1,612 research outputs found
Noncommutative Uncertainty Principles
The classical uncertainty principles deal with functions on abelian groups.
In this paper, we discuss the uncertainty principles for finite index
subfactors which include the cases for finite groups and finite dimensional Kac
algebras. We prove the Hausdorff-Young inequality, Young's inequality, the
Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty
principle. We characterize the minimizers of the uncertainty principles. We
also prove that the minimizer is uniquely determined by the supports of itself
and its Fourier transform. The proofs take the advantage of the analytic and
the categorial perspectives of subfactor planar algebras. Our method to prove
the uncertainty principles also works for more general cases, such as Popa's
-lattices, modular tensor categories etc.Comment: 41 pages, 71 figure
Spacetime and Fields, a Quantum Texture
We report on joint work, past and in progress, with K.Fredenhagen and with
J.E,Roberts, on the quantum structure of spacetime in the small which is
dictated by the principles of Quantum Mechanics and of General Relativity; we
comment on how these principles point to a deep link between coordinates and
fields. This is an expanded version of a lecture delivered at the 37th Karpacz
School in Theoretical Physics, February 2001.Comment: LaTeX, 15 pages. Misprints and wording corrected, references added;
change in section 3. Related references: hep-th/0303037, hep-th/0201222,
hep-th/030110
The noncommutative standard model, post- and predictions
I try to assess the weak and strong points of the standard model of
electro-magnetic, weak and strong forces, how it can be derived from general
relativity by generalizing Riemannian to noncommutative geometry and what post-
and predictions this unification of all four forces entails in particle
physics.Comment: contribution to Moriond '10 electro-wea
Doubly-Special Relativity: Facts, Myths and Some Key Open Issues
I report, emphasizing some key open issues and some aspects that are
particularly relevant for phenomenology, on the status of the development of
"doubly-special" relativistic ("DSR") theories with both an
observer-independent high-velocity scale and an observer-independent
small-length/large-momentum scale, possibly relevant for the
Planck-scale/quantum-gravity realm. I also give a true/false characterization
of the structure of these theories. In particular, I discuss a DSR scenario
without modification of the energy-momentum dispersion relation and without the
-Poincar\'e Hopf algebra, a scenario with deformed Poincar\'e
symmetries which is not a DSR scenario, some scenarios with both an invariant
length scale and an invariant velocity scale which are not DSR scenarios, and a
DSR scenario in which it is easy to verify that some observable relativistic
(but non-special-relativistic) features are insensitive to possible nonlinear
redefinitions of symmetry generators.Comment: This is the preprint version of a paper prepared for a special issue
"Feature Papers: Symmetry Concepts and Applications" of the journal Symmetr
Worldvolume Uncertainty Relations for D-Branes
By quantizing an open string ending on a D-brane in a nontrivial supergravity
background, we argue that there is a new kind of uncertainty relation on a
D-brane worldvolume. Furthermore, we fix the form of the uncertainty relations
and their dependence on the string coupling constant by requiring them to be
consistent with various string theory and M theory dualities. In this way we
find a web of uncertainties of spacetime for all kinds of brane probes,
including fundamental strings, D-branes of all dimensions as well as M theory
membranes and fivebranes.Comment: 19 pages, minor modification on p.
Self-Completeness and the Generalized Uncertainty Principle
The generalized uncertainty principle discloses a self-complete
characteristic of gravity, namely the possibility of masking any curvature
singularity behind an event horizon as a result of matter compression at the
Planck scale. In this paper we extend the above reasoning in order to overcome
some current limitations to the framework, including the absence of a
consistent metric describing such Planck-scale black holes. We implement a
minimum-size black hole in terms of the extremal configuration of a neutral
non-rotating metric, which we derived by mimicking the effects of the
generalized uncertainty principle via a short scale modified version of
Einstein gravity. In such a way, we find a self-consistent scenario that
reconciles the self-complete character of gravity and the generalized
uncertainty principle.Comment: 20 pages, 6 figures, v2: additional references, version in press on
JHE
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