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Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 22 (3 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Differential calculi and linear connections
 J. Math. Phys
, 1996
"... A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a onetoone correspondence, between the module structure of ..."
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Cited by 21 (13 self)
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A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a onetoone correspondence, between the module structure of the 1forms and the metric torsionfree connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1forms.
Some Aspects of Noncommutative Differential Geometry
"... We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finall ..."
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Cited by 13 (2 self)
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We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1
Sitarz, "Deformations of Differential Calculi
 Mod. Phys. Lett. A
"... It has been suggested that quantum fluctuations of the gravitational field could give rise in the lowest approximation to an effective noncommutative version of KaluzaKlein theory which has as extra hidden structure a noncommutative geometry. It would seem however from the Standard Model, at least ..."
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Cited by 3 (3 self)
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It has been suggested that quantum fluctuations of the gravitational field could give rise in the lowest approximation to an effective noncommutative version of KaluzaKlein theory which has as extra hidden structure a noncommutative geometry. It would seem however from the Standard Model, at least as far as the weak interactions are concerned, that a doublesheeted structure is the phenomenologically appropriate one at present accelerator energies. We examine here to what extent this latter structure can be considered as a singular limit of the former.
Fuzzy Surfaces of Genus Zero
 Preprint LPTHE Orsay 97/26, grqc/9706047
"... A fuzzy version of the ordinary round 2sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large numb ..."
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Cited by 2 (2 self)
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A fuzzy version of the ordinary round 2sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere.
MPI–PhT–2001/31 gk–mp–0108/71 Noncommutative Geometry: Calculation of the Standard Model Lagrangian
, 2001
"... The calculation of the standard model Lagrangian of classical field theory within the framework of noncommutative geometry is sketched using a variant with 18 parameters. Improvements compared with the traditional formulation are contrasted with remaining deviations from the requirements of physics. ..."
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The calculation of the standard model Lagrangian of classical field theory within the framework of noncommutative geometry is sketched using a variant with 18 parameters. Improvements compared with the traditional formulation are contrasted with remaining deviations from the requirements of physics. This paper is based on a talk given at the Euroconference “Brane New World and Noncommutative
Laboratoire de Physique Théorique et Hautes Energies*
, 1994
"... Abstract: A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric. ..."
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Abstract: A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
Noncommutative Geometry: Calculation of the Standard Model Lagrangian
, 2001
"... The calculation of the standard model Lagrangian of classical field theory within the framework of noncommutative geometry is sketched using a variant with 18 parameters. Improvements compared with the traditional formulation are contrasted with remaining deviations from the requirements of physics. ..."
Abstract
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The calculation of the standard model Lagrangian of classical field theory within the framework of noncommutative geometry is sketched using a variant with 18 parameters. Improvements compared with the traditional formulation are contrasted with remaining deviations from the requirements of physics. This paper is based on a talk given at the Euroconference “Brane New World and Noncommutative
On Connes ’ new principle of general relativity Can spinors hear the forces of spacetime?
, 1996
"... Connes has extended Einstein’s principle of general relativity to noncommutative geometry. The new principle implies that the Dirac operator is covariant with respect to Lorentz and internal gauge transformations and the Dirac operator must include Yukawa couplings. It further implies that the actio ..."
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Connes has extended Einstein’s principle of general relativity to noncommutative geometry. The new principle implies that the Dirac operator is covariant with respect to Lorentz and internal gauge transformations and the Dirac operator must include Yukawa couplings. It further implies that the action for the metric, the gauge potentials and the Higgs scalar is coded in the spectrum of the covariant Dirac operator. This universal action has been computed by Chamseddine & Connes, it is the coupled EinsteinHilbert and YangMillsHiggs action. This result is rederived and we discuss the physical consequences. PACS92: 11.15 Gauge field theories MSC91: 81T13 YangMills and other gauge theories
NONCOMMUTATIVE YANGMILLS AND NONCOMMUTATIVE RELATIVITY: A BRIDGE OVER TROUBLE WATER
"... Connes ’ view at YangMills theories is reviewed with special emphasis on the gauge invariant scalar product. This landscape is shown to contain Chamseddine and Connes ’ noncommutative extension of general relativity restricted to flat spacetime, if the top mass is between 172 and 204 GeV. Then the ..."
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Connes ’ view at YangMills theories is reviewed with special emphasis on the gauge invariant scalar product. This landscape is shown to contain Chamseddine and Connes ’ noncommutative extension of general relativity restricted to flat spacetime, if the top mass is between 172 and 204 GeV. Then the Higgs mass is between 188 and 201 GeV. PACS92: 11.15 Gauge field theories MSC91: 81T13 YangMills and other gauge theories