Weyl Equation Research Articles

On a generalization of the Weyl equations

Equations for massless spin-one-half particle fields of form between the Dirac equation and the Weyl equations are shown to be covariant even though they contain non-helical in addition to the usual helical functions. The analogy to the homogenous Maxwell equations and their integrals is pointed out.

Huygens' Principle on Petrov Type D Space‐Times

AbstractGeneralizing the recently by J. Carminati and R. McLenaghan [2] obtained results it is shown that there exist no Petrov type D space‐times on which the conformally invariant scalar wave equation, or Maxwell's equations, or Weyl's equations satisfy Huygens' principle.

Exact scalar and spinor solutions in some rotating universes

In this paper exact solutions of the Klein–Gordon and Weyl equations in some rotating universes are presented.

Notizen: On Electromagnetic Spinors and Quantum Theory

Abstract The Maxwell theory is related to the Dirac theory by two heuristic arguments based on electromagnetic spinors. First, given the condition to have spinors, if the free field Maxwell's equations are satisfied, then the spinors satisfy the Weyl equations. Second, a sufficient condition for electromagnetic spinors directly gives the Weyl equations with the usual substitutions for momentum and energy. It is proposed that the heuristic nature of the first argument is related to the idea of a point having a more general symmetry than a sphere.

Grassmann-valued processes for the Weyl and the Dirac equations.

The time evolution of a massless particle satisfying the Weyl equation is described as a stochastic process on a space of Grassmann variables, in close formal analogy with the use of Brownian motion for Schr\"odinger evolution. The Grassmann process is then combined with a Poisson process previously used for the evolution of Dirac electrons. (In that process electrons propagate as massless left- or right-handed particles with random changes in direction occurring at an average rate given by the particle mass.) Electron motion is thus given as an expectation over the two processes and spatial position supersymmetrically acquires contributions from sums of products of Grassmann variables.

Group-theoretic derivation of equations for massive and massless spin-one and spin-one-half fields

With direct-product reduction of Lorentz-group representations first-order field equations are shown to be divisible into two, physically significant, classes, whose solutions of the corresponding equations are only in part related bym→0, even though the classes correspond to existing rest frame and lack thereof. For spin-one-half and mass-zero generalized Weyl equations come about, which include potentiallike functions. These nonhelical functions do not exist for Majorana neutrinos. For spin-one fields of mass zero the potential term assumes quite naturally the form of a necessary condition for the existence of Dirac-like monopoles.

On gauge fields with sources

It is shown that in an algebraically special space-time that admits a congruence of null strings, the Yang–Mills equations with sources reduce to a pair of nonlinear first-order differential equations for two matrices, provided that the gauge field is aligned with the congruence. In the case where the current is tangent to the null strings, the gauge field is determined by a matrix potential that has to satisfy a second-order differential equation with quadratic nonlinearities. As an example of this case, the Yang–Mills–Weyl equations are reduced, assuming that the multiplet of Weyl neutrino fields are also aligned with the congruence, and a reduced form of the Einstein–Yang–Mills–Weyl equations is also given.

Explanation of parity nonconservation

Space inversion and other discrete symmetries are treated within the frame of a theory of fundamental forces based only on general considerations of causality, symmetry, and stability, without ad hoc differential equations. The basic space-time M is the Einstein universe R(1) x S(3) as a causal (or conformal) rather than a pseudo-Riemannian manifold. Its connected symmetry group is then a 15-parameter group G locally equivalent to SO(2, 4), while the isometry group K of the Einstein universe is a 7-parameter subgroup. Correlation with conventional relativistic theory is based on a canonical imbedding of Minkowski space M(0) into M, together with the unique extendability of all transformations of the scaling-extended Poincaré group P from M(0) to global transformations on M. The fundamental fermion field F and boson field B are here restricted to be real and are fully invariant under G(e), where the superscript e denotes the inclusion of space and time inversions. The role of C on F is taken over by a real matrix having the eigenvalues +/-i, that commutes with G but anticommutes with space inversion. The spin space for B consists of the real linear transformations on that for F. There is a corresponding natural total Lagrangian that is both G(e) and O(2)-gauge invariant, the latter leading to lepton and baryon number conservation, and which is nonparametric except for scale. The Weyl and Maxwell equations are deduced, and compelling identifications made for neutrinos and the photon. The e and mu neutrino pairs occur in strikingly inequivalent positions in F, appearing symmetric only in the conventional relativistic limit R --> infinity, where R is the ( G-invariant) fundamental length interpretable as the radius of the space S(3). The photon occurs as the lowest member of a coherent subfamily of B that includes natural candidates for bare versions of the W and Z particles. In the relativistic limit the interaction Lagrangian becomes a sum over all elementary processes, one of which appears as quantum electrodynamics with Majorana-type electrons.

Maximal symmetry groups of quantum relativistic equations

The author has used Lie's extended group method to obtain the maximal symmetry groups of the Dirac equation for finite mass spin 1/2 particles and the Weyl equation for zero mass spin 1/2 particles. In both cases the maximal symmetry group is an infinite parameter Lie group having an invariant subgroup also of an infinite number of generators. The corresponding factor group for the Dirac equation is an eleven-parameter Lie group isomorphic to the Weyl group. In the case of the Weyl equation the corresponding factor group is a sixteen-parameter Lie group containing a proper subgroup isomorphic to the conformal group.

Wavelike solutions to the Einstein equations coupled to neutrino and gauge fields

Starting from the plane-wave metric, solutions to the Einstein field equations coupled to a Weyl neutrino field and to a Yang–Mills field are found. These solutions can be superposed to yield a solution with both sources if the direct interaction between them is neglected. A solution to the coupled Einstein–Yang–Mills–Weyl equations that represents a multiplet of neutrino fields interacting with the gauge field and the gravitational field is also found. All these solutions contain arbitrary functions.

Klein-Gordon and Weyl equations in the Gödel universe

Klein-Gordon and Weyl equations are investigated in the Gödel universe as the background metric. Both equations are solved, the Klein-Gordon equation for arbitrary coupling to the gravitational field.

Remarks on pure spinors

General remarks on the significance of spinors are followed by a brief description of spinor connections on low-dimensional spheres and their interpretation as gauge configurations. Cartan's notion of pure spinors is related to the general problem of classification of orbits of the spin group in projective spinor space. There is a nontrivial bundle of pure spinor directions over the conformal compactification of any space with a metric of suitable signature. In higher dimensions, pure spinors introduce natural nonlinearities and lead to topologically nontrivial configurations. It is shown how the constraint defining pure spinors may induce a ‘mass term’ in the Weyl equation for such spinors in a space of signature (3, 4).

Field theory onR�S 3 topology. II: The Weyl equation

A Weyl-type equation onR×S 3 topology is derived, as a generalization to previously obtained Klein-Gordon- and Schrodinger-type equations for the same topology. The general solution of the new equation is given as an expansion in the matrix elements of the irreducible representations of the groupSU 2. The properties of the solutions are discussed.

Static solutions of the coupled Yang–Mills–Weyl equations

It is shown that static, spherically symmetric solutions of the massless Yang–Mills–Dirac equation which carry finite energy are trivial.

Covariant Quantization of Spinor Fields in a Given Gravitational Field

AbstractIn coupling gravity with the quantum field theory, unitary transformations, depending on space‐time‐points, were considered and derivatives were introduced, which imply a nonintegrable parallel transport of the state vectors of Hilbert space [1]. The Dirac equation, built with these generalized derivatives, is quantized in a prescribed classical gravitational field. The quantization can be performed in complete analogy to the usual procedure in Minkowski space, but the quantum state vector becomes path dependent. In carrying out the quantization, two two‐component classical spinor fields necessarily occur, which obey Weyl's equation. The considered quantized Dirac equations are also picture‐covariant, that is they have the same from in each physical picture, especially in the Heisenberg picture and the Schrödinger picture.

Null strings and Bianchi identities

The results of the theory of complexified V4’s which admit a null string are reexamined by using systematically the Bianchi identities. The formal reasons which permit the integration of the Gμν=0 equations to one differential constraint in the case of the ℋℋ structures are exhibited. Some new results concerning complexified Einstein–Weyl equations in the presence of a null string are given as well.

Gruppentheoretische aspekte der signalübertragung und der kardinalen interpolationssplines I

AbstractAs is well known the real Heisenberg nilpotent group A(R) constitutes the group‐theoretic embodiment of the Heisenberg canonical commutation relations (CCR) of classical quantum mechanics. In this connection, quantum mechanics stands for the quantum‐mechanical description, at a given instant of time, of a non‐relativistic microparticle moving in the one‐dimensional configuration space R and having the plane R2 as its (flat) phase space. In fact, the (nilpotent) Lie algebra n of Ã(R) reflects the Weyl equations which are global versions of the Heisenberg CCR. Unfortunately, the subject of Heisenberg nilpotent groups is outside quantum mechanics and all the more outside mathematical physics not as commonly known as it should be considering its wide range of applications in a variety of different fields. The present paper which has two parts aims to develop a central topic of nilpotent harmonic analysis, to wit, the microparticle model, the lattice model which will be realized on the Heisenberg compact nilmanifold, and the complex wave model (or Bargmann‐Fock‐Segal model) of the linear Schrödinger representation of Ã(R) in order to examine geometrically several applications which are governed by the real Heisenberg nilpotent group Ã(R). These applications are in Part I the classical Whittaker‐Shannon sampling theorem which is of basic importance in signal processing, to wit, for the transmission of digital signals as well as analog signals, and in Part II the Subbotin‐Schoenberg existence and uniqueness theorem of cardinal spline interpolation. Moreover, Part II indicates briefly some connections of the aforementioned models to the Wigner phase‐space quasiprobability density function of quantum statistical mechanics via the Schwartz kernels theory on unimodular Lie groups, an approach to the cross‐ and autoambiguity functions of radar synthesis, and to the Zak transform of solid state physics. The second part also points out some relations of harmonic analysis of the finite nilpotent group A(Z/NZ) to periodic spline interpolants admitting N equidistant knots on the one‐dimensional compact torus group T. These last examples should serve mainly as hints for some further lines of investigations in the field of applications of nilpotent harmonic analysis.

On the nonexistence of static solutions of the Einstein–Weyl equations in general relativity

It is proved that the combined gravitational-neutrino equations in general relativity admit no nontrivial solutions in a static space-time provided that the energy density of the neutrino field is nonnull for all observers.

Exact solutions of the Bach field equations of general relativity

Conformally invariant gravitational field equations on the hand and fourth order field equations on the other were discussed in the early history of general relativity (Weyl Einstein, Bach et al.) and have recently gained some new interest (Deser, P. Günther, Treder, et al.). The equations B αβ =0 or B αβ = ϰT αβ , where B αβ denotes the Bach tensor and T αβ a suitable energy-momentum tensor, possess both the mentioned properties. We construct exact solutions ds 2= g αβ dx α dx β of the Bach equations: (2, 2)-decomposable, centrally symmetric and pp-wave solutions. The gravitational field g αβ is coupled by B αβ = ϰT αβ to an electromagnetic field F αβ =− F αβ obeying the Maxwell equations or to a neutrino field ϕ A obeying the Weyl equations respectively. Among interesting new metrics ds 2 there appear some physically well-known ones, such as the De Sitter universe, the Weyl-Trefftz metric. and the plane-fronted gravitational waves with parallel rays (pp-waves) known from Einstein's theory. The solutions are built up by means of special techniques: A separation method for (2, 2)-decomposable solutions, simplification of centrally symmetric metrics by a suitable conformal transformation, and complex function methods for pp-wave solutions.

Neutrino-generated static space-times

We consider the interaction between neutrinos and gravitation by studying the Einstein-Dirac (Weyl) equations. For simplicity we discuss only plane symmetric space-times. We derive exact, static solutions to the Einstein-Dirac equations. Although the models we study are not of direct astrophysical interest, they are simple and point out explicit properties of the theory. These are the first static, neutrinogenerated space-times, as far as we are aware.