The Fuzzy Disc

Abstract

We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and theta. It is composed of functions which decay exponentially outside a disc. In the limit in which the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a Laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two--points correlation function for a free massless theory (Green's function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kurkcuoglu.

17 pages, 8 figures, references added and corrected

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Cited by in corpus (28)

• Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces
• Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere
• Noncommutative field theory on ${\mathbb{R}}_{\lambda }^3$
• Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom
• Duality symmetry for star products
• Quantum gauge theories on noncommutative 3-d space
• Spectral geometry with a cut-off: topological and metric aspects
• Exact operator bosonization of finite number of fermions in one space dimension
• Fuzzy Toric Geometries
• Curing the UV/IR mixing for field theories with translation-invariant $\star$ products
• Matrix Bases for Star Products: a Review
• Entanglement Entropy on Fuzzy Spaces
• Self-intersecting fuzzy extra dimensions from squashed coadjoint orbits in $\mathcal{N}=4$ SYM and matrix models
• Edge Currents in Non-commutative Chern-Simons Theory from a New Matrix Model
• Significantly super-Chandrasekhar mass-limit of white dwarfs in noncommutative geometry
• Quantum (Matrix) Geometry and Quasi-Coherent States
• The beat of a fuzzy drum: fuzzy Bessel functions for the disc
• The Phase Diagram of Scalar Field Theory on the Fuzzy Disc
• Boundaries in the Moyal plane
• Regular Black Holes and Noncommutative Geometry Inspired Fuzzy Sources
• Non-constant Non-commutativity in 2d Field Theories and a New Look at Fuzzy Monopoles
• A Perturbative Approach to Fuzzifying Field Theories
• Asymptotic commutativity of quantized spaces: the case of ${\mathbb{CP}}^{p,q}$
• Noncommutative Spherically Symmetric Spaces
• Angles in Fuzzy Disc and Angular Noncommutative Solitons
• Truncated quantum observables and their semiclassical limit
• The semiclassical limit of a quantum Zeno dynamics
• Noncommutative Q-Lumps