Results 1  10
of
240
On conformal field theories
 in fourdimensions,” Nucl. Phys. B533
, 1998
"... We review the generalization of field theory to spacetime with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last ..."
Abstract

Cited by 366 (1 self)
 Add to MetaCart
(Show Context)
We review the generalization of field theory to spacetime with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.
Noncommutative manifolds, the instanton algebra and isospectral deformations
 Comm. Math. Phys
"... We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the ..."
Abstract

Cited by 172 (29 self)
 Add to MetaCart
(Show Context)
We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the NC4spheres S4 θ. We construct the noncommutative algebras A = C ∞ (S4 θ) of functions on NCspheres as solutions to the vanishing, chj(e) = 0,j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M4(A), e2 = e, e = e ∗. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC3sphere intimately related to quantum group deformations SUq(2) of SU(2) but for unusual values (complex values of modulus one) of the parameter q of qanalogues, q = exp(2πiθ). We then construct the noncommutative geometry of S4 θ as given by a spectral triple (A, H,D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric gµν on S4 whose volume form √ g d4x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation, e − 1
Metrics on states from actions of compact groups
 Doc. Math
, 1998
"... Abstract. Let a compact Lie group act ergodically on a unital C ∗algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the correspon ..."
Abstract

Cited by 70 (7 self)
 Add to MetaCart
Abstract. Let a compact Lie group act ergodically on a unital C ∗algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak∗ topology. Connes [Co1, Co2, Co3] has shown us that Riemannian metrics on noncommutative spaces (C ∗algebras) can be specified by generalized Dirac operators. Although in this setting there is no underlying manifold on which one then obtains an ordinary metric, Connes has shown that one does obtain in a simple way an ordinary metric on the state space of the C ∗algebra, generalizing the MongeKantorovich metric on probability measures [Ra] (called the “Hutchinson metric ” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees with the underlying weak ∗ topology on the state space. Note that for locally compact spaces their topology agrees with the weak ∗ topology coming from viewing points as linear functionals (by evaluation) on the algebra of continuous functions vanishing at infinity.
Combinatorial Hopf algebras in quantum field theory I
 Reviews of Mathematical Physics
, 2005
"... This manuscript collects and expands for the most part a series of lectures on the interface between combinatorial Hopf algebra theory (CHAT) and renormalization theory, delivered by the secondnamed author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Li ..."
Abstract

Cited by 60 (3 self)
 Add to MetaCart
(Show Context)
This manuscript collects and expands for the most part a series of lectures on the interface between combinatorial Hopf algebra theory (CHAT) and renormalization theory, delivered by the secondnamed author in the framework of the joint mathematical physics seminar of the Universités d’Artois and Lille 1, from late January till midFebruary 2003. The plan is as follows: Section 1 is the introduction, and Section 2 contains an elementary invitation to the subject. Sections 3–7 are devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Section 8 contains a first, direct approach to the Faà di Bruno Hopf algebra. Section 9 gives applications of that to quantum field theory and Lagrange reversion. Section 10 rederives the Connes– Moscovici algebras. In Section 11 we turn to Hopf algebras of Feynman graphs. Then in Section 12 we give an extremely simple derivation of (the properly combinatorial part of) Zimmermann’s method, in its original diagrammatic form. In Section 13 general incidence algebras are introduced. In the next section the Faà di Bruno bialgebras
GromovHausdorff distance for quantum metric spaces
 Mem. Amer. Math. Soc
"... Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distanc ..."
Abstract

Cited by 57 (7 self)
 Add to MetaCart
(Show Context)
Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ. 1.
The Dirac operator on SUq(2)
, 2005
"... We construct a 3 +summable spectral triple (A(SUq(2)), H,D) over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Uq(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operat ..."
Abstract

Cited by 44 (8 self)
 Add to MetaCart
We construct a 3 +summable spectral triple (A(SUq(2)), H,D) over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Uq(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and firstorder properties need be satisfied only modulo infinitesimals of arbitrary high order.
On Groupoid C∗Algebras, Persistent Homology and TimeFrequency Analysis
"... We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in tim ..."
Abstract

Cited by 44 (1 self)
 Add to MetaCart
(Show Context)
We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in timefrequency analysis. The main result of our work is to illustrate how noncommutative C ∗algebras and the concept of Morita equivalence can be applied as a new type of analysis layer in signal processing. From a conceptual point of view, we use groupoid C∗algebras constructed with timefrequency data in order to study a given signal. From a computational point of view, we consider persistent homology as an algorithmic tool for estimating topological properties in timefrequency analysis. The usage of C∗algebras in our environment, together with the problem of designing computational algorithms, naturally leads to our proposal of using AFalgebras in the persistent homology setting. Finally, a computational toy example is presented, illustrating some elementary aspects of our framework. Due to the interdisciplinary nature
Combinatorics of rooted trees and Hopf algebras
, 2002
"... We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of nonroot vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the op ..."
Abstract

Cited by 40 (5 self)
 Add to MetaCart
(Show Context)
We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of nonroot vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators to Kreimer’s Hopf algebra and relate them via the inner product. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the GrossmanLarson Hopf algebra with the graded dual of Kreimer’s Hopf algebra, correcting an earlier result of Panaite. 1
Between classical and quantum
, 2008
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
Quantum symmetry groups of noncommutative spheres
 Commun. Math. Phys
, 2001
"... We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. 1 ..."
Abstract

Cited by 34 (2 self)
 Add to MetaCart
(Show Context)
We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. 1