Weyl Equation Research Articles

A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type

HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can in principle be constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures.

Exact general relativistic thick disks

A method to construct exact general relativistic thick disks that is a simple generalization of the ``displace, cut and reflect'' method commonly used in Newtonian, as well as, in Einstein theory of gravitation is presented. This generalization consists in the addition of a new step in the above mentioned method. The new method can be pictured as a ``displace, cut, {\it fill} and reflect'' method. In the Newtonian case, the method is illustrated in some detail with the Kuzmin-Toomre disk. We obtain a thick disk with acceptable physical properties. In the relativistic case two solutions of the Weyl equations, the Weyl gamma metric (also known as Zipoy-Voorhees metric) and the Chazy-Curzon metric are used to construct thick disks. Also the Schwarzschild metric in isotropic coordinates is employed to construct another family of thick disks. In all the considered cases we have non trivial ranges of the involved parameter that yield thick disks in which all the energy conditions are satisfied.

SOME REMARKS ON THE GRAVITATIONAL AHARONOV–BOHM EFFECT

A massless spinor particle is considered in the background spacetimes generated by a moving mass current and by a spinning cosmic string. In the weak field approximation it is shown that the solution of the Weyl equations depends on the velocity of the source, which does not affect the curvature in this approximation in the case of a moving mass current. In the case of a spinning cosmic string, the solution of the Weyl equations depends on the deficit angle and on the angular momentum of the string. These effects may be viewed as examples of the gravitational analogues of the Aharonov–Bohm effect in electrodynamics.

Delocalized spinors

Solutions to the four-dimensional Euclidean Weyl equation in the background of a general JNR N-instanton are known to be normalisable and regular throughout four-space. We show that these solutions are asymptotically given by a linear combination of simple singular solutions to the free Weyl equation, which can be interpreted as localised spinors. The `spinorial' data parameterising the asymptotics of the delocalised solutions to the Weyl equation in the presence of the instanton almost determines the background instanton, yet not completely. However, it captures the geometry and symmetry of the underlying instanton configuration.

Transport in nanostructures and nanotubes

In two-dimensional honeycomb lattices, electronic states are described by Weyl's equation for a massless neutrino when each site is occupied by an electron on average. The system has a topological singularity at the origin k=0 of the wave vector, giving rise to nontrivial Berry's phase when k is rotated around the origin. The singularity causes discrete jumps in the conductivity at the energy corresponding to k=0 in honeycomb lattices and leads to the absence of backscattering and a perfect conductance in carbon nanotubes.

General Solutions of Relativistic Wave Equations

General solutions of relativistic wave equations are studied in terms of functions on the Lorentz group. A close relationship between hyperspherical functions and matrix elements of irreducible representations of the Lorentz group is established. A generalization of the Gel'fand-Yaglom theory for higher-spin equations is given. A two-dimensional complex sphere is associated with each point of Minkowski spacetime. The separation of variables in a general relativistically invariant system is obtained via the hyperspherical functions defined on the surface of the two-dimensional complex sphere. In virtue of this the wave functions are represented in the form of series in hyperspherical functions. Such a description allows one to consider all the physical fields on an equal footing. General solutions of the Dirac and Weyl equations, and also the Maxwell equations in the Majorana-Oppenheimer form, are given in terms of functions on the Lorentz group.

ON THE SUPERSYMMETRIC DIRAC EQUATION AND THE HETEROTIC SUPERSTRING THEORY

The supersymmetry transformations under which the four-dimensional massless Dirac equation for a two-component, spin-1/2 fermion field ψ (the Weyl equation) remains invariant were obtained by Volkov and Akulov, who used the result to construct the action S = a-1 ∫ |W| d4x in terms of the energy-momentum tensor [Formula: see text], where Wij = δij + aTij and a is a constant. Here, we show, in the approximation [Formula: see text], that the terms linear, quadratic and quartic in Tij are contained in the bosonic sector of the dimensionally reduced, heterotic superstring action, including higher-derivative gravitational terms up to order ℛ4. By comparison of coefficients, we derive the value B r ≈ 3.5 for the radius squared of the internal space in units of the Regge slope parameter α′, slightly greater than the Hagedorn radius squared [Formula: see text]. The cubic terms are also discussed.

De Donder-Weyl equations and multisymplectic geometry

Multisymplectic geometry is an adequate formalism to geometrically describe first order classical field theories. The De Donder—Weyl equations are treated in the framework of multisymplectic geometry, solutions are identified as integral manifolds of Hamiltonian multi-vector fields. In contrast to mechanics, solutions cannot be described by points in the multisymplectic phase space. Foliations of the configuration space by solutions and a multisymplectic version of Hamilton—Jacobi theory are also discussed.

INTEGRABILITY OF EINSTEIN-WEYL EQUATIONS FOR SPATIALLY HOMOGENEOUS MODELS OF TYPE III BY BIANCHI

This paper deals with nonisotropic spatially homogeneous models for a self-consistent system of Einstein–Weyl equations with a spinor field. It is shown that for spaces of type III by Bianchi the system of Einstein–Weyl equations is integrable.

A family of integrable nonlinear equations of hyperbolic type

A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional reduction of the anti-self-dual Yang–Mills equations, is presented. It represents a generalization of the Ernst–Weyl equation of General Relativity related to colliding neutrino and gravitational waves, as well as of the fourth order equation of Schwarzian type related to the KdV hierarchies, which was introduced by Nijhoff, Hone, and Joshi recently. An auto-Bäcklund transformation of the new system is constructed, leading to a superposition principle remarkably similar to the one connecting four solutions of the KdV equation. At the level of the Ernst–Weyl equation, this Bäcklund transformation and the associated superposition principle yield directly a generalization of the single and double Harrison transformations of the Ernst equation, respectively. The very method of construction also allows for revealing, in an essentially algorithmic fashion, other integrability features of the main subsystems, such as their reduction to the Painlevé transcendents.

ADHM construction of instantons on the torus

We apply the ADHM instanton construction to SU(2) gauge theory on T^n x R^(4-n)for n=1,2,3,4. To do this we regard instantons on T^n x R^(4-n) as periodic (modulo gauge transformations) instantons on R^4. Since the R^4 topological charge of such instantons is infinite the ADHM algebra takes place on an infinite dimensional linear space. The ADHM matrix M is related to a Weyl operator (with a self-dual background) on the dual torus tilde T^n. We construct the Weyl operator corresponding to the one-instantons on T^n x R^(4-n). In order to derive the self-dual potential on T^n x R^(4-n) it is necessary to solve a specific Weyl equation. This is a variant of the Nahm transformation. In the case n=2 (i.e. T^2 x R^2) we essentially have an Aharonov Bohm problem on tilde T^2. In the one-instanton sector we find that the scale parameter, lambda, is bounded above, (lambda)^2 tv<4 pi, tv being the volume of the dual torus tilde T^2.

Exact Solutions to the Dirac Equation for Neutrinos Propagating in a Particular Vaidya Background

In this paper I show that the solutions of the Weyl equation reported by Christodoulakis et al. (Journal of Mathematical Physics (1994), 35, 2430) are not single-valued and therefore are unphysical.

Einstein–Weyl structures from hyper–Kähler metrics with conformal Killing vectors

We consider four (real or complex) dimensional hyper-Kähler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein–Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein–Weyl equations reduce down to a single second order PDE for one function. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with this PDE are found, and some group invariant solutions are considered.

Einstein–Weyl geometry, the dKP equation and twistor theory

It is shown that Einstein–Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev–Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector then it is locally given by h= dy 2−4 dx dt−4u dt 2 , ν=−4u x dt , where u= u( x, y, t) satisfies the dKP equation ( u t − uu x ) x = u yy . Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-Kähler metrics in signature (++−−) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP 1 -bundle over a Weyl space. It is proved that complex solutions to the dKP equations, modulo a certain coordinate freedom, are in a one-to-one correspondence with mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2) ) that admit a section of κ −1/4, where κ is the canonical bundle of Z . Real solutions are obtained if the mini-twistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ −1/4 that are invariant under the involution.

On Laplace–Darboux-type sequences of generalized Weingarten surfaces

Two novel classes of integrable surfaces (“generalized Weingarten surfaces of Class 1 and 2”) which may be defined in a coordinate-independent manner or, equivalently, via an extension of the classical Lelieuvre formulas are introduced. The classes 1 and 2 generalize Bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the Ernst–Weyl equation descriptive of gravitational and neutrino fields in general relativity. The latter connection is exploited to derive Laplace–Darboux-type transformations for Class 2 via the associated Loewner–Konopelchenko–Rogers representation. The Laplace–Darboux-type transformations are shown to have remarkable geometric properties in terms of sphere congruences. Connections with the Pohlmeyer–Lund–Regge vortex model and an inhomogeneous Heisenberg spin equation are recorded. A Bäcklund transformation for a particular Painlevé III equation is derived in a purely geometric manner and a geometric interpretation of an associated discrete Painlevé III equation is given.

Why odd-space and odd-time dimensions in even-dimensional spaces?

We are answering the question why 4-dimensional space has the metric 1+3 by making a general argument from a certain type of equations of motion linear in momentum for any spin (except spin zero) in any even dimension d. All known free equations of motion for non-zero spin for massless fields belong to this type of equations. Requiring Hermiticity 1 With respect to requiring Hermiticity, this is a generalization of an earlier work which shows that without assuming the Lorentz invariance – which in the present work is assumed – the Weyl equation follows using Hermiticity. 1 of the equations of motion operator as well as irreducibility with respect to the Lorentz group representation, we prove that only metrics with the signature corresponding to q time + ( d− q) space dimensions with q being odd exist. Correspondingly, in four dimensional space, Nature could only make the realization of 1+3-dimensional space.

The Painlevé III, V and VI transcendents as solutions of the Einstein–Weyl equations

We demonstrate that the integrable Ernst–Weyl equation governing neutrino and gravitational fields in axially symmetric space–times of general relativity admits symmetry reductions to the Painlevé III, V and VI equations with arbitrary constants. In particular, the matrix form of PVI is shown to be a canonical symmetry reduction of the integrable Loewner–Konopelchenko–Rogers (LKR) system.

Local Constraints on Einstein-Weyl Geometries: The 3-Dimensional Case

Following an earlier study [3], we consider the Einstein–Weyl equations on a fixed (complex) background metric as an equation for a 1-form and its first few derivatives. If the background is flat then we conclude that the only solutions are conformal rescalings of constant curvature metrics. If the background is a homogeneous 3-geometry in Bianchi class A (i.e., with unimodular isometry group), we find necessary and sufficient conditions on the 3-geometry for solutions of the Einstein–Weyl equations to exist. The solutions we find are complexifications of known ones. In particular, we find that the general left-invariant metric on S3 and the metric 'Sol' admit no local solutions of the Einstein–Weyl equations.

Exact solutions of the Dirac equation in a nonfactorizable metric

We present the covariant generalization of the Dirac equation in a nonfactorizable metric and give the corresponding exact solutions in terms of special functions as well as the explicit form of the spinor solution. Then we treat the particular case of the Weyl equation for the neutrinos.

Stimulated scattering of massless fermionic neutrinos by plasmas

The Weyl equation has been used to study the interaction of a neutrino flux with a plasma in a stellar medium. The resulting collective stimulated scattering of the neutrino by the plasma is described by a cubic dispersion relation, which is solved numerically and analytically which gives a maximum growth rate structurally identical to growth rate obtained by Bingham et al. [Phys. Lett. A 193 (1994) 279] by using a Klein–Gordon like equation model, in which the neutrinos are considered as bosons, to describe the linear regime of the collective interaction.