Robin Horan

Charges in gauge theories

In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green’s functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions.

Asymptotic dynamics in quantum field theory

A crucial element of scattering theory and the LSZ reduction formula is the assumption that the coupling vanishes at large times. This is known not to hold for the theories of the Standard Model and in general such asymptotic dynamics is not well understood. We give a description of asymptotic dynamics in field theories which incorporates the important features of weak convergence and physical boundary conditions. Applications to theories with three and four point interactions are presented and the results are shown to be completely consistent with the results of perturbation theory.

A rigidity theorem for quaternionic Kähler manifolds

Abstract We investigate the cohomology groups H 1 ( Z , O ( L ⊗ m )), where Z is the twistor space of a compact quaternionic-Kahler manifold M , of dimension 4 k , for k ⩾ 1, L is the holomorphic contact line bundle on Z , and m ⩾ 0. The Penrose transform is used to prove a vanishing theorem for this cohomology group when M has negative scalar curvature. This theorem implies that if ( M , g ) is a compact quaternionic-Kahler manifold of dimension 4 k , for k ⩾ 1, then ( M , g ) has no nontrivial deformations through quaternionic-Kahler manifolds. The vanishing theorem is also used to show that the first Betti number of M is zero, when k > 1 and M has negative scalar curvature.

On the Hamiltonian reduction of geodesic motion on SU(3) to SU(3)/SU(2)

The reduced Hamiltonian system on T*(SU(3)∕SU(2)) is derived from a Riemannian geodesic motion on the SU(3) group manifold parametrized by the generalized Euler angles and endowed with a bi-invariant metric. Our calculations show that the metric defined by the derived reduced Hamiltonian flow on the orbit space SU(3)∕SU(2)≃S5 is not isometric or even geodesically equivalent to the standard Riemannian metric on the five-sphere S5 embedded into R6.

Charged Particles: A Builder's Guide

It is sometimes claimed that one cannot describe charged particles in gauge theories. We identify the root of the problem and present an explicit construction of charged particles. This is shown to have good perturbative properties and, asymptotically before and after scattering, to recover particle modes.

The glue around quarks and the interquark potential

Abstract The quarks of quark models cannot be identified with the quarks of the QCD Lagrangian. We review the restrictions that gauge field theories place on any description of physical (colour) charges. A method to construct charged particles is presented. The solutions are applied to a variety of applications. Their Greens functions are shown to be free of infra-red divergences to all orders in perturbation theory. The interquark potential is analysed and it is shown that the interaction responsible for anti-screening results from the force between two separately gauge invariant constituent quarks. A fundamental limit on the applicability of quark models is identified.

Charges, Monopoles and Gauge Invariance

A systematic approach to the description of gauge invariant charges is presented and applied to the construction of both the static colour charge configuration in QCD and the monopole solution in pure SU (2). The gauge invariant non-abelian monopole offers a new style of order parameter for monopole condensation.

Cohomology of Elementary States in Twistor Conformal Field Theory

In Twistor Conformal Field Theory the Riemann surfaces and holomorphic functions of two-dimensional conformal field theory are replaced by “flat” twistor spaces (arising from conformally-flat four-manifolds) and elements of the holomorphic first cohomology. The analogue of a Laurent Series is the expansion of a cohomology element in “elementary states” and we calculate the dimension of the space of these states for twistor spaces of compact hyperbolic manifolds. Our method follows the strategy used in the classical problem of calculating the number of meromorphic functions with prescribed poles on Rieiemann surface. We express the problem globally (in terms of the cohomology of a blown-up twistor space), calculate the holomorphic Euler characteristic of this blown-up space, and then use some vanishing theorems to isolate the first cohomology term.