Papers by Joseph C Varilly
Communications in Mathematical Physics, 2005
Journal of Mathematical Physics, 1994
Systematic use of the infinite-dimensional spin representation simplifies and rigorizes several q... more Systematic use of the infinite-dimensional spin representation simplifies and rigorizes several questions in Quantum Field Theory. This representation permutes "Gaussian" elements in the fermion Fock space, and is necessarily projective: we compute its cocycle at the group level, and obtain Schwinger terms and anomalies from infinitesimal versions of this cocycle. Quantization, in this framework, depends on the choice of the "right" complex structure on the space of solutions of the Dirac equation. We show how the spin representation allows one to compute exactly the S-matrix for fermions in an external field; the cocycle yields a causality condition needed to determine the phase. II.1. Orthogonal complex structures We start with a real vector space V and a symmetric bilinear form d, given a priori. We lose nothing by supposing V to be complete in the metric induced by d, so we take (V, d) to be a real Hilbert space, either infinite-dimensional or of finite even dimension. An orthogonal complex structure J is a real-linear operator on V satisfying: J 2 = −I, and d(Ju, Jv) = d(u, v) for u, v ∈ V. Now, regarding V as a complex vector space via the rule (α + iβ)v := αv + βJv for α, β real, the hermitian form u | v J := d(u, v) + id(Ju, v) makes (V, d, J) a complex Hilbert space. The orthogonal group O(V) is { g ∈ GL R (V) : d(gu, gv) = d(u, v), for all u, v ∈ V }. Note that g is orthogonal iff g t g = I where the transpose is with respect to d.
International Journal of Modern Physics A, 1999
By exploiting the relation between Fredholm modules and the Segal–Shale–Stinespring version of ca... more By exploiting the relation between Fredholm modules and the Segal–Shale–Stinespring version of canonical quantization, and taking as starting point the first-quantized fields described by Connes' axioms for noncommutative spin geometries, a Hamiltonian framework for fermion quantum fields over noncommutative manifolds is introduced. We analyze the ultraviolet behavior of second-quantized fields over noncommutative three-tori, and discuss what behavior should be expected on other noncommutative spin manifolds.
Physics Reports, 1998
We render a thorough, physicist's account of the formulation of the Standard Model (SM) of partic... more We render a thorough, physicist's account of the formulation of the Standard Model (SM) of particle physics within the framework of noncommutative differential geometry (NCG). We work in Minkowski spacetime rather than in Euclidean space. We lay the stress on the physical ideas both underlying and coming out of the noncommutative derivation of the SM, while we provide the necessary mathematical tools. Postdiction of most of the main characteristics of the SM is shown within the NCG framework. This framework, plus standard renormalization technique at the one-loop level, suggest that the Higgs and top masses should verify 1.3 m top m H 1.73 m top .
Communications in Mathematical Physics, 2001
We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for ce... more 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.
K-Theory, 2005
We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometr... more We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU (2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.
International Journal of Quantum Chemistry, 2011
Forty-five years after the point de départ [1] of density functional theory, its applications in ... more Forty-five years after the point de départ [1] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the electron density still eludes us-and possibly will do so forever [2]. In what follows we examine a formulation in the same spirit with phase space variables. The validity of Hohenberg-Kohn-Levy-type theorems on phase space is recalled. We study the representability problem for reduced Wigner functions, and proceed to analyze properties of the new functional. Along the way, new results on states in the phase space formalism of quantum mechanics are established. Natural Wigner orbital theory is developed in depth, with the final aim of constructing accurate correlation-exchange functionals on phase space. A new proof of the overbinding property of the Müller functional is given. This exact theory supplies its home at long last to that illustrious ancestor, the Thomas-Fermi model.
Communications in Mathematical Physics, 2004
Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, a... more Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R 2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes-Lott functional action, are given for these noncommutative hyperplanes.
Communications in Mathematical Physics, 1998
Modulo the moment asymptotic expansion, the Cesàro and parametric behaviours of distributions at ... more Modulo the moment asymptotic expansion, the Cesàro and parametric behaviours of distributions at infinity are equivalent. On the strength of this result, we construct the asymptotic analysis for spectral densities, arising from elliptic pseudodifferential operators. We show how Cesàro developments lead to efficient calculations of the expansion coefficients of counting number functionals and Green functions. The bosonic action functional proposed by Chamseddine and Connes can more generally be validated as a Cesàro asymptotic development.
Journal of Physics A, Mar 21, 1990
In this paper, we obtain the phase-space quantization for relativistic spinning particles. The ma... more In this paper, we obtain the phase-space quantization for relativistic spinning particles. The main tool is what we call a "Stratonovich-Weyl quantizer" which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Our phase spaces are coadjoint orbits of (the covering group of) the restricted Poincaré group; we compute and explicitly coordinatize the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincaré group; we show that there is a unique Stratonovich-Weyl quantizer from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly. We develop the formalism by computing relativistic Wigner functions and twisted products for Klein-Gordon particles; these Wigner functions are supported on the mass shell. We thereby obtain an expression for the position probability density which is local, that is, free from the difficulty of supraluminal propagation of the usual position probability density. It is shown explicitly how observables on phase space may be quantized; for example, we prove that the canonical position coordinate corresponds to the Newton-Wigner position operator, irrespective of spin. We show how relativistic phase-space quantization applies to particles governed by the Dirac equation. In effect, we construct a Stratonovich-Weyl quantizer whose associated Hilbert space is the space of positive-energy solutions of the Dirac equation.
Letters in Mathematical Physics, 1981
Journal of the Australian Mathematical Society, 2008
In this note we show that the crucial orientation condition for commutative geometries fails for ... more In this note we show that the crucial orientation condition for commutative geometries fails for the natural commutative spectral triple of an orbifold M/G.
CONTENTS 7. Equivalence of Geometries Unitary equivalence of geometries Morita equivalence and He... more CONTENTS 7. Equivalence of Geometries Unitary equivalence of geometries Morita equivalence and Hermitian connections Vector bundles over the noncommutative torus Morita-equivalent toral geometries Gauge potentials 8. Action Functionals Automorphisms of the algebra The fermionic action The spectral action principle Spectral densities and asymptotics References
We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for ce... more 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.
We show that the algebra A of a commutative unital spectral triple (A, H, D) satisfying several a... more We show that the algebra A of a commutative unital spectral triple (A, H, D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.
Connes''noncommutative di erential geometry and the Standard Model
We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several add... more We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.
The lecture notes of this course at the EMS Summer School on Noncommutative Geometry and Applicat... more The lecture notes of this course at the EMS Summer School on Noncommutative Geometry and Applications in September, 1997 are now published by the EMS. Here are the contents, preface and updated bibliography from the published book. Contents 1 Commutative Geometry from the Noncommutative Point of View 1.1 The Gelfand-Naȋmark cofunctors 1.2 The Γ functor 1.3 Hermitian metrics and spin c structures 1.4 The Dirac operator and the distance formula 2 Spectral Triples on the Riemann Sphere 2.1 Line bundles and the spinor bundle 2.2 The Dirac operator on the sphere S 2 2.3 Spinor harmonics and the spectrum of D / 2.4 Twisted spinor modules 2.5 A reducible spectral triple 3 Real Spectral Triples: the Axiomatic Foundation 3.1 The data set 3.2 Infinitesimals and dimension 3.3 The first-order condition 3.4 Smoothness of the algebra 3.5 Hochschild cycles and orientation 3.6 Finiteness of the K-cycle 3.7 Poincaré duality and K-theory 3.8 The real structure Geometries on the Noncommutative Torus 4.1 Algebras of Weyl operators 4.2 The algebra of the noncommutative torus 4.3 The skeleton of the noncommutative torus 4.4 A family of spin geometries on the torus The Noncommutative Integral 5.1 The Dixmier trace on infinitesimals 5.2 Pseudodifferential operators 5.3 The Wodzicki residue 5.4 The trace theorem 5.5 Integrals and zeta residues Quantization and the Tangent Groupoid 6.1 Moyal quantizers and the Moyal deformation 6.2 Smooth groupoids 6.3 The tangent groupoid 6.4 Moyal quantization as a continuity condition 6.5 The hexagon and the analytical index 6.6 Quantization and the index theorem Equivalence of Geometries 7.1 Unitary equivalence of spin geometries 7.2 Morita equivalence and connections 7.3 Vector bundles over noncommutative tori 7.4 Morita-equivalent toral geometries 7.5 Gauge potentials Action Functionals 8.1 Algebra automorphisms and the metric 8.2 The fermionic action 8.3 The spectral action principle 8.4 Spectral densities and asymptotics Epilogue: New Directions 9.1 Noncommutative field theories 9.2 Isospectral deformations 9.3 Geometries with quantum group symmetry 9.4 Other developments
Connes' Tangent Groupoid and Strict Quantization
We address one of the open problems in quantization theory recently listed by Rieffel. By develop... more We address one of the open problems in quantization theory recently listed by Rieffel. By developing in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.
Journal of Geometry and Physics
Title: Connes' noncommutative differential geometry and the standard model. ... Origin: ELSE... more Title: Connes' noncommutative differential geometry and the standard model. ... Origin: ELSEVIER. Keywords: noncommutative geometry, Standard Model, Dixmier trace, K-cycle, Dirac operator, Yang-Mills functional, 46 L 87, 81 T 13, 18 G 60, 12.10.Dm, 02.40.+m. ...
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Papers by Joseph C Varilly