11,004 research outputs found
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
Quark confinement and color transparency in a gauge-invariant formulation of QCD
We examine a nonlocal interaction that results from expressing the QCD
Hamiltonian entirely in terms of gauge-invariant quark and gluon fields. The
interaction couples one quark color-charge density to another, much as electric
charge densities are coupled to each other by the Coulomb interaction in QED.
In QCD, this nonlocal interaction also couples quark color-charge densities to
gluonic color. We show how the leading part of the interaction between quark
color-charge densities vanishes when the participating quarks are in a color
singlet configuration, and that, for singlet configurations, the residual
interaction weakens as the size of a packet of quarks shrinks. Because of this
effect, color-singlet packets of quarks should experience final state
interactions that increase in strength as these packets expand in size. For the
case of an SU(2) model of QCD based on the {\em ansatz} that the
gauge-invariant gauge field is a hedgehog configuration, we show how the
infinite series that represents the nonlocal interaction between quark
color-charge densities can be evaluated nonperturbatively, without expanding it
term-by-term. We discuss the implications of this model for QCD with SU(3)
color and a gauge-invariant gauge field determined by QCD dynamics.Comment: Revtex, 23 pages; contains additional references with brief comments
on sam
LCS Tool : A Computational Platform for Lagrangian Coherent Structures
We give an algorithmic introduction to Lagrangian coherent structures (LCSs)
using a newly developed computational engine, LCS Tool. LCSs are most
repelling, attracting and shearing material lines that form the centerpieces of
observed tracer patterns in two-dimensional unsteady dynamical systems. LCS
Tool implements the latest geodesic theory of LCSs for two-dimensional flows,
uncovering key transport barriers in unsteady flow velocity data as explicit
solutions of differential equations. After a review of the underlying theory,
we explain the steps and numerical methods used by LCS Tool, and illustrate its
capabilities on three unsteady fluid flow examples
Gauge equivalence in QCD: the Weyl and Coulomb gauges
The Weyl-gauge ( 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 -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
Pairwise Interaction on Random Graphs
We analyze dynamic local interaction in population games where the local interaction structure (modeled as a graph) can change over time: A stochastic process generates a random sequence of graphs. This contrasts with models where the initial interaction structure (represented by a deterministic graph or the realization of a random graph) cannot change over time.
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
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