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

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

Addendum to "Coherent Lagrangian vortices: The black holes of turbulence"

In Haller and Beron-Vera (2013) we developed a variational principle for the detection of coherent Lagrangian vortex boundaries. The solutions of this variational principle turn out to be closed null-geodesics of the Lorentzian metric associated with a generalized Green-Lagrange strain tensor family. This metric interpretation implies a mathematical analogy between coherent Lagrangian vortex boundaries and photon spheres in general relativity. Here we give an improved discussion on this analogy.Comment: Revised 27 June 201

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

Kierkegaard’s ‘Repetition’ and Pilgrimage

In 1843, the Danish philosopher Kierkegaard wondered whether it was possible to repeat an experience. He attempted to relive experiences he once had in Berlin by revisiting haunts of his earlier self. After several days, he concluded that his repetition of experience was unsuccessful. Many people make similar attempts at repetition when they make, for example, the pilgrimage to Camino de Santiago multiple times. What could a person hope to gain by this repetition? What prevents successful repetition, suggests Kierkegaard, is beginning with the end in mind rather than traveling merely to collect random impressions. Repetition fails, argues Kierkegaard, when it is tried as some kind of experiment rather than a commitment, and this failure of immersion makes us a passive observer of ourselves. An authentic repetition, he argues, can only happen after one surrenders control of events. One must give something up to get anything back. To experience Berlin again, or to repeat a pilgrimage, one must give up expectations and surrender to whatever unfolds. In this way, the repeat traveler would surrender the objective stance of someone comparing events with earlier ones. Instead, they would be actively engaged, and thus able to actually re-experience something like what happened before. In contrast to passively ‘recollecting’ past events, Kierkegaard advises that a successful ‘repetition’ is to live with a repeatedly renewed commitment to living in the present. Repetition is, paradoxically, not about the past but, rather, about ‘the earnestness of existence.’ Pilgrims, however, often try on identities and roles in attempts to experience what they have heard of in other pilgrim’s stories. It may be that some repeat pilgrims are not so much nostalgically reliving their previous experiences, as trying to experience what others have experienced, and that they themselves did not experience the first time. It will be argued that this imitation of life can never be fully engaged in, and thus, it will always be disappointing because it necessarily involves self-consciously observing oneself in the role

Pilgrimage and Paradigm Shifts: The Role of Experience in Identity Transformations

Some kinds of knowledge can only be acquired by direct experience. Engaging in a pilgrimage can transform a person by allowing them to inhabit different worldviews, or paradigms, that would otherwise remain inaccessible to them. The pilgrim then learns to see the world in a new and different way; has ideas not imagined before the experience, and; may even change their life in ways unforeseen before the pilgrimage. It is uncertain just how, and in what unexpected ways, one might change. Most often these changes are welcomed; however, it is unclear whether one’s prepilgrim- self, before the change, would approve of the end-result. How can one rationally choose to undertake a pilgrimage that might cause change, when one cannot really appreciate whether that change will be for the better until after one has leapt into it? This makes the choice seem irrational and arbitrary. This paper argues that the most likely changes that come about on pilgrimage are a matter of reorientation, rather than a complete metamorphosis (although those happen). Religious experiences that change one’s identity are most often a matter of reorientation of commitment, rather than radical transformation of self. The permanence of the change depends on the commitment to that identity after the change is made

The Possibility of Pilgrimage in a Scientific World

The paper is a philosophical argument about whether a pilgrimage can be meaningful in a scientific age. Since a scientific world-view rules out many ideas which are traditionally associated with pilgrimage, such as miracles and the effectiveness of prayer, it seems that pilgrimage might be a practice inconsistent with the modern scientific age. Attempts have been made to reconcile this conflict by arguing that science and religion do not conflict, but are non-overlapping spheres of inquiry. Thus, it is possible to make sense of pilgrimage in a scientific age, if one strips their pilgrimage of all aspects to which science might object. However, this move makes it possible for almost any activity to be interpreted as a pilgrimage activity, including many activities usually associated with consumption, tourism, materialism, consumerism, and entertainment. Using ideas developed by the philosopher William James, and others, one can argue that many of these apparently secular activities can indeed be seen as religious experiences. By identifying what all religious experiences have in common, one can interpret many different kinds of experiences as lying on a continuum with traditional pilgrimage experiences

A Changed Understanding of Miracles in Religious Tourism

In this modern age, an unsceptical acceptance of supernatural events–those which cannot be explained as part of the natural order of things–is less common than it once was. This trend is reflected in the declining frequency of miracle-cures certified by the Medical Bureau at Lourdes. Yet miracles past, and the promise of possible miracles in the present, still attract multitudes of religious pilgrims and tourists to sacred sites all over the world. While the frequency of miracles goes down, the appeal of miracles goes on, and the number of religious visitors has not declined. What role do miracles now play in religious tourism? The miracles associated with religious pilgrimage and tourism will be distinguished into two categories. Archaic Miracles are those that occurred in pre-scientific, often medieval, times. These often involve very implausible stories, and have the air of folklore and fairy-tales. Modern Age Miracles occur after the development of science and the Enlightenment commitment to understanding things through reason. This paper will conclude with a ‘compatibilist solution’ between two seemingly contradictory positions–miracles and science. A miraculous event is often taken as one that is contrary to the laws of nature; while religious sceptics reject miracles as unscientific. Yet the scientific demand for complete explanations is too demanding and may be impossible to satisfy. Inspired by a physicist, Marcel Glieser, I explain that there are fundamental limits to our understanding of the universe, which implies that mysteries will always remain. However, an inescapable mystery is no support for supernatural explanations. A modern-day pilgrim need not believe in the supernatural to find meaning in unexplained events, but merely needs to recognise that even ordinary things remain fundamentally unexplained. I defend this ‘wonder of existence’ solution to the problem of miracles, and provide examples, and show how this is relevant to religious tourism

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