Results 1  10
of
27
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 ..."
Abstract

Cited by 64 (9 self)
 Add to MetaCart
(Show Context)
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. ..."
Abstract

Cited by 49 (5 self)
 Add to MetaCart
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.
Discrete Differential Calculus, Graphs, Topologies and Gauge Theory
 TO APPEAR IN J. MATH. PHYS.
, 1994
"... Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘reduction ’ of the ‘universal differential algebra ’ and this allows a systematic exploration of differential algebras on a given set. Assoc ..."
Abstract

Cited by 28 (4 self)
 Add to MetaCart
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘reduction ’ of the ‘universal differential algebra ’ and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a ‘Hasse diagram’ determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the twopoint space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an ‘internal’ discrete space (à la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, also a ‘symmetric lattice’ is studied which (in a certain continuum limit) turns out to be related to a ‘noncommutative differential calculus’ on manifolds.
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 ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
(Show Context)
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
Modular Theory, NonCommutative Geometry and Quantum Gravity
, 2010
"... 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 state ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
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.