Conformally covariant field equations II. First-order massless equations

The necessary and sufficient conditions for the finite as well as infinite component field equation

$\text{L}{}_{\mathrm{\mu }}\text{?}{}^{\mathrm{\mu }}\text{ψ(x) = 0}$

to be conformally covariant are derived and their consequences discussed. The derived results are specified for massless field equations treated in literature.     

Conformally covariant nonlinear equations on tensor-spinors

We prove that to each conformally covariant equation on tensor-spinors on a Riemannian or pseudo-Riemannian manifold with spin structure one can add a nonlinear term without losing the property of conformal covariance. It follows in particular that, on a manifold of dimension n, the nonlinear Dirac equation, Pψ + λ|ψ|^1/(n?1)ψ = 0, where P is the Dirac operator and λ is a constant, is conformally covariant. This generalizes a result of Gürsey [1]. Some results of Ørsted [2], concerning a nonlinear equation associated with the Laplacian on function, and of Branson, concerning distinguished nonlinearities associated with his modified Laplacian on differential forms [3] are also derived as particular cases of this general result.     

On conformally covariant spinor field equations

A spinor field equation, covariant with respect to the general conformal group (including reflections), should consist in general of not less than eight linear equations and then, in Minkowski space, could be represented by not less than two massless Dirac equations. Their reduction through projectors to only one equation, while not spoiling conformal covariance implies unphysical consequences. It is shown instead that two Dirac equations may be brought unambiguously through a stereographic projection to a manifestly conformal covariant form inE _4,2 space. The physical implications are discussed and it is shown that if the fundamental elementary interactions are expressed in terms of conformal semispinors (which can never appear as free particles), then the corresponding physical Dirac spinors appear in the elementary interactions in terms of their chiral projections. This could indicate both the conformally invariant origin of weak interactions and their fundamental character. The possibility of constructing unified models from conformally invariant Lagrangians is envisaged.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.A preliminary version was issued as Internal Report IC/78/43, ICTP Trieste May 1978, see also Lett. Nuovo Cim.21 (1978), 473.I am indebted to Prof. I. T.Todorov for interesting discussions.     

Conformally invariant equations of the quantum field theory

The equations invariant under the transformation of the conformal algebra are obtained using the Casimir operators. The connections among the components of the field are explicitly derived in the case of indecomposable representations of the conformal algebra which give rise to e.g. the Maxwell equations with currents. The free field equations are also incorporated in the conformally covariant scheme.     

Meronlike solutions of conformally covariant coupled nonlinear field equations

We derive a new class of solutions of conformally covariant coupled spinor and scalar equations including a nonlinear spinor self-coupling term, for which the energy-momentum density is nonzero, but the total energy is zero (“meronlike” solutions).     

Reggeon field with non-vanishing vacuum expectation value

It is argued that if the vacuum expectation value of the reggeon field does not vanish, the solution of the reggeon calculus corresponds to the leading singularity of intercept one. The physical interpretation of the reggeon field theory with unstable vacuum is discussed.     

Complete sets of solutions of linear Lorentz covariant field equations with an infinite number of field components

We study field equations of the Gelfand-Yaglom type where transforms as a unitary representation of the inhomogeneous Lorentz group. We construct a complete set of solutions of this equation. This set includes solutions with spacelike momentum. Our method makes use of the decomposition of unitary representations of the homogeneous Lorentz group into unitary representations of the little groupsS U (2) andS U (1, 1). The covariant operators ^µ are written as differential operators on homogeneous spaces. For some classes of equations we calculate the mass spectrum explicitly.     

On the Harish-Chandra condition for first-order relativistically-invariant free field equations

It has hitherto been accepted that the degree of the Harish-Chandra condition applying to single-mass equations of arbitrary spin is determined by the maximum spin appearing in the representation ofSL(2,C) which acts on the field. The present paper demonstrates a fallacy in the published arguments which lead to the above conclusion, and gives the correct conclusion which can be deduced from the hypotheses. A counter-example of an irreducible, single-mass, spin 3/2 equation which does not satisfy the accepted theory is provided in an appendix.Part of this research was carried out at the University of British Columbia with the support of the National Research Council of Canada.     

Conformally symmetric vacuum solutions of the gravitational field equations in the brane-world models

A class of exact solutions of the gravitational field equations in the vacuum on the brane are obtained by assuming the existence of a conformal Killing vector field, with non-static and non-central symmetry. In this case, the general solution of the field equations can be obtained in a parametric form in terms of the Bessel functions. The behavior of the basic physical parameters describing the non-local effects generated by the gravitational field of the bulk (dark radiation and dark pressure) is also considered in detail, and the equation of state satisfied at infinity by these quantities is derived. As a physical application of the obtained solutions we consider the behavior of the angular velocity of a test particle moving in a stable circular orbit. The tangential velocity of the particle is a monotonically increasing function of the radial distance and, in the limit of large values of the radial coordinate, tends to a constant value, which is independent on the parameters describing the model. Therefore, a brane geometry admitting a one-parameter group of conformal motions may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter.     

Derivation and uniqueness of vacuum solutions of conformally covariant coupled nonlinear field equations

We derive the solutions of conformally covariant coupled Dirac and scalar fields including a nonlinear fermion self-coupling term for which the conformally covariant (not the canonical, nor the symmetric) energy-momentum tensor θ_μν vanishes. This “vacuum” state is degenerate.     

Geometrization of linear perturbation theory for diffeomorphism-invariant covariant field equations. II. Basic gauge-invariant variables with applications to de sitter space-time

In a companion paper, a systematic treatment of linearized perturbations and a new geometric definition of gauge-invariant variables, based on the theory of vector bundles and applicable to the case of an arbitrary system of covariant field equations, were carefully presented. One of the purposes of the present paper is to specify a necessary and sufficient condition that a given, finite set of gaugeinvariant variables, denoted collectively by ω and referred to as the complete set of basic variables, can be used to extract the equivalence classes of perturbations from ω in a unique way. The above set is complete because it has the following property: a knowledge of ω is all one needs in the sense that ifx represents an arbitrary point of the “space-time” manifoldX andG denotes any gauge-invariant tensor field onX, then the value ofG atx∈X is uniquely specified by giving the germs of basic gauge-invariant variables atx∈X. Arguments are proposed that ω also has a stronger property which is more immediately useful: anyG is obtainable directly from the basic variables through purely algebraic and differential operations. These results are of practical interest, and one concrete setting where one is led to the explicit definition of ω occurs when considering the infinitesimal perturbation of the metric tensor itself (pure gravity) defined on a fixed background de Sitter space-time and obeying the linearized empty-space Einstein equations with nonnegative cosmological constant Λ; the case Λ=0 corresponds to linear perturbation theory in Minkowski space-time.     

Conformally flat space-times of locally constant connection. I

Conformally flat space-times of locally constant connection are studied. The constant connection defines a global vector field which is assumed timelike. The general solution of the geodesic equations is presented and several theorems characterizing the geometry of such space-times are proved.     

Causal first-order equations for wave field modeling in a horizontally inhomogeneous ocean

A mathematical model is developed for acoustic fields in a horizontally inhomogeneous oceanic medium with allowance for prominent inhomogeneities. On the basis of the system of equations of linear acoustics and the ideas of the method of transverse sections, causal equations of the first order that exactly describe the horizontal components of acoustic modes are derived. These equations offer a possibility to model the situations with nonsmooth inhomogeneities, when the known approximate methods are inappropriate. The equations are characterized by a number of important advantages, which include relative simplicity, physical clarity, and suitability for numerical simulations with the use of the existing algorithms designed for solving wave problems in layered media.     

Gauge covariant theory of the generating operator. I

A gauge covariant formulation of the generating operator (-operator) theory for the Zakharov-Shabat system is proposed. The operator , corresponding to the gauge equivalent system in the pole gauge is explicitly calculated. Thus the unified approach to the nonlinear Schrödinger-type equations based on is automatically reformulated with the help of for the Heisenberg ferromagnet-type equations. Consequently, it is established that the conserved densities for the Heisenberg-ferromagnet-type equations are polynomial inS(x) and itsx-derivatives. Special attention is paid to the interrelation between the hierarchies of symplectic structures corresponding to the above mentioned families of gauge-equivalent equations. It is shown that the geometrical properties of the conjugated operator * are gaugeindependent.     

Conformally Ricci flat Einstein-Maxwell solutions with a null electromagnetic field

A proof is given that a conformally Ricci-flat Einstein-Maxwell field is null if and only if the conformai scalar field has a null gradient. The solutions belong then necessarily to the family ofpp waves.     

Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system,I

A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented. A number of sharp regularity and smoothness properties of the solutions are obtained. The present paper is devoted to the case ofR ⁿ with suitable asymptotic conditions imposed. As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz. These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques. Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature.Partially supported by AEC Contract AT(04-3)-34.Partially Supported by NSF Contract GP-8257.