S-matrix elements for gauge theories with and without implemented constraints

We derive an expression for the relation between two scattering transition amplitudes which reflect the same dynamics, but which differ in the description of their initial and final state vectors. In one version, the incident and scattered states are elements of a perturbative Fock space, and solve the eigenvalue problem for the `free' part of the Hamiltonian --- the part that remains after the interactions between particle excitations have been `switched off'. Alternatively, the incident and scattered states may be coherent states that are transforms of these Fock states. In earlier work, we reported on the scattering amplitudes for QED, in which a unitary transformation relates perturbative and non-perturbative sets of incident and scattered states. In this work, we generalize this earlier result to the case of transformations that are not necessarily unitary and that may not have unique inverses. We discuss the implication of this relationship for Abelian and non-Abelian gauge theories in which the `transformed', non-perturbative states implement constraints, such as Gauss's law.Comment: 8 pages. Invited contribution to Foundation of Physics for an issue honoring Prof. Lawrence Horwitz on his 65th Birthda

Quantum Gauge Equivalence in QED

We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gauss's law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb and spatial axial gauges to what we call ``common form,'' in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons. By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another.Comment: Contribution for a special issue of Foundations of Physics honoring Fritz Rohrlich; edited by Larry P. Horwitz, Tel-Aviv University, and Alwyn van der Merwe, University of Denver (Plenum Publishing, New York); 40 pages, REVTEX, Preprint UCONN-93-3, 1 figure available upon request from author

Gauge equivalence in QCD: the Weyl and Coulomb gauges

The Weyl-gauge ($A_0^a=0)$ QCD Hamiltonian is unitarily transformed to a representation in which it is expressed entirely in terms of gauge-invariant quark and gluon fields. In a subspace of gauge-invariant states we have constructed that implement the non-Abelian Gauss's law, this unitarily transformed Weyl-gauge Hamiltonian can be further transformed and, under appropriate circumstances, can be identified with the QCD Hamiltonian in the Coulomb gauge. We demonstrate an isomorphism that materially facilitates the application of this Hamiltonian to a variety of physical processes, including the evaluation of $S$-matrix elements. This isomorphism relates the gauge-invariant representation of the Hamiltonian and the required set of gauge-invariant states to a Hamiltonian of the same functional form but dependent on ordinary unconstrained Weyl-gauge fields operating within a space of ``standard'' perturbative states. The fact that the gauge-invariant chromoelectric field is not hermitian has important implications for the functional form of the Hamiltonian finally obtained. When this nonhermiticity is taken into account, the ``extra'' vertices in Christ and Lee's Coulomb-gauge Hamiltonian are natural outgrowths of the formalism. When this nonhermiticity is neglected, the Hamiltonian used in the earlier work of Gribov and others results.Comment: 25 page

Gauge-invariant fields in the temporal gauge, Coulomb-gauge fields, and the Gribov ambiguity

We examine the relation between Coulomb-gauge fields and the gauge-invariant fields constructed in the temporal gauge for two-color QCD by comparing a variety of properties, including their equal-time commutation rules and those of their conjugate chromoelectric fields. We also express the temporal-gauge Hamiltonian in terms of gauge-invariant fields and show that it can be interpreted as a sum of the Coulomb-gauge Hamiltonian and another part that is important for determining the equations of motion of temporal-gauge fields, but that can never affect the time evolution of ``physical'' state vectors. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We argue that these multiplicities of gauge-invariant fields are manifestations of the Gribov ambiguity. We show that the differential equation that bases the multiplicities of gauge-invariant fields on their underlying gauge-dependent temporal-gauge fields has nonlinearities identical to those of the ``Gribov'' equation, which demonstrates the non-uniqueness of Coulomb-gauge fields. These multiplicities of gauge-invariant fields --- and, hence, Gribov copies --- appear in the temporal gauge, but only with the imposition of Gauss's law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss's law is left unimplemented.Comment: 27 pages, 1 figure; text has been revised and references adde

Topology of the gauge-invariant gauge field in two-color QCD

We investigate solutions to a nonlinear integral equation which has a central role in implementing the non-Abelian Gauss's Law and in constructing gauge-invariant quark and gluon fields. Here we concern ourselves with solutions to this same equation that are not operator-valued, but are functions of spatial variables and carry spatial and SU(2) indices. We obtain an expression for the gauge-invariant gauge field in two-color QCD, define an index that we will refer to as the ``winding number'' that characterizes it, and show that this winding number is invariant to a small gauge transformation of the gauge field on which our construction of the gauge-invariant gauge field is based. We discuss the role of this gauge field in determining the winding number of the gauge-invariant gauge field. We also show that when the winding number of the gauge field is an integer $\ell{\neq}0$, the gauge-invariant gauge field manifests winding numbers that are not integers, and are half-integers only when $\ell=0$.Comment: 26 pages including 6 encapsulated postscript figures. Numerical errors have been correcte

Developing Seismogenic Source Models Based on Geologic Fault Data

Calculating seismic hazard usually requires input that includes seismicity associated with known faults, historical earthquake catalogs, geodesy, and models of ground shaking. This paper will address the input generally derived from geologic studies that augment the short historical catalog to predict ground shaking at time scales of tens, hundreds, or thousands of years (e.g., SSHAC 1997). A seismogenic source model, terminology we adopt here for a fault source model, includes explicit three-dimensional faults deemed capable of generating ground motions of engineering significance within a specified time frame of interest. In tectonically active regions of the world, such as near plate boundaries, multiple seismic cycles span a few hundred to a few thousand years. In contrast, in less active regions hundreds of kilometers from the nearest plate boundary, seismic cycles generally are thousands to tens of thousands of years long. Therefore, one should include sources having both longer recurrence intervals and possibly older times of most recent rupture in less active regions of the world rather than restricting the model to include only Holocene faults (i.e., those with evidence of large-magnitude earthquakes in the past 11,500 years) as is the practice in tectonically active regions with high deformation rates. During the past 15 years, our institutions independently developed databases to characterize seismogenic sources based on geologic data at a national scale. Our goal here is to compare the content of these two publicly available seismogenic source models compiled for the primary purpose of supporting seismic hazard calculations by the Istituto Nazionale di Geofisica e Vulcanologia (INGV) and the U.S. Geological Survey (USGS); hereinafter we refer to the two seismogenic source models as INGV and USGS, respectively. This comparison is timely because new initiatives are emerging to characterize seismogenic sources at the continental scale (e.g., SHARE in the Euro- Mediterranean, http://www.share-eu.org/; EMME in the Middle East, http://www.emmegem. org/) and global scale (e.g., GEM, http://www.globalquakemodel.org/; Anonymous 2008). To some extent, each of these efforts is still trying to resolve the level of optimal detail required for this type of compilation. The comparison we provide defines a common standard for consideration by the international community for future regional and global seismogenic source models by identifying the necessary parameters that capture the essence of geological fault data in order to characterize seismogenic sources. In addition, we inform potential users of differences in our usage of common geological/seismological terms to avoid inappropriate use of the data in our models and provide guidance to convert the data from one model to the other (for detailed instructions, see the electronic supplement to this article). Applying our recommendations will permit probabilistic seismic hazard assessment codes to run seamlessly using either seismogenic source input. The USGS and INGV database schema compare well at a first-level inspection. Both databases contain a set of fields representing generalized fault three-dimensional geometry and additional fields that capture the essence of past earthquake occurrences. Nevertheless, there are important differences. When we further analyze supposedly comparable fields, many are defined differently. These differences would cause anomalous results in hazard prediction if one assumes the values are similarly defined. The data, however, can be made fully compatible using simple transformations

Persistent Transport Barrier on the West Florida Shelf

Analysis of drifter trajectories in the Gulf of Mexico has revealed the existence of a region on the southern portion of the West Florida Shelf (WFS) that is not visited by drifters that are released outside of the region. This so-called ``forbidden zone'' (FZ) suggests the existence of a persistent cross-shelf transport barrier on the southern portion of the WFS. In this letter a year-long record of surface currents produced by a Hybrid-Coordinate Ocean Model simulation of the WFS is used to identify Lagrangian coherent structures (LCSs), which reveal the presence of a robust and persistent cross-shelf transport barrier in approximately the same location as the boundary of the FZ. The location of the cross-shelf transport barrier undergoes a seasonal oscillation, being closer to the coast in the summer than in the winter. A month-long record of surface currents inferred from high-frequency (HF) radar measurements in a roughly 60 km $\times$ 80 km region on the WFS off Tampa Bay is also used to identify LCSs, which reveal the presence of robust transient transport barriers. While the HF-radar-derived transport barriers cannot be unambiguously linked to the boundary of the FZ, this analysis does demonstrate the feasibility of monitoring transport barriers on the WFS using a HF-radar-based measurement system. The implications of a persistent cross-shelf transport barrier on the WFS for the development of harmful algal blooms on the shoreward side of the barrier are considered.Comment: Submitted to Geophysical Research Letter

The global electroweak fit at NNLO and prospects for the LHC and ILC

For a long time, global fits of the electroweak sector of the Standard Model (SM) have been used to exploit measurements of electroweak precision observables at lepton colliders (LEP, SLC), together with measurements at hadron colliders (Tevatron, LHC), and accurate theoretical predictions at multi-loop level, to constrain free parameters of the SM, such as the Higgs and top masses. Today, all fundamental SM parameters entering these fits are experimentally determined, including information on the Higgs couplings, and the global fits are used as powerful tools to assess the validity of the theory and to constrain scenarios for new physics. Future measurements at the Large Hadron Collider (LHC) and the International Linear Collider (ILC) promise to improve the experimental precision of key observables used in the fits. This paper presents updated electroweak fit results using newest NNLO theoretical predictions, and prospects for the LHC and ILC. The impact of experimental and theoretical uncertainties is analysed in detail. We compare constraints from the electroweak fit on the Higgs couplings with direct LHC measurements, and examine present and future prospects of these constraints using a model with modified couplings of the Higgs boson to fermions and bosons.Comment: 26 pages, 9 figure

Spin Susceptibility of Noncentrosymmetric Heavy-fermion Superconductor CeIrSi3 under Pressure: 29Si-Knight Shift Study on Single Crystal

We report 29Si-NMR study on a single crystal of the heavy-fermion superconductor CeIrSi3 without an inversion symmetry along the c-axis. The 29Si-Knight shift measurements under pressure have revealed that the spin susceptibility for the ab-plane decreases slightly below Tc, whereas along the c-axis it does not change at all. The result can be accounted for by the spin susceptibility in the superconducting state being dominated by the strong antisymmetric (Rashba-type) spin-orbit interaction that originates from the absence of an inversion center along the c-axis and it being much larger than superconducting condensation energy. This is the first observation which exhibits an anisotropy of the spin susceptibility below Tc in the noncentrosymmetric superconductor dominated by strong Rashba-type spin-orbit interaction.Comment: 4 pages, 4 figures, Accepted for publication in Phys. Rev. Let

Precision Measurements of Stretching and Compression in Fluid Mixing

The mixing of an impurity into a flowing fluid is an important process in many areas of science, including geophysical processes, chemical reactors, and microfluidic devices. In some cases, for example periodic flows, the concepts of nonlinear dynamics provide a deep theoretical basis for understanding mixing. Unfortunately, the building blocks of this theory, i.e. the fixed points and invariant manifolds of the associated Poincare map, have remained inaccessible to direct experimental study, thus limiting the insight that could be obtained. Using precision measurements of tracer particle trajectories in a two-dimensional fluid flow producing chaotic mixing, we directly measure the time-dependent stretching and compression fields. These quantities, previously available only numerically, attain local maxima along lines coinciding with the stable and unstable manifolds, thus revealing the dynamical structures that control mixing. Contours or level sets of a passive impurity field are found to be aligned parallel to the lines of large compression (unstable manifolds) at each instant. This connection appears to persist as the onset of turbulence is approached.Comment: 5 pages, 5 figure