Volume 16, Issue 2, February 1975
Index of content:

The construction of solution of nonlinear relativistic wave equation in λ:Φ^{4} _{4}: theory
View Description Hide DescriptionThe nonlinear equation (⧠+m ^{2}) Φ (x) =λ:Φ^{3}: for the quantum scalar field with special form of ordering of the interaction term is considered. The unique soultion Φ (x) of this equation is constructed. It satisfies the relativistic covariance, asymptotic, irredicibility and the relativistic primitive causality conditions in the sense of sesquilinear form. The spectral condition is also satisfied.

Application of nonstandard analysis to quantum mechanics
View Description Hide DescriptionQuantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.

An algebraic representation of continuous superselection rules
View Description Hide DescriptionFrom the logic approach to quantum and classical mechanics, the W*−algebraic approach is deduced in dependence of a suitable ’’prestate.’’ An algebraic representation of the logic description is in fact constructed in a framework in which continuous superselection rules can be present. Logic propositions, observables, and states are represented by decomposable projections, decomposable self−adjoint operators, and normal states in a direct integral of Hilbert spaces. In this representation each algebraic term becomes the representative of a homologous logic one and the expectation values as well as the superselection rules are conserved. When a principle of ’’undistinguishability’’ is taken into account, the representation is faithful. In the classical case, the representation results in Koopman’s formalism.

Scattering of a finite beam in a random medium with a nonhomogeneous background
View Description Hide DescriptionWe consider the scattering of a finite beam of radiation in a random medium with a nonhomogeneous background that contains linear and quadratic variation. We assume that both the fluctuations in the random medium and the background nonhomogeneity are weak. We obtain general expressions for the coherence function and intensity distribution. We present explicit solutions in the multiple scattering region. Results are compared to the case in which the background is homogeneous.

Riemann−Green’s functions for solving electromagnetic problems exhibiting rotational symmetry in media moving with superluminal velocities
View Description Hide DescriptionA new method of investigation of the electromagnetic field configuration in media moving with superluminal velocities is illustrated. The basic assumption is the rotational symmetry of the field. It is shown that a Volterra integral equation can be written for the magnetic field on the metallic boundary surface. This equation rests on the knowledge of the Riemann−Green’s functions of two differential operators. Some expressions for these functions are obtained.

Closed gravitational−wave universes: Analytic solutions with two−parameter symmetry
View Description Hide DescriptionEinstein’s vacuum field equations are solved for spacetimes with two−parameter spacelike symmetry, a space−reflection symmetry, and space sections homeomorphic to either S ^{1}×S ^{2} or S ^{3}. All integrals are evaluated, and the spacetime metrics are presented in analytic form.

Clebsch−Gordan coefficients for crystal space groups
View Description Hide DescriptionA practical method for calculating Clebsch−Gordan coefficients for crystal space groups is presented. It is based on properties of the space group irreducible representations as induced from ray representations of subgroups. Using this method, we obtain all Clebsch−Gordan coefficients for a family of representations in a single calculation: For space groups, for a given triangle of stars *k, *k′, *k ^{″}, where *k ⊗ *k′ ‐ *k ^{″}, the coefficients for all allowable little group representations l, l′, l ^{″} are obtained. In the following paper this is applied to rocksalt O ^{5} _{ h }−F m3m and diamondO ^{7} _{ h }−F d3m space groups.

Clebsch−Gordan coefficients for *X ⊗ *X in diamond O ^{7} _{ h }−Fd3m and rocksalt O ^{5} _{ n }−Fm3m
View Description Hide DescriptionBy using the method described in the previous paper, based on properties of space group irreducible representations as induced from ray representations of subgroups, Clebsch−Gordan coefficients are calculated for *X ⊗ *X in diamondO ^{7} _{ h }−F d3m and rocksalt O ^{5} _{ h }−F m3m structures. Tables of coefficients for these stars are presented. An example of explicit calculation of the coefficients is given for these symmorphic and nonsymmorphic groups with multiplicity included in the former.

On the spinning axis representation
View Description Hide DescriptionA new spinning axis representation is introduced. It allows us to calculate the evolution operator of a system with slowly varying time−dependent Hamiltonian with the desired degree of approximation in the parameter used for describing its dynamical evolution. The procedure is compared with a previously existing one and applied to a simple example.

An exact solution of the spin−spin autocorrelation function for a one−dimensional system of hard rods
View Description Hide DescriptionIt is shown exactly that, in a one−dimensional system of hard rods with spins, the autocorrelation function of any function of spin F(w) decays as t ^{−1} at long times provided that 〈F〉_{eq} exists and that g (0) ≠ 0, where g (v) is the linear velocity distribution function. As a consequence of this, when F (w) = w, the spin diffusion coefficient defined by the Kubo relation D _{ s } = F^{∞} _{0} 〈w (0) w (t) 〉 d t does not exist. The results are true for arbitrary initial equilibrium velocity and spin distributions, the only restriction being that they be symmetric.

Ray and explosive solutions of nonlinear evolutional equations in Hilbert spaces
View Description Hide DescriptionMany nonlinear phenomena occurring in physical systems are describable by a set of ordinary, partial or functional differential equations which can be regarded as an evolutional equation in a suitable abstract vector space. In this paper, we consider nonlinear evolutional equations defined on Hilbert spaces. Attention is focused on developing conditions for the existence of solutions which lie along half−rays emanating from the origin of the space. The results are used to establish sufficient conditions for the existence or nonexistence of explosive solutions or solutions having finite escape time. The paper concludes with a discussion of the application of some of the results to specific classes of evolutional equations arising from physical situations.

Construction of spin−orbit potentials from the phase shifts at fixed energy
View Description Hide DescriptionThe nonrelativistic scattering of spin−1/2 particles by central and spin−orbit potentials is considered. The form of central and spin−orbit potentials is deduced from a knowledge of the S matrix as a function of angular momentum at a fixed energy. Similar to the case of central potentials, the problem of constructing central and spin−orbit potentials from information on the phase shifts at a fixed energy has an infinity of solutions, depending on an infinite number of parameters.

A theorem on the adiabatic scaling of classical orbits
View Description Hide DescriptionThe theorem describes the effect on the motion when the Hamiltonian is scaled, that is, when dimensioned parameters in the Hamiltonian are varied without changing any dimensionless parameters in it. The motion changes in a way that is not in general simple or predictable via dimensional analysis. But under certain conditions the motion scales with the Hamiltonian: Classical orbits scale in coordinate space and constants of the motion scale according to their dimensions. The theorem is based on the observation that the observable D which generates dilations may be expressed as a time derivative of a simple quantity. We find it most direct to use quantum mechanics to describe dynamics in the adiabatic limit. In so doing we develop methods that may be useful for other classical adiabatic problems.

Functional integration, Padé approximants, and the Schrödinger equation
View Description Hide DescriptionA method for finding the eigenvalues and the eigenvectors of the Schrödinger Schrödinger equation is presented. If H = T + V is an M−body Hamiltonian, we use Trotter’s formula in the form e ^{−βT/n } e ^{−βV/n } ∼ e ^{−βH/n } (for n large). This allows the computation of the matrix elements of e ^{−βH } in the configuration representation, and the moments μ_{ r } = (ψ,(e ^{−βH })^{ r }ψ) (r=1,2,⋅⋅⋅) for any wavefunction ψ. From the moments μ_{ r } we compute the [N − 1/N] Padé approximant, whose poles are the approximate eigenvalues of e ^{−βH }. The convergence of the method is proved and asymptotic formulas for the matrix elements of e ^{−βT } projected on states of given angular momentum are derived.

Particles and simple scattering theory
View Description Hide DescriptionA necessary and sufficient condition for a pair of subspaces to be the in and out scatteringsubspaces for a simple scattering system is obtained. It involves the existence of certain representations of the Galilean presymmetry group on these subspaces. The physical interpretation is that, in scattering, particles remain as particles even in the presence of the interaction.

Variational methods and nonlinear forced oscillations
View Description Hide DescriptionAn approximate, direct variational method, simple in concept, and straightforward in application is presented to deal with the problem of forced oscillation of nonlinear systems. The general procedure is illustrated in detail by treating a particular example, i.e., the Duffing’s equation. The same procedure is also applied to some other examples in mathematical physics.

Degeneracies in energy levels of quantum systems of variable dimensionality
View Description Hide DescriptionIntroduction of variable dimensionality to the Schrödinger equation gives rise to interdimensional degeneracies in the energy levels of one−, two−, and three−electron atoms and molecules. In all cases the degeneracies result from a factorization of the wavefunction into a product of a ’’radial’’ type function times an ’’angular’’ type function. Scaling of the orbital angular momentum quantum number in the one−electron radial equation to obtain ’’excess angular momentum’’ is shown to be equivalent to a variation in dimensionality.

Electrostatic screening
View Description Hide DescriptionUsing the methods of partial differential equations and functional analysis, we investigate the electric field in the presence of a screen composed of wires of radius r spaced at distance R spread over a surface S. In the limit as r and R converge to zero if [R lnr]^{−1} → −∞, the field in the presence of the screen converges to the field with a conducting sheet spread over S. If [R lnr]^{−1} → 0, the field

A generalized theory of multiplicative stochastic processes using cumulant techniques
View Description Hide DescriptionThe rules for the construction of the nth order cumulant for time−dependent, stochastic, matrices or operators which do not commute with themselves at unequal times are derived. The results are identical with van Kampen’s rules. In the Gaussian case, Kubo’s concept of a generalized Gaussian process is criticized. Under certain conditions Kubo’s idea becomes asymptotically valid, while the same conditions justify use of the author’s earlier delta functiontheory. A generalized density matrix equation is presented and its behavior during the approach to equilibrium is discussed. A finite correlation time, τ_{ c }, does not necessarily invalidate a monotonic approach to equilibrium.

The unitary irreducible representations of (3,R)
View Description Hide DescriptionThe unitary irreducible representations of the universal covering group, (3,R), of the S L (3,R) group are analyzed by means of the methods developed by Harish−Chandra and Kihlberg. We have found a single closed expression for the matrix elements of the noncompact generators for an arbitrary unitary representation of the (3,R) group. The irreducibility of the representations is achieved by using the little group technique and the scalar product for each irreducible Hilbert space is explicitly given. Contraction (in the Inönü and Wigner sense) of the (3,R) unitary irreducible representations to the corresponding representations of the T _{5} σ S U (2) group is discussed.