Chern-Simons Theory with Sources and Dynamical Quantum Groups I: Canonical Analysis and Algebraic Structures

We study the quantization of Chern-Simons theory with group $G$ coupled to dynamical sources. We first study the dynamics of Chern-Simons sources in the Hamiltonian framework. The gauge group of this system is reduced to the Cartan subgroup of $G.$ We show that the Dirac bracket between the basic dynamical variables can be expressed in term of dynamical $r-$matrix of rational type. We then couple minimally these sources to Chern-Simons theory with the use of a regularisation at the location of the sources. In this case, the gauge symmetries of this theory split in two classes, the bulk gauge transformation associated to the group $G$ and world lines gauge transformations associated to the Cartan subgroup of $G$. We give a complete hamiltonian analysis of this system and analyze in detail the Poisson algebras of functions invariant under the action of bulk gauge transformations. This algebra is larger than the algebra of Dirac observables because it contains in particular functions which are not invariant under reparametrization of the world line of the sources. We show that the elements of this Poisson algebra have Poisson brackets expressed in term of dynamical $r-$matrix of trigonometric type. This algebra is a dynamical generalization of Fock-Rosly structure. We analyze the quantization of these structures and describe different star structures on these algebras, with a special care to the case where $G=SL(2,{\mathbb R})$ and $G=SL(2,{\mathbb C})_{\mathbb R},$ having in mind to apply these results to the study of the quantization of massive spinning point particles coupled to gravity with a cosmological constant in 2+1 dimensions.Comment: 32 pages and 1 eps figur

Harmonic Analysis on the quantum Lorentz group

This work begins with a review of complexification and realification of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of different classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of $6j$ symbols of $SU_q (2)$ with complex spins, we give a new description of the unitary representations of SL_q (2,\CC)_{\RR} and find explicit expressions for the characters of SL_q (2,\CC)_{\RR}. The major theorem of this article is the Plancherel theorem for the Quantum Lorentz Group.Comment: 60 pages, tared gzipped Postscript file, major revision of the previous version, the Plancherel theorem is established in the more general sense and we delay the study of Fusion theory to the next part of this pape

Applicability of Boussinesq approximation in a turbulent fluid with constant properties

The equations of motion describing buoyant fluids are often simplified using a set of approximations proposed by J. Boussinesq one century ago. To resume, they consist in assuming constant fluid properties, incompressibility and conservation of calories during heat transport. Assuming fulfilment of the first requirement (constant fluid properties), we derive a set of 4 criteria for assessing the validity of the two other requirements in turbulent Rayleigh-B\'enard convection. The first criterion $\alpha \Delta \ll 1$ simply results from the incompressibility condition in the thermal boundary layer ($\alpha$ and $\Delta$ are the thermal expansion coefficient and the temperature difference driving the flow). The 3 other criteria are proportional or quadratic with the density stratification or, equivalently with the temperature difference resulting from the adiabatic gradient across the cell $\Delta_{h}$. Numerical evaluations with air, water and cryogenic helium show that most laboratory experiments are free from such Boussinesq violation as long as the first criterion is fulfilled. In ultra high Rayleigh numbers ($Ra>10^{16}$) experiments in He, one of the stratification criteria, scaling with $\alpha \Delta_{h}$, could be violated. This criterion garanties that pressure fluctuations have a negligible influence both on the density variation and on the heat transfer equation through compression/expansion cycles. Extrapolation to higher $Ra$ suggests that strong violation of Boussinesq approximation could occur in atmospheric convection.Comment: Submitted to Phys.Fluids (oct 2007

Shot-noise statistics in diffusive conductors

We study the full probability distribution of the charge transmitted through a mesoscopic diffusive conductor during a measurement time T. We have considered a semi-classical model, with an exclusion principle in a discretized single-particle phase-space. In the large T limit, numerical simulations show a universal probability distribution which agrees very well with the quantum mechanical prediction of Lee, Levitov and Yakovets [PRB {51} 4079 (1995)] for the charge counting statistics. Special attention is given to its third cumulant, including an analysis of finite size effects and of some experimental constraints for its accurate measurement.Comment: Submitted to Eur. Phys. J. B (Jan. 2002

Universal Solutions of Quantum Dynamical Yang-Baxter Equations

We construct a universal trigonometric solution of the Gervais-Neveu-Felder equation in the case of finite dimensional simple Lie algebras and finite dimensional contragredient simple Lie superalgebras.Comment: 12 pages, LaTeX2e with packages vmargin, wasysym, amsmath, amssym

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical Yang--Baxter Equation can be built from the standard R--matrix and from the solution F(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical coCycle Equation as $R(x)=F^{-1}_{21}(x) R F_{12}(x).$ It has been conjectured that, in the case where g=sl(n+1) n greater than 1 only, there could exist an element M(x) in $U_q(sl(n+1))$ such that $F(x)=\Delta(M(x)){J} M_2(x)^{-1}(M_1(xq^{h_2}))^{-1},$ in which $J\in U_q(sl(n+1))^{\otimes 2}$ is the universal cocycle associated to the Cremmer--Gervais's solution. The aim of this article is to prove this conjecture and to study the properties of the solutions of the Quantum Dynamical coBoundary Equation. In particular, by introducing new basic algebraic objects which are the building blocks of the Gauss decomposition of M(x), we construct M(x) in $U_q(sl(n+1))$ as an explicit infinite product which converges in every finite dimensional representation. We emphasize the relations between these basic objects and some Non Standard Loop algebras and exhibit relations with the dynamical quantum Weyl group.Comment: 46 page

Vortex spectrum in superfluid turbulence: interpretation of a recent experiment

We discuss a recent experiment in which the spectrum of the vortex line density fluctuations has been measured in superfluid turbulence. The observed frequency dependence of the spectrum, $f^{-5/3}$, disagrees with classical vorticity spectra if, following the literature, the vortex line density is interpreted as a measure of the vorticity or enstrophy. We argue that the disagrement is solved if the vortex line density field is decomposed into a polarised field (which carries most of the energy) and an isotropic field (which is responsible for the spectrum).Comment: Submitted for publication http://crtbt.grenoble.cnrs.fr/helio/GROUP/infa.html http://www.mas.ncl.ac.uk/~ncfb