Volume 27, Issue 5, May 1986
Index of content:

A general method of generating and classifying Clifford algebras
View Description Hide DescriptionA general construction program is given for the generation of higher‐order Clifford algebras of various signatures together with their faithful representations in terms of Pauli‐type operators. The analysis is based upon several isomorphism theorems that provide a simpler understanding of the standard classification scheme of these algebras, and an improved signature index notation is suggested that identifies each of the five types of Clifford algebras

On Diophantine equations and nontrivial Racah coefficients
View Description Hide DescriptionSome families of zeros of weight‐1 6j coefficients are given, each in terms of four parameters. They arise from a geometrical investigation of certain Diophantine equations. Some general remarks on the solutions of Diophantine equations are also made.

On the exceptional equivalence of complex Dirac spinors and complex space‐time vectors
View Description Hide DescriptionIt is well known that there exists an equivalence of R^{8} vectors and spinors, which has its roots in the rotational symmetry of the Dynkin diagram for D _{4}. We endow R^{8} with a real action of ∼(SO(4,4)), and restrict to ∼(SO(3,1)). Under this restriction the R^{8} spinor decomposes into the direct sum of two real Dirac spinors, while the R^{8} vector decomposes into a real space‐time vector plus four real scalars. The equivalence is preserved under this restriction; it is shown that it is realized in the (exceptional) equivalence of a complex Dirac spinor and a complex space‐time vector.

Adapted slicings of space‐times possessing simply transitive similarity groups
View Description Hide DescriptionThe relationship between the hypersurface‐homogeneous slicing of an exact power law metric space‐time and slicings adapted to spatial self‐similarities is discussed in a group theoretical setting.

The most degenerate representation matrix elements of finite rotations of SO(n−2, 2)
View Description Hide DescriptionUsing a technique of Strom and Boyer [S. Strom, Ark. Fys. 3 3, 465 (1966) and C. P. Boyer, J. Math. Phys. 1 2, 1599 (1971)], the matrix elements of finite rotations of the group SO(n−2, 2) have been computed in the most degenerate principal series of continuous representations.

Relativistic plasma dispersion functions
View Description Hide DescriptionThe known properties of plasmadispersion functions (PDF’s) for waves in weakly relativistic, magnetized, thermal plasmas are reviewed and a large number of new results are presented. The PDF’s required for the description of waves with small wave number perpendicular to the magnetic field (Dnestrovskii and Shkarofsky functions) are considered in detail; these functions also arise in certain quantum electrodynamical calculations involving strongly magnetized plasmas. Series, asymptotic series, recursion relations, integral forms, derivatives, differential equations, and approximations for these functions are discussed as are their analytic properties and connections with standard transcendental functions. In addition a more general class of PDF’s relevant to waves of arbitrary perpendicular wave number is introduced and a range of properties of these functions are derived.

On the integrability of systems of nonlinear ordinary differential equations with superposition principles
View Description Hide DescriptionA new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally n o t derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincaré surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.

Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra
View Description Hide DescriptionThe Kadomtsev–Petviashvili (KP) equation (u _{ t }+3u u _{ x }/2+ 1/4 u _{ x x x })_{ x } +3σu _{ y y }/4=0 allows an infinite‐dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group depends on three arbitrary functions of time ‘‘t’’ and is shown to be related to a subalgebra of the loop algebraA ^{(1)} _{4}. Low‐dimensional subalgebras of the symmetry algebra are identified, specifically all those of dimension n≤3, and also a physically important six‐dimensional Lie algebra containing translations, dilations, Galilei transformations, and ‘‘quasirotations.’’ New solutions of the KP equation are obtained by symmetry reduction, using the one‐dimensional subalgebras of the symmetry algebra. These solutions contain up to three arbitrary functions of t.

Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two‐center problem in molecular quantum mechanics
View Description Hide DescriptionThe differential equation,x(x−x _{0})(d ^{2} y/d x ^{2})+(B _{1}+B _{2} x) (d y/d x)+[ω^{2} x(x−x _{0}) −[2ηω(x−x _{0})+B _{3}]y=0, arises both in the quantum scattering theory of nonrelativistic electrons from polar molecules and ions, and, in the guise of Teukolsky’s equations, in the theory of radiation processes involving black holes. This article discusses analytic representations of solutions to this equation. Previous results of Hylleraas [E. Hylleraas, Z. Phys. 7 1, 739 (1931)], Jaffé [G. Jaffé, Z. Phys. 8 7, 535 (1934)], Baber and Hassé [W. G. Baber and H. R. Hassé, Proc. Cambridge Philos. Soc. 2 5, 564 (1935)], and Chu and Stratton [L. J. Chu and J. A. Stratton, J. Math. Phys. (Cambridge, Mass.) 2 0, 3 (1941)] are reviewed, and a rigorous proof is given for the convergence of Stratton’s spherical Bessel function expansion for the ordinary spheroidal wave functions. An integral is derived that relates the eigensolutions of Hylleraas to those of Jaffé. The integral relation is shown to give an integral equation for the scalar field quasinormal modes of black holes, and to lead to irregular second solutions to the equation. New representations of the general solutions are presented as series of Coulomb wave functions and confluent hypergeometric functions. The Coulomb wave‐function expansion may be regarded as a generalization of Stratton’s representation for ordinary spheroidal wave functions, and has been fully implemented and tested on a digital computer. Both solutions given by the new algorithms are analytic in the variable x and the parameters B _{1}, B _{2}, B _{3}, ω, x _{0}, and η, and are uniformly convergent on any interval bounded away from x _{0}. They are the first representations for generalized spheroidal wave functions that allow the direct evaluation of asymptotic magnitude and phase.

Prolongation structures and Lie algebra real forms
View Description Hide DescriptionA technique useful as an aid for finding finite‐dimensional representations of a Wahlquist–Estabrook prolongation algebra is introduced. The technique is illustrated with an examination of equations of the form ∂^{2} z ^{ a }/∂x ^{1} ∂x ^{2}=f ^{ a }(z ^{ b }).

Painless nonorthogonal expansions
View Description Hide DescriptionIn a Hilbert spaceH, discrete families of vectors {h _{ j }} with the property that f=∑_{ j }〈h _{ j }‖ f〉h _{ j } for every f in H are considered. This expansion formula is obviously true if the family is an orthonormal basis of H, but also can hold in situations where the h _{ j } are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non‐Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

The Thomas precession and velocity‐space curvature
View Description Hide DescriptionThe motion of a physical system acted upon by external torqueless forces causes the relativistic Thomas precession of the system’s spin vector, relative to an inertial frame. A time‐dependent force that returns the system to its initial velocity is considered. The precession accumulates to become a finite rotation of the final spin vector, relative to its initial value. This rotation is commonly explained as the Wigner rotation due to the sequence of pure boosts caused by the force. An alternative interpretation is presented here: The rotation is due to the change of the spin vector as it is parallel‐transported around the closed trajectory described by the system in hyperbolic three‐velocity space. As an application, the angle of precession for a planar motion is shown to be equal to the area enclosed by the trajectory in velocity space.

Simulation of classical particle trajectories in a complex two‐dimensional space
View Description Hide DescriptionClassical particles of arbitrary rest mass and spin are modeled in a two‐dimensional space C _{2} (which has two complex coordinates ξ^{ A }, A=1, 2) in the following way. It is first shown that a preferred set of trajectories ξ^{ A }=ζ^{ A }(s), designated geodesics, can be introduced from a variation principle that makes stationary the real variable s. The latter plays the role of proper time and serves to parametrize the trajectories. The geodesics so defined are then shown to be associated with a nonlinear representation of the Poincaré group. A set of Poincaré vectors and tensors are constructed from ζ^{ A } and its proper time derivatives and these simulate the properties of the position, momentum, angular momentum, and internal angular momentum variables of a classical massive particle of arbitrary spin.

Bäcklund transformation and the Painlevé property
View Description Hide DescriptionWhen a differential equation possesses the Painlevé property it is possible (for specific equations) to define a Bäcklund transformation (by truncating an expansion about the ‘‘singular’’ manifold at the constant level term). From the Bäcklund transformation, it is then possible to derive the Lax pair, modified equations and Miura transformations associated with the ‘‘completely integrable’’ system under consideration. In this paper completely integrable systems are considered for which Bäcklund transformations (as defined above) may not be directly defined. These systems are of two classes. The first class consists of equations of Toda lattice type (e.g., sine–Gordon, Bullough–Dodd equations). We find that these equations can be realized as the ‘‘minus‐one’’ equation of sequences of integrable systems. Although the ‘‘Bäcklund transformation’’ may or may not exist for the ‘‘minus‐one’’ equation, it is shown, for specific sequences, that the Bäcklund transformation does exist for the ‘‘positive’’ equations of the sequence. This, in turn, allows the derivation of Lax pairs and the recursion operation for the entire sequence. The second class of equations consists of sequences of ‘‘Harry Dym’’ type. These equations have branch point singularities, and, thus, do not directly possess the Painlevé property. Yet, by a process similar to the ‘‘uniformization’’ of algebraic curves, their solutions may be parametrically’’ represented by ‘‘meromorphic’’ functions. For specific systems, this is shown to provide a natural extension of the Painlevé property.

General Lorentz transformations and applications
View Description Hide DescriptionIt is known that the most general proper orthochronous vector Lorentz (transformation) operator can be generated by a skew‐symmetric 4×4 matrix containing an antisymmetric tensor of the second rank. The corresponding Lorentz operator for the two‐component spinor is presented and, as can be expected, it contains the same tensor as the vector operator. Since the Pauli matrices of the spinor operator have very simple multiplication properties, the behavior of the tensor under multiplication of spinor operators is easily obtained. By comparison the corresponding properties of the tensor in vector operators can be obtained without multiplying 4×4 matrices. The physical meaning of the tensor contained in a Lorentz operator is discussed. Apart from the usual or regular operator a singular operator is discussed. Still other types of Lorentz operators are possible.

Lorentz transformations in terms of initial and final vectors
View Description Hide DescriptionGiven arbitrary initial vector(s) and their final vector(s) in a Lorentz transformation, the problem is to determine the operator of the transformation. The solution presented here consists of expressing the tensor contained in the Lorentz operator in terms of the given vectors and then the operator itself follows by the exponentiation of a generating matrix. This is possible in certain special cases. In the general case a number of simultaneous nonlinear equations have to be solved. The analytical solution of these equations is elusive while attempts at numerical solution indicate that supplementary information is required. The corresponding procedure for a singular operator is also presented.

A brief study on the transformation of Maxwell equations in Euclidean four‐space
View Description Hide DescriptionElectromagnetic‐type fields in Euclidean four‐space are studied by changing the sign of the time differential term in Faraday’s law of induction. Although a covariant set of field equations can be derived, difficulties arise in the case of time‐dependent fields.

Geometric quantization and constrained systems
View Description Hide DescriptionThe problem of obtaining the quantum theory of systems with first class constraints is discussed in the context of geometric quantization. The precise structure needed on the constraint surface of the full phase space to obtain a polarization on the reduced phase space is displayed in a form that is particularly convenient for applications. For unconstrained systems, a n ypolarization on the phase space leads to a mathematically consistent quantum description, although not all of these descriptions may be viable from a physical standpoint. It is pointed out that the situation is worse in the presence of constraints: a general polarization on the full phase space need not lead to even a mathematically consistent quantum theory. Examples are given to illustrate the general constructions as well as the subtle difficulties.

Time‐dependent invariants and quantum mechanics in two dimensions
View Description Hide DescriptionA class of time‐dependent problems in two space dimensions possessing time‐dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time‐dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 1 0, 1458 (1969)] is discussed.

Dynamic‐group approach to the x ^{2}+λx ^{2}/(1+g x ^{2}) potential
View Description Hide DescriptionA modified operator method based upon the SO (2,1) dynamic group is applied to the one‐dimensional anharmonic oscillator potential V(x)=x ^{2}+λx ^{2}/(1+g x ^{2}). A tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series. Very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g‐values.