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Lectures On Alain Connes' Non Commutative Geometry And Applications To Fundamental Interactions
, 1994
"... We introduce the reader to Alain Connes non commutative differential geometry, and sketch the applications made to date to (the lagrangian level of) fundamental physical interactions. ..."
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We introduce the reader to Alain Connes non commutative differential geometry, and sketch the applications made to date to (the lagrangian level of) fundamental physical interactions.
CLASSIFICATION OF INJECTIVE FACTORS: THE WORK OF ALAIN CONNES
, 1981
"... ABSTRACT: The fundamental results of A. Connes which determine a complete set of isomorphism classes for most injectlve factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines of the ..."
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ABSTRACT: The fundamental results of A. Connes which determine a complete set of isomorphism classes for most injectlve factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines
A detailed account of Alain Connes’ version of the standard model in noncommutative differential geometry
 I. and II., Rev. Math. Phys. Vol 5, N
, 1993
"... We give a detailed account of the computation of the YangMills action for the ConnesLott model with general coupling constant in the commutant of the Kcycle. This leads to treeapproximation results amazingly compatible with experiment, yielding a first indication on the Higgs mass. PACS92: 11.15 ..."
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Cited by 32 (5 self)
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We give a detailed account of the computation of the YangMills action for the ConnesLott model with general coupling constant in the commutant of the Kcycle. This leads to treeapproximation results amazingly compatible with experiment, yielding a first indication on the Higgs mass. PACS92: 11
CPT93/P.2960
, 1993
"... Alain Connes ’ applications of noncommutative geometry to interaction physics ..."
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Alain Connes ’ applications of noncommutative geometry to interaction physics
On the BaumConnes Conjecture
, 2005
"... In the early 1980s Paul Baum and Alain Connes conjectured a link between the Ktheory of the reduced C∗algebra of a group and the Khomology of the corresponding classifying space of proper actions of that group. This statement, also known as the BaumConnes Conjecture, is not ver ..."
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In the early 1980s Paul Baum and Alain Connes conjectured a link between the Ktheory of the reduced C∗algebra of a group and the Khomology of the corresponding classifying space of proper actions of that group. This statement, also known as the BaumConnes Conjecture, is not ver
Relating the ConnesKreimer and GrossmanLarson Hopf algebras built on rooted trees
"... In [8], Dirk Kreimer discovered the striking fact that the process of renormalization in quantum field theory may be described, in a conceptual manner, by means of certain Hopf algebras (which depend on the chosen renormalization scheme). A toy model was studied in detail by Alain Connes and Dirk Kr ..."
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Cited by 19 (0 self)
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In [8], Dirk Kreimer discovered the striking fact that the process of renormalization in quantum field theory may be described, in a conceptual manner, by means of certain Hopf algebras (which depend on the chosen renormalization scheme). A toy model was studied in detail by Alain Connes and Dirk
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 ..."
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Cited by 366 (1 self)
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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.
Gravity coupled with matter and the foundation of non commutative geometry
, 1996
"... We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×— × = D −1 where D i ..."
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Cited by 354 (18 self)
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We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = ×— × = D −1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita’s involution J. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.
Renormalization in quantum field theory and the RiemannHilbert problem. II. The βfunction, diffeomorphisms and the renormalization group
 Comm. Math. Phys
"... We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann–Hilbert problem. Given a loop γ(z), z  = 1 of elements of a complex Lie group G the general procedure is given by evalu ..."
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Cited by 344 (39 self)
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We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann–Hilbert problem. Given a loop γ(z), z  = 1 of elements of a complex Lie group G the general procedure is given by evaluation of γ+(z) at z = 0 after performing the Birkhoff decomposition γ(z) = γ−(z) −1 γ+(z) where γ±(z) ∈ G are loops holomorphic in the inner and outer domains of the Riemann sphere (with γ−(∞) = 1). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the MilnorMoore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme. 1
Results 1  10
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