José M. Gracia-Bondía

Algebras of distributions suitable for phase‐space quantum mechanics. I

The twisted product of functions on R2N is extended to a *‐algebra of tempered distributions that contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.

Moyal Planes are Spectral Triples

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 R2N 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.

The Moyal representation for spin

Abstract The phase-space approach to spin is developed from two basic principles, SU(2)-covariance and traciality, as a theory of Wigner functions on the sphere. The twisted product of phase-space functions is related to group convolution on SU(2) by means of a Fourier transform theory on the coadjoint orbits, which yields the Plancherel-Parseval formula. Coherent spin states provide an alternative route to the same phase-space description of spin. The Wigner functions for spin states and transitions are exhibited up to j = 2. It is shown that for Hamiltonians such as arise from time-dependent magnetic fields, the quantum spin dynamics is given entirely by the classical motion on the sphere. The Majorana formula becomes transparent in the Moyal representation.

Chiral gauge anomalies on noncommutative R4

Abstract We discuss the noncommutative counterparts of chiral gauge theories and compute the associated anomalies.

THE STANDARD MODEL AS A NONCOMMUTATIVE GEOMETRY : THE LOW-ENERGY REGIME

Abstract We render a thorough, physicists 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 .

On asymptotic expansions of twisted products

The series development of the quantum‐mechanical twisted product is studied. The series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product, either pointwise or in the distributional sense depending on the nature of the factors. A condition is given that ensures convergence and is stronger than previously known results. Possible applications are examined.

COMBINATORIAL HOPF ALGEBRAS IN QUANTUM FIELD THEORY I

This paper stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Sec. 1.1 is the introduction, and contains an elementary invitation to the subject as well. The rest of Sec. 1 is devoted to the basics of Hopf algebra theory and examples in ascending level of complexity. Section 2 turns around the all-important Faa di Bruno Hopf algebra. Section 2.1 contains a first, direct approach to it. Section 2.2 gives applications of the Faa di Bruno algebra to quantum field theory and Lagrange reversion. Section 2.3 rederives the related Connes–Moscovici algebras. In Sec. 3, we turn to the Connes–Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Sec. 3.1, we describe the first. Then in Sec. 3.2, we give a simple derivation of (the properly combinatorial part of) Zimmermanns cancellation-free method, in its original diagrammatic form. In Sec. 3.3, general incidence algebras are introduced, and the Faa di Bruno bialgebras are described as incidence bialgebras. In Sec. 3.4, deeper lore on Rotas incidence algebras allows us to reinterpret Connes–Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained. The structure results for commutative Hopf algebras are found in Sec. 4. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota–Baxter map in renormalization.

Algebras of distributions suitable for phase‐space quantum mechanics. II. Topologies on the Moyal algebra

The topology of the Moyal *‐algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the *‐algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. The equivalence of the three topologies thereby obtained is proved. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, new sufficient conditions are given for a phase‐space function to correspond to a trace‐class operator via the Weyl correspondence rule.

On Summability of Distributions and Spectral Geometry

Abstract:Modulo the moment asymptotic expansion, the Cesáro 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 Cesáro 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 Cesáro asymptotic development.

A Lie Theoretic Approach to Renormalization

Motivated by recent work of Connes and Marcolli, based on the Connes–Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on properties of Hopf algebras encapsulated in the notion of associated descent algebras. Besides leading very directly to proofs of the main combinatorial aspects of the renormalization procedures, the new techniques give rise to an algebraic approach to the Galois theory of renormalization. In particular, they do not depend on the geometry underlying the case of dimensional regularization and the Riemann–Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.