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49
Quantum field theory on noncommutative spaces
"... A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommuta ..."
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Cited by 396 (26 self)
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A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative YangMills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an indepth study of the gauge group of noncommutative YangMills theory. Some of the more mathematical ideas and
Moyal planes are spectral triples
, 2003
"... Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R 2N endowed with Moyal products are intensively investigated. Some physical applications, ..."
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Cited by 75 (20 self)
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Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R 2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes.
The spectral action for Moyal planes
 J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 43 (9 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the ConnesLott action [15] previously computed by Gayral [23] for symplectic Θ.
Dixmier traces on noncompact isospectral deformations
 J. FUNCT. ANAL
, 2005
"... We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of fu ..."
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Cited by 18 (8 self)
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We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of R l, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
Noncommutative differential calculus for Moyal subalgebras
 J. Geom. Phys
"... subalgebras ..."
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The action functional for Moyal planes
, 2003
"... Modulo some natural generalizations to noncompact spaces, we show in this letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the noncommutative YangMills action associated with the Moyal product. In parti ..."
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Cited by 10 (1 self)
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Modulo some natural generalizations to noncompact spaces, we show in this letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the noncommutative YangMills action associated with the Moyal product. In particular, we show that the rigorous framework of noncommutative geometry is suitable for Moyal gauge theory.
On the hermiticity of qdifferential operators and forms on the quantum Euclidean spaces R N q
"... We show that the complicated ⋆structure characterizing for positive q the Uqso(N)covariant differential calculus on the noncommutative manifold RN q boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ˜v. Subare made ..."
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Cited by 7 (7 self)
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We show that the complicated ⋆structure characterizing for positive q the Uqso(N)covariant differential calculus on the noncommutative manifold RN q boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ˜v. Subare made into Hilbert spaces of the spaces of functions and of pforms on RN q spaces by introducing nonconventional “weights ” in the integrals defining the corresponding scalar products, namely suitable positivedefinite qpseudodifferential operators ˜v ′±1 realizing the action of ˜v ±1; this serves to make the partial qderivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the ‘radial coordinate ’ r. Unless we choose a constant m, then the squareintegrables functions/forms must fulfill an additional condition, namely their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are qspecial functions with ‘quantized ’ free parameters.