238 research outputs found
Boundary maps for -crossed products with R with an application to the quantum Hall effect
The boundary map in K-theory arising from the Wiener-Hopf extension of a
crossed product algebra with R is the Connes-Thom isomorphism. In this article
the Wiener Hopf extension is combined with the Heisenberg group algebra to
provide an elementary construction of a corresponding map on higher traces (and
cyclic cohomology). It then follows directly from a non-commutative Stokes
theorem that this map is dual w.r.t.Connes' pairing of cyclic cohomology with
K-theory. As an application, we prove equality of quantized bulk and edge
conductivities for the integer quantum Hall effect described by continuous
magnetic Schroedinger operators.Comment: to appear in Commun. Math. Phy
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
An Obstruction to Quantization of the Sphere
In the standard example of strict deformation quantization of the symplectic
sphere , the set of allowed values of the quantization parameter
is not connected; indeed, it is almost discrete. Li recently constructed a
class of examples (including ) in which can take any value in an
interval, but these examples are badly behaved. Here, I identify a natural
additional axiom for strict deformation quantization and prove that it implies
that the parameter set for quantizing is never connected.Comment: 23 page. v2: changed sign conventio
An Introduction to Quantum Computing for Non-Physicists
Richard Feynman's observation that quantum mechanical effects could not be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used quantum effects. This speculation appeared justified when Peter Shor described a polynomial time quantum algorithm for factoring integers. In quantum systems, the computational space increases exponentially with the size of the system which enables exponential parallelism. This parallelism could lead to exponentially faster quantum algorithms than possible classically. The catch is that accessing the results, which requires measurement, proves tricky and requires new non-traditional programming techniques. The aim of this paper is to guide computer scientists and other non-physicists through the conceptual and notational barriers that separate quantum computing from conventional computing. We introduce basic principles of quantum mechanics to explain where the power of quantum computers comes from and why it is difficult to harness. We describe quantum cryptography, teleportation, and dense coding. Various approaches to harnessing the power of quantum parallelism are explained, including Shor's algorithm, Grover's algorithm, and Hogg's algorithms. We conclude with a discussion of quantum error correction
Endomorphism Semigroups and Lightlike Translations
Certain criteria are demonstrated for a spatial derivation of a von Neumann
algebra to generate a one-parameter semigroup of endomorphisms of that algebra.
These are then used to establish a converse to recent results of Borchers and
of Wiesbrock on certain one-parameter semigroups of endomorphisms of von
Neumann algebras (specifically, Type III_1 factors) that appear as lightlike
translations in the theory of algebras of local observables.Comment: 9 pages, Late
Deformations of quantum field theories on de Sitter spacetime
Quantum field theories on de Sitter spacetime with global U(1) gauge symmetry
are deformed using the joint action of the internal symmetry group and a
one-parameter group of boosts. The resulting theory turns out to be wedge-local
and non-isomorphic to the initial one for a class of theories, including the
free charged Dirac field. The properties of deformed models coming from
inclusions of CAR-algebras are studied in detail.Comment: 26 pages, no figure
Unstable solitons on noncommutative tori and D-branes
We describe a class of exact solutions of super Yang-Mills theory on
even-dimensional noncommutative tori. These solutions generalize the solitons
on a noncommutative plane introduced in hep-th/0009142 that are conjectured to
describe unstable D2p-D0 systems. We show that the spectrum of quadratic
fluctuations around our solutions correctly reproduces the string spectrum of
the D2p-D0 system in the Seiberg-Witten decoupling limit. In particular the
fluctuations correctly reproduce the 0-0 string winding modes. For p=1 and p=2
we match the differences between the soliton energy and the energy of an
appropriate SYM BPS state with the binding energies of D2-D0 and D4-D0 systems.
We also give an example of a soliton that we conjecture describes branes of
intermediate dimension on a torus such as a D2-D4 system on a four-torus.Comment: 22 pages, Latex; v.2: references adde
A Deformation Quantization Theory for Non-Commutative Quantum Mechanics
We show that the deformation quantization of non-commutative quantum
mechanics previously considered by Dias and Prata can be expressed as a Weyl
calculus on a double phase space. We study the properties of the star-product
thus defined, and prove a spectral theorem for the star-genvalue equation using
an extension of the methods recently initiated by de Gosson and Luef.Comment: Submitted for publicatio
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
Noncommutativity in (2+1)-dimensions and the Lorentz group
In this article we considered models of particles living in a
three-dimensional space-time with a nonstandard noncommutativity induced by
shifting canonical coordinates and momenta with generators of a unitary
irreducible representation of the Lorentz group. The Hilbert space gets the
structure of a direct product with the representation space, where we are able
to construct operators which realize the algebra of Lorentz transformations. We
study the modified Landau problem for both Schr\"odinger and Dirac particles,
whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of
the dynamical variables from the ones of the usual problem in the normal space.
The spectrum of these models are considered in perturbation theory, both for
small and large noncommutativity parameters. We find no constraint between the
parameters referring to no-commutativity in coordinates and momenta but they
rather play similar roles. Since the representation space of the unitary
irreducible representations SL(2,R) can be realized in terms of spaces of
square-integrable functions, we conclude that these models are equivalent to
quantum mechanical models of particles living in a space with an additional
compact dimension.Comment: PACS: 03.65.-w; 11.30.Cp; 02.40.Gh, 19 pages, no figures. Version to
appear in Physical Review
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