152 research outputs found
Points. Lack thereof
I will discuss some aspects of the concept of "point" in quantum gravity,
using mainly the tool of noncommutative geometry. I will argue that at Planck's
distances the very concept of point may lose its meaning. I will then show how,
using the spectral action and a high momenta expansion, the connections between
points, as probed by boson propagators, vanish. This discussion follows closely
[1] (Kurkov-Lizzi-Vassilevich Phys. Lett. B 731 (2014) 311, [arXiv:1312.2235
[hep-th]].Comment: Proceedings of the XXII Krakow Methodological Conference: Emergence
of the Classical, Copernicus Centre, 11-12 October 2018. Mostly based on
arXiv:1312.2235. V2 corrects several typo
The Structure of Spacetime and Noncommutative Geometry
We give a general and nontechnical review of some aspects of noncommutative
geometry as a tool to understand the structure of spacetime. We discuss the
motivations for the constructions of a noncommutative geometry, and the passage
from commutative to noncommutative spaces. We then give a brief description of
Connes approach to the standard model, of the noncommutative geometry of
strings and of field theory on noncommutative spaces. We also discuss the role
of symmetries and some possible consequences for cosmology.Comment: 30 pages, Talk given at the workshop: Geometry, Topology, QFT and
Cosmology, Paris, 28-30 May 2008. To appear in the proceeding
Strings, Noncommutative Geometry and the Size of the Target Space
We describe how the presence of the antisymmetric tensor (torsion) on the
world sheet action of string theory renders the size of the target space a
gauge non invariant quantity. This generalizes the R 1/R symmetry in which
momenta and windings are exchanged, to the whole O(d,d,Z). The crucial point is
that, with a transformation, it is possible always to have all of the lowest
eigenvalues of the Hamiltonian to be momentum modes. We interpret this in the
framework of noncommutative geometry, in which algebras take the place of point
spaces, and of the spectral action principle for which the eigenvalues of the
Dirac operator are the fundamental objects, out of which the theory is
constructed. A quantum observer, in the presence of many low energy eigenvalues
of the Dirac operator (and hence of the Hamiltonian) will always interpreted
the target space of the string theory as effectively uncompactified.Comment: 19 pages, LateX, instructions for older LateX version
Dimensional Deception from Noncommutative Tori: An alternative to Horava-Lifschitz
We study the dimensional aspect of the geometry of quantum spaces.
Introducing a physically motivated notion of the scaling dimension, we study in
detail the model based on a fuzzy torus. We show that for a natural choice of a
deformed Laplace operator, this model demonstrates quite non-trivial behaviour:
the scaling dimension flows from 2 in IR to 1 in UV. Unlike another model with
the similar property, the so-called Horava-Lifshitz model, our construction
does not have any preferred direction. The dimension flow is rather achieved by
a rearrangement of the degrees of freedom. In this respect the number of
dimensions is deceptive. Some physical consequences are discussed.Comment: 20 pages + extensive appendix. 3 figure
Matrix Bases for Star Products: a Review
We review the matrix bases for a family of noncommutative products
based on a Weyl map. These products include the Moyal product, as well as the
Wick-Voros products and other translation invariant ones. We also review the
derivation of Lie algebra type star products, with adapted matrix bases. We
discuss the uses of these matrix bases for field theory, fuzzy spaces and
emergent gravity
Projective Systems of Noncommutative Lattices as a Pregeometric Substratum
We present an approximation to topological spaces by {\it noncommutative}
lattices. This approximation has a deep physical flavour based on the
impossibility to fully localize particles in any position measurement. The
original space being approximated is recovered out of a projective limit.Comment: 30 pages, Latex. To appear in `Quantum Groups and Fundamental
Physical Applications', ISI Guccia, Palermo, December 1997, D. Kastler and M.
Rosso Eds., (Nova Science Publishers, USA
Noncommutative Geometry, Strings and Duality
In this talk, based on work done in collaboration with G. Landi and R.J
Szabo, I will review how string theory can be considered as a noncommutative
geometry based on an algebra of vertex operators. The spectral triple of
strings is introduced, and some of the string symmetries, notably target space
duality, are discussed in this framework.Comment: Latex, 18 pages, Talk delivered at the Arbeitstagung: "The standard
Model of Elementary particle Physics, Mathematical and Geometrical Aspects",
Hesselberg, March 14-19 199
Electric-magnetic Duality in Noncommutative Geometry
The structure of S-duality in U(1) gauge theory on a 4-manifold M is examined
using the formalism of noncommutative geometry. A noncommutative space is
constructed from the algebra of Wilson-'t Hooft line operators which encodes
both the ordinary geometry of M and its infinite-dimensional loop space
geometry. S-duality is shown to act as an inner automorphism of the algebra and
arises as a consequence of the existence of two independent Dirac operators
associated with the spaces of self-dual and anti-selfdual 2-forms on M. The
relations with the noncommutative geometry of string theory and T-duality are
also discussed.Comment: 13 pages LaTeX, no figure
Higgs-Dilaton Lagrangian from Spectral Regularization
In this letter we calculate the full Higgs-Dilaton action describing the Weyl
anomaly using the bosonic spectral action. This completes the work we started
in our previous paper (JHEP 1110 (2011) 001). We also clarify some issues
related to the dilaton and its role as collective modes of fermions under
bosonization
Noncommutative Geometry and String Duality
A review of the applications of noncommutative geometry to a systematic
formulation of duality symmetries in string theory is presented. The spectral
triples associated with a lattice vertex operator algebra and the corresponding
Dirac-Ramond operators are constructed and shown to naturally incorporate
target space and discrete worldsheet dualities as isometries of the
noncommutative space. The target space duality and diffeomorphism symmetries
are shown to act as gauge transformations of the geometry. The connections with
the noncommutative torus and Matrix Theory compactifications are also
discussed.Comment: 17 pages, Latex2e, uses JHEP.cls (included); Based on talk given by
the first author at the 6th Hellenic School and Workshop on Elementary
Particle Physics, Corfu, Greece, September 6-26 1998. To be published in JHEP
proceeding
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