81 research outputs found
Mapping spaces and automorphism groups of toric noncommutative spaces
We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application we study the 'internalized' automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations
Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras
The assignment (nonstable K_0-theory), that to a ring R associates the monoid
V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices
with only finitely nonzero entries over R, extends naturally to a functor. We
prove the following lifting properties of that functor: (1) There is no functor
F, from simplicial monoids with order-unit with normalized positive
homomorphisms to exchange rings, such that VF is equivalent to the identity.
(2) There is no functor F, from simplicial monoids with order-unit with
normalized positive embeddings to C*-algebras of real rank 0 (resp., von
Neumann regular rings), such that VF is equivalent to the identity. (3) There
is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be
lifted, with respect to the functor V, by exchange rings and by C*-algebras of
real rank 1, but not by semiprimitive exchange rings, thus neither by regular
rings nor by C*-algebras of real rank 0. By using categorical tools from an
earlier paper (larders, lifters, CLL), we deduce that there exists a unital
exchange ring of cardinality aleph three (resp., an aleph three-separable
unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence
2, such that V(R) is the positive cone of a dimension group and V(R) is not
isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0
or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea
Quantized reduction as a tensor product
Symplectic reduction is reinterpreted as the composition of arrows in the
category of integrable Poisson manifolds, whose arrows are isomorphism classes
of dual pairs, with symplectic groupoids as units. Morita equivalence of
Poisson manifolds amounts to isomorphism of objects in this category.
This description paves the way for the quantization of the classical
reduction procedure, which is based on the formal analogy between dual pairs of
Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with
correspondences between von Neumann algebras. Further analogies are drawn with
categories of groupoids (of algebraic, measured, Lie, and symplectic type). In
all cases, the arrows are isomorphism classes of appropriate bimodules, and
their composition may be seen as a tensor product. Hence in suitable categories
reduction is simply composition of arrows, and Morita equivalence is
isomorphism of objects.Comment: 44 pages, categorical interpretation adde
Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization
For arbitrary compact quantizable Kähler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential manner. It is shown that the star product is null on constants and fulfills parity. A trace is constructed and the relation to deformation quantization by geometric quantization is given
Analyzing D-wave quantum macro assembler security
As we enter the quantum computing era, security becomes of at most importance. With the release of D-Wave One in 2011 and most recently the 2000Q, with 2,000 qubits, and with NASA and Google using D-wave Systems quantum computers, a thorough examination of quantum computer security is needed. Quantum computers underlying hardware is not compatible with classical boolean and binary-based computer systems and software. Assemblers and compliers translate modern programming languages and problems into quantum-annealing methods compatible with quantum computers. This paper presents a vulnerability assessment utilizing static source code analysis on Qmasm Python tool. More specifically, we use flow-sensitive, inter-procedural and context-sensitive data flow analysis to uncover vulnerable points in the program. We demonstrate the Qmasm security flaws that can leave D-Wave 2X system vulnerable to severe threats
Analyzing D-wave quantum macro assembler security
As we enter the quantum computing era, security becomes of at most importance. With the release of D-Wave One in 2011 and most recently the 2000Q, with 2,000 qubits, and with NASA and Google using D-wave Systems quantum computers, a thorough examination of quantum computer security is needed. Quantum computers underlying hardware is not compatible with classical boolean and binary-based computer systems and software. Assemblers and compliers translate modern programming languages and problems into quantum-annealing methods compatible with quantum computers. This paper presents a vulnerability assessment utilizing static source code analysis on Qmasm Python tool. More specifically, we use flow-sensitive, inter-procedural and context-sensitive data flow analysis to uncover vulnerable points in the program. We demonstrate the Qmasm security flaws that can leave D-Wave 2X system vulnerable to severe threats
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds
We develop quantization techniques for describing the nonassociative geometry
probed by closed strings in flat non-geometric R-flux backgrounds M. Starting
from a suitable Courant sigma-model on an open membrane with target space M,
regarded as a topological sector of closed string dynamics in R-space, we
derive a twisted Poisson sigma-model on the boundary of the membrane whose
target space is the cotangent bundle T^*M and whose quasi-Poisson structure
coincides with those previously proposed. We argue that from the membrane
perspective the path integral over multivalued closed string fields in Q-space
is equivalent to integrating over open strings in R-space. The corresponding
boundary correlation functions reproduce Kontsevich's deformation quantization
formula for the twisted Poisson manifolds. For constant R-flux, we derive
closed formulas for the corresponding nonassociative star product and its
associator, and compare them with previous proposals for a 3-product of fields
on R-space. We develop various versions of the Seiberg-Witten map which relate
our nonassociative star products to associative ones and add fluctuations to
the R-flux background. We show that the Kontsevich formula coincides with the
star product obtained by quantizing the dual of a Lie 2-algebra via convolution
in an integrating Lie 2-group associated to the T-dual doubled geometry, and
hence clarify the relation to the twisted convolution products for topological
nonassociative torus bundles. We further demonstrate how our approach leads to
a consistent quantization of Nambu-Poisson 3-brackets.Comment: 52 pages; v2: references adde
Graphene and non-Abelian quantization
In this article we employ a simple nonrelativistic model to describe the low
energy excitation of graphene. The model is based on a deformation of the
Heisenberg algebra which makes the commutator of momenta proportional to the
pseudo-spin. We solve the Landau problem for the resulting Hamiltonian which
reduces, in the large mass limit while keeping fixed the Fermi velocity, to the
usual linear one employed to describe these excitations as massless Dirac
fermions. This model, extended to negative mass, allows to reproduce the
leading terms in the low energy expansion of the dispersion relation for both
nearest and next-to-nearest neighbor interactions. Taking into account the
contributions of both Dirac points, the resulting Hall conductivity, evaluated
with a -function approach, is consistent with the anomalous integer
quantum Hall effect found in graphene. Moreover, when considered in first order
perturbation theory, it is shown that the next-to-leading term in the
interaction between nearest neighbor produces no modifications in the spectrum
of the model while an electric field perpendicular to the magnetic field
produces just a rigid shift of this spectrum.
PACS: 03.65.-w, 81.05.ue, 73.43.-fComment: 23 pages, 4 figures. Version to appear in the Journal of Physics A.
The title has been changed into "Graphene and non-Abelian quantization". The
motivation and presentation of the paper has been changed. An appendix and
Section 6 on the evaluation of the Hall conductivity have been added.
References adde
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