Weyl Field Research Articles

The Evolution of the Weyl and Maxwell Fields in Curved Space—Times

AbstractThe covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space‐time (M, g) are considered. It is shown that the condition g00(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δtφ = L(s)φ. Here L(s) is a linear first order differential operator on the pre—Hilbert space (C (Ut, 2s+1). (…)), where Ut ⊂ IR3 is the image of the coordinate map of the spacelike hyper‐surface t = const, and (φ, C) = ƒUt ϕ *Q d(3)x with a suitable Hermitian n × n‐ matrix Q = Q(t,x). The total energy of the spinor field ϕ with respect to Ut is then simply given by E = 〈ϕ,ϕ〉. In this way inequalities for the energy change rate with respect to time, δt|ϕ|2 = 2Re (ϕ, L(s)ϕ) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.

Trace anomalies and cocycles of the Weyl group

The response of the partition function of six-dimensional classically Weyl-invariant theories on a finite Weyl transformation is calculated (this is a cocycle of the Weyl group, connected with the trace anomaly), and it is shown that this cocycle can be chosen quadratic over the Weyl field, in exact analogy with the two-dimensional Liouville action: S = ∫ d 6x g (σΔ 6σ+σ × Anomaly ) where Δ 6 is the zero weight conformal-invariant operator of order six, derived in the present letter. The structure of the trace anomaly as a direct consequence of the Wess-Zumino consistensy condition is discussed.

The wave equation for the Lanczos potential. I

The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor C abcd of Weyl. For this field Lanczos found a tensor potential L abc with 16 independent components. We can make L abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions L ab s :s = 0. Both fields C abcd , L abc satisfy wave equations. The wave equation satisfied by C abcd is nonlinear, even in vacuo . However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for L abc has yet been published. Here, we derive a correct tensor wave equation for L abc and when it is simplified with the aid of some four­-dimensional identities it is equivalent to Illge’s wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge’s spinor wave equation. A set of analogues of well-known results of classical electromagnetic radiation theory can now be given. We indicate how a Green’s function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.

First order formalism off(R) gravity

The first order formalism is applied to study the field equations of a general Lagrangian density for gravity of the form $$\mathcal{L}_G = \sqrt { - g} f(R)$$ . These field equations correspond to theories which are a subclass of conformally metric theories in which the derivative of the metric is proportional to the metric by a Weyl vector field. The resulting geometrical structure is unique, except whenf(R)=aR 2, in the sense that the Weyl field is identifiable in terms of the trace of the energy-momentum tensor and its derivatives. In the casef(R)=aR 2 the metric is only defined up to a conformai factor. We discuss the matter conservation equations which are implied by the invariance of the theories under diffeomorphisms. We apply the results to the case of dust and obtain that in general the dust particles will not follow geodesic Unes. We consider the linearized field equations and apply them to obtain the weak field slow motion limit. It is found that the gravitational potential acquires a new term which depends linearly on the mass density. The importance of these new equations is briefly discussed.

Generalizations of the Kerr and Kerr-Newman metrics possessing an arbitrary set of mass-multipole moments

The authors present in a concise analytical form two asymptotically flat metrics describing the superposition of the Kerr solution with an arbitrary static vacuum Weyl field which differ in their angular momentum distributions. They are then used for the construction of two asymptotically flat generalizations of the Kerr-Newman spacetime possessing the full set of mass-multipole moments able to describe the exterior gravitational field of a charged rotating arbitrary axisymmetric mass.

Auxiliary field in conformal gauge theory.

When the full conformal algebra is gauged there arises a gauge field for inverse translations in addition to the usual translational gauge field. By considering every scale-invariant action constructible from the curvatures of the conformal group and the metric, we show that when the gauge field of the usual translations is identified as the vierbein, the gauge field of inverse translations may always be eliminated by its own field equation. After this elimination, every torsion-free, scale-invariant action reduces to a linear combination of the square of the conformal curvature tensor ${C}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\mu}\ensuremath{\nu}}$ and the square of the Weyl field strength ${C}_{\ensuremath{\alpha}\ensuremath{\beta}}$.

Theory of matter in Weyl spacetime.

Weyl's geometry of spacetime is reconsidered based on a novel geometric coupling between the Weyl vector and fermions. Starting from a manifestly Weyl-invariant action for the metric, Weyl field, spinors, and other fields, we introduce and define this coupling: the square root of the scalar Weyl curvature. An important consideration in this development is the reduction of the Weylian to an effective Riemannian structure for spacetime, without trivializing the role of the Weyl field. The aim of this reduction is to make contact with Einstein gravity for the metric sector of the theory. This is achieved by appealing to a dynamical-symmetry-breaking mechanism. The resulting spacetime is Riemannian, the Weyl field survives as a massive vector, and the geometric coupling to fermions decomposes into an admixture of vector and pseudovector couplings. Internal consistency of the field equations further narrows down the class of fermions which can couple to the Weyl vector: these can only be spinors of a fixed chirality. We tentatively identify them with the standard-model neutrinos. Analysis of couplings to other types of fields and particles indicates that the Weyl field is a form of dark matter.

On the gravitational interaction of plane symmetric clouds of null dust

Plane symmetric solutions of the Einstein field equations are considered, solutions that represent the collision of oppositely moving clouds of initially unidentified massless particles—clouds of ‘‘null dust.’’ In terms of specific examples it is shown that the corresponding Cauchy problem remains ambiguous even when the Riemann tensor is free of any kind of discontinuity. By solving the corresponding Einstein–Klein–Gordon and Einstein–Maxwell–Weyl field equations the ambiguity is resolved and space-time models are constructed representing, among others, various types of collisions between a pair of scalar wave pulses as well as of a pulse of electromagnetic waves with a cloud of neutrinos.

Dynamical generation of Majorana masses.

We address the general question of the dynamical generation of Majorana masses through quartic interactions of the Nambu--Jona-Lasinio (NJL) type that have both chiral and lepton-number invariances. We make composite the Higgs field of the schemes of spontaneous breaking of the leptonic number; we can thus assign to it a leptonic number \ensuremath{\Vert}L\ensuremath{\Vert}=2 in a natural way. We consider a Weyl field and write a quartic self-interaction for this field that dynamically breaks chiral and fermion-number invariances and exhibits a whole spectrum of composite particles with different quantum numbers, in addition to a Goldstone Majoron. We compare in detail the Dirac and the Majorana cases. The vacuum degeneracy is the same in both cases, but the vacuum invariances are not. For a single fermion species, we have for the Dirac case a U(1${)}_{\mathit{V}\mathrm{\ensuremath{-}}\mathit{A}}$\ifmmode\times\else\texttimes\fi{}U(1${)}_{\mathit{V}+\mathit{A}}$ invariance that breaks down to U(1${)}_{\mathit{V}}$ and for the Majorana case a single U(1) invariance that breaks down to the identity open1. In general the Schwinger-Dyson equation is not the same for both cases, since for Majorana fermions we have propagators of several types. However, in the particular case of a NJL contact interaction (for Majorana fermions this is the only nonvanishing contact quartic/B interaction), and with a convenient convention for the coupling, the Schwinger-Dyson equation turns out to have the same form for Dirac and for Majorana fermions. The bound-state boson spectrum is quite different in both cases: for the Dirac case, one has a spectrum $^{2\mathit{S}+1}$${\mathit{L}}_{\mathit{J}}$(S=0,1) (${\mathit{J}}^{\mathit{P}\mathit{C}}$=${0}^{\mathrm{\ensuremath{-}}+}$,${1}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{-}}}$,${0}^{++}$,${1}^{++}$,${1}^{+\mathrm{\ensuremath{-}}}$,${2}^{++}$,. . . ) with all L, S allowed, while for the Majorana case only the quantum numbers $^{1}$(L=even${)}_{\mathit{J}}$, $^{3}$(L=odd${)}_{\mathit{J}}$ are allowed (${\mathit{J}}^{\mathit{P}\mathit{C}}$=${0}^{\mathrm{\ensuremath{-}}+}$,${0}^{++}$,${1}^{++}$,${2}^{++}$,. . . ). The position of these remaining levels is not the same in both cases, even for a NJL contact interaction.

Quantum measurement and geometry.

A model for the interpretation of spacetime as a Weyl geometry is proposed, based on the hypothesis that a system moves on any given path with a probability which is inversely proportional to the resulting change in length of the system. The results of physical measurements are calculated as the product of Weyl-conjugate gauge-dependent probabilities for the detection of conjugate objects. Each probability, expressed as a Wiener integral, is the Green's function for a diffusion equation. If the line integral of the Weyl field equals the action functional divided by $\ensuremath{\hbar}$ this equation provides the stochastic equivalent of the Schr\"odinger equation, with expanding and contracting states replacing the oscillating states of standard quantum theory. These dilatations are not directly measurable within the theory, but the rates of expansion and contraction are. We establish the classical limit of the theory by proving that the most general form of the Weyl field which gives an extremum of the action is restricted to be of a special form. Particles obeying the equations of motion for these extremal Weyl fields experience no measurable dilatation. However, the Weyl field itself is classically detectable, and gives a correct description of the electromagnetic interaction of a point particle.

Quantum integrability and Kac-Moody algebras

We introduce a quantum “linear” problem, associated to the integrability of certain current algebras. In the conformal invariant limit we interpret the quantum fields of the quantum “linear” problem as Weyl fields of a field theory generated by a Kac-Moody algebra.

The Dirac equation in the de Broglie-Bohm theory

If spacetime is assumed to have a scale covariant Finsler structure then part of the action due to the intrinsic Weyl field can be reinterpreted as the Dirac lagrangian. The rest contributes mass to the Dirac particle and its fluctuations, the quantum potential.

Weyl fields, quantum integrability and conformal invariant field theories

We show that some Weyl field theories arise as a quantum “linear” problem associated to some Kac-Moody algebras. We relate this quantum “linear” problem to the conformal invariant field theories studied by Dashen and Frishman and to the WZW field theory.

Weyl field strength symmetries for arbitrary helicity and gauge invariant Fierz-Pauli and Rarita-Schwinger wave equations

The algebra of the restricted Lorentz group and the Weyl formulation of massless Poincare irreducible fields have been developed by primarily Lorentz-covariant Pauli matrix methods which facilitate the generation of both indexed Weyl spinor identities and Dirac identities. The results have been used to display a complex set of symmetries for the (anti)self-dual Weyl field strengths of arbitrary helicity j(>or=1). These symmetries and the requirement of Poincare irreducibility have then been used to give a direct and uniform determination of the forms of the gauge invariant (Lagrangian) free field wave equations for the Fierz-Pauli and Rarita-Schwinger potentials of spin 1, 3/2 and 2. The procedures set out indicate that it should be possible to establish the gauge invariant Lagrangian free field wave equations of arbitrary helicity in a uniform and direct manner from the corresponding much simpler equations governing the field strengths in unmixed spin representations.

Anomalies and Weyl Modes in Chiral Gauge Theories

Using canonical operator formalism, the chiral U(1) anomalies are investigated in model field theories with 1+1 dimensions. In some special cases, these theories have those particle spectra which behave effectively as two dimensional Weyl fields. This must be taken into account in defining charge operators with normal ordering. It is shown how such structure of the theories produces chiral U(1) anomalies.

A comment on the Nielsen-Ninomiya theorem

An example of a local lattice hamiltonian in three space dimensions whose continuum limit spectrum consists of a single free Weyl particle and a set of non-propagating modes is presented. Our model satisfies all the hypotheses of the Nielsen-Ninomiya theorem. However, when locally coupled to a background abelian gauge field, it fails to exhibit the spectrum and anomalies expected for a charged Weyl field.

Some simple type D solutions to the Einstein equations with sources

Starting from a particular metric of the Kerr–Schild form, we find solutions to the Einstein equations coupled to a Weyl field, to an electromagnetic field, to a nonlinear (Born–Infeld) electromagnetic field, to a Yang–Mills field, and to a cosmological constant. These solutions can be superposed to construct others with any combination of the sources considered.

Introduction to Majorana masses

We present a pedogogical review of Majorana masses and Majorana's theory of two-component massive fermions. We discuss the difference between Majorana and Dirac masses and show that Majorana masses are formion-number violating. We discuss the connection between Majorana and, Weyl spinors and show that the massive Majorana and Weyl field theories are equivalent. We study the second quantization of the massive Weyl theory in detail.

The coupled einstein-weyl field equations with cosmological constant and role of two higgs phenomena in weyl’s gauge model coupled to a higgs field

We study, un.der the full action of gauge invariancc including two Itiggs phenomena, Weyl’s gauge model coupled to a Higgs field. As a result, we obtain the coupled Einsteia- Weyl field equations with the cosmological constant, which are analogous to the Einstein-Maxwell equations apart from difficulties inherent to Weyl’s geometry. The vacuum solution of a Higgs field will be discussed in connection with a manifestly gaugc-invariant formulation of the Lagrangian density of gravitation. A generalized action which is a function of the gauge-invariant scalar curvature is examined.

Colour and hypercharge—Are they structural symmetries?

By starting from a left-handed non-Hermitian isodoublet Weyl field χ without hypercharge and colour degrees of freedom, the possibility is investigated if the effective leptons and quarks, arising in the composite χχχ and χχ*χ* level, allow the introduction of additional «structural» symmetries. Those structural degrees of freedom are introduced as symmetric consequences (not Higgs-like) of the spontaneous breakdown of the basicU1×SU2,loc group and are carried by dressing operators associated with the broken degrees of freedom.