86 research outputs found
A Renormalization Group Analysis of the NCG constraints m_{top} = 2\,m_W},
We study the evolution under the renormalization group of the restrictions on
the parameters of the standard model coming from Non-Commutative Geometry,
namely and . We adopt the point of
view that these relations are to be interpreted as {\it tree level} constraints
and, as such, can be implemented in a mass independent renormalization scheme
only at a given energy scale . We show that the physical predictions on
the top and Higgs masses depend weakly on .Comment: 7 pages, FTUAM-94/2, uses harvma
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 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.Comment: Latex, 54 pages. Version 3 with Moyal-Wick section update
Quasi-Dirac Operators and Quasi-Fermions
We investigate examples of quasi-spectral triples over two-dimensional
commutative sphere, which are obtained by modifying the order-one condition. We
find equivariant quasi-Dirac operators and prove that they are in a
topologically distinct sector than the standard Dirac operator.Comment: 11 page
On summability of distributions and spectral geometry
Modulo the moment asymptotic expansion, the CesĂ ro 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 CesĂ ro 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 CesĂ ro asymptotic development
Local covariant quantum field theory over spectral geometries
A framework which combines ideas from Connes' noncommutative geometry, or
spectral geometry, with recent ideas on generally covariant quantum field
theory, is proposed in the present work. A certain type of spectral geometries
modelling (possibly noncommutative) globally hyperbolic spacetimes is
introduced in terms of so-called globally hyperbolic spectral triples. The
concept is further generalized to a category of globally hyperbolic spectral
geometries whose morphisms describe the generalization of isometric embeddings.
Then a local generally covariant quantum field theory is introduced as a
covariant functor between such a category of globally hyperbolic spectral
geometries and the category of involutive algebras (or *-algebras). Thus, a
local covariant quantum field theory over spectral geometries assigns quantum
fields not just to a single noncommutative geometry (or noncommutative
spacetime), but simultaneously to ``all'' spectral geometries, while respecting
the covariance principle demanding that quantum field theories over isomorphic
spectral geometries should also be isomorphic. It is suggested that in a
quantum theory of gravity a particular class of globally hyperbolic spectral
geometries is selected through a dynamical coupling of geometry and matter
compatible with the covariance principle.Comment: 21 pages, 2 figure
Noncommutative spacetime symmetries: Twist versus covariance
We prove that the Moyal product is covariant under linear affine spacetime
transformations. From the covariance law, by introducing an -space
where the spacetime coordinates and the noncommutativity matrix components are
on the same footing, we obtain a noncommutative representation of the affine
algebra, its generators being differential operators in -space. As
a particular case, the Weyl Lie algebra is studied and known results for Weyl
invariant noncommutative field theories are rederived in a nutshell. We also
show that this covariance cannot be extended to spacetime transformations
generated by differential operators whose coefficients are polynomials of order
larger than one. We compare our approach with the twist-deformed enveloping
algebra description of spacetime transformations.Comment: 19 pages in revtex, references adde
The Moyal Sphere
We construct a family of constant curvature metrics on the Moyal plane and
compute the Gauss-Bonnet term for each of them. They arise from the conformal
rescaling of the metric in the orthonormal frame approach. We find a particular
solution, which corresponds to the Fubini-Study metric and which equips the
Moyal algebra with the geometry of a noncommutative sphere.Comment: 16 pages, 3 figure
On the ultraviolet behaviour of quantum fields over noncommutative manifolds
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
behaviour of second-quantized fields over noncommutative 3-tori, and discuss
what behaviour should be expected on other noncommutative spin manifolds.Comment: 10 pages, RevTeX version, a few references adde
Geometrical origin of the *-product in the Fedosov formalism
The construction of the *-product proposed by Fedosov is implemented in terms
of the theory of fibre bundles. The geometrical origin of the Weyl algebra and
the Weyl bundle is shown. Several properties of the product in the Weyl algebra
are proved. Symplectic and abelian connections in the Weyl algebra bundle are
introduced. Relations between them and the symplectic connection on a phase
space M are established. Elements of differential symplectic geometry are
included. Examples of the Fedosov formalism in quantum mechanics are given.Comment: LaTeX, 39 page
On a Classification of Irreducible Almost-Commutative Geometries V
We extend a classification of irreducible, almost-commutative geometries
whose spectral action is dynamically non-degenerate, to internal algebras that
have six simple summands. We find essentially four particle models: An
extension of the standard model by a new species of fermions with vectorlike
coupling to the gauge group and gauge invariant masses, two versions of the
electro-strong model and a variety of the electro-strong model with Higgs
mechanism
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