The geodesic approximation for the Yang-Mills-Higgs equations

In this paper we consider the dynamics of the monopole solutions of Yang-Mills-Higgs theory on Minkowski space. The monopoles are solutions of the Yang-Mills-Higgs equations on three dimensional Euclidean space. It is of interest to understand how they evolve in time when considered as solutions of the Yang-Mills-Higgs equations on Minkowski space-i.e. the time dependent equations. It was suggested by Manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain, metric on the space of guage equivalence classes of monopoles-the moduli space. The metric is defined by taking theL ² inner product of tangent vectors to this space. In this paper we will prove that Manton's approximation is indeed valid in the right circumstances, which correspond to the slow motion of monopoles. The metric on the moduli space of monopoles was analysed in a book by Atiyah and Hitchin, so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in Yang-Mills-Higgs theory is obtained. The strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions, and then to use energy estimates to prove their validity. For the case of monopoles to be considered here there is a technical obstacle to be overcome-when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin. This is overcome by using a norm introduced by Taubes in a discussion of index, theory for the Yang-Mills-Higgs functional.Supported by grant DMS-9214067 from the National Science Foundation.     

Infinite-Energy dyon-like solutions for Yang-Mills-Higgs theory

Two dyon-like solutions to theSU(2) Yang-Mills-Higgs system are presented. These solutions are obtained from the BPS dyon solution by allowing the gauge fields to be complex, or by letting the free parameter of the new solution be imaginary. In both cases the measurable quantities connected with these new solutions are real. Although the new solutions are mathematically simple variations of the BPS solution, they have distinct characteristics.     

On abelian solutions to the Yang-Mills-Higgs equations

The topology of some reducible connections in the (first) Georgi-Glashow model is considered. It is shown that pairs of U(1) connections with non-null Hopf index are minima of the Euclidean action under severe symmetry restrictions. Their physical meaning and some generalizations are outlined.     

Gauge invariance and formal integrability of the Yang-Mills-Higgs equations

In the framework of the formal theory of overdetermined systems of partial differential equations, it is shown that the Yang-Mills-Higgs equations are an involutive, and hence formally integrable, system. To this end a key role is played by the gauge invariance of the theory and the resulting differential identities involving the field equations themselves. By applying a theorem of Malgrange, an existence theorem for the solutions of the Yang-Mills-Higgs field equations in the analytic context is thus obtained. The approach is within differential geometry.     

A finite-action solution to generalized Yang-Mills-Higgs theory

We study the seven-dimensional generalized Yang-Mills-Higgs theory which results by dimensional reduction from eight-dimensional generalized SO(8) Yang-Mills theory. Using an SO(7)-symmetric ansatz, we prove the existence of a weak topologically nontrival finite-action solution.     

Yang-Mills-Higgs versus Connes-Lott

By a suitable choice of variables we show that every Connes-Lott model is a Yang-Mills-Higgs model. The contrary is far from being true. Necessary conditions are given. Our analysis is pedestrian and illustrated by examples.     

The existence of non-minimal solutions of the Yang-Mills-Higgs equations overR 3 with arbitrary positive coupling constant

This paper proves the existence of a non-trivial critical point of theSU(2) Yang-Mills-Higgs functional onR ³ with arbitrary positive coupling constant. The critical point lies in the zero monopole class but has action bounded strictly away from zero.Supported in part by NSF Grant DMS-9200576.Supported in part by NSF Grant DMS-9109491.     

Transposition-invariant equations for the unified field theory

We discuss, within the framework provided by a recently developed variational method, transposition-invariant field equations for unified field theories. Systems that are, in addition, invariant under Weyl-type gauge transformations or lambda transformations are derived. It is found that in a weak field limit two of the systems contain the equations of general relativity and the covariant Maxwell equations for a charge-free region.     

Scaling theory for homogenization of the Maxwell equations

A scaling theory for homogenization of the Maxwell equations is developed upon the representation of any field as a sum of its dipole, quadrupole, and magnetic dipole moments. This representation is exact and is connected neither with multipole expansion nor with the Helmholtz theorem. A chain of hierarchical equations is derived to calculate the moments. It is shown that the resulting macroscopic fields are governed by the homogenized Maxwell equations. Generally, these fields differ from the mean values of microscopic fields.     

A Yang-Mills-Higgs monopole of charge 2

A new static, purely magnetic Yang-Mills-Higgs monopole solution is presented. It is axisymmetric and has a topological charge of 2; the charge is located at a single point.     

Quantum chaos in the Yang-Mills-Higgs system at finite temperature

Quantum chaos in the finite-temperature Yang-Mills-Higgs system is studied. The energy spectrum of a spatially homogeneous SU(2) Yang-Mills-Higgs system is calculated within thermofield dynamics. Level statistics of the spectra is studied by plotting nearest-level spacing distribution histograms. It is found that finite-temperature effects lead to a strengthening of chaotic effects, i.e. a spectrum which has the Poissonian distribution at zero temperature has the Gaussian distribution at finite temperature.Received: 18 August 2004, Revised: 23 March 2005, Published online: 22 June 2005PACS: 05.45.-a,11.10.Wx, 11.15.-q     

Multidimensional perturbation theory for nonseparable wave equations

We show that powerful procedures can be developed for deriving explicit asymptotic expansions for the solutions and eigenvalues of a large class of nonseparable wave equations. In particular, these expansions allow an investigation of the large-order behaviour.     

Yang-Mills-Higgs monopole solutions of arbitrary topological charge

We propose a construction of static magnetic Yang-Mills-Higgs monopole solutions of arbitrary topological charge. They are axially symmetric and contain no free parameters except for their position. The regularity of the solutions has yet be proved; doing so would complete the constructive proof of existence.     

Hamilton-Jacobi functional theory for the integration of classical field equations

By employing the terminology of functional differential calculus, Hamilton-Jacobi theory is extended to apply to classical field equations. It is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an asymptotic general solution to the associated nonlinear classical field equations.Work supported by a National Science Foundation grant.     

Renormalized perturbation theory flow equations for the Anderson impurity model

We apply the renormalized perturbation theory (RPT) to the symmetric Anderson impurity model. Within the RPT framework exact results for physical observables such as the spin and charge susceptibility can be obtained in terms of the renormalized values \(\tilde \mu = (\tilde \Delta ,\tilde U)\) of the hybridization Δ and Coulomb interaction U of the model. The main difficulty in the RPT approach usually lies in the calculation of the renormalized values themselves. In the present work we show how this can be accomplished by deriving differential flow equations describing the evolution of \(\tilde \mu = (\tilde \Delta )\) with Δ. By exploiting the fact that \(\tilde \mu = (\tilde \Delta )\) can be determined analytically in the limit Δ → ∞ we solve the flow equations numerically to obtain estimates for the renormalized parameters in the range 0 <U/πΔ< 3.5.     

Global existence of time-dependent Yang-Mills-Higgs monopoles

We study the Cauchy problem for non-abelian Yang-Mills-Higgs theory in (3+1)-dimensional Minkowski spacetime. With suitable conditions on the background fields and a suitable choice of a Sobolev space for the subtracted gauge potentials and the Higgs field, we establish local existence. We then prove global existence by showing that an appropriate norm of the solutions cannot blow up in a finite time.