Volume 22, Issue 7, July 1981
Index of content:

Comments on a paper by L. L. Lee [J. Math. Phys. 2 1, 1055 (1980)]
View Description Hide DescriptionAnalytic expressions are given for the eigenvalues of some matrices arising in connection with Feynman path integrals.

Clifford algebras in 2ω dimensions
View Description Hide DescriptionWe consider the generalization of the Dirac (gamma) algebra to a space‐time with dimension 2ω, which is required for the use of the dimensional regularization scheme applied to fermions. In particular we introduce several tricks to facilitate the manipulation of expressions in such algebras, including a polynomial algorithm for taking traces and explicit formulas for Fierz matrices.

On the structure of phase space
View Description Hide DescriptionWe consider phase space as the carrier space of canonical transformations and see that this implies, for nonbijective ones, a much subtler structure than the one commonly assumed. Discussing only problems of one degree of freedom, i.e., a phase plane, we are able to clarify this structure by analogy with the better‐known situation of conformal mapping in the complex plane. Apart from the usual Riemann sheet concept, an alternate method is developed that involves the irreducible representations of the ambiguity group, i.e., the group of transformations that connects all points that are mapped on a single point by the conformal or canonical transformation. The algebra of variables then becomes a matrix algebra for which bijectiveness is retrieved.

Quantization on the sphere
View Description Hide DescriptionWhen phase‐space is a sphere, then conventional quantization is not available and the Heisenberg algebra is most naturally replaced by so(3). Quantization may be carried out in terms of invariant star‐products. This leads to the study of an interesting family of polynomials that are defined in a natural and intrinsic way on the enveloping algebra of sl(2). These polynomials are related to Legendre polynomials by a reordering rule that resembles the relation between Hermite polynomials and normal ordered monomials; they are identified as Pasternack polynomials and related to Jacobi polynomials. New orthogonality properties are found and interpreted in terms of unitary representations of SL(2,R) and SO(3).

Irreducible linear–antilinear representations and internal symmetries
View Description Hide DescriptionWe reconsider a well‐known classification, due to Wigner, of the unitary–antiunitary finite‐dimensional irreducible group representations within a somewhat generalized mathematical framework, where, in particular, any algebraically closed field K with an involutory automorphism j is considered in place of the complex field C. We show that each case of the classification can be characterized by the set U′^{l} of the linear mappings that commute with the given set U, which is now assumed to be an irreducible semigroup of linear and antilinear (i.e., j‐semilinear) mappings, and explicitly exhibit U′^{l}. Then, this classification is crossed with the classification of the sets of linear and antilinear mappings that has been obtained in some previous work and that generalizes the old classification of the unitary representations introduced by Frobenius and Schur. We obtain a new classification in which every case can be characterized by the group U^{c} of the invertible linear and antilinear mappings which commute with the semigroup U. Whenever U represents the symmetry group of some physical system, U^{c} may represent ’’internal’’ symmetries, so that the problem of finding these symmetries is now related to some natural classification of U. In particular, it turns out that in four cases U^{c} reduces to its linear part U^{cl} = U′^{l}\{0}, whereas in the remaining nine cases U^{c} is an extension of U^{cl} through G _{2}, which reduces to a semidirect product, but not to a direct one, in three cases and to a direct product in three cases.

Matrix orthogonal polynomials on the unit circle
View Description Hide DescriptionThe properties of matrix orthogonal polynomials on the unit circle are investigated beginning with their recurrence formulas. The techniques of scattering theory and Banach algebras are used in the investigation. A matrix generalization of a theorem of Baxter is proved.

Universal formats for nonlinear ordinary differential systems
View Description Hide DescriptionIt is shown that very general nonlinear ordinary differential systems (embracing all that arise in practice) may, first, be brought down to polynomialsystems (where the nonlinearities occur only as polynomials in the dependent variables) by introducing suitable new variables into the original system; second, that polynomialsystems are reducible to ’’Riccati systems,’’ where the nonlinearities are quadratic at most; third, that Riccati systems may be brought to elemental universal formats containing purely quadratic terms with simple arrays of coefficients that are all zero or unity. The elemental systems have representations as novel types of matrix Riccati equations. Different starting systems and their associated Riccati systems differ from one another, at the final elemental level, in order and in initial data, but not in format.

Treatment of some singular potentials by change of variables in Wiener integrals
View Description Hide DescriptionUsing a regularization of the classical Kepler problem, we show that a change of process and a change of time in stochastic differential equations allow a treatment of singular potentials in Wiener integrals.

Jump conditions for fields that have infinite, integrable singularities at an interface
View Description Hide DescriptionJump conditions are derived for a basic set of first order partial differential equations whose fields have infinite, integrable singularities, as well as finite discontinuities, at a moving and deforming interface. These basic formulas are then applied to Maxwell’s electrodynamic equations and yield the jump conditions that hold when the electric and magnetic fields have such singularities. These jump conditions are shown to be generalizations of formulas previously derived for double charge layers in electrostatics and for an interface with surface magnetization density in magnetostatics. The basic formulas are also used to obtain jump conditions for the wave equation and other second order partial differential equations whose fields have finite discontinuities at an interface. A similar application to second order vector identities yields a new set of j u m p i d e n t i t i e s. These identities show that the normal and tangential components of the jump in a vector field are kinematically interdependent; e.g., shock waves and vortex sheets are kinematically linked—a fact that may be significant for shock‐slip flows in aerodynamics. The jump identities also indicate the fields that must be measured at an epoch in order to calculate the instantaneous growth/decay rate of a propagating discontinuity, such as an atmospheric front. A feature of the derivation is that the field jumps and surface densities are mathematically defined as continuous and differentiable functions of three‐dimensional space and time that assume physical values on the physical interface. This approach is simpler and more general than previous approaches that define jumps and surface densities only on the physical interface because the brackets (jumps) now commute with a l l derivatives, instead of with only the tangential and displacement derivatives. Boundary value problems with propagating infinite singularities in the electrodynamic fields are presented as examples.

Distributionlike representations of *‐algebras
View Description Hide DescriptionNon‐self‐adjoint representations of *‐algebras in a Hilbert space give rise by an extension and transposition procedure to representations in larger spaces, such as distribution spaces. Those new representations provide examples of operators of nested Hilbert spaces which would be very singular operators when considered in the Hilbert space only.

A class of solutions of Calapso–Guichard equations
View Description Hide DescriptionA class of solutions is indicated for Calapso–Guichard equations of surface theory.

New nonlinear evolution equations from surface theory
View Description Hide DescriptionWe point out that the connection between surfaces in three‐dimensional flat space and the inverse scattering problem provides a systematic way for constructing new nonlinear evolution equations. In particular we study the imbedding for Guichard surfaces which gives rise to the Calapso–Guichard equations generalizing the sine‐Gordon (SG) equation. Further, we investigate the geometry of surfaces and their imbedding which results in the Korteweg–deVries (KdV) equation. Then by constructing a family of applicable surfaces we obtain a generalization of the KdV equation to a compressible fluid.

Numerical evaluation of integrals containing a spherical Bessel function by product integration
View Description Hide DescriptionA method is developed for numerical evaluation of integrals with k‐integration range from 0 to ∞ that contain a spherical Bessel function j _{ l }(k r) explicitly. The required quadrature weights are easily calculated and the rate of convergence is rapid—only a relatively small number of quadrature points is needed—for an accurate evaluation even when r is large. The quadrature rule is obtained by the method of product integration. With the abscissas chosen to be those of Clenshaw–Curtis and the Chebyshev polynomials as the interpolating polynomials, quadrature weights are obtained that depend on the spherical Bessel function. An inhomogenous recurrence relation is derived from which the weights can be calculated without accumulation of roundoff error. The procedure is summarized as an easily implementable algorithm. Questions of convergence are discussed and the rate of convergence demonstrated for several test integrals. Alternative procedures are given for generating the integration weights and an error analysis of the method is presented.

Equivalent Lagrangians: Multidimensional case
View Description Hide DescriptionWe generalize a theorem known for one‐dimensional nonsingular equivalent Lagrangians (L and ?) to the multidimensional case. In particular, we prove that the matrix Λ, which relates the left‐hand sides of the Euler–Lagrange equations obtained from L and ?, is such that the trace of all its integer powers are constants of the motion. We construct several multidimensional examples in which the elements of Λ are functions of position, velocity, and time, and prove that in some cases equivalence prevails even if detΛ = 0.

Decomposition of vector fields and mixed dynamics
View Description Hide DescriptionSome theorems are proved concerning the decomposition of vector fields into gradient and Hamiltonian components. A constructive method to carry out one of the decompositions is applied to some three‐ and four‐ dimensional dynamical models.

An alternative derivation of transition amplitudes for time‐dependent harmonic oscillators
View Description Hide DescriptionIt is shown that the relations which link the x and p operators of a time‐dependent harmonic oscillator to the x and p operators of the corresponding time‐independent oscillator can be cast into a generalized Bogoliubov transformation on the standard harmonic oscillator boson creation and annihilation operators. Once this fact has been recognized, various techniques can be invoked to derive in a short way transition amplitudes.

Symmetries of the Schrödinger bundle
View Description Hide DescriptionWe exhibit in this paper the invariance of the Schrödinger Lagrangian density under the eleven‐parameter group G_{(m)}, central extension of the Galilei group G. As a result, the quantum mechanical probability Fρ d ^{3} x turns out to be the conserved charge associated with the central generator of G_{(m)}, and the continuity equation is simply the expression of the conservation of the corresponding Noether current.

Bound state perturbation theory for the one‐space and one‐time dimension Klein–Gordon equation
View Description Hide DescriptionWe present a perturbation theory for an arbitrary bound state in the one‐space and one‐time dimension Klein–Gordon equation in the presence of a scalar potential and a vector (fourth component only) potential by reducing it to a Ricatti equation with the method of logarithmic perturbation expansions. All corrections to the energies and wavefunctions, including corrections to the positions of the nodes in excited states, are expressed in quadratures in a hierarchical scheme, without the use of either the Green’s function or the sum over intermediate states.

Formulation of variational principles via Lagrange multipliers
View Description Hide DescriptionRecent work on variational principles in mathematical physics enables one to construct, in a novel and systematic way, stationary expressions for a wide class of functionals P(E), where E is an unknown (vector) function whose defining equation cannot be solved exactly. The method involves the use of Lagrange multipliers, which can be a constant λ, a function F(x,y,z), and a dyadic operator Γ, to account for each of the equations (constraints) that define E. As illustrations, we consider vector and scalar scattering problems, and eigenvalue problems such as the determination of resonant frequencies and propagation constants of waveguides.

Integrated rotational parameter as a constraint for the variational lower bound of the diffraction peak
View Description Hide DescriptionThe contributions of the imaginary parts of the partial waves to an angle integral of the rotation parameter R is taken as a third constraint in addition to the total cross section σ^{ T } and σ_{el,im}, in the variational calculation of the lower bound for the diffraction peak.