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Non-commutative Renormalization

Vincent Rivasseau

TL;DR

This paper reviews renormalization in non-commutative quantum field theory on Moyal space, focusing on how UV/IR mixing initially obstructed renormalizability and how the Grosse–Wulkenhaar vulcanization (a harmonic term respecting Langmann–Szabo duality) cures these issues. It develops both matrix-basis and direct-space multi-scale analyses, derives Ward identities that underpin a ghost-free renormalization flow, and demonstrates that the beta function can vanish at a self-dual point, yielding a perturbatively renormalizable (and potentially non-perturbatively constructible) NCQFT. The work extends to fermionic models like the non-commutative Gross–Neveu and to covariant/gauge-like theories, showing renormalizability across several NCQFT classes via Mehler-like kernels and hyperbolic parametric representations. Overall, the framework suggests non-commutativity can tame ultraviolet behavior without introducing new particles, offering a possible alternative to supersymmetry and guiding future exploration of NC gauge theories, quantum Hall physics, and beyond.

Abstract

A new version of scale analysis and renormalization theory has been found on the non-commutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on non-commutative than on ordinary space: indeed it has no Landau ghost. Noncommutativity might therefore be an alternative to supersymmetry. We review this rapidly growing subject.

Non-commutative Renormalization

TL;DR

This paper reviews renormalization in non-commutative quantum field theory on Moyal space, focusing on how UV/IR mixing initially obstructed renormalizability and how the Grosse–Wulkenhaar vulcanization (a harmonic term respecting Langmann–Szabo duality) cures these issues. It develops both matrix-basis and direct-space multi-scale analyses, derives Ward identities that underpin a ghost-free renormalization flow, and demonstrates that the beta function can vanish at a self-dual point, yielding a perturbatively renormalizable (and potentially non-perturbatively constructible) NCQFT. The work extends to fermionic models like the non-commutative Gross–Neveu and to covariant/gauge-like theories, showing renormalizability across several NCQFT classes via Mehler-like kernels and hyperbolic parametric representations. Overall, the framework suggests non-commutativity can tame ultraviolet behavior without introducing new particles, offering a possible alternative to supersymmetry and guiding future exploration of NC gauge theories, quantum Hall physics, and beyond.

Abstract

A new version of scale analysis and renormalization theory has been found on the non-commutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on non-commutative than on ordinary space: indeed it has no Landau ghost. Noncommutativity might therefore be an alternative to supersymmetry. We review this rapidly growing subject.

Paper Structure

This paper contains 53 sections, 21 theorems, 216 equations, 29 figures.

Table of Contents

  1. Introduction
  2. Commutative Renormalization, a Blitz Review
  3. Functional integral
  4. Feynman Rules
  5. Scale Analysis and Renormalization
  6. The BPHZ Theorem
  7. The Landau ghost and Asymptotic Freedom
  8. Non-commutative field theory
  9. Field theory on Moyal space
  10. The Moyal space
  11. The Algebra ${\mathcal{A}}_{\Theta}$
  12. Extension by Duality
  13. The phi4-theory on Moyal space
  14. UV/IR mixing
  15. The Grosse-Wulkenhaar breakthrough
  16. ...and 38 more sections

Key Result

Proposition 3.1

For all $f,g\in{\mathcal{S}}_{D}$,

Figures (29)

  • Figure 1: A possible contraction scheme with $n=N=4$.
  • Figure 2: A high energy subgraph S seen from lower energies looks quasi-local.
  • Figure 3: A family of graphs $P_n$ producing a renormalon.
  • Figure 4: A family of convergent graphs $Q_n$, that do not produce any renormalon.
  • Figure 5: The $\phi^4$ connected graphs with $n=2$, $N=4$.
  • ...and 24 more figures

Theorems & Definitions (27)

  • Proposition 3.1: Trace
  • proof
  • Lemma 3.2
  • Corollary 3.3
  • Definition 3.1
  • Definition 3.2: Multipliers algebra
  • Lemma 3.4: Propagator toolbox05
  • Definition 4.1
  • Proposition 4.1
  • Proposition 4.2
  • ...and 17 more