Volume 21, Issue 1, January 1980
Index of content:

Explicit evaluation of the representation functions of ISO(n)
View Description Hide DescriptionAll representation functions of ISO(n) have been found in explicitly closed form. They can all be expressed in terms of Bessel functions and Clebsch–Gordan coefficients of SO(n) involving the most degenerate representation [k,0].

Asymptotic approximations for modified Bessel functions
View Description Hide DescriptionThe behavior of a q _{ν}(x) =I _{ν}(x)/I _{ν} ^{ a }(x), where I _{ν} is a modified Bessel function with integral or half‐integral index ν and I _{ν} ^{ a } the leading term of its asymptotic series, is investigated for x≫1. It is shown that q _{ν}(x) may be approximated by e _{ν}(x) =exp(−ν^{2}/2x), the difference r _{ν}(x) =q _{ν}(x)−e _{ν}(x) being of order x ^{−1/4}. Bounds for r _{ν}(x) depending only on x are derived for each of the two classes of ν’s and an application of these results in scattering theory is indicated.

The Planck integral cannot be evaluated in terms of a finite series of elementary functions
View Description Hide DescriptionIt is shown that the Planck integral cannot be evaluated in terms of a finite series of elementary functions. A relation is first established between the Planck and dilogarithm integrals. To prove nonintegrability the Risch decision procedure for elementary functions is then applied to the dilogarithm integral.

Linearization stability and a globally singular gange of variables
View Description Hide DescriptionAn example is given showing that global properties of a change of field variables can affect the validity of linear perturbation theory.

Conservation laws of the Benjamin–Ono equation
View Description Hide DescriptionWe report here an empirical algorithm to construct conservation laws of the Benjamin–Ono equation.

Asymptotic regularizations as an alternative to distributions for the study of singular hypersurfaces
View Description Hide DescriptionA technique designed to give shock wave surface and propagation equations of quasilinear differential systems is given which can be used when the distribution method does not lead to any practicable results. The technique, called ’’asymptotic regularizations,’’ uses a smoothing process of the discontinuous functions which becomes negligible as a parameter ω is made arbitrarily large, thus revealing the behavior of the discontinuities on the singular hypersurface. This paper is also, in a way, the answer to a conjecture formulated by Lichnérowicz to the effect that there should be a theorem relating distribution theory to asymptotic expansions on manifolds (as defined by Choquet‐Bruhat) which could explain the formal similarity of their respective results when they are applied to the differential systems of some relativistic fluids.

A note on completeness
View Description Hide DescriptionCompleteness relationships for eigenfunctions of second order differential equations are presented in a form which employs a contour integration rather than the usual integration and summation over eigenvalues. This technique which is particularly applicable for scattering problems simplifies the usual procedures and the proper weight functions are easily obtained. Some examples are given.

Construction of doubly orthogonal functions
View Description Hide DescriptionIt is shown that the extremal functions associated with a general functional form a set of biorthogonal and doubly orthogonal functions. The theory is applied to antenna theory to find the doubly orthogonal functions of the E and H plane strip source antennas.

Quadratic Hamiltonians: The four classes of quadratic invariants, their interrelations and symmetries
View Description Hide DescriptionThe quadratic invariants of the three basic quadratic Hamiltonian systems‐attractive oscillator, repulsive oscillator, and free particle‐ are shown to be the same. These invariants are divided into two categories, useful and nonuseful. The definition of useful is in terms of contributing to the (quadratic invariant based) symmetry group of the appropriate Hamiltonian. Usefulness is not invariant under a time‐dependent linear canonical transformation. Hence different classes of invariants produce the different symmetry groups for the three different types of quadratic Hamiltonian considered here. The paper concludes with a consideration of a useful transformation of arbitrary quadratic Hamiltonians.

A note on the Hénon–Heiles problem
View Description Hide DescriptionThe Hamiltonian for the Hénon–Heiles problem, H= (1/2)(p _{1} ^{2}+p _{2} ^{2}+q _{1} ^{2}+q _{2} ^{2})+q _{1} ^{2} q _{2} −(1/3) q _{2} ^{3}, is a particular example of time‐independent Hamiltonians for two‐dimensional oscillator systems with third degree anharmonicity. It has been used as a model for galactic motion. There has been much discussion of the possible existence of an integral other than the Hamiltonian. In this note we show that the Hénon–Heiles Hamiltonian in particular and the class in general does not possess an invariant series which is explicitly time‐independent other than the Hamiltonian itself.

Mobility of nonlinear systems
View Description Hide DescriptionGlobal mobility is defined and is found to be decisive for the structure of a physical system. The structure of some simple nonlinear systems is elucidated by studying their dynamical mobility. It is shown that due to an excessive mobility a nonlinear system may acquire some classical features.

A phase space approach to generalized Hamilton–Jacobi theory
View Description Hide DescriptionGeneralizations of H–J theory have been discussed before in the literature. The present approach differs from others in that it employs geometrical ideas on phase space and classical transformation theory to derive the basic equations. The relation between constants of motion and symmetries of the generalized H–J equations is then clarified.

On electromagnetic multipole fields in a finite, spherically symmetric region
View Description Hide DescriptionThe electromagnetic eigenfields for the region bounded by two concentric spheres are discussed and compared with the corresponding eigenfields of a spherical cavity. These characteristic fields are the solenoidal and irrotational multipole solutions of the vector Helmholtz equation that satisfy the source‐free boundary conditions. They constitute a complete set for the expansion of an arbitrary, square‐integrable electromagnetic field, which may be generated by surface and volume sources. The frequencies of the solenoidal and irrotational eigenfields for the annular region are analyzed as functions of the radius ratio, α=r _{1}/r _{2} (r _{1}<r _{2} =constant), of the two concentric spheres. The results are illustrated by graphs and tables. Two relations obtained by applying the implicit function theorem to the transcendental eigenfrequency equations are also derived by calculating the work performed against the radiation pressure as the electromagnetic field is compressed adiabatically. The multipole fields are expressed in terms of vector spherical harmonics and vector spherical multipoles. Two formulas for the reduction of vector products of multipole fields to sums of vector spherical harmonics are derived.

Relativistic Brownian motion and the space–time approach to quantum mechanics
View Description Hide DescriptionAn attempt has been made to extend the stochastic quantization procedures introduced by Nelson in the nonrelativistic case to the relativistic case in the four‐dimensional Finsler space. The space–time in the microdomain is considered to be quantized and a more general concept of probability is needed to have a consistent and complete theory of quantum mechanics.

Generalized Stern–Gerlach experiments and the observability of arbitrary spin operators
View Description Hide DescriptionUnder the assumption that we can create in the laboratory any electromagnetic field consistent with Maxwell’sequations, it is shown that an arbitrary Hermitian operator on a spin system can be measured using a suitable generalization of the Stern–Gerlach experiment. In particular, it is shown that every proposition about the spin‐1 system is verifiable, answering the challenge of Hultgren and Shimony. The analysis also reveals complications in the standard Stern–Gerlach experiment of which many physicists are apparently not aware.

A kernel of Gel’fand–Levitan type for the three‐dimensional Schrödinger equation
View Description Hide DescriptionIn a previous paper we introduced a Green’s function for the three‐dimensional Schrödinger equation analogous to the Green’s function used to obtain the integral equation for the Jost wave functions in one dimension. The three‐dimensional Green’s function was used to define Jost wave functions for the three‐dimensional problem and the completeness relations for these wave functions were obtained. In the present paper we use the three‐dimensional Green’s function to construct influence functions for the 3+3 ultrahyperbolic partial differential equation which have analogs to the causal properties of the corresponding influence functions for the 1 + 1 hyperbolic partial differential equation. Just as the 1 + 1 influence function can be used to obtain an integral equation for the one‐dimensional Gel’fand–Levitan kernel in terms of the scattering potential, we use the 3 + 3 influence function to obtain an analogous integral equation for our proposed Gel’fand–Levitan kernel for the three‐dimensional problem. Though much of the formalism for finding the properties of the kernel for the three‐dimensional problem can be carried out in a straightforward manner, the interpretation of the triangularity properties is more difficult than in the one‐dimensional case because of the complicated geometrical picture associated with the notion of causality. In addition to its use in obtaining a Gel’fand–Livitan kernel, the 3+3 influence function can be used to simplify the second term in an expansion of the potential in terms of the minimal scattering data. This simplification is also given. In the Appendix the asymptotic form of the three‐dimensional Jost wave function is given in a form which is analogous to the asymptotic form for the one‐dimensional Jost wave function and which is compatible with our notion of triangularity for the Gel’fand–Levitan kernel.

Five‐term WKBJ approximation
View Description Hide DescriptionAn expression is derived for the five‐term WKBJ approximation.

The rosette of rosettes of Hilbert spaces in the indefinite metric state space of the quantized Maxwell field
View Description Hide DescriptionThe indefinite metric spaceO_{ M } of the covariant form of the quantized Maxwell field M is analyzed in some detail. S_{ M } contains not only the pre‐Hilbert space X^{0} of states of transverse photons which occurs in the Gupta–Bleuler formalism of the f r e e M, but a whole rosette of continuously many, isomorphic, complete, p r e‐H i l b e r t s p a c e sL^{ q } disjunct up to the zero element o of S_{ M }. The L^{ q } are the maximal subspaces of S_{ M } which allow the usual statistical interpretation. Each L^{ q } corresponds uniquely to one square integrable, spatial distribution j ^{ o }(x) of the total charge Q=0. If M is in a n y state from L^{ q }, the bare charge j ^{0}(x) appears to be inseparably dressed by the quantum equivalent of its proper, classical Coulomb field E(x). The vacuum occurs only in the state space L^{0} of the free Maxwell field. Each L^{ q } contains a secondary rosette of continuously many, up to o disjunct, isomorphic H i l b e r t s p a c e sH_{ g } ^{ q } related to different electromagnetic gauges. The space H_{ o } ^{ q }, which corresponds to the Coulomb gauge within the Lorentz gauge, plays a physically distinguished role in that only it leads to the usual concept of energy. If M is in any state from H_{ g } ^{ q }, the bare 4‐current j ^{0}(x), j(x), where j(x) is any square integrable, transverse current density in space, is endowed with its proper 4‐potential which depends on the chosen gauge, and with its proper, gauge independent, Coulomb–Oersted field E(x), B(x). However, these fields exist only in the sense of quantum mechanical expectation values equipped with the corresponding field fluctuations. So they are basically different from c l a s s i c a lelectromagnetic fields.

Summation of partial wave expansions in the scattering by short‐range potentials
View Description Hide DescriptionPrevious theorems on the convergence of the punctual Padé approximant to the scattering amplitude are extended. The new proofs correspond to the case of potentials having a shortrange tail of the type V (r)_{ r→∞}∼V _{ o } r ^{−ρ−1} exp[−μr], where V _{0} is a constant, ρ an integer and μ≳0, and are restricted to within the Lehmann ellipse, in the complex cosϑ plane, where the partial wave expansion converges. Asymptotic estimates are obtained for the error of the approximants.

Einstein spaces and homothetic motions. I
View Description Hide DescriptionAlgebraically special, nonflat vacuum Einstein spaces with an expanding and/or twisting geodesic principal null congruence are considered. These spaces are assumed to possess locally a homothetic symmetry as well as two or more Killing vectors. All metrics of such spaces are determined along with the form of the homothetic Killing vector admitted. All but one of the metrics are twist free. It is proved that two of the NUT metrics do not admit a homothetic motion.