A short survey of noncommutative geometry

DOI10.1063/1.533329zbMath0974.58008arXivhep-th/0003006OpenAlexW2032656674MaRDI QIDQ2738026

Alain Connes

Publication date: 30 August 2001

Published in: Journal of Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/hep-th/0003006

zbMATH Keywords

surveynoncommutative geometryRiemann-Hilbert problemrenormalizationstring theory

Mathematics Subject Classification ID

Selfadjoint operator algebras ((C^*)-algebras, von Neumann ((W^*)-) algebras, etc.) (46L99) Feynman diagrams (81T18) Noncommutative geometry methods in quantum field theory (81T75) Noncommutative geometry (à la Connes) (58B34) Noncommutative global analysis, noncommutative residues (58J42)

Related Items (42)

Geometry and the quantum: basics ⋮ Scale invariance in the spectral action ⋮ The minimal length uncertainty and the nonextensive thermodynamics ⋮ Unnamed Item ⋮ Perturbations, deformations, and variations (and “near-misses") in geometry, physics, and number theory ⋮ Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics ⋮ Non-commutative and commutative vacua effects in a scalar torsion scenario ⋮ Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space ⋮ The \(F\) model on dynamical quadrangulations ⋮ Twisted supersymmetry in a deformed Wess-Zumino model in (2 + 1) dimensions ⋮ Noncommutative geometry and structure of space-time ⋮ STABILIZATION OF INTERNAL SPACE IN NONCOMMUTATIVE MULTIDIMENSIONAL COSMOLOGY ⋮ Optimal estimation of parameters for scalar field in an expanding spacetime exhibiting Lorentz invariance violation ⋮ Understanding truncated non-commutative geometries through computer simulations ⋮ Noncommutative geometry and the stability of circular orbits in a central force potential ⋮ Spontaneous localisation from a coarse-grained deterministic and non-unitary dynamics ⋮ Dynamical noncommutative spheres ⋮ Quantization: Deformation and/or functor? ⋮ A fundamental length as a candidate for dark energy: a DSR inspired FRW spacetime ⋮ UV/IR MIXING AND BLACK HOLE THERMODYNAMICS ⋮ Multiverse effects on the CMB angular correlation function in the framework of NCG ⋮ (Non)commutative isotropization in Bianchi I with barotropic perfect fluid and \(\Lambda\) cosmological ⋮ Noncommutativity, generalized uncertainty principle and FRW cosmology ⋮ THE GEOMETRY OF THE HIGGS FIELD ⋮ Advances in Dixmier traces and applications ⋮ Non-linear σ-models in noncommutative geometry: fields with values in finite spaces ⋮ Matroid theory and Chern–Simons ⋮ Some consequences of spacetime fuzziness ⋮ NONCOMMUTATIVE U(1) SUPER-YANG–MILLS THEORY: PERTURBATIVE SELF-ENERGY CORRECTIONS ⋮ Particle creation from vacuum by Lorentz violation ⋮ A cosmological viewpoint on the correspondence between deformed phase-space and canonical quantization ⋮ Matrix theory compactification on noncommutative T4/Z2 ⋮ Introduction to M(atrix) theory and noncommutative geometry ⋮ Quantum noncommutative multidimensional cosmology ⋮ Peierls Brackets in Field Theory ⋮ Editorial note to: Erwin Schrödinger, ``Dirac electron in the gravitational field I ⋮ The non-relativistic limit of the DKP equation in non-commutative phase-space ⋮ NONCOMMUTATIVE BLACK HOLES, THE FINAL APPEAL TO QUANTUM GRAVITY: A REVIEW ⋮ Noncommutative geometry and arithmetics ⋮ NONLOCAL REGULARISATION OF NONCOMMUTATIVE FIELD THEORIES ⋮ Combinatorics of (perturbative) quantum field theory ⋮ Instanton algebras and quantum 4-spheres

Cites Work

• Gravity coupled with matter and the foundation of non-commutative geometry
• Duality for crossed products and the structure of von Neumann algebras of type III
• Divergencies in a field theory on quantum space.
• Non-commutative differential geometry
• The cyclic homology of the group rings
• Connes' bicentralizer problem and uniqueness of the injective factor of type \(III_ 1\)
• Quantum K-theory. I: The Chern character
• Entire cyclic cohomology of Banach algebras and characters of \(\theta\)- summable Fredholm modules
• Quasi homomorphismes, cohomologie cyclique et positivité. (Quasi homomorphisms, cyclic cohomology and positivity)
• C*-algebras associated with irrational rotations
• Morita equivalence for C\(^*\)-algebras and W\(^*\)-algebras
• On ergodic flows and the isomorphism of factors
• Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
• A geometric construction of the discrete series for semisimple Lie groups
• Extensions of \(C^*\)-algebras and \(K\)-homology
• The flow of weights on factors of type III
• Hopf algebras, renormalization and noncommutative geometry
• On the Hopf algebra strucutre of perturbative quantum field theories
• Hopf algebras, cyclic cohomology and the transverse index theorem
• Operads and motives in deformation quantization
• Cyclic cohomology and Hopf algebras
• The Dirac operator and gravitation
• Renormalization in quantum field theory and the Riemann-Hilbert problem
• On overlapping divergences
• Cyclic cohomology, the Novikov conjecture and hyperbolic groups
• Gravity, non-commutative geometry and the Wodzicki residue
• The Cancellation Theorem for Projective Modules Over Irrational Rotation C*-Algebras
• THE OPERATORK-FUNCTOR AND EXTENSIONS OFC*-ALGEBRAS
• ON THE ULTRAVIOLET BEHAVIOR OF QUANTUM FIELDS OVER NONCOMMUTATIVE MANIFOLDS
• PERTURBATIVE QUANTUM GAUGE FIELDS ON THE NONCOMMUTATIVE TORUS
• Universal Formula for Noncommutative Geometry Actions: Unification of Gravity and the Standard Model
• Noncommutative geometry and reality
• K-THEORY AND REALITY
• Noncommutative perturbative dynamics
• Renormalization of quantum field theories on non-commutative \(\mathbb{R}^d\). I: Scalars

This page was built for publication: A short survey of noncommutative geometry