Volume 22, Issue 10, October 1981
Index of content:

Algebras with three anticommuting elements. I. Spinors and quaternions
View Description Hide DescriptionA general construction of alternative algebras with three anticommuting elements and a unit is given. As an exhaustive result over the real and complex fields, we obtain the Clifford algebrasH (quaternions), N _{1} (dihedral Clifford algebra which is related to real 2‐spinors), and S _{1}(algebra of Pauli matrices which is related to complex 2‐spinors). What is important is that the algebrasN _{1} and S _{1} possess inverses everywhere except on a region akin to the light cone of the Minkowski space, while the quaternion algebraH has inverses everywhere except at the zero element. We discuss the reasons why the three algebrasN _{1}, H, and S _{1} are so difficult to distinguish in the representation space of 2×2 complex matrices.

Algebras with three anticommuting elements. II. Two algebras over a singular field
View Description Hide DescriptionThe Clifford algebra Ω generated by the elements {1,ω} with (ω)^{2} = +1, is an abelian ring of dimension two with properties analogous to the complex field C. The ring Ω has a string of singular inverses, and may be regarded as a ’’singular field’’ which circumvents both the fundamental theorem of algebra and the Frobenius theorem. We construct two associative algebras of dimension four over Ω: the Clifford algebra Ω_{1} and the biquaternions of Clifford Ω_{2}, and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the Minkowski space. Matrix representations are discussed, as well as the importance of the algebras Ω_{1} and Ω_{2}, in the description of physical vector fields.

A unified theory of the point groups and their general irreducible representations
View Description Hide DescriptionPoint groups and their general irreducible (vector and projective) representations are characterized by the subgroup conditions for SU(2) in a unified scheme: these conditions are given by simple polynomialequations imposed on the matrix elements of the 2×2 unitary matrices of SU(2). The general irreducible representations for the point groups D _{∞},D _{ n } (with arbitrary integer n?2), O, and T are given by four simple and effective tables.

On the parametrization of certain finite groups and their representations
View Description Hide DescriptionWe discuss a convenient way of parametrizing certain finite groups. They comprise groups with abelian normal subgroups and central extensions thereof; they include the groups D _{ n }, T, O, and their double groups, as well as the SU(3) subgroups T _{ n }, Δ(3n ^{2}), Δ(6n ^{2}), Σ(72), Σ(216), Σ(36) and their Z _{3}‐extensions. The method allows for a rapid calculation of representations and coupling coefficients; in particular, it solves the ’’multiplicity problem’’ of nonsimply reducible groups. We illustrate the method treating the Hessian group Σ(216) and its central extension.

The Racah algebra for groups with time reversal symmetry .II
View Description Hide DescriptionIn a previous paper, J. Math. Phys, 22, 233 (1981) it was shown that a Racah algebra could be developed for groups containing the antilinear operator of time reversal. Here the 1j m and 2j m symbols are constructed explicitly, and it is shown that the 3j m symbols may be found in terms of those of the linear subgroup. Thus the Racah algebra of these groups is known once the Racah algebra of the linear subgroup is known.

Generalized Bessel series and multiplicity problem in complex semisimple Lie algebra theory
View Description Hide DescriptionThe connection between Bessel series and SU(2) is reviewed from the standpoint of the outer multiplicity problem for this group. Its extension to any complex semisimple Lie algebras allows one to introduce new objects called ’’generalized Bessel series.’’ Some applications concerning the special functiontheory (addition theorems) are given.

Canonical realizations of Lie superalgebras: Ladder representations of the Lie superalgebra A(m,n)
View Description Hide DescriptionA simple formula for realizations of Lie superalgebras in terms of Bose and Fermi creation and annihilation operators is given. The essential new feature is that Bose and Fermi operators mutually anticommute. The Fock representation of these operators is used in order to construct a class of irreducible finite‐dimensional representations of the simple Lie superalgebraA(m,n). The matrix elements of the generators are written down. For m≳0 all representations turn out to be nontypical.

Fredholm determinants associated with Wiener integrals
View Description Hide DescriptionWe present a method for evaluating a large class of Fredholm determinants that are associated with the evaluation of certain Wiener integrals. The infinite‐dimensional determinant is shown to be equal to a single finite‐dimensional determinant.

Modified singular perturbation method for a stiff system of linear evolution equations
View Description Hide DescriptionA stiff system of linear evolution equations in Banach space is investigated. It is shown that the standard singular perturbation algorithm can be considerably simplified. In this modified algorithm the asymptotic solutions can be obtained in each order independently. The initial conditions are given explicitly so there is no need to solve the so‐called ’’inner’’ equations. Two examples of application are considered.

Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold
View Description Hide DescriptionWe consider the limiting case λ→0 of the Cauchy problem, ∂g _{λ}(x,t)/∂t = (1/2) λΔ_{ x } g _{λ}(x,t)+(V(x)/λ) g _{λ}( x,t), with g _{λ} (x,0) = exp{−S _{0}(x)/λ}T _{0}(x), V, S _{0} being real‐valued functions on N, T _{0} a complex‐valued function on N; V, S _{0}, T _{0} being independent of λ, Δ_{ x } being the Laplace–Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem D ^{2} Z(s)/∂s ^{2} = −∇_{ Z } V[Z(s)], s∈[0,t], with Z(t) = x and Z(0) = ∇S _{0}(Z(0)).One of our results is equivalent to the fact that for short times Schrödinger quantum mechanics on the Riemannian manifoldN tends to classical Newtonian mechanics on N as h/ tends to zero.

On rotations in a pseudo‐Euclidean space and proper Lorentz transformations
View Description Hide DescriptionIt is shown that in a general pseudo‐Euclidean space E ^{ p } _{ n }, 2‐flats (planes) passing through the origin of the coordinate system may be classified into six invariant types and explicit formulas for ’’planar rotations’’ in these flats are obtained. In the physically important case of the Minkowski World E ^{3} _{4}, planar rotations are characterized as r o t a t i o n l i k e, b o o s t l i k e, and s i n g u l a r transformations and an invariant classification of proper Lorentz transformations into these types is given. It is shown that a general nonsingular proper Lorentz transformation may be resolved as a c o m m u t i n g product of two transformations one of which is rotationlike and the other boostlike while a singular transformation may be written as a product of two rotationlike transformations, each with a rotation angle π. Such a rotationlike transformation with angle π is called ’’exceptional’’ following Weyl’s terminology for similar transformations of SO(3). In all cases, explicit formulas for the angles and planes of rotations in terms of the elements of a given Lorentz matrix are obtained and the procedure yields in a natural manner an explicit formula for the image of L in the D ^{10}(D ^{01}) representation of SO(3,1) which in turn leads to two more classification schemes in terms of the character χ of L in the D ^{10}(D ^{01}) and the D ^{ (1)/(2) 0}(D ^{0 (1)/(2) }) representations.

Extension of inverse scattering method to nonlinear evolution equation in nonuniform medium
View Description Hide DescriptionBy allowing the entire spectrum of certain linear eigenvalue problems to evolve with time a general type of nonlinear evolution equation in nonuniform medium which is exactly integrable by the inverse scattering method has been derived. The derivative nonlinear Schrödinger equation or the nonlinear Schrödinger equation with linear or parabolic density profiles are special cases of this generalized form.

The behavior at infinity of isotropic vortices and monopoles
View Description Hide DescriptionWe derive detailed asymptotic formulae for the behavior at infinity of isotropic vortex solutions of the abelian Higgs model and monopole solutions of the Yang–Mills Higgs model. In particular we find that the classical mass of the Higgs field is the smaller of m and twice the mass of the gauge field, where (m c/h/)^{2} is the curvature of the Higgs self‐interaction potential at the classical vacuum.

Inverse scattering. III. Three dimensions, continued
View Description Hide DescriptionThis paper presents further progress in the solution of the three‐dimensional inverse scattering problem for the Schrödinger equation. We prove that if the potential is in a specified class and produces no bound states, then the kernel of the generalized Marchenko equation defines a compact operator and the equation has a unique solution unless the operator has the eigenvalue 1. A partial characterization of scattering amplitudes associated with underlying local potentials without bound states is given and the potential is constructed without assuming its existence. An improved generalization of the Marchenko equation is presented for the case with bound states. The generalized Gel’fand–Levitan equation is critically reviewed.

A new direct proof of the expansion of a Coulomb‐distorted plane wave in Coulomb‐distorted spherical waves
View Description Hide DescriptionBy employing specific properties of confluent hypergeometric functions, it has been directly proved that a Coulomb‐distorted plane wave is expressible in superposition of Coulomb‐distorted spherical waves.

Algebraic and geometric structure of linear filters and scattering systems. I
View Description Hide DescriptionSome of the problems of linear filter and scattering theory are interpreted in terms of the theory of algebraic and analytical subvarieties of Grassman manifolds.

A perturbative look at the dynamics of extended systems in quantum field theory
View Description Hide DescriptionThe structure of a quantum field theory with an extended object is explored perturbatively. The perturbative expansion consists of coupled differential equations which are derived from the Heisenberg field equation. These equations are used to reduce the zero mode problem to a choice of boundary conditions. They are then integrated and the constraint of the equal time commutation relations is used to set the boundary conditions and derive commutation relations for the physical fields and the quantum coordinate. Using this information the quantal Hilbert space is constructed.

Incompressibility in relativistic continuous media
View Description Hide DescriptionA formalism of continuous media in general relativity is given and the concept of shock waves is defined using manifolds with boundary and the transversality theory of submanifolds. A definition of relativistic incompressibility is proposed and one gets that the shock waves are longitudinal with the speed of light.

Classical solutions of the equations of supergravity
View Description Hide DescriptionWe describe an algorithm to solve the classical equations of supergravity for the coupled fields of helicity 2 and 3/2. The algorithm depends on an expansion in the associated Grassman algebra and leads to a sequence of coupled equations that may be solved in a step by step manner. The procedure begins with solutions of the usual empty space Einstein equations but the subsequent equations are all linear differential equations. To complete the method, it is necessary to generalize the supplementary conditions on the vector spinor field from flat to curved space. The algorithm also permits a classification of the complete solutions in terms of the associated gravitational fields. It is shown that vanishing curvature implies Minkowski space and does not permit other conceivable possibilities such as a Clifford space. Generalizations of the vanishing curvature space are suggested.

Charged Demianski metric
View Description Hide DescriptionA solution of Einstein–Maxwell equations is presented. This solution is the charged version of the Demianski metric.