Asymptotic Freedom: From Paradox to Paradigm

Asymptotic freedom was developed as a response to two paradoxes: the weirdness of quarks, and in particular their failure to radiate copiously when struck; and the coexistence of special relativity and quantum theory, despite the apparent singularity of quantum field theory. It resolved these paradoxes, and catalyzed the development of several modern paradigms: the hard reality of quarks and gluons, the origin of mass from energy, the simplicity of the early universe, and the power of symmetry as a guide to physical law.Comment: 26 pages, 10 figures. Lecture on receipt of the 2004 Nobel Prize. v2: typo (in Ohm's law) correcte

Normalization of the covariant three-body bound state vertex function

The normalization condition for the relativistic three nucleon Bethe-Salpeter and Gross bound state vertex functions is derived, for the first time, directly from the three body wave equations. It is also shown that the relativistic normalization condition for the two body Gross bound state vertex function is identical to the requirement that the bound state charge be conserved, proving that charge is automatically conserved by this equation.Comment: 24 pages, 9 figures, published version, minor typos correcte

Two-pion exchange potential and the πN\pi N amplitude

We discuss the two-pion exchange potential which emerges from a box diagram with one nucleon (the spectator) restricted to its mass shell, and the other nucleon line replaced by a subtracted, covariant $\pi N$ scattering amplitude which includes $\Delta$, Roper, and $D_{13}$ isobars, as well as contact terms and off-shell (non-pole) dressed nucleon terms. The $\pi N$ amplitude satisfies chiral symmetry constraints and fits $\pi N$ data below $\sim$ 700 MeV pion energy. We find that this TPE potential can be well approximated by the exchange of an effective sigma and delta meson, with parameters close to the ones used in one-boson-exchange models that fit $NN$ data below the pion production threshold.Comment: 9 pages (RevTex) and 7 postscript figures, in one uuencoded gzipped tar fil

Covariant spectator theory of np scattering: Phase shifts obtained from precision fits to data below 350 MeV

Using the covariant spectator theory (CST), we present two one boson exchange kernels that have been successfully adjusted to fit the 2007 world np data (containing 3788 data) below 350 MeV. One model (which we designate WJC-1) has 27 parameters and fits with a chi2/N = 1.06. The other model (designated WJC-2) has only 15 parameters and fits with a chi2/N = 1.12. Both of these models also reproduce the experimental triton binding energy without introducing additional irreducible three-nucleon forces. One result of this work is a new phase shift analysis, updated for all data until 2006, which is useful even if one does not work within the CST. In carrying out these fits we have reviewed the entire data base, adding new data not previously used in other high precision fits and restoring some data omitted in previous fits. A full discussion and evaluation of the 2007 data base is presented.Comment: 43 pages, 27 figures, and 13 table

Enumerative aspects of the Gross-Siebert program

We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming Fields Institute volume. 81 page

Gauging the three-nucleon spectator equation

We derive relativistic three-dimensional integral equations describing the interaction of the three-nucleon system with an external electromagnetic field. Our equations are unitary, gauge invariant, and they conserve charge. This has been achieved by applying the recently introduced gauging of equations method to the three-nucleon spectator equations where spectator nucleons are always on mass shell. As a result, the external photon is attached to all possible places in the strong interaction model, so that current and charge conservation are implemented in the theoretically correct fashion. Explicit expressions are given for the three-nucleon bound state electromagnetic current, as well as the transition currents for the scattering processes \gamma He3 -> NNN, Nd -> \gamma Nd, and \gamma He3 -> Nd. As a result, a unified covariant three-dimensional description of the NNN-\gamma NNN system is achieved.Comment: 23 pages, REVTeX, epsf, 4 Postscript figure

Formation of nanosized strontium substituted hydroxyapatites

Incorporation of specific elements into calcium phosphates offers the combination of a bioactive material and a therapeutic effect. This is important for improving the integration of implants as well as treating medical conditions. Strontium is a suitable candidate and displays the ability to stimulate bone growth and reducing bone resorption. This study investigated the formation of strontium carbonated hydroxyapatite nanoparticles from an amorphous phase. Crystallization of carbonated hydroxyapatite occurred at 585 oC, but samples with an intended 25% and 75% replacement of calcium with strontium crystallized at 624 oC. Heat treatment at the crystallization temperature revealed that strontium free apatite does not crystallize in 5 minutes, but an increasing strontium concentration leads to a higher rate of crystallization. X-ray diffraction patterns suggest that it may be difficult to include strontium, but higher strontium concentrations are possibly included with ease in the lattice. This work has produced a nanosized apatite accompanied by an amorphous phase after a short heat-treatment time. This offers a range of features that collectively show great promise for significantly enhancing the release of strontium for improved bone therapeutic effects

Beyond Mean-Field Theory for Attractive Bosons under Transverse Harmonic Confinement

We study a dilute gas of attractive bosons confined in a harmonic cylinder, i.e. under cylindric confinement due to a transverse harmonic potential. We introduce a many-body wave function which extends the Bethe ansatz proposed by McGuire (J. Math. Phys. {\bf 5}, 622 (1964)) by including a variational transverse Gaussian shape. We investigate the ground state properties of the system comparing them with the ones of the one-dimensional (1D) attractive Bose gas. We find that the gas becomes ultra 1D as a consequence of the attractive interaction: the transverse width of the Bose gas reduces by increasing the number of particles up to a critical width below which there is the collapse of the cloud. In addition, we derive a simple analytical expression for the simmetry-breaking solitonic density profile of the ground-state, which generalize the one deduced by Calogero and Degasperis (Phys. Rev. A {\bf 11}, 265 (1975)). This bright-soliton analytical solution shows near the collapse small deviations with respect to the 3D mean-field numerical solution. Finally, we show that our variational Gauss-McGuire theory is always more accurate than the McGuire theory. In addition, we prove that for small numbers of particles the Gauss-McGuire theory is more reliable than the mean-field theory described by the 3D Gross-Pitaevskii equation.Comment: To be published in J. Phys. B.: At. Mol. Opt. Phy