nelson Hamiltonian with zero total momentum has a ground state for arbitrary values of coupling constants in two dimension. Furthermore the ultraviolet renormalization of the polaron model is also studied.