78 research outputs found
The odd nilHecke algebra and its diagrammatics
We introduce an odd version of the nilHecke algebra and develop an odd
analogue of the thick diagrammatic calculus for nilHecke algebras. We
graphically describe idempotents which give a Morita equivalence between odd
nilHecke algebras and the rings of odd symmetric functions in finitely many
variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent
to rings which are odd analogues of the cohomology rings of Grassmannians. Like
their even counterparts, odd nilHecke algebras categorify the positive half of
quantum sl(2).Comment: 48 pages, eps and xypic diagram
Categorical formulation of quantum algebras
We describe how dagger-Frobenius monoids give the correct categorical
description of certain kinds of finite-dimensional 'quantum algebras'. We
develop the concept of an involution monoid, and use it to construct a
correspondence between finite-dimensional C*-algebras and certain types of
dagger-Frobenius monoids in the category of Hilbert spaces. Using this
technology, we recast the spectral theorems for commutative C*-algebras and for
normal operators into an explicitly categorical language, and we examine the
case that the results of measurements do not form finite sets, but rather
objects in a finite Boolean topos. We describe the relevance of these results
for topological quantum field theory.Comment: 34 pages, to appear in Communications in Mathematical Physic
2-Vector Spaces and Groupoids
This paper describes a relationship between essentially finite groupoids and
2-vector spaces. In particular, we show to construct 2-vector spaces of
Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding
to functors between groupoids in both a covariant and contravariant way, which
are ambidextrous adjoints. This is used to construct a representation--a weak
functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids)
into 2Vect. In this paper we prove this and give the construction in detail.Comment: 44 pages, 5 figures - v2 adds new theorem, significant changes to
proofs, new sectio
Astrophysical and Cosmological Tests of Quantum Theory
We discuss several proposals for astrophysical and cosmological tests of
quantum theory. The tests are motivated by deterministic hidden-variables
theories, and in particular by the view that quantum physics is merely an
effective theory of an equilibrium state. The proposed tests involve searching
for nonequilibrium violations of quantum theory in: primordial inflaton
fluctuations imprinted on the cosmic microwave background, relic cosmological
particles, Hawking radiation, photons with entangled partners inside black
holes, neutrino oscillations, and particles from very distant sources.Comment: 25 pages. Amendment to section 7. Contribution to: "The Quantum
Universe", special issue of Journal of Physics A, dedicated to Prof. G.-C.
Ghirardi on the occasion of his seventieth birthda
Picturing classical and quantum Bayesian inference
We introduce a graphical framework for Bayesian inference that is
sufficiently general to accommodate not just the standard case but also recent
proposals for a theory of quantum Bayesian inference wherein one considers
density operators rather than probability distributions as representative of
degrees of belief. The diagrammatic framework is stated in the graphical
language of symmetric monoidal categories and of compact structures and
Frobenius structures therein, in which Bayesian inversion boils down to
transposition with respect to an appropriate compact structure. We characterize
classical Bayesian inference in terms of a graphical property and demonstrate
that our approach eliminates some purely conventional elements that appear in
common representations thereof, such as whether degrees of belief are
represented by probabilities or entropic quantities. We also introduce a
quantum-like calculus wherein the Frobenius structure is noncommutative and
show that it can accommodate Leifer's calculus of `conditional density
operators'. The notion of conditional independence is also generalized to our
graphical setting and we make some preliminary connections to the theory of
Bayesian networks. Finally, we demonstrate how to construct a graphical
Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture
Canonical basis for quantum osp(1|2)
We introduce a modified quantum enveloping algebra as well as a (modified)
covering quantum algebra for the ortho-symplectic Lie superalgebra osp(1|2).
Then we formulate and compute the corresponding canonical bases, and relate
them to the counterpart for sl(2). This provides a first example of canonical
basis for quantum superalgebras.Comment: v2, minor corrections and one reference added, to appear in Lett.
Math. Phys. v3, very minor corrections, final versio
Turning big bang into big bounce: II. Quantum dynamics
We analyze the big bounce transition of the quantum FRW model in the setting
of the nonstandard loop quantum cosmology (LQC). Elementary observables are
used to quantize composite observables. The spectrum of the energy density
operator is bounded and continuous. The spectrum of the volume operator is
bounded from below and discrete. It has equally distant levels defining a
quantum of the volume. The discreteness may imply a foamy structure of
spacetime at semiclassical level which may be detected in astro-cosmo
observations. The nonstandard LQC method has a free parameter that should be
fixed in some way to specify the big bounce transition.Comment: 14 pages, no figures, version accepted for publication in Class.
Quant. Gra
Bicategories for boundary conditions and for surface defects in 3-d TFT
We analyze topological boundary conditions and topological surface defects in
three-dimensional topological field theories of Reshetikhin-Turaev type based
on arbitrary modular tensor categories. Boundary conditions are described by
central functors that lift to trivializations in the Witt group of modular
tensor categories. The bicategory of boundary conditions can be described
through the bicategory of module categories over any such trivialization. A
similar description is obtained for topological surface defects. Using string
diagrams for bicategories we also establish a precise relation between special
symmetric Frobenius algebras and Wilson lines involving special defects. We
compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong
on boundary conditions and surface defects in abelian Chern-Simons theories and
in Turaev-Viro type TFTs, respectively.Comment: 34 pages, some figures. v2: references added. v3: typos corrected and
biliography update
Contrasting disease patterns in seropositive and seronegative neuromyelitis optica: A multicentre study of 175 patients
BACKGROUND: The diagnostic and pathophysiological relevance of antibodies to aquaporin-4 (AQP4-Ab) in patients with neuromyelitis optica spectrum disorders (NMOSD) has been intensively studied. However, little is known so far about the clinical impact of AQP4-Ab seropositivity. OBJECTIVE: To analyse systematically the clinical and paraclinical features associated with NMO spectrum disorders in Caucasians in a stratified fashion according to the patients' AQP4-Ab serostatus. METHODS: Retrospective study of 175 Caucasian patients (AQP4-Ab positive in 78.3%). RESULTS: Seropositive patients were found to be predominantly female (p 1 myelitis attacks in the first year were identified as possible predictors of a worse outcome. CONCLUSION: This study provides an overview of the clinical and paraclinical features of NMOSD in Caucasians and demonstrates a number of distinct disease characteristics in seropositive and seronegative patients
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