Volume 21, Issue 9, September 1980
Index of content:

Shift operators for discrete representations of O(p, q)
View Description Hide DescriptionShift operators for discrete representations of O( p,q) are found in product form. Their normalization is found and conditions are derived for the weight components of such representations.

The character generator of SU(n)
View Description Hide DescriptionA simple combinatorial method for writing the character generator of SU(n) is described.

Classification of three‐particle states according to an orthonormal SU(3)⊇SO(3) basis
View Description Hide DescriptionIn this paper we generalize Dragt’s approach to classifying three‐particle states. Using his formalism of creation and annihilation operators, we obtain explicitly a complete set of orthonormal functions Y ^{λμR L } _{ M } on S _{5}. This set of functions carries all the irreducible representations of the group SU(3) reduced according to SO(3). The Y ^{λμR L } _{ M }, which are eigenvectors of the togetherness and angular momentum operators, have very simple properties under three‐particle permutations. We obtain also explicitly the coefficients ’’3ν’’ which reduce the products of these functions.

On a new relation between semisimple Lie algebras
View Description Hide DescriptionA recently discovered relation between pairs of semisimple Lie algebras is further investigated. This relation, which is called subjoining and denoted by ’’≳’’, is a generalization of inclusion, where a subalgebra is embedded in an algebra. Nontrivial subjoinings of two algebras of the same type are described. New chains of algebras involving proper inclusions and subjoinings can be formed. Infinite families of maximal subjoinings C _{ n }≳B _{ n } and B _{ n }≳C _{ n } are shown.

Schrödinger spectral problems with energy–dependent potentials as sources of nonlinear Hamiltonian evolution equations
View Description Hide DescriptionWe develop a method to derive infinite families of completely integrable nonlinear Hamiltonian evolution equations associated with Schrödinger spectral problems whose potential functions depend on the spectral parameter.

Gauge invariance and the Helmholtz conditions
View Description Hide DescriptionIn a previous work I have outlined a new formalism of theoretical mechanics, extending the traditional subject in a new way to elementary systems whose second‐order dynamical equations do not satisfy the classical Helmholtz conditions for the existence of a Lagrange function. In the present paper I determine the role of a new gauge invariance (’’dynamical’’ gauge invariance), satisfied by the equations of motion of systems covered by the formalism, and the relationship this bears to the Helmholtz conditions. A certain subgroup, of ’’kinematic’’ gauge transformations, is singled out: the kinematic gauge transformations correspond for the Lorentz force law to the usual gauge transformation of electromagnetism, general dynamical gauge transformations correspond to minimal substitutions. An analogue for the nonalbelian gauge force law is discussed briefly. An implication of the present work is the result, in a sense made explicit within, that (the usual) gauge invariance is an i n t r i n s i c p r o p e r t y of classical Hamiltonian systems.

Relativistic covariance and rotational electrodynamics
View Description Hide DescriptionThe following article demonstrates how the logical coherence of relativistic electrodynamics is maintained for a particular family of rotational paradoxes. The internal computational unity, for rotation, is preserved through the manifestation of a commonly unrecognized geometrical property of tensorcalculus.

Fock–Tani representation for composite particles in a soluble model
View Description Hide DescriptionThe transformation from the usual Fock representation to the Fock–Tani representation for composite particles is carried out for a simplified model of the composite which allows closed‐form expressions for the transformed quantities. A ’’statistical renormalization’’, in which single particle energies and interactions become dependent on the composite particle occupation number, plays an essential rôle in the solution, allowing absorption of unlinked terms into the definition of the renormalized energies.

A class of explicitly soluble, local, many‐center Hamiltonians for one‐particle quantum mechanics in two and three dimensions. I
View Description Hide DescriptionWe derive an explicit formula for the resolvent of a class of one‐particle, many‐center, local Hamiltonians. This formula gives, in particular, a full description of a model molecule given by point interactions at n arbitrarily placed fixed centers in three dimensions. It also gives a three−dimensional analog of the Kronig–Penney model.

Representations and properties of para‐Bose oscillator operators. I. Energy position and momentum eigenstates
View Description Hide DescriptionPara‐Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation D_{α} of the para‐Bose system is obtained as the direct sum D _{β}⊕D _{β+1/2} of the representations of the SL(2,R) Lie algebra. The position and momentum eigenstates are then obtained in this representation D_{α}, using the matrix mechanical method. The orthogonality, completeness, and the overlap of these eigenstates are derived. The momentum eigenstates are also derived using the wave mechanical method by specifying the domain of the definition of the momentum operator in addition to giving it a formal differential expression. By a careful consideration in this manner we find that the two apparently different solutions obtained by Ohnuki and Kamefuchi in this context are actually unitarily equivalent.

A new proof of absence of positive discrete spectrum of the Schrödinger operator
View Description Hide DescriptionA new method to prove the absence of positive discrete spectrum of the Schrödinger operator is given.

Upper and lower bounds in nonrelativistic scattering theory
View Description Hide DescriptionWe consider the problem of determining rigorous upper and lower bounds to the difference between the exact and approximate scattering phase shift, for the case of central potential scattering. The present work is based on the Kato identities and the phase‐amplitude formalism of potential scattering developed by Calogero. For nonstationary approximations, a new first‐order (in small quantities) bound is established which is particularly useful for partial waves other than s waves. Similar, but second‐order, bounds are established for approximations which are stationary. Some previous results, based on the use of the Lippman–Schwinger equation are generalized, and some new bounds are established. These are illustrated, and compared to previous results, by a simple example. We discuss the advantages and disadvantages of the present results in comparison to those derived previously. Finally, we present the generalization of some of the present formalism to the case of many‐channel scattering involving many‐particle systems, and discuss some of the difficulties of their practical implementation.

The existence of wave operators for oscillating potentials
View Description Hide DescriptionThe existence of the Mo/ller wave operators is proved for Hamiltonians of the form H=−Δ+a sinb r ^{α}/r ^{β}+V(x), where V is a short range potential, generally noncentral, and α and β take on suitable values including the case α=1, β≳ 1/2 .

Properties of a covering space defined by Hawking
View Description Hide DescriptionWe investigate the topological properties of a covering space introduced by Hawking in 1967 to assist in his derivation of some singularitytheorems.

Relationships between the group‐theoretic and soliton‐theoretic techniques for generating stationary axisymmetric gravitational solutions
View Description Hide DescriptionWe investigate the precise interrelationships between several recently developed solution‐generating techniques capable of generating asymptotically flat gravitational solutions with arbitrary multipole parameters. The transformations we study in detail here are the Lie groups Q and Q̃ of Cosgrove, the Hoenselaers–Kinnersley–Xanthopoulos (HKX) transformations and their SL(2) tensor generalizations, the Neugebauer–Kramer discrete mapping, the Neugebauer Bäcklund transformations I _{1} and I _{2}, the Harrison Bäcklund transformation, and the Belinsky–Zakharov (BZ) one‐ and two‐soliton transformations. Two particular results, among many reported here, are that the BZ soliton transformations are essentially equivalent to Harrison transformations and that the generalized HKX transformation may be deduced as a confluent double soliton transformation. Explicit algebraic expressions are given for the transforms of the Kinnersley–Chitre generating functions under all of the above transformations. In less detail, we also study the Kinnersley–Chitre β transformations, the non‐null HKX transformations, and the Hilbert problems proposed independently by Belinsky and Zakharov, and Hauser and Ernst. In conclusion, we describe the nature of the exact solutions constructible in a finite number of steps with the available methods.

On the singular eigenfunctions for linear transport in an exponential atmosphere
View Description Hide DescriptionWe prove that under an explicit condition on the parameters in the isotropic‐scattering linear transport equation for an exponential atmosphere, the continuum eigensolutions developed by Millikin and Siewert are complete on the half range 0<μ⩽1. We also treat numerically the equation for the outgoing flux, which can be derived using these eigenfunctions, and we show that excellent numerical results are obtained if the above condition is satisfied, while poor results are obtained if the condition is sufficiently violated. Finally, we describe a method for constructing elementary solutions of the anisotropic‐scattering transport equation for an exponential atmosphere.

Interaction of isovector scalar mesons with simple sources. II. Isobars and meson scattering
View Description Hide DescriptionFor the case of a static source, appropriate special coherent states are defined so as to give a relatively simple treatment of states of any isospin in the no‐meson and one‐meson approximations. The analogous localized coherent states for nonstatic sources are described and shown to be suitable for use in translated‐localized‐state calculations.

Quantum field theory of particles of indefinite mass. I
View Description Hide DescriptionA quantum field theory of particles of indefinite mass is derived using rigorous correspondence arguments starting with a classical theory of particles of indefinite mass. The classical theory can be recovered in a suitable limiting case as h/→0 by means of Ehrenfest’s theorem. Our deviation leads to a quantum mechanical waveequation which turns out to be basically the same equation investigated earlier by Fock, Nambu, and others, but differs from this earlier equation in our use of a new evolution parameter—herein called ’’evolution‐time’’—defined as proper time divided by the classical mass. Owing to our use of evolution time as the development parameter of the system our Fock equation is without any reference to a mass parameter, in contrast to the older Fock equation. The indefiniteness of the particle mass frees the ordinary time, t—herein called ’’observer’s time’’—of any fixed relationship to the evolution time, and the two times becomes quite independent parameters. The Hamiltonian of our system turns out to be (minus half) the total mass squared of the system and is a constant of the motion. In order to guarantee negative definiteness of the Hamiltonian of the second quantized system, it is necessary to quantize our Fock equation using Fermi–Dirac statistics. A real scalar field is described by an equation which is second‐order in the evolution time, obtained by iterating our first‐order Fock equation. The second‐order Fock equation must be second quantized using Bose–Einstein statistics, in order to preserve the interpretation of the Fourier amplitudes as creation and annihilation operators. The propagator for the second‐order Fock equation has a Pauli–Villars type sum of terms, in which one term describe the propagation of timelike states, the other term describes space‐like states.

On computing eigenvalues in radiative transfer
View Description Hide DescriptionThe Wiener–Hopf factorization of the dispersion function is used to deduce explicit expressions for the discrete eigenvalues in the theory of radiative transfer.