36,999 research outputs found
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
The action functional for Moyal planes
Modulo some natural generalizations to noncompact spaces, we show in this
letter that Moyal planes are nonunital spectral triples in the sense of Connes.
The action functional of these triples is computed, and we obtain the expected
result, ie the noncommutative Yang-Mills action associated with the Moyal
product. In particular, we show that Moyal gauge theory naturally fit into the
rigorous framework of noncommutative geometry.Comment: latex, 10 page
The spectral distance on the Moyal plane
We study the noncommutative geometry of the Moyal plane from a metric point
of view. Starting from a non compact spectral triple based on the Moyal
deformation A of the algebra of Schwartz functions on R^2, we explicitly
compute Connes' spectral distance between the pure states of A corresponding to
eigenfunctions of the quantum harmonic oscillator. For other pure states, we
provide a lower bound to the spectral distance, and show that the latest is not
always finite. As a consequence, we show that the spectral triple [20] is not a
spectral metric space in the sense of [5]. This motivates the study of
truncations of the spectral triple, based on M_n(C) with arbitrary integer n,
which turn out to be compact quantum metric spaces in the sense of Rieffel.
Finally the distance is explicitly computed for n=2.Comment: Published version. Misprints corrected and references updated;
Journal of Geometry and Physics (2011
Connes distance by examples: Homothetic spectral metric spaces
We study metric properties stemming from the Connes spectral distance on
three types of non compact noncommutative spaces which have received attention
recently from various viewpoints in the physics literature. These are the
noncommutative Moyal plane, a family of harmonic Moyal spectral triples for
which the Dirac operator squares to the harmonic oscillator Hamiltonian and a
family of spectral triples with Dirac operator related to the Landau operator.
We show that these triples are homothetic spectral metric spaces, having an
infinite number of distinct pathwise connected components. The homothetic
factors linking the distances are related to determinants of effective Clifford
metrics. We obtain as a by product new examples of explicit spectral distance
formulas. The results are discussed.Comment: 23 pages. Misprints corrected, references updated, one remark added
at the end of the section 3. To appear in Review in Mathematical Physic
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
Examples of derivation-based differential calculi related to noncommutative gauge theories
Some derivation-based differential calculi which have been used to construct
models of noncommutative gauge theories are presented and commented. Some
comparisons between them are made.Comment: 22 pages, conference given at the "International Workshop in honour
of Michel Dubois-Violette, Differential Geometry, Noncommutative Geometry,
Homology and Fundamental Interactions". To appear in a special issue of
International Journal of Geometric Methods in Modern Physic
On Radon transforms on tori
We show injectivity of the X-ray transform and the -plane Radon transform
for distributions on the -torus, lowering the regularity assumption in the
recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of
the X-ray transform on the -torus for tensor fields of any order, allowing
the tensors to have distribution valued coefficients. These imply new
injectivity results for the periodic broken ray transform on cubes of any
dimension.Comment: 13 page
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