4,198 research outputs found
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
Towards Spinfoam Cosmology
We compute the transition amplitude between coherent quantum-states of
geometry peaked on homogeneous isotropic metrics. We use the holomorphic
representations of loop quantum gravity and the
Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at
first order in the vertex expansion, second order in the graph (multipole)
expansion, and first order in 1/volume. We show that the resulting amplitude is
in the kernel of a differential operator whose classical limit is the canonical
hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an
indication that the dynamics of loop quantum gravity defined by the new vertex
yields the Friedmann equation in the appropriate limit.Comment: 8 page
Linear Finite-Field Deterministic Networks With Many Sources and One Destination
We find the capacity region of linear finite-field deterministic networks
with many sources and one destination. Nodes in the network are subject to
interference and broadcast constraints, specified by the linear finite-field
deterministic model. Each node can inject its own information as well as relay
other nodes' information. We show that the capacity region coincides with the
cut-set region. Also, for a specific case of correlated sources we provide
necessary and sufficient conditions for the sources transmissibility. Given the
"deterministic model" approximation for the corresponding Gaussian network
model, our results may be relevant to wireless sensor networks where the
sensing nodes multiplex the relayed data from the other nodes with their own
data, and where the goal is to decode all data at a single "collector" node.Comment: 5 pages, 3 figures, submitted to ISIT 201
Quantisation of twistor theory by cocycle twist
We present the main ingredients of twistor theory leading up to and including
the Penrose-Ward transform in a coordinate algebra form which we can then
`quantise' by means of a functorial cocycle twist. The quantum algebras for the
conformal group, twistor space CP^3, compactified Minkowski space CMh and the
twistor correspondence space are obtained along with their canonical quantum
differential calculi, both in a local form and in a global *-algebra
formulation which even in the classical commutative case provides a useful
alternative to the formulation in terms of projective varieties. We outline how
the Penrose-Ward transform then quantises. As an example, we show that the
pull-back of the tautological bundle on CMh pulls back to the basic instanton
on S^4\subset CMh and that this observation quantises to obtain the
Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the
tautological bundle on our \theta-deformed CMh. We likewise quantise the
fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed
CP^3 that maps over under the transform to the \theta-deformed instanton.Comment: 68 pages 0 figures. Significant revision now has detailed formulae
for classical and quantum CP^
A note on quantum structure constants
The Cartan-Maurer equations for any -group of the
series are given in a convenient form, which allows their direct computation
and clarifies their connection with the case. These equations, defining
the field strengths, are essential in the construction of -deformed gauge
theories. An explicit expression \om ^i\we \om^j= -\Z {ij}{kl}\om ^k\we \om^l
for the -commutations of left-invariant one-forms is found, with \Z{ij}{kl}
\om^k \we \om^l \qonelim \om^j\we\om^i.Comment: 9 pp., LaTe
Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
We show that the -product for , group Fourier transform and
effective action arising in [1] in an effective theory for the integer spin
Ponzano-Regge quantum gravity model are compatible with the noncommutative
bicovariant differential calculus, quantum group Fourier transform and
noncommutative scalar field theory previously proposed for 2+1 Euclidean
quantum gravity using quantum group methods in [2]. The two are related by a
classicalisation map which we introduce. We show, however, that noncommutative
spacetime has a richer structure which already sees the half-integer spin
information. We argue that the anomalous extra `time' dimension seen in the
noncommutative geometry should be viewed as the renormalisation group flow
visible in the coarse-graining in going from to . Combining our
methods we develop practical tools for noncommutative harmonic analysis for the
model including radial quantum delta-functions and Gaussians, the Duflo map and
elements of `noncommutative sampling theory'. This allows us to understand the
bandwidth limitation in 2+1 quantum gravity arising from the bounded
momentum and to interpret the Duflo map as noncommutative compression. Our
methods also provide a generalised twist operator for the -product.Comment: 53 pages latex, no figures; extended the intro for this final versio
On the Fock space for nonrelativistic anyon fields and braided tensor products
We realize the physical N-anyon Hilbert spaces, introduced previously via
unitary representations of the group of diffeomorphisms of the plane, as N-fold
braided-symmetric tensor products of the 1-particle Hilbert space. This
perspective provides a convenient Fock space construction for nonrelativistic
anyon quantum fields along the more usual lines of boson and fermion fields,
but in a braided category. We see how essential physical information is thus
encoded. In particular we show how the algebraic structure of our anyonic Fock
space leads to a natural anyonic exclusion principle related to intermediate
occupation number statistics, and obtain the partition function for an
idealised gas of fixed anyonic vortices.Comment: Added some references, more explicit formulae for the discrete case
and remark on partition function. 25 pages latex, no figure
Cosmological constant from quantum spacetime
http://dx.doi.org/10.1103/PhysRevD.91.124028© 2015, Physical Review
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