46 research outputs found
Laminar Spirals in the Outer Stationary Cylinder Couette-Taylor System
We present numerical simulations to demonstrate the existence of laminar spiral flows between both finite and infinite length concentric cylinders and finite truncated cones where only the inner wall rotates. The velocities and pressure are calculated by a spectral element/Fourier method. Different gap ratios are investigated. Convergence of the numerical results is shown with reference to flows between infinite cylinders. The presence of top and bottom endplates results in vortex dislocations that are observed at the frontiers between the Ekman vortices present at each end and the spiral vortices
Droplets Transport in a Microfluidic Chip for In Vitro Compartmentalisation
In vitro compartmentalisation is an emerging technology for protein evolution and selection. In this presentation, we will report the development of a microdrop-based microfluidic platform for in vitro enzyme evolution and selection applications. A microfluidic chip has been developed and fabricated using the standard photolithography method in conjunction with electroplating and hot embossing techniques. A cross channel geometry was used to focus liquid flows for droplet generation. To realize on-chip compartmentalised bio-reactions, two droplet generators were fabricated on the same chip. Experiments have been carried out to measure droplet size, generation rate and speed using a photographic technique. Droplet size was found to be decreasing with increasing focusing oil flow rate for a given aqueous phase flow rate. When two droplet generators are used in the same chip, the droplets may be generated asynchronously due to different flow conditions. If the droplets were significantly smaller than channel size, the faster moving droplets could pass the slower moving droplets with little coalescence. If the droplets were of the channel size, the faster moving droplets would break or fuse with the slow droplets. To achieve high rate of droplet fusion, active control should be in place for synchronous generation and fusion
Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
In this work we investigate the canonical quantization of 2+1 gravity with
cosmological constant in the canonical framework of loop quantum
gravity. The unconstrained phase space of gravity in 2+1 dimensions is
coordinatized by an SU(2) connection and the canonically conjugate triad
field . A natural regularization of the constraints of 2+1 gravity can be
defined in terms of the holonomies of . As a first step
towards the quantization of these constraints we study the canonical
quantization of the holonomy of the connection on the
kinematical Hilbert space of loop quantum gravity. The holonomy operator
associated to a given path acts non trivially on spin network links that are
transversal to the path (a crossing). We provide an explicit construction of
the quantum holonomy operator. In particular, we exhibit a close relationship
between the action of the quantum holonomy at a crossing and Kauffman's
q-deformed crossing identity. The crucial difference is that (being an operator
acting on the kinematical Hilbert space of LQG) the result is completely
described in terms of standard SU(2) spin network states (in contrast to
q-deformed spin networks in Kauffman's identity). We discuss the possible
implications of our result.Comment: 19 pages, references added. Published versio
Loop quantum gravity without the Hamiltonian constraint
We show that under certain technical assumptions, including the existence of
a constant mean curvature (CMC) slice and strict positivity of the scalar
field, general relativity conformally coupled to a scalar field can be
quantised on a partially reduced phase space, meaning reduced only with respect
to the Hamiltonian constraint and a proper gauge fixing. More precisely, we
introduce, in close analogy to shape dynamics, the generator of a local
conformal transformation acting on both, the metric and the scalar field, which
coincides with the CMC gauge condition. A new metric, which is invariant under
this transformation, is constructed and used to define connection variables
which can be quantised by standard loop quantum gravity methods. While it is
hard to address dynamical problems in this framework (due to the complicated
'time' function), it seems, due to good accessibility properties of the CMC
gauge, to be well suited for problems such as the computation of black hole
entropy, where actual physical states can be counted and the dynamics is only
of indirect importance. The corresponding calculation yields the surprising
result that the usual prescription of fixing the Barbero-Immirzi parameter beta
to a constant value in order to obtain the well-known formula S = a(Phi) A/(4G)
does not work for the black holes under consideration, while a recently
proposed prescription involving an analytic continuation of beta to the case of
a self-dual space-time connection yields the correct result. Also, the
interpretation of the geometric operators gets an interesting twist, which
exemplifies the deep relationship between observables and the choice of a time
function and has consequences for loop quantum cosmology.Comment: 8 pages. v2: Journal version. Black hole state counting based on
physical states added. Applications to loop quantum cosmology discussed.
Gauge condition used shown to coincide with CMC gauge. Minor clarifications.
v3: Erroneous topology dependence of the entropy in journal version
corrected, conclusions fixed accordingly. Main results unaffecte
Cosmological Deformation of Lorentzian Spin Foam Models
We study the quantum deformation of the Barrett-Crane Lorentzian spin foam
model which is conjectured to be the discretization of Lorentzian Plebanski
model with positive cosmological constant and includes therefore as a
particular sector quantum gravity in de-Sitter space. This spin foam model is
constructed using harmonic analysis on the quantum Lorentz group. The
evaluation of simple spin networks are shown to be non commutative integrals
over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that
the introduction of the cosmological constant removes all the infrared
divergences: for any fixed triangulation, the integration over the area
variables is finite for a large class of normalization of the amplitude of the
edges and of the faces.Comment: 37 pages, 7 figures include
Quantum symmetry, the cosmological constant and Planck scale phenomenology
We present a simple algebraic argument for the conclusion that the low energy
limit of a quantum theory of gravity must be a theory invariant, not under the
Poincare group, but under a deformation of it parameterized by a dimensional
parameter proportional to the Planck mass. Such deformations, called
kappa-Poincare algebras, imply modified energy-momentum relations of a type
that may be observable in near future experiments. Our argument applies in both
2+1 and 3+1 dimensions and assumes only 1) that the low energy limit of a
quantum theory of gravity must involve also a limit in which the cosmological
constant is taken very small with respect to the Planck scale and 2) that in
3+1 dimensions the physical energy and momenta of physical elementary particles
is related to symmetries of the full quantum gravity theory by appropriate
renormalization depending on Lambda l^2_{Planck}. The argument makes use of the
fact that the cosmological constant results in the symmetry algebra of quantum
gravity being quantum deformed, as a consequence when the limit \Lambda
l^2_{Planck} -> 0 is taken one finds a deformed Poincare invariance. We are
also able to isolate what information must be provided by the quantum theory in
order to determine which presentation of the kappa-Poincare algebra is relevant
for the physical symmetry generators and, hence, the exact form of the modified
energy-momentum relations. These arguments imply that Lorentz invariance is
modified as in proposals for doubly special relativity, rather than broken, in
theories of quantum gravity, so long as those theories behave smoothly in the
limit the cosmological constant is taken to be small.Comment: LaTex, 19 page
Background Independent Quantum Gravity: A Status Report
The goal of this article is to present an introduction to loop quantum
gravity -a background independent, non-perturbative approach to the problem of
unification of general relativity and quantum physics, based on a quantum
theory of geometry. Our presentation is pedagogical. Thus, in addition to
providing a bird's eye view of the present status of the subject, the article
should also serve as a vehicle to enter the field and explore it in detail. To
aid non-experts, very little is assumed beyond elements of general relativity,
gauge theories and quantum field theory. While the article is essentially
self-contained, the emphasis is on communicating the underlying ideas and the
significance of results rather than on presenting systematic derivations and
detailed proofs. (These can be found in the listed references.) The subject can
be approached in different ways. We have chosen one which is deeply rooted in
well established physics and also has sufficient mathematical precision to
ensure that there are no hidden infinities. In order to keep the article to a
reasonable size, and to avoid overwhelming non-experts, we have had to leave
out several interesting topics, results and viewpoints; this is meant to be an
introduction to the subject rather than an exhaustive review of it.Comment: 125 pages, 5 figures (eps format), the final version published in CQ
Laplacians on discrete and quantum geometries
We extend discrete calculus for arbitrary (-form) fields on embedded
lattices to abstract discrete geometries based on combinatorial complexes. We
then provide a general definition of discrete Laplacian using both the primal
cellular complex and its combinatorial dual. The precise implementation of
geometric volume factors is not unique and, comparing the definition with a
circumcentric and a barycentric dual, we argue that the latter is, in general,
more appropriate because it induces a Laplacian with more desirable properties.
We give the expression of the discrete Laplacian in several different sets of
geometric variables, suitable for computations in different quantum gravity
formalisms. Furthermore, we investigate the possibility of transforming from
position to momentum space for scalar fields, thus setting the stage for the
calculation of heat kernel and spectral dimension in discrete quantum
geometries.Comment: 43 pages, 2 multiple figures. v2: discussion improved, references
added, minor typos correcte
The Spin Foam Approach to Quantum Gravity
This article reviews the present status of the spin foam approach to the
quantization of gravity. Special attention is payed to the pedagogical
presentation of the recently introduced new models for four dimensional quantum
gravity. The models are motivated by a suitable implementation of the path
integral quantization of the Plebanski formulation of gravity on a simplicial
regularization. The article also includes a self-contained treatment of the 2+1
gravity. The simple nature of the latter provides the basis and a perspective
for the analysis of both conceptual and technical issues that remain open in
four dimensions.Comment: To appear in Living Reviews in Relativit
Spin Foam Models for Quantum Gravity
In this article we review the present status of the spin foam formulation of
non-perturbative (background independent) quantum gravity. The article is
divided in two parts. In the first part we present a general introduction to
the main ideas emphasizing their motivations from various perspectives.
Riemannian 3-dimensional gravity is used as a simple example to illustrate
conceptual issues and the main goals of the approach. The main features of the
various existing models for 4-dimensional gravity are also presented here. We
conclude with a discussion of important questions to be addressed in four
dimensions (gauge invariance, discretization independence, etc.).
In the second part we concentrate on the definition of the Barrett-Crane
model. We present the main results obtained in this framework from a critical
perspective. Finally we review the combinatorial formulation of spin foam
models based on the dual group field theory technology. We present the
Barrett-Crane model in this framework and review the finiteness results
obtained for both its Riemannian as well as its Lorentzian variants.Comment: Topical review, to appear in CQG. Typos corrected and new references
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