Time-Dependent Observables in Heavy Ion Collisions I: Setting up the Formalism

We adapt the Schwinger-Keldysh formalism to study heavy-ion collisions in perturbative QCD. Employing the formalism, we calculate the two-point gluon correlation function $G_{22}^{a\mu,b\nu}$ due to the lowest-order classical gluon fields in the McLerran-Venugopalan model of heavy ion collisions and observe an interesting transition from the classical fields to the quasi-particle picture at later times. Motivated by this observation, we push the formalism to higher orders in the coupling and calculate the contribution to $G_{22}^{a\mu,b\nu}$ coming from the diagrams representing a single rescattering between two of the produced gluons. We assume that the two gluons go on mass shell both before and after the rescattering. The result of our calculation depends on the ordering between the proper time of the rescattering $\tau_Z$ and the proper time $\tau$ when the gluon distribution is measured. For (i) $\tau_Z\gg 1/Q_s$ and $\tau-\tau_Z\gg 1/Q_s$ (with $Q_s$ the saturation scale) we obtain the same results as from the Boltzmann equation. For (ii) $\tau-\tau_Z\gg \tau_Z\gg 1/Q_s$ we end up with a result very different from kinetic theory and consistent with a picture of "free-streaming" particles. Due to the approximations made, our calculation is too coarse to indicate whether the ordering (i) or (ii) is the correct one: to resolve this controversy, we shall present a detailed diagrammatic calculation of the rescattering correction in the $\varphi^4$ theory in the second paper of this duplex.Comment: 45 pages, 9 figure

Time-Dependent Observables in Heavy Ion Collisions II: in Search of Pressure Isotropization in the φ4\varphi^4 Theory

To understand the dynamics of thermalization in heavy ion collisions in the perturbative framework it is essential to first find corrections to the free-streaming classical gluon fields of the McLerran-Venugopalan model. The corrections that lead to deviations from free streaming (and that dominate at late proper time) would provide evidence for the onset of isotropization (and, possibly, thermalization) of the produced medium. To find such corrections we calculate the late-time two-point Green function and the energy-momentum tensor due to a single $2 \to 2$ scattering process involving two classical fields. To make the calculation tractable we employ the scalar $\varphi^4$ theory instead of QCD. We compare our exact diagrammatic results for these quantities to those in kinetic theory and find disagreement between the two. The disagreement is in the dependence on the proper time $\tau$ and, for the case of the two-point function, is also in the dependence on the space-time rapidity $\eta$: the exact diagrammatic calculation is, in fact, consistent with the free streaming scenario. Kinetic theory predicts a build-up of longitudinal pressure, which, however, is not observed in the exact calculation. We conclude that we find no evidence for the beginning of the transition from the free-streaming classical fields to the kinetic theory description of the produced matter after a single $2 \to 2$ rescattering.Comment: 46 pages, 9 figure

Brunnian Braids and Lie Algebras

Brunnian braids have interesting relations with homotopy groups of spheres. In this work, we study the graded Lie algebra of the descending central series related to Brunnian subgroup of the pure braid group. A presentation of this Lie algebra is obtained.Comment: 21 page

Real-time multiple-look synthetic aperture radar processor for spacecraft applications

A spaceborne synthetic aperture radar (SAR) having pipeline multiple-look data processing is described which makes use of excessive azimuth bandwidth in radar echo signals to produce multiple-looking images. Time multiplexed single-look image lines from an azimuth correlator go through an energy analyzer which analyzes the mean energy in each separate look to determine the radar antenna electric boresight for use in generating the correct reference functions for the production of high quality SAR images. The multiplexed single look image lines also go through a registration delay to produce multi-look images

Apparatus for determining changes in limb volume

Measuring apparatus for determining changes in the volume of limbs or other boty extremities by determining the cross-sectional area of such limbs many comprise a transmitter including first and second transducers for positioning on the surface of the limb at a predetermined distance there between, and a receiver including a receiver crystal for positioning on the surface of the limb. The distance between the receiver crystal and the first and second transducers are represented by respective first and second chords of the cross-section of the limb and the predetermined distance between the first and second transducers is represented by a third chord of the limb cross section

Basis-conjugating automorphisms of a free group and associated Lie algebras

Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the integers. In particular among them are the automorphisms chi_{k,i} which conjugate the generator x_k by the generator x_i leaving the x_j fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by chi_{k,i} for i<k is given here. Partial results are obtained for the group generated by all chi_{k,i}.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

Regulation and the neo-Wicksellian approach to monetary policy

Laubach and Williams (2003) employ a Kalman filter approach to jointly estimate the neutral real federal funds rate and trend output growth using an IS relationship and an output gap based inflation equation. They find a positive link between these two variables, but also much error surrounding neutral real rate estimates. We modify their approach by including variables for regulations on deposit interest rates and on wages and prices. These variables are statistically significant and notably affect estimates of two policy relevant coefficients: the sensitivity of output to the real interest rate and that of inflation to the output gap.Monetary policy ; Federal funds rate