123,185 research outputs found
Quantum mechanics on non commutative spaces and squeezed states: a functional approach
We review here the quantum mechanics of some noncommutative theories in which
no state saturates simultaneously all the non trivial Heisenberg uncertainty
relations. We show how the difference of structure between the Poisson brackets
and the commutators in these theories generically leads to a harmonic
oscillator whose positions and momenta mean values are not strictly equal to
the ones predicted by classical mechanics.
This raises the question of the nature of quasi classical states in these
models. We propose an extension based on a variational principle. The action
considered is the sum of the absolute values of the expressions associated to
the non trivial Heisenberg uncertainty relations. We first verify that our
proposal works in the usual theory i.e we recover the known Gaussian functions.
Besides them, we find other states which can be expressed as products of
Gaussians with specific hyper geometrics.
We illustrate our construction in two models defined on a four dimensional
phase space: a model endowed with a minimal length uncertainty and the non
commutative plane. Our proposal leads to second order partial differential
equations. We find analytical solutions in specific cases. We briefly discuss
how our proposal may be applied to the fuzzy sphere and analyze its
shortcomings.Comment: 15 pages revtex. The title has been modified,the paper shortened and
misprints have been corrected. Version to appear in JHE
Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis
We consider the Lie group generated by the Lie algebra
of -Minkowski space. Imposing the invariance of the metric under the
pull-back of diffeomorphisms induced by right translations in the group, we
show that a unique right invariant metric is associated with
. This metric coincides with the metric of de Sitter
space-time. We analyze the structure of unitary representations of the group
relevant for the realization of the non-commutative
-Minkowski space by embedding into -dimensional Heisenberg
algebra. Using a suitable set of generalized coherent states, we select the
particular Hilbert space and realize the non-commutative -Minkowski
space as an algebra of the Hilbert-Schmidt operators. We define dequantization
map and fuzzy variant of the Laplace-Beltrami operator such that dequantization
map relates fuzzy eigenvectors with the eigenfunctions of the Laplace-Beltrami
operator on the half of de Sitter space-time.Comment: 21 pages, v3 differs from version published in Fortschritte der
Physik by a note and references added and adjuste
Non-commutative geometry of 4-dimensional quantum Hall droplet
We develop the description of non-commutative geometry of the 4-dimensional
quantum Hall fluid's theory proposed recently by Zhang and Hu. The
non-commutative structure of fuzzy appears naturally in this theory.
The fuzzy monopole harmonics, which are the essential elements in this
non-commutative geometry, are explicitly constructed and their obeying the
matrix algebra is obtained. This matrix algebra is associative. We also propose
a fusion scheme of the fuzzy monopole harmonics of the coupling system from
those of the subsystems, and determine the fusion rule in such fusion scheme.
By products, we provide some essential ingredients of the theory of SO(5)
angular momentum. In particular, the explicit expression of the coupling
coefficients, in the theory of SO(5) angular momentum, are given. It is
discussed that some possible applications of our results to the 4-dimensional
quantum Hall system and the matrix brane construction in M-theory.Comment: latex 22 pages, no figures. some references added. some results are
clarifie
Self-intersecting fuzzy extra dimensions from squashed coadjoint orbits in SYM and matrix models
We find new vacuum solutions of super-Yang-Mills with totally
anti-symmetric cubic soft SUSY breaking terms, or equivalently solutions of the
IKKT matrix model of type with flux
terms. The solutions can be understood in terms of 4- and 6- dimensional fuzzy
branes in extra dimensions, describing self-intersecting
projections of compact flag manifolds of . The 6-dimensional solutions
provide a 6-fold covering of the internal space near the origin, while the
4-dimensional branes have a triple self-intersections spanning all 6 internal
directions. The solutions have lower energy than the trivial vacuum, and we
prove that there are no negative modes. The massless modes are identified
explicitly. In particular there are chiral fermionic zero modes, linking the
coincident sheets with opposite flux at the origin. They have a
family symmetry, originating from the Weyl group rotations.Comment: 28+8 pages, 2 figures. V2: improved discussion, published versio
Higher Dimensional Geometries from Matrix Brane constructions
Matrix descriptions of even dimensional fuzzy spherical branes in
Matrix Theory and other contexts in Type II superstring theory reveal, in the
large limit, higher dimensional geometries , which have an
interesting spectrum of harmonics and can be up to 20 dimensional,
while the spheres are restricted to be of dimension less than 10. In the case
, the matrix description has two dual field theory formulations. One
involves a field theory living on the non-commutative coset which
is a fuzzy fibre bundle over a fuzzy . In the other, there is a U(n)
gauge theory on a fuzzy with instantons. The two
descriptions can be related by exploiting the usual relation between the fuzzy
two-sphere and U(n) Lie algebra. We discuss the analogous phenomena in the
higher dimensional cases, developing a relation between fuzzy
cosets and unitary Lie algebras.Comment: 28 pages (Harvmac big) ; version 2 : minor typos fixed and ref. adde
Branes, Quantization and Fuzzy Spheres
We propose generalized quantization axioms for Nambu-Poisson manifolds, which
allow for a geometric interpretation of n-Lie algebras and their enveloping
algebras. We illustrate these axioms by describing extensions of
Berezin-Toeplitz quantization to produce various examples of quantum spaces of
relevance to the dynamics of M-branes, such as fuzzy spheres in diverse
dimensions. We briefly describe preliminary steps towards making the notion of
quantized 2-plectic manifolds rigorous by extending the groupoid approach to
quantization of symplectic manifolds.Comment: 18 pages; Based on Review Talk at the Workshop on "Noncommutative
Field Theory and Gravity", Corfu Summer Institute on Elementary Particles and
Physics, September 8-12, 2010, Corfu, Greece; to be published in Proceedings
of Scienc
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