Running couplings and triviality of field theories on non-commutative spaces

TL;DR

This paper investigates the renormalizability of asymptotically free field theories on non-commutative spaces, demonstrating that the non-commutative O(N) Gross-Neveu model becomes trivial and non-renormalizable when the ultraviolet cutoff is removed.

Contribution

It provides a detailed analysis of the non-commutative Gross-Neveu model at large N, revealing the triviality and non-renormalizability issues in non-commutative space.

Findings

Non-commutative Gross-Neveu model is non-renormalizable with translation invariant ground state.

Removing the ultraviolet cutoff leads to a trivial, non-interacting theory.

The model's behavior differs significantly from the commutative case.

Abstract

We examine the issue of renormalizability of asymptotically free field theories on non-commutative spaces. As an example, we solve the non-commutative O(N) invariant Gross-Neveu model at large N. On commutative space this is a renormalizable model with non-trivial interactions. On the noncommutative space, if we take the translation invariant ground state, we find that the model is non-renormalizable. Removing the ultraviolet cutoff yields a trivial non-interacting theory.