Volume 26, Issue 2, February 1985
Index of content:

Morphisms of crystallographic groups: Kernels and images
View Description Hide DescriptionThe crystallographic groups play an important role in solid state physics, where their representations are of particular interest. In this paper we classify the kernels of all possible representations by deriving necessary and sufficient conditions for a subgroup H of a crystallographic group G to be invariant. The structure of G/H is also discussed. A list of the one‐ and two‐dimensional invariant subgroups of the two‐dimensional crystallographic groups is appended as Table I; it includes the structural features of these subgroups needed for determining their settings, relative to the parent groups, and identifies the corresponding images. Table II is a list of the commutator subgroups of the two‐dimensional groups.

An infinite hierarchy of conservation laws and nonlinear superposition principles for self‐dual Einstein spaces
View Description Hide DescriptionSelf‐dual Einstein spaces are shown to admit an infinite hierarchy of conservation laws, and this hierarchy is then used to derive a formal version of Penrose’s twistor construction. The set of formal holomorphic bundles of fiber dimension 2 over the Riemann sphere P ^{1} is shown to form a formal infinite group which is used to derive nonlinear superposition principles. As an example of our methods a new self‐dual Einstein space is obtained as the result of a ‘‘collision’’ of complex p p‐waves ‘‘traveling in opposite directions.’’

Elementary representations and intertwining operators for SU(2, 2). I
View Description Hide DescriptionThe structure of the group SU(2, 2) and of its Lie algebra is studied in detail. The results will be applied in subsequent parts devoted to the explicit construction of elementary representations of SU(2, 2) induced from different parabolic subgroups and of the intertwining operators between these representations. A summary of some results of Parts II and III is given.

Markov‐type Lie groups in GL(n,R)
View Description Hide DescriptionThe general linear group GL(n,R) is decomposed into a Markov‐type Lie group and an abelian scale group. The Markov‐type Lie group basis is shown to generate all singly stochastic matrices which are continuously connected to the identity when non‐negative parameters are used. A basis is found which shows that it in turn contains a Lie subgroup which corresponds to doubly stochastic matrices, which basis, over the complex field, is shown to give the symmetric group for certain discrete values of the complex parameters. The basis of the Markov algebra is shown to give the negative of the corresponding M‐matrices with property ‘‘C’’ (for non‐negative combinations). These stochastic Lie groups are shown to be isomorphic to the affine group and the general linear group in one less dimension. The basis generates transformations with a natural interpretation for physical applications.

The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations
View Description Hide DescriptionWe investigate the sequence of Boussinesq equations by the method of singular manifolds. For the Boussinesq equation, which is known to possess the Painlevé property, a Bäcklund transformation is defined. This Bäcklund transformation, which is formulated in terms of the Schwarzian derivative, obtains the system of modified Boussinesq equations and the resulting Miura‐type transformation. The modified Boussinesq equations are found to be invariant under a discrete group of symmetries, acting on the dependent variables. By linearizing the Miura transformation (and modified equations) the Lax pair is readily obtained. Furthermore, by a result of Fokas and Anderson, the recursion operators defining the sequence of (higher‐order) Boussinesq equations may be constructed from the Miura transformation. This allows the (recursive) definition of Bäcklund transformations for this sequence of equations. The recursion operator is found to preserve the discrete symmetries of the modified Boussinesq equations. This leads to the conclusion that the sequences of Boussinesq and modified Boussinesq equations identically possess the Painlevé property (are meromorphic). We also find that, by a simple reduction, the sequences of Caudrey–Dodd–Gibbon and Kuperschmidt equations are contained within the Boussinesq sequence. Rational solutions are iteratively constructed for the Boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems.

Solution of the Cauchy problem for a generalized sine–Gordon equation
View Description Hide DescriptionWe consider a spectral problem generating a hierarchy of nonlinear evolution equations including the sine‐Gordon equation and a physically interesting generalization in the laboratory coordinates. The direct and inverse problems are treated. The time evolution of the spectral data is explicitly given and, therefore, the Cauchy problem for the related equations is solved.

Weakly convergent expansions of a plane wave and their use in Fourier integrals
View Description Hide DescriptionThe Fourier transform of an irreducible spherical tensor is normally computed with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. The angular integrations are then trivial. However, the remaining radial integral containing a spherical Bessel function may be so complicated that the applicability of Fourier transformation is severely restricted. As an alternative, the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested. The weakly convergent expansions of a plane wave are constructed in such a way that their application in Fourier integrals leads to expansions of the Fourier or inverse Fourier transform that converge with respect to the norm of either the Hilbert spaceL ^{2}(R^{3}) or the Sobolev space W ^{(1)} _{2}(R^{3}). Accordingly, these weakly convergent expansions may be viewed as distributions that are defined on either L ^{2}(R^{3}) or W ^{(1)} _{2}(R^{3}). The properties of some complete orthonormal sets of functions, in particular their Fourier transforms, are also studied. Shibuya and Wulfman [Proc. R. Soc. London Ser. A 2 8 6, 376 (1965)] derived an expansion of a plane wave involving the four‐dimensional spherical harmonics. It is shown that this Shibuya–Wulfman expansion is also a distribution which is defined on the Sobolev space W ^{(1)} _{2}(R^{3}). Finally, as an application it is shown how weakly convergent expansions can be used profitably for the construction of addition theorems.

Structural invariance of the Schrödinger equation and chronoprojective geometry
View Description Hide DescriptionWe describe an extension of the chronoprojective geometry and show how its automorphisms are related to the invariance properties of the Schrödinger equation describing a quantum test particle in any Newton–Cartan structure.

Maxwell’s equations and the bundle of null directions on Minkowski space
View Description Hide DescriptionWe reexpress the Maxwell field as a cross section of a line bundle over M×S ^{2}, the six‐dimensional space of null directions on Minkowski space. Maxwell’sequations then become a pair of linear equations for a Herz‐like scalar on M×S ^{2}. We obtain a deeper understanding of the simple, yet nontrivial relationship between the self‐dual and the anti‐self‐dual parts of a real Maxwell field. Our results are then applied to study solutions which are globally regular (on M×S ^{2}) namely, the pure radiation solutions, as well as solutions associated with discrete sources (the Lienard–Wiechert fields).

The dequantization programme for stochastic quantum mechanics
View Description Hide DescriptionThe classical limit ℏ→0 of stochastic quantum mechanics is investigated. An algebra of stochastic classical observables is canonically associated to the algebra of stochastic quantum observables with a product inherited from sequential instruments and also with a Lie commutator. It is shown that the sharp‐point limit (limit of no stochasticity), which implies ℏ→0, yields the usual algebra of classical observables.

Inverse scattering by a local impurity in a periodic potential in one dimension. II
View Description Hide DescriptionThis paper continues and completes the solution to the inverse scattering problem initiated in a recent paper. It allows for the existence of bound states in the band gaps and corrects a number of errors in the first paper.

An electromagnetic inverse problem for dispersive media
View Description Hide DescriptionThe dispersion of transient electromagnetic waves in a homogeneous medium can be characterized by expressing either the complex permittivity as a function of frequency or the susceptibility kernel as a function of time. In this paper, a time domain technique is used to derive a nonlinear integrodifferential equation which relates the susceptibility kernel for a one‐dimensional homogeneous slab to the reflection operator for the medium. Thus, the susceptibility kernel (which is a function of time) can be determined from reflection data. A numerical implementation of this technique is shown. The more general case of a medium consisting of a stack of homogeneous dispersive layers is also addressed.

Solitary wave solutions to the Einstein equations
View Description Hide DescriptionThe soliton solutions to the vacuum Einstein equations generated by the special class of Einstein–Rosen metrics described by linear combinations of homogeneous solutions to the usual cylindrically symmetric wave equation are studied.

The well‐posedness of (N=1) classical supergravity
View Description Hide DescriptionIn this paper we investigate whether classical (N=1) supergravity has a well‐posed locally causal Cauchy problem. We define well‐posedness to mean that any choice of initial data (from an appropriate function space) which satisfies the supergravity constraint equations and a set of gauge conditions can be continuously developed into a space‐time solution of the supergravity field equations around the initial surface. Local causality means that the domains of dependence of the evolution equations coincide with those determined by the light cones. We show that when the fields of classical supergravity are treated as formal objects, the field equations are (under certain gauge conditions) equivalent to a coupled system of quasilinear nondiagonal second‐order partial differential equations which is formally nonstrictly hyperbolic (in the sense of Leray–Ohya). Hence, if the fields were numerical valued, there would be an applicable existence theorem leading to well‐posedness. We shall observe that well‐posedness is assured if the fields are taken to be Grassmann (i.e., exterior algebra) valued, for then the second‐order system decouples into the vacuum Einstein equation and a sequence of numerical valued linear diagonal strictly hyperbolic partial differential equations which can be solved successively.

On the Cauchy problem for the nonlinear Boltzmann equation global existence uniqueness and asymptotic stability
View Description Hide DescriptionThe analysis of the initial value problem for the nonlinear Boltzmann equation is considered in this paper. A theorem defining global existence and uniqueness for initial data which decay at infinity with an inverse power law is the main result of this work and is obtained by suitable application of fixed point theorems in Banach spaces. The theorem also defines the asymptotic stability of the solutions.

Representations of the current algebra of a charged massless Dirac field
View Description Hide DescriptionIt is shown that the current algebraA _{0} of a charged, massless Dirac particle has representations with positive energy of all types I_{∞}, II_{∞}, and III in the classification of Araki–Woods.

Superfield actions for N=2 degenerate central charges
View Description Hide DescriptionWe construct the superfield actions for degenerate (spin‐reducing) multiplets of N=2 extended supersymmetry as integrals over all superspace. This requires the integration over the two available central charges as well. We evaluate the detailed component contributions to these actions and show they are total derivatives with respect to central charge dimensions. The resulting spectrum of the theories are analyzed in four dimensions in terms of various boundary conditions in the higher dimensions and the nature of the integration domain.

Planar factors of proper homogeneous Lorentz transformations
View Description Hide DescriptionThis article discusses two constructions factoring proper homogeneous Lorentz transformations H into the product of two planar transformations. A planar transformation is a proper homogeneous Lorentz transformation changing vectors in a two‐flat through the origin, called the transformation two‐flat, into new vectors in the same two‐flat and which leaves unchanged vectors in the orthogonal two‐flat, called the pointwise invariant two‐flat. The first construction provides two planar factors such that a given timelike vector lies in the transformation two‐flat of one and in the pointwise invariant two‐flat of the other; it leads to several basic conditions on the trace of H and to necessary and sufficient conditions for H to be planar. The second construction yields explicit formulas for the orthogonal factors of H when they exist and are unique, where two planar transformations are orthogonal if the transformation two‐flat of one is the pointwise invariant two‐flat of the other.