Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy Sphere

Abstract

We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.

This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (54)

• Noncommutative Field Theory
• Quantum Field Theory on Noncommutative Spaces
• The quantum structure of spacetime at the Planck scale and quantum fields
• Noncommutative Perturbative Dynamics
• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
• An Introduction to Noncommutative Spaces and their Geometry
• Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres
• Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory
• Noncommutative Gauge Theory on Fuzzy Sphere from Matrix Model
• Brane Dynamics in Background Fluxes and Non-commutative Geometry
• Gravity in Non-Commutative Geometry
• Duality in Scalar Field Theory on Noncommutative Phase Spaces
• Finite N Matrix Models of Noncommutative Gauge Theory
• Review of the Phenomenology of Noncommutative Geometry
• Nonperturbative Dynamics of Noncommutative Gauge Theory
• Quantized Gauge Theory on the Fuzzy Sphere as Random Matrix Model
• Noncommutative Geometry and Gauge Theory on Fuzzy Sphere
• Scaling Limits of the Fuzzy Sphere at one Loop
• Coherent State Induced Star-Product on $R_{\lambda }^3$ and the Fuzzy Sphere
• Renormalization of noncommutative phi 4-theory by multi-scale analysis
• Fuzzy CP2
• Stripes from (noncommutative) stars
• Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term
• Finite Gauge Theory on Fuzzy CP^2
• Matrix ${\varphi }^4$ Models on the Fuzzy Sphere and their Continuum Limits
• A Non-Perturbative Study of Gauge Theory on a Non-Commutative Plane
• Phase diagram and dispersion relation of the non-commutative λϕ^{4} model in d=3
• A Gauge-Invariant UV-IR Mixing and The Corresponding Phase Transition For U(1) Fields on the Fuzzy Sphere
• Effective Actions of Matrix Models on Homogeneous Spaces
• Scalar Field Theory on Fuzzy S^4
• Spacetime and Fields, a Quantum Texture
• Renormalization of the noncommutative phi^3 model through the Kontsevich model
• Gauge theories on the noncommutative sphere
• Fuzzy Supersphere and Supermonopole
• Towards Finite Quantum Field Theory in Non-Commutative Geometry
• D-brane Solutions in Non-Commutative Gauge Theory on Fuzzy Sphere
• Nonuniform symmetry breaking in noncommutative $\lambda {\Phi }^4$ theory
• Dynamical aspects of the fuzzy CP${}^2$ in the large $N$ reduced model with a cubic term
• Fuzzy spaces, the M(atrix) model and the quantum Hall effect
• Simulating non-commutative field theory
• On the Origin of the UV-IR Mixing in Noncommutative Matrix Geometry
• Gauge Theory on Noncommutative Supersphere from Supermatrix Model
• Perturbative versus nonperturbative dynamics of the fuzzy S^2*S^2
• Numerical Results for U(1) Gauge Theory on 2d and 4d Non-Commutative Spaces
• Impact of supersymmetry on the nonperturbative dynamics of fuzzy spheres
• Absence of a fuzzy $S^4$ phase in the dimensionally reduced 5d Yang-Mills-Chern-Simons model
• Non-Linear Sigma Model on the Fuzzy Supersphere
• Causality Violation in Non-local Quantum Field Theory
• Inherent Holography In Fuzzy Spaces and An N-tropic Approach To The Cosmological Constant Problem
• Note on Gauge Theory on Fuzzy Supersphere
• SUSY anomalies break N=2 to N=1: The supersphere and the fuzzy supersphere
• On Critical Phenomena in a Noncommutative Space
• D-branes, Cyclic Symmetry and Noncommutative Geometry

Cited by in corpus (15)

• Simulation of a scalar field on a fuzzy sphere
• Noncommutative geometry, gauge theory and renormalization
• Noncommutative Field Theory: Numerical Analysis with the Fuzzy Disc
• The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n
• Matrix geometries and Matrix Models
• The lowest modes around Gaussian solutions of tensor models and the general relativity
• On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory
• Matrix model approximations of fuzzy scalar field theories and their phase diagrams
• UV/IR mode mixing and the CMB
• Noncommutative Fluids
• The instability of intersecting fuzzy spheres
• SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry
• Complex (super)-matrix models with external sources and $q$-ensembles of Chern-Simons and ABJ(M) type
• Spectrum of Spin 1 Dirac Operators on the Fuzzy 2-Sphere
• Unusual Thermodynamics on the Fuzzy 2-Sphere