[Paper Review] Non-Commutative Renormalization

Vincent Rivasseau, Fabien Vignes-Tourneret|arXiv (Cornell University)|Sep 30, 2004

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper proposes a constructive approach to non-commutative field theory by extending the Grosse-Wulkenhaar renormalization framework to non-perturbative settings. It identifies the $φ^4_4$ theory at $Ω=1$ as a candidate for non-perturbative existence due to vanishing $β$-function and asymptotic safety, despite challenges from matrix structure in Feynman diagrams.

ABSTRACT

We review the recent approach of Grosse and Wulkenhaar to the perturbative renormalization of non commutative field theory and suggest a related constructive program. This paper is dedicated to J. Bros on his 65th birthday.

Motivation & Objective

• To extend perturbative non-commutative field theory to a non-perturbative, constructive framework.
• To address the UV-IR mixing problem in non-commutative $φ^4_4$ theory using the Grosse-Wulkenhaar approach.
• To investigate whether non-commutative field theories can be rigorously constructed at the non-perturbative level, especially at $Ω=1$.
• To overcome challenges in applying constructive RG techniques to matrix models arising in non-commutative field theories.
• To assess the viability of non-commutative $φ^4_4$ and Gross-Neveu models as candidates for rigorous non-perturbative construction.

Proposed method

• Adopt the Grosse-Wulkenhaar renormalization procedure, which introduces a harmonic oscillator term to restore symmetry between UV and IR scales.
• Use momentum-space formulation with the Moyal star product to derive vertex factors involving oscillatory exponential terms dependent on $θ^{\mu\nu}$.
• Apply multi-scale cluster expansions and Hadamard's bound instead of Gram's bound to handle matrix indices in Fermionic models.
• Analyze the $β$-function of $φ^4_4$ at $Ω=1$, showing it vanishes at one-loop level, implying no running coupling and asymptotic safety.
• Leverage the fact that at $Ω=1$, the theory exhibits perfect duality between UV and IR regimes, enabling potential non-perturbative control.
• Consider the Gross-Neveu model on non-commutative $ρ^2$ as a simpler candidate for constructive analysis due to asymptotic freedom and stability.

Experimental results

Research questions

• RQ1 Can the Grosse-Wulkenhaar renormalization procedure be extended to a non-perturbative, constructive framework for non-commutative field theories?
• RQ2 What are the implications of UV-IR mixing for the non-perturbative existence of non-commutative $φ^4_4$ theory?
• RQ3 Why does the $β$-function vanish at $Ω=1$, and what does this imply for the renormalized coupling and asymptotic safety?
• RQ4 How do matrix indices in non-commutative field theories complicate the application of standard constructive techniques like Gram's bound?
• RQ5 Can the Gross-Neveu model on non-commutative $ρ^2$ serve as a viable candidate for rigorous non-perturbative construction despite increased combinatorial complexity?

Key findings

• The $φ^4_4$ theory at $Ω=1$ has a vanishing $β$-function at one-loop order, indicating no running of the coupling constant and asymptotic safety.
• At $Ω=1$, the theory exhibits perfect duality between ultraviolet and infrared scales, suggesting a potential path to non-perturbative construction.
• The non-commutative Gross-Neveu model on $ρ^2$ is identified as a promising candidate for constructive field theory due to asymptotic freedom and stability.
• Matrix structure in non-commutative field theories introduces an extra sum over indices at each vertex, which challenges standard constructive techniques like Gram’s bound.
• The use of Hadamard’s bound and multi-scale cluster expansions is proposed as a necessary alternative to overcome the combinatorial explosion in matrix models.
• The model is not expected to satisfy Wightman’s axioms, particularly Lorentz invariance, due to the explicit breaking by the $θ^{\mu\nu}$ tensor.