Wave and Dirac operators,and representations of the conformal group

Let M be the flat Minkowski space. The solutions of the wave equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U_s, of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H_s, such that the representation U_s, will act unitarily on H_s. We prove that the mass 0 equations give intertwining operators between representations of principal series. We relate these representations to the Segal-Shale-Weil (or “ladder”) representation of U(2, 2).     

Dirac equation and spin 1 representations,a connection with symmetries of the Maxwell equations

A unitary operator on the space of spinors that makes it possible to associate each transformation in this space with a transformation in the space of electromagnetic field strengths is found. A connection is established by means of this operator between representations in the space of spinors and the space of field strengths for the Lorentsz, Poincaré, and conformal groups. Unusual symmetries of the Dirac equation are found on this basis. It is noted that the Pauli—Gürsey symmetry operators (without the ⁵ operator) of the Dirac equation withm=0 form the same representation D(1/2, 0)D(0, 1/2) of the O(1, 3) algebra of the Lorentz group as the spin matrices of the standard spinor representation. It is shown that besides the standard (spinor) representation of the Poincaré group, the massless Dirac equation is invariant with respect to two other representations of this group, namely, the vector and tensor representations specified by the generators of the representations D(1/2, 1/2) and D(1, 0) D(0, 0) of the Lorentz group, respectively. Unusual families of representations of the conformal algebra associated with these representations of the group O(1, 3) are investigated. Analogous O(1, 2) and P(1, 2) invariance algebras are established for the Dirac equation withm>0.Institute of Nuclear Research, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 388–406, March, 1992.     

Global solvability of massless Dirac–Maxwell systems

We consider the Cauchy problem for massless Dirac–Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac–Higgs–Yang–Mills theories.     

Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

We study two interesting new bundles over the universal cosmos M̃ (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III (S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983), 18–22)) of conventional bundles over M̃. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries.Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically.The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.     

Asymptotic behavior of the solutions of maxwell—klein—gordon field equations in 4—dimensional minkowski space

Abstract. This paper studies the asymptotic behavior of the solutions of massive coupled Maxwell—Klein—Gordon field equations in the 4—dimensional Minkowski space for the case of small initial data with charge. The proof relies on gauge invariant energy estimates and geometric properties of the fields equations. The presence of charge together with the mass term in the Klein—Gordon equation are the novelties of this project and provide us with a situation which cannot be accomodated by standard methods such as the conformal transformation. A covariant Lie derivative operator for the Klein—Gordon field is introduced and allows us to handle the most troublesome terms in the error estimates.     

Solution globale des equations de Maxwell-Dirac-Klein-Gordon

We prove the global existence on Minkowski space time of a solution of the Cauchy problem for the non linear system of coupled Maxwell, Dirac and Klein-Gordon equations, for small data with appropriate decay at space-like infinity. The method uses the conformal mapping of Minkowski space time onto a bounded open set of the Einstein cylinder.     

Analysis in space-time bundles,II. The spinor and form bundles

The structures of the spin and form bundles over the universal cosmos M?, and their relations with corresponding bundles over the Minkowski space M₀ canonically imbedded in M?, are treated. Wave equations covariant with respect to the causal group G of M? are studied, their solution manifolds and other stable (essentially positive-energy) invariant subspaces of the section spaces of the bundles are determined, and the indecomposability of relevant invariant subspace chains is shown. Explicit parallelizations of the bundles are applied to the Dirac and Maxwell equations on M?. A basis for spinor fields that diagonalizes a complete set of K?-covariant quantum numbers (K? = maximal essentially compact subgroup of G?) is developed. Local multilinear invariants of bundles over M? are treated and specialized to convergent mathematical versions of the Fermi and Yukawa interaction Lagrangians that are G?-invariant for the appropriate conformal weights.     

The BRST-BV approach to massless fields adapted for the AdS/CFT correspondence

In the framework of the BRST-BV approach to the formulation of relativistic mechanics, we consider massless and massive fields of arbitrary spin propagating in a flat space and massless fields propagating in the AdS space. For such fields, we obtain BRST-BV Lagrangians invariant under gauge transformations. The Lagrangians and gauge transformations are constructed in terms of traceless gauge fields and traceless parameters of the gauge transformations. We consider the fields in the AdS space using the Poincaré parameterization of this space, which leads to a simple form of the BRST-BV Lagrangian. We show that in the Siegel gauge, the Lagrangian of the massless AdS fields leads to a decoupling of the equations of motion, and this substantially simplifies the study of the AdS/CFT correspondence. In a conformal algebra basis, we find a realization of the relativistic symmetries of fields and antifields in the AdS space.     

Maxwell–Dirac Equations in Four-Dimensional Minkowski Space

Abstract

I derive the global existence and asymptotic behavior of small amplitude solutions to the system of massive coupled classical Maxwell–Dirac equations in the four-dimensional Minkowski space. Because the physically defined energy of the system is not positive definite, I transform it into an equivalent system of Maxwell–Klein–Gordon equations, which I study with a method based on gauge invariant energy estimates and geometric properties of the equations.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

Analysis in space-time bundles. III. Higher spin bundles

The universal cosmos M? is the unique four-dimensional globally causal space-time manifold to which the Dirac and Maxwell equations (among others) maximally and covariantly extend. A systematic treatment is presented of general fields over M?, of arbitrary spin; considered are fields induced from all irreducible representation of the isotropy group (scale-extended Poincaré group) to G?, the connected causal group of M?. Restricted to any species of such fields, the K?-invariant canonical Dirac operator (K? = maximal essentially compact subgroup of G?) is shown G?-covariant for a unique conformal weight. A normalized K?-finite basis for such fields is constructed. The basis actions thereon of the Dirac operator, infinitesimal generators of G?, discrete symmetries, second-order Casimir, and the essentially unique third-order noncentral quantum number (enveloping algebra element) invariant under K? are derived. Composition series under G? of a class of these field spaces—namely, the extension to M? of the relativistic fields considered by Bargman and Wigner, or arbitrary spin and conformal weight—are determined, distinguishing by invariance and causality features alone the essentially conventional positive-energy mass 0 subspaces and massive invariant sub-quotient spaces, whose unitarity under G? is given a new proof. The “completely positive” subclass (cf. below) of representations is determined. A more detailed treatment of spin one bundles (vector and two-form, of arbitrary conformal weight) is included; the exterior derivative transformations are diagonalized, and the conformally invariant massive spin one scalar product is identified with a mathematical version of the conventional electromagnetic field Lagrangian.     

Dirac-Lu space with pseudo-Riemannian metric of constant curvature

In this paper, we will discuss the geometries of the Dirac-Lu space whose boundary is the third conformal space N defined by Dirac and Lu. We firstly give the SO(3, 3) invariant pseudo-Riemannian metric with constant curvature on this space, and then discuss the timelike geodesics. Finally we get a solution of the Yang-Mills equation on it by using the reduction theorem of connections.     

Electrodynamics with charged strings

We show that in a four-dimensional space–time a complex scalar field can be associated with a one-dimensionally extended object, called a charged string. The string is said to be charged because the complex scalar field describing it interacts with an electromagnetic field. A charged string is characterized by an extension of the symmetry group of the charge space to a group of stretch rotations. We propose relativistically invariant and gauge-invariant equations describing the interaction of a complex scalar field with an electromagnetic field, and each solution of them corresponds to a charged string. We achieve this by introducing the notion of a charged string index, which, as verified, takes only integer values. We establish equations from which it follows that charged strings fit naturally into the framework of the Maxwell–Dirac electrodynamics.     

Global strong solution to a nonlinear Dirac type equation in one dimension

This paper studies a class of nonlinear massless Dirac equations in one dimension, which include the equations for the massless Thirring model and the massless Gross–Neveu model. Under the assumptions of the initial data having small charge and being bounded, the global existence of the strong solution is established. The decay of the local charge is also proved.     

Massless Relativistic Wave Equations and Quantum Field Theory

We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincaré group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fiber degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between (the two-fold cover of the 2-dimensional Euclidean group) and We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localization. Communicated by Klaus FredenhagenSubmitted 16/03/03, accepted 11/12/03     

Closely embedded Kre?n spaces and applications to Dirac operators

Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Kre?n spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L. Schwartz, and they are special representations of induced Kre?n spaces. In this article we present a canonical representation of closely embedded Kre?n spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness.     

Generalised Maxwell Equations in Higher Dimensions

This paper deals with the generalisation of the classical Maxwell equations to arbitrary dimension \(m\) and their connections with the Rarita–Schwinger equation. This is done using the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the spin group can be realised in terms of polynomials satisfying a system of differential equations. This allows the construction of generalised wave equations in terms of the unique conformally invariant second-order operator acting on harmonic-valued functions. We prove the ellipticity of this operator and use this to investigate the kernel, focusing on both polynomial solutions and the fundamental solution.     

Existence of dichotomies and invariant splittings for linear differential systems,II

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.