Poisson Lie Group Symmetries for the Isotropic Rotator

We find a new Hamiltonian formulation of the classical isotropic rotator where left and right $SU(2)$ transformations are not canonical symmetries but rather Poisson Lie group symmetries. The system corresponds to the classical analog of a quantum mechanical rotator which possesses quantum group symmetries. We also examine systems of two classical interacting rotators having Poisson Lie group symmetries.Comment: 22pp , Latex fil

Edge Currents and Vertex Operators for Chern-Simons Gravity

We apply elementary canonical methods for the quantization of 2+1 dimensional gravity, where the dynamics is given by E. Witten's $ISO(2,1)$ Chern-Simons action. As in a previous work, our approach does not involve choice of gauge or clever manipulations of functional integrals. Instead, we just require the Gauss law constraint for gravity to be first class and also to be everywhere differentiable. When the spatial slice is a disc, the gravitational fields can either be unconstrained or constrained at the boundary of the disc. The unconstrained fields correspond to edge currents which carry a representation of the $ISO(2,1)$ Kac-Moody algebra. Unitary representations for such an algebra have been found using the method of induced representations. In the case of constrained fields, we can classify all possible boundary conditions. For several different boundary conditions, the field content of the theory reduces precisely to that of 1+1 dimensional gravity theories. We extend the above formalism to include sources. The sources take into account self- interactions. This is done by punching holes in the disc, and erecting an $ISO(2,1)$ Kac-Moody algebra on the boundary of each hole. If the hole is originally sourceless, a source can be created via the action of a vertex operator $V$. We give an explicit expression for $V$. We shall show that when actingComment: 42 pages, UAHEP 925, SU-4240-508, INFN-NA-IV-92/1

The Chern-Simons Source as a Conformal Family and Its Vertex Operators

In a previous work, a straightforward canonical approach to the source-free quantum Chern-Simons dynamics was developed. It makes use of neither gauge conditions nor functional integrals and needs only ideas known from QCD and quantum gravity. It gives Witten's conformal edge states in a simple way when the spatial slice is a disc. Here we extend the formalism by including sources as well. The quantum states of a source with a fixed spatial location are shown to be those of a conformal family, a result also discovered first by Witten. The internal states of a source are not thus associated with just a single ray of a Hilbert space. Vertex operators for both abelian and nonabelian sources are constructed. The regularized abelian Wilson line is proved to be a vertex operator. We also argue in favor of a similar nonabelian result. The spin-statistics theorem is established for Chern-Simons dynamics even though the sources are not described by relativistic quantum fields. The proof employs geometrical methods which we find are strikingly transparent and pleasing. It is based on the research of European physicists about ``fields localized on cones.'

Scale Transformations on the Noncommutative Plane and the Seiberg-Witten Map

We write down three kinds of scale transformations {\tt i-iii)} on the noncommutative plane. {\tt i)} is the analogue of standard dilations on the plane, {\tt ii)} is a re-scaling of the noncommutative parameter $\theta$, and {\tt iii)} is a combination of the previous two, whereby the defining relations for the noncommutative plane are preserved. The action of the three transformations is defined on gauge fields evaluated at fixed coordinates and $\theta$. The transformations are obtained only up to terms which transform covariantly under gauge transformations. We give possible constraints on these terms. We show how the transformations {\tt i)} and {\tt ii)} depend on the choice of star product, and show the relation of {\tt ii)} to Seiberg-Witten transformations. Because {\tt iii)} preserves the fundamental commutation relations it is a symmetry of the algebra. One has the possibility of implementing it as a symmetry of the dynamics, as well, in noncommutative field theories where $\theta$ is not fixed.Comment: 20 page

Fabrication and characterization of high current-density, submicron, NbN/MgO/NbN tunnel junctions

At near-millimeter wavelengths, heterodyne receivers based on SIS tunnel junctions are the most sensitive available. However, in order to scale these results to submillimeter wavelengths, certain device properties should be scaled. The tunnel-junction's current density should be increased to reduce the RC product. The device's area should be reduced to efficiently couple power from the antenna to the mixer. Finally, the superconductor used should have a large energy gap to minimize RF losses. Most SIS mixers use Nb or Pb-alloy tunnel junctions; the gap frequency for these materials is approximately 725 GHz. Above the gap frequency, these materials exhibit losses similar to those in a normal metal. The gap frequency in NbN films is as-large-as 1440 GHz. Therefore, we have developed a process to fabricate small area (down to 0.13 sq microns), high current density, NbN/MgO/NbN tunnel junctions

Lie-Poisson Deformation of the Poincar\'e Algebra

We find a one parameter family of quadratic Poisson structures on ${\bf R}^4\times SL(2,C)$ which satisfies the property {\it a)} that it is preserved under the Lie-Poisson action of the Lorentz group, as well as {\it b)} that it reduces to the standard Poincar\'e algebra for a particular limiting value of the parameter. (The Lie-Poisson transformations reduce to canonical ones in that limit, which we therefore refer to as the `canonical limit'.) Like with the Poincar\'e algebra, our deformed Poincar\'e algebra has two Casimir functions which we associate with `mass' and `spin'. We parametrize the symplectic leaves of ${\bf R}^4\times SL(2,C)$ with space-time coordinates, momenta and spin, thereby obtaining realizations of the deformed algebra for the cases of a spinless and a spinning particle. The formalism can be applied for finding a one parameter family of canonically inequivalent descriptions of the photon.Comment: Latex file, 26 page

Lorentz Transformations as Lie-Poisson Symmetries

We write down the Poisson structure for a relativistic particle where the Lorentz group does not act canonically, but instead as a Poisson-Lie group. In so doing we obtain the classical limit of a particle moving on a noncommutative space possessing $SL_q(2,C)$ invariance. We show that if the standard mass shell constraint is chosen for the Hamiltonian function, then the particle interacts with the space-time. We solve for the trajectory and find that it originates and terminates at singularities.Comment: 18 page