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Noncommutative Renormalization
 SÉMINAIRE POINCARÉ X (2007) 1 – 81
, 2007
"... A new version of scale analysis and renormalization theory has been found on the noncommutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on noncommutative tha ..."
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Cited by 169 (28 self)
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A new version of scale analysis and renormalization theory has been found on the noncommutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on noncommutative than on ordinary space: indeed it has no Landau ghost. Noncommutativity might therefore be an alternative to supersymmetry. We review this rapidly growing subject.
Induced Gauge Theory on a Noncommutative Space
, 2007
"... We consider a scalar φ4 theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field fr ..."
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Cited by 53 (16 self)
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We consider a scalar φ4 theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field from the divergent terms of the 1loop effective action using a matrix basis and propose an action for the noncommutative gauge theory, which is a candidate for a renormalisable model.
Noncommutative Induced Gauge Theory
, 2007
"... We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4dimensional Moyal space and compute in position space the oneloop YangMillstype effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action ..."
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Cited by 38 (10 self)
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We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4dimensional Moyal space and compute in position space the oneloop YangMillstype effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure YangMills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ 4theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.
THE DIXMIER TRACE AND ASYMPTOTICS OF ZETA FUNCTIONS
, 2006
"... We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semifinite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function det ..."
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Cited by 21 (11 self)
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We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semifinite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p, ∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.
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.
Heat Kernel and Number Theory on NCTorus
 Commun. Math. Phys
, 2007
"... The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the numbertheoretical aspect of the deformation parameters. The central condition we ..."
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Cited by 18 (11 self)
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The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the numbertheoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find nonlocal counterterms in the NC φ 4 theory on T 4, we show that this theory can be made renormalizable at least at one loop, and may be even beyond.
Spectral action on noncommutative torus
 J. Noncommut. Geom
"... Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series ..."
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Cited by 18 (9 self)
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Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.
Noncommutative QFT and Renormalization
, 2006
"... Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use the heat kernel ..."
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Cited by 15 (5 self)
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Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to φ 3 models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a θdeformed space and derive noncommutative gauge field actions.
Noncommutative Induced Gauge Theories on Moyal Spaces
, 2007
"... Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, ..."
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Cited by 13 (4 self)
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Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4D of the oneloop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure YangMills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative ϕ 4theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.
Non Commutative Field Theory on Rank One Symmetric Spaces,” arXiv:0806.4255 [hepth
"... Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary spacetime. Some models at least should be completely finite, even beyond perturbation theory. In this paper a first step is taken to extend such theories to nonflat backgrounds such ..."
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Cited by 7 (0 self)
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Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary spacetime. Some models at least should be completely finite, even beyond perturbation theory. In this paper a first step is taken to extend such theories to nonflat backgrounds such as solvable symmetric spaces. 1