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Novel Phenomena in Noncommutative Field Theory: Emergent Geometry

Badis Ydri

TL;DR

The work develops a comprehensive framework where geometry and gauge theories emerge from matrix-model formulations of noncommutative field theories. By combining multitrace matrix models with Wilsonian RG analyses and YM matrix-model dynamics, it reveals robust phases (disordered, uniform-ordered, non-uniform stripe) and fixed-point structures that encode emergent two-dimensional geometries such as fuzzy spheres and NC gauge theories. The stripe phase captures noncommutative UV-IR mixing, while the uniform phase signals emergent geometry whose dimension is inferred from critical exponents and Wigner semicircle behavior; expanding around uniform order yields noncommutative gauge theories with Myers terms. The YM matrix-model approach demonstrates geometry-to-matrix transitions and the persistence of UV-IR mixing in gauge sectors, offering a quantitative path from first-quantized geometry to a second-quantized, emergent geometric framework with potential links to emergent gravity.

Abstract

Noncommutative field theory (NCFT) is an extension of quantum field theory (QFT) that redefines spacetime, replacing commuting coordinates with a noncommutative structure. This shift fundamentally alters the way fields, interactions, and symmetries are understood. NCFT uniquely integrates with supersymmetry, making it a natural framework for unifying quantum mechanics and gravity. It also provides a consistent mechanism for spontaneous supersymmetry breaking. Unlike conventional QFT, which quantizes fields on a fixed spacetime, NCFT begins by quantizing spacetime itself. This perspective reveals novel phenomena, such as ultraviolet-infrared mixing and a natural transition from discrete to continuous geometries. It offers insights into quantum gravity at the Planck scale. Mathematically, NCFT bridges quantum mechanics and geometry through operator algebras, enabling the exploration of new theories unattainable in traditional frameworks. This dual role cements NCFT as a cornerstone of modern theoretical physics. This work is published as Chapter 18, "Novel phenomena in noncommutative field theory: emergent geometry", in A Modern Course in Quantum Field Theory, Volume 2 (Second Edition), IOP Publishing (2025). ISBN 978-0-7503-5834-7. DOI 10.1088/978-0-7503-5834-7. The published chapter is available at the publisher's website: https://iopscience.iop.org/book/mono/978-0-7503-5834-7/chapter/bk978-0-7503-5834-7ch18

Novel Phenomena in Noncommutative Field Theory: Emergent Geometry

TL;DR

The work develops a comprehensive framework where geometry and gauge theories emerge from matrix-model formulations of noncommutative field theories. By combining multitrace matrix models with Wilsonian RG analyses and YM matrix-model dynamics, it reveals robust phases (disordered, uniform-ordered, non-uniform stripe) and fixed-point structures that encode emergent two-dimensional geometries such as fuzzy spheres and NC gauge theories. The stripe phase captures noncommutative UV-IR mixing, while the uniform phase signals emergent geometry whose dimension is inferred from critical exponents and Wigner semicircle behavior; expanding around uniform order yields noncommutative gauge theories with Myers terms. The YM matrix-model approach demonstrates geometry-to-matrix transitions and the persistence of UV-IR mixing in gauge sectors, offering a quantitative path from first-quantized geometry to a second-quantized, emergent geometric framework with potential links to emergent gravity.

Abstract

Noncommutative field theory (NCFT) is an extension of quantum field theory (QFT) that redefines spacetime, replacing commuting coordinates with a noncommutative structure. This shift fundamentally alters the way fields, interactions, and symmetries are understood. NCFT uniquely integrates with supersymmetry, making it a natural framework for unifying quantum mechanics and gravity. It also provides a consistent mechanism for spontaneous supersymmetry breaking. Unlike conventional QFT, which quantizes fields on a fixed spacetime, NCFT begins by quantizing spacetime itself. This perspective reveals novel phenomena, such as ultraviolet-infrared mixing and a natural transition from discrete to continuous geometries. It offers insights into quantum gravity at the Planck scale. Mathematically, NCFT bridges quantum mechanics and geometry through operator algebras, enabling the exploration of new theories unattainable in traditional frameworks. This dual role cements NCFT as a cornerstone of modern theoretical physics. This work is published as Chapter 18, "Novel phenomena in noncommutative field theory: emergent geometry", in A Modern Course in Quantum Field Theory, Volume 2 (Second Edition), IOP Publishing (2025). ISBN 978-0-7503-5834-7. DOI 10.1088/978-0-7503-5834-7. The published chapter is available at the publisher's website: https://iopscience.iop.org/book/mono/978-0-7503-5834-7/chapter/bk978-0-7503-5834-7ch18

Paper Structure

This paper contains 38 sections, 255 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Aspects of noncommutative phi-four theory on the Moyal-Weyl space
  3. UV-IR mixing and stripe phase on noncommutative ${\bf R}^d_{\theta}$
  4. UV-IR mixing
  5. $4-$Dimensions:
  6. $2-$Dimensions:
  7. Stripe phase
  8. Dispersion relations on the noncommutative torus
  9. The Grosse-Wulkenhaar model
  10. Wilson renormalization group recursion formula
  11. The Wilson-Fisher fixed point in noncommutative phi-four
  12. The noncommutative $O(N)$ Wilson-Fisher fixed point
  13. The $\theta=\infty$ matrix model fixed point
  14. Noncommutative phi-four on the fuzzy sphere ${\bf S}^2_N$
  15. UV-IR mixing and renormalizability
  16. ...and 23 more sections

Figures (26)

  • Figure 1: The phase diagrams of causal dynamical triangulation, Lifshitz scalar field theory and multitrace matrix model.
  • Figure 2: The one-loop planar and non-planar contributions.
  • Figure 3: The phase diagram of noncommutative phi-four in $d=4$ at fixed $\lambda\sim g^2$.
  • Figure 4: The phase diagram of phi-four theory on the fuzzy sphere.
  • Figure 5: The termination point of the doubletrace matrix model as the triple point.
  • ...and 21 more figures