3,066 research outputs found
Almost commutative Riemannian geometry: wave operators
Associated to any (pseudo)-Riemannian manifold of dimension is an
-dimensional noncommutative differential structure (\Omega^1,\extd) on
the manifold, with the extra dimension encoding the classical Laplacian as a
noncommutative `vector field'. We use the classical connection, Ricci tensor
and Hodge Laplacian to construct (\Omega^2,\extd) and a natural
noncommutative torsion free connection on . We show
that its generalised braiding \sigma:\Omega^1\tens\Omega^1\to
\Omega^1\tens\Omega^1 obeys the quantum Yang-Baxter or braid relations only
when the original is flat, i.e their failure is governed by the Riemann
curvature, and that \sigma^2=\id only when is Einstein. We show that if
has a conformal Killing vector field then the cross product algebra
viewed as a noncommutative analogue of has a
natural -dimensional calculus extending and a natural spacetime
Laplacian now directly defined by the extra dimension. The case
recovers the Majid-Ruegg bicrossproduct flat spacetime model and the
wave-operator used in its variable speed of light preduction, but now as an
example of a general construction. As an application we construct the wave
operator on a noncommutative Schwarzschild black hole and take a first look at
its features. It appears that the infinite classical redshift/time dilation
factor at the event horizon is made finite.Comment: 39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a
separate paper (now ArXived) to meet referee length requirements.
Corresponding slight restructure but no change to remaining conten
Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
Braided groups and braided matrices are novel algebraic structures living in
braided or quasitensor categories. As such they are a generalization of
super-groups and super-matrices to the case of braid statistics. Here we
construct braided group versions of the standard quantum groups . They
have the same FRT generators but a matrix braided-coproduct \und\Delta
L=L\und\tens L where , and are self-dual. As an application, the
degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices
; it is a braided-commutative bialgebra in a braided category. As a
second application, we show that the quantum double D(\usl) (also known as
the `quantum Lorentz group') is the semidirect product as an algebra of two
copies of \usl, and also a semidirect product as a coalgebra if we use braid
statistics. We find various results of this type for the doubles of general
quantum groups and their semi-classical limits as doubles of the Lie algebras
of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction
Some Remarks on Producing Hopf Algebras
We report some observations concerning two well-known approaches to
construction of quantum groups. Thus, starting from a bialgebra of
inhomogeneous type and imposing quadratic, cubic or quartic commutation
relations on a subset of its generators we come, in each case, to a q-deformed
universal enveloping algebra of a certain simple Lie algebra. An interesting
correlation between the order of initial commutation relations and the Cartan
matrix of the resulting algebra is observed. Another example demonstrates that
the bialgebra structure of sl_q(2) can be completely determined by requiring
the q-oscillator algebra to be its covariant comodule, in analogy with Manin's
approach to define SL_q(2) as a symmetry algebra of the bosonic and fermionic
quantum planes.Comment: 6 pages, LATEX, no figures, Contribution to the Proceedings of the
4th Colloquium "Quantum Groups and Integrable Systems" (Prague, June 1995
Generalized exclusion and Hopf algebras
We propose a generalized oscillator algebra at the roots of unity with
generalized exclusion and we investigate the braided Hopf structure. We find
that there are two solutions: these are the generalized exclusions of the
bosonic and fermionic types. We also discuss the covariance properties of these
oscillatorsComment: 10 pages, to appear in J. Phys.
Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
A noncommutative geometric generalisation of the quantum field theoretical
framework is developed by generalising the Heisenberg commutation relations.
There appear nonzero minimal uncertainties in positions and in momenta. As the
main result it is shown with the example of a quadratically ultraviolet
divergent graph in theory that nonzero minimal uncertainties in
positions do have the power to regularise. These studies are motivated with the
ansatz that nonzero minimal uncertainties in positions and in momenta arise
from gravity. Algebraic techniques are used that have been developed in the
field of quantum groups.Comment: 52 pages LATEX, DAMTP/93-33. Revised version now includes a chapter
on the Poincare algebra and curvature as noncommutativity of momentum spac
Stack-run adaptive wavelet image compression
We report on the development of an adaptive wavelet image coder based on stack-run representation of the quantized coefficients. The coder works by selecting an optimal wavelet packet basis for the given image and encoding the quantization indices for significant coefficients and zero runs between coefficients using a 4-ary arithmetic coder. Due to the fact that our coder exploits the redundancies present within individual subbands, its addressing complexity is much lower than that of the wavelet zerotree coding algorithms. Experimental results show coding gains of up to 1:4dB over the benchmark wavelet coding algorithm
Comment on "Fermionic entanglement ambiguity in noninertial frames"
In this comment we show that the ambiguity of entropic quantities calculated
in Physical Review A 83, 062323 (2011) for fermionic fields in the context of
Unruh effect is not related to the properties of anticommuting fields, as
claimed in Physical Review A 83, 062323 (2011), but rather to wrong
mathematical manipulations with them and not taking into account a fundamental
superselection rule of quantum field theory.Comment: To appear in Physical Review A. Some of the problems discussed in
this comment can also be found in other previously published papers studying
the Unruh effect for fermions (in the context of quantum information theory).
An extended version of the comment can be found here
http://arxiv.org/abs/1108.555
Induced Representations of Quantum Kinematical Algebras and Quantum Mechanics
Unitary representations of kinematical symmetry groups of quantum systems are
fundamental in quantum theory. We propose in this paper its generalization to
quantum kinematical groups. Using the method, proposed by us in a recent paper
(olmo01), to induce representations of quantum bicrossproduct algebras we
construct the representations of the family of standard quantum inhomogeneous
algebras . This family contains the quantum
Euclidean, Galilei and Poincar\'e algebras, all of them in (1+1) dimensions. As
byproducts we obtain the actions of these quantum algebras on regular co-spaces
that are an algebraic generalization of the homogeneous spaces and --Casimir
equations which play the role of --Schr\"odinger equations.Comment: LaTeX 2e, 20 page
On the construction of generalized Grassmann representatives of state vectors
Generalized -graded Grassmann variables are used to label coherent
states related to the nilpotent representation of the q-oscillator of
Biedenharn and Macfarlane when the deformation parameter is a root of unity.
These states are then used to construct generalized Grassmann representatives
of state vectors.Comment: 8 page
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