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48
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 397 (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
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 44 (10 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 19 (9 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
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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 (3 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.
The κMinkowski Spacetime: Trace, Classical Limit and Uncertainty Relations
, 2009
"... Starting from a discussion of the concrete representations of the coordinates of the κMinkowski spacetime (in 1+1 dimensions, for simplicity), we explicitly compute the associated Weyl operators as functions of a pair of Schrödinger operators. This allows for explicitly computing the trace of a qua ..."
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Cited by 7 (1 self)
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Starting from a discussion of the concrete representations of the coordinates of the κMinkowski spacetime (in 1+1 dimensions, for simplicity), we explicitly compute the associated Weyl operators as functions of a pair of Schrödinger operators. This allows for explicitly computing the trace of a quantised function of spacetime. Moreover, we show that in the classical (i.e. large scale) limit the origin of space is a topologically isolated point, so that the resulting classical spacetime is disconnected. Finally we show that there exist states with arbitrarily sharp simultaneous localisation in all the coordinates; in other words, an arbitrarily high energy density can be transferred to spacetime by means of localisation alone, which amounts to say that the model is not stable under localisation. 1
Nonrenormalizability of θexpanded noncommutative QED
"... Computing all divergent oneloop Green’s functions of θexpanded noncommutative quantum electrodynamics up to first order in θ, we show that this model is not renormalizable. The reason is a divergence in the electron fourpoint function which cannot be removed by field redefinitions. Ignoring this ..."
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Cited by 7 (0 self)
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Computing all divergent oneloop Green’s functions of θexpanded noncommutative quantum electrodynamics up to first order in θ, we show that this model is not renormalizable. The reason is a divergence in the electron fourpoint function which cannot be removed by field redefinitions. Ignoring this problem, we find however clear hints for new symmetries in massless θexpanded noncommutative QED: Four additional divergences which would be compatible with gauge and Lorentz symmetries and which are not reachable by field redefinitions are absent. raimar@doppler.thp.univie.ac.at, MarieCurie Fellow. 1
Twisted noncommutative field theory with the Wick–Voros and Moyal products
 REV. D
"... We present a comparison of the noncommutative field theories built using two different star products: Moyal and WickVoros (or normally ordered). For the latter we discuss both the classical and the quantum field theory in the quartic potential case, and calculate the Green’s functions up to one loo ..."
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Cited by 7 (3 self)
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We present a comparison of the noncommutative field theories built using two different star products: Moyal and WickVoros (or normally ordered). For the latter we discuss both the classical and the quantum field theory in the quartic potential case, and calculate the Green’s functions up to one loop, for the two and four points cases. We compare the two theories in the context of the noncommutative geometry determined by a Drinfeld twist, and the comparison is made at the level of Green’s functions and Smatrix. We find that while the Green’s functions are different for the two theories, the Smatrix is