Dynamics of Abelian Higgs vortices in the near Bogomolny regime
type: Item
instance of: scholarly article
title: Dynamics of Abelian Higgs vortices in the near Bogomolny regime
zbMATH Open document ID: 0807.35141
DOI: 10.1007/BF02100485
author: V. Pereyra
published in: Communications in Mathematical Physics
publication date: 1995-03-01T00:00:00Z
review text: The paper treats a known model of the classical field theory: two- dimensional Maxwell's equations coupled to an equation for a complex scalar Higgs field with the cubic nonlinearity, which lends an effective mass to a ``photon'' in this model via the mechanism called spontaneous symmetry breaking. The model's Hamiltonian is \[ H = {1 \over 2} \int_{\mathbb{R}^ 2} [E_ 1^ 2 + E_ 2^ 2 + B^ 2 + | D_ A \Phi |^ 2 + {1 \over 4} \lambda (1- | \Phi |^ 2)^ 2], \tag{1} \] where the electric and magnetic fields \(E_{1,2}\) and \(B\) are expressed through the components \(A_{1,2}\) and \(A_ 0\) of the electromagnetic vector potential \(E_{1,2} = \partial_ t A_{1,2} - \partial_{1,2} A_ 0\), \(B = \partial_ 1 A_ 2 - \partial_ 2A_ 1\), \(\Phi\) is the Higgs field, \(D_ A = \nabla - iA\) is the gauge- invariant gradient operator, and \(\lambda\) is a real coupling constant. It is known that the model (1) admits a vortex solution with the local amplitude of the Higgs field vanishing at the central point of the vortex and taking an asymptotically constant value, 1, at infinity. It was also known that, at \(\lambda = 1\) (the Bogomolny point), the equations of motion ensuing from the Hamiltonian (1) had a class of remarkable exact static solutions in the form of a system of vortices with their centers constituting an arbitrary set of points on the plane. In this work, the case of small but finite values of \(\lambda-1\) is considered by means of the perturbation theory. It is demonstrated that, in this case, the model's dynamics in an \(N\)-vortex sector may be reduced, on a basis of a well-controlled approximation, to a finite-dimensional dynamical system in the so-called moduli space, i.e., the space of parameters of the \(N\)- vortex configuration modulo the gauge transformations. In the course of the derivation, estimates of the approximation errors are based on using the energy conservation. The main result obtained is that, while the interaction forces between the vortices are a…
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 35Q51
zbMATH DE Number: 509891
zbMATH Keywords: moduli space
zbMATH Keywords: spontaneous symmetry breaking
zbMATH Keywords: scattering
reviewed by: B. A. Malomed
MaRDI profile type: Publication
cites work: Q3994593
cites work: Some remarks on the paper of Callias
cites work: Stability theory for solitary-wave solutions of scalar field equations
cites work: Q4119838
cites work: A remark on the scattering of BPS monopoles.
cites work: Vortex scattering
cites work: Perturbation theory for kinks
cites work: The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on \(R^ 3.\) I
cites work: Gelfand Theory of Pseudo Differential Operators and Hypoelliptic Operators
description: scientific article; zbMATH DE number 509891