Invariants of 2+1 Quantum Gravity

In [1,2] we established and discussed the algebra of observables for 2+1 gravity at both the classical and quantum level. Here our treatment broadens and extends previous results to any genus $g$ with a systematic discussion of the centre of the algebra. The reduction of the number of independent observables to $6g-6 (g > 1)$ is treated in detail with a precise classification for $g = 1$ and $g = 2$.Comment: 10 pages, plain TEX, no figures, DFTT 46/9

Extension of geodesic algebras to continuous genus

Using the Penner--Fock parameterization for Teichmuller spaces of Riemann surfaces with holes, we construct the string-like free-field representation of the Poisson and quantum algebras of geodesic functions in the continuous-genus limit. The mapping class group acts naturally in the obtained representation.Comment: 16 pages, submitted to Lett.Math.Phy

Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant

We relate the geometrical construction of (2+1)-spacetimes via grafting to phase space and Poisson structure in the Chern-Simons formulation of (2+1)-dimensional gravity with vanishing cosmological constant on manifolds of topology $R\times S_g$, where $S_g$ is an orientable two-surface of genus $g>1$. We show how grafting along simple closed geodesics \lambda is implemented in the Chern-Simons formalism and derive explicit expressions for its action on the holonomies of general closed curves on S_g. We prove that this action is generated via the Poisson bracket by a gauge invariant observable associated to the holonomy of $\lambda$. We deduce a symmetry relation between the Poisson brackets of observables associated to the Lorentz and translational components of the holonomies of general closed curves on S_g and discuss its physical interpretation. Finally, we relate the action of grafting on the phase space to the action of Dehn twists and show that grafting can be viewed as a Dehn twist with a formal parameter $\theta$ satisfying $\theta^2=0$.Comment: 43 pages, 10 .eps figures; minor modifications: 2 figures added, explanations added, typos correcte

Field-scale evaluation of phosphorus leaching in acid sandy soils receiving swine waste

Accurate descriptions of P leaching are important because excess P applied to soils can enter surface water via leaching and subsurface transport, thereby negatively impacting water quality. The objectives of this study were to monitor P leaching in soils with a long-term history of waste application, relate soil solution P concentrations to soil P status, and quantify P leaching losses. Soil solution was monitored for 20 mo with samplers installed at 45-, 90-, and 135-cm depths in two pits (1 X 3 X L5 m) in Autryville (loamy, siliceous, thermic Arenic Paleudults) and Blanton (loamy, siliceous, semiactive, thermic Grossarenic Paleudults) soils located in a grazed pasture in Sampson County, NC, which had received swine waste for >20 yr. Maximum soil solution P concentrations at 45 cm exceeded 18 mg L' in both soils. Soil solution P concentrations at 90 cm in the Blanton soil were similar to that at 45 cm indicating low P sorption. Soil solution P concentrations at 90 cm in the Autryville soil averaged 0.05 mg L-' compared to 10 mg L' at 45 cm. A split-line model related soil solution P concentration to the degree of phosphorus saturation (DPS), identifying a change point at 45% DPS. Phosphorus movement past 45 cm equaled or exceeded surplus P additions for both soils. Longterm waste applications resulted in DPS > 90%, high soil solution P concentrations, and substantial vertical P movement. Phosphorus leaching should be considered when assessing long-term risk of P loss from waste-amended soils

Effective dynamics for particles coupled to a quantized scalar field

We consider a system of N non-relativistic spinless quantum particles (``electrons'') interacting with a quantized scalar Bose field (whose excitations we call ``photons''). We examine the case when the velocity v of the electrons is small with respect to the one of the photons, denoted by c (v/c= epsilon << 1). We show that dressed particle states exist (particles surrounded by ``virtual photons''), which, up to terms of order (v/c)^3, follow Hamiltonian dynamics. The effective N-particle Hamiltonian contains the kinetic energies of the particles and Coulomb-like pair potentials at order (v/c)^0 and the velocity dependent Darwin interaction and a mass renormalization at order (v/c)^{2}. Beyond that order the effective dynamics are expected to be dissipative. The main mathematical tool we use is adiabatic perturbation theory. However, in the present case there is no eigenvalue which is separated by a gap from the rest of the spectrum, but its role is taken by the bottom of the absolutely continuous spectrum, which is not an eigenvalue. Nevertheless we construct approximate dressed electrons subspaces, which are adiabatically invariant for the dynamics up to order (v/c)\sqrt{\ln (v/c)^{-1}}. We also give an explicit expression for the non adiabatic transitions corresponding to emission of free photons. For the radiated energy we obtain the quantum analogue of the Larmor formula of classical electrodynamics.Comment: 67 pages, 2 figures, version accepted for publication in Communications in Mathematical Physic