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Strong connections on quantum principal bundles
 Commun. Math. Phys
, 1996
"... A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and nonstrong ..."
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Cited by 37 (7 self)
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A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and nonstrong connections are provided. In particular, such connections are constructed on a quantum deformation of the Hopf fibration S 2 → RP 2. A certain class of strong Uq(2)connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the qdependent hermitian metric. A particular form of the Yang–Mills action on a trivial Uq(2)bundle is investigated. It is proved to coincide with the Yang–Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.
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.
Gravity on fuzzy spacetime
, 1992
"... Dedicated to Walter Thirring on the occasion of his 70th birthday A review is made of recent efforts to add a gravitational field to noncommutative models of spacetime. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable spacetime. It i ..."
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Cited by 15 (0 self)
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Dedicated to Walter Thirring on the occasion of his 70th birthday A review is made of recent efforts to add a gravitational field to noncommutative models of spacetime. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable spacetime. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the spacetime on the one hand and the nature of the gravitational field which remains as a ‘shadow ’ in the commutative limit on the other. ESI Preprint 478 (1997). Lecture given at the International Workshop “Mathematical
Differential calculus and gauge theory on finite sets, Göttingen preprint GOETP 33/93
"... We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes ’ twopoint model (which is an essential ingredient of his reformulation of the standard model of elementary ..."
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Cited by 13 (6 self)
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We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes ’ twopoint model (which is an essential ingredient of his reformulation of the standard model of elementary particle physics) is recovered in our approach. Reductions of the universal differential calculus to ‘lowerdimensional’ differential calculi are considered. The ‘complete reduction ’ leads to a differential calculus on a periodic lattice.
Noncommutative Geometry for Pedestrians
"... A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of spacetime and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to ..."
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Cited by 3 (0 self)
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A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of spacetime and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature.
Differential Calculus and Discrete Structures
, 1994
"... There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of differential calculus on discrete sets. This framework gener ..."
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Cited by 2 (0 self)
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There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of differential calculus on discrete sets. This framework generalizes the usual (lattice) discretization.
Gauge Theories of Dirac Type
, 2005
"... A specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of YangMills gauge theories without making use of a Higgs potential. In more physical terms, it is show ..."
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Cited by 2 (2 self)
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A specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of YangMills gauge theories without making use of a Higgs potential. In more physical terms, it is shown that the Yukawa coupling of fermions, together with gravity, necessarily yields a symmetry reduction provided the fermionic mass is considered as a globally welldefined concept. The structure of this symmetry breaking is shown to be compatible with the symmetry breaking that is induced by the Higgs potential of the minimal Standard Model. As a consequence, it is shown that the fermionic mass has a simple geometrical interpretation in terms of curvature and that the (semiclassical) “fermionic vacuum ” determines the intrinsic geometry of spacetime. We also discuss the issue of “fermion doubling ” in some detail and introduce a specific projection onto the “physical subspace ” that is motivated from the Standard Model.
Regular FréchetLie groups of invertible elements in some inverse limits of unital involutive Banach algebras
 Georgian Math. J
"... Abstract. We consider a wide class of unital involutive topological algebras provided with a C ∗norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebras are taking a prominent position in noncommutative differential geometry, where they are often called un ..."
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Cited by 2 (0 self)
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Abstract. We consider a wide class of unital involutive topological algebras provided with a C ∗norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebras are taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet–Lie groups of Campbell–Baker–Hausdorff type and fulfill a nice infinitedimensional version of Lie’s second fundamental theorem.
Particle and Field Symmetries and Noncommutative Geometry
, 2003
"... The development of Noncommutative geometry is creating a reworking and new possibilities in physics. This paper identifies some of the commutation and derivation structures that arise in particle and field interactions and fundamental symmetries. The requirements of coexisting structures, and their ..."
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Cited by 1 (1 self)
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The development of Noncommutative geometry is creating a reworking and new possibilities in physics. This paper identifies some of the commutation and derivation structures that arise in particle and field interactions and fundamental symmetries. The requirements of coexisting structures, and their consistency, produce a mathematical framework that underlies a fundamental physics theory. Among other developments in Quantum theory of particles and fields are the symmetries of gauge fields and the FermiBose symmetry of particles. These involve a gauge covariant derivation and the action functionals; and commutation algebras and Bogoliubov transforms. The non commutative Theta form introduces an additional and fundamental structure. This paper obtains the interrelations of the various structures; and the conditions for the symmetries of Fermionic/Bosonic particles interacting with YangMills gauge fields. Many example physical systems are being solved, and the mathematical formalism is being created to understand the fundamental basis of physics. The mathematical structures of the physics of particles and fields were developed using commutative and non commutative algebra, and Euclidean and non Euclidean Geometry. This led to Quantum Mechanics and General
Modular Theory, NonCommutative Geometry and Quantum Gravity ⋆
"... doi:10.3842/SIGMA.2010.067 Abstract. This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita–Takesaki modular theory and A. Connes noncommutative geometry aiming at the reconstruction of spectral geometries fr ..."
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doi:10.3842/SIGMA.2010.067 Abstract. This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita–Takesaki modular theory and A. Connes noncommutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in noncommutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.