Volume 15, Issue 5, May 1974

Regular perturbations, Brillouin‐Wigner expansion, and continued fractions
View Description Hide DescriptionThe least eigenvalue of an operator H = H _{0} + g V is considered, where H _{0} is a semibounded self‐adjoint operator in a Hilbert space and V is symmetric. It is shown that the J‐type continued fraction (i.e., the sequence of the [N‐1, N] Padé approximants) to the Brillouin‐Wigner perturbation expansion converges to the eigenvalue, provided V is a regular perturbation of H _{0}. An application of this result to some quantum mechanical systems, such as the helium atom, is briefly discussed.

Solving linear stochastic differential equations
View Description Hide DescriptionThe aim of this paper is to provide the user with tools for the solution of linear differential equations with random coefficients. Only analytic methods which lead to expressions in closed form for first and second order moments and probability distributions of the solution are considered. The paper deals both with approximate methods which require the existence of a small (or large) dimensionless parameter and with the method of model coefficients, where the true coefficients of the stochastic equation are replaced by random step functions with the same first and second order moments and probability distributions, chosen in such a way that the equation can be solved analytically. The second procedure does not rely on the existence of a small parameter.

Nonradiative algebraically special space‐times
View Description Hide DescriptionA recently characterized class of (in general nonvacuum) algebraically special space‐times with twisting rays is studied. The Weyl tensor satisfies the peeling‐off property along the repeated principal null congruence, and the Ricci tensor exhibits an equally simple asymptotic behavior, which is in fact compatible (via the Einstein field equations) with the presence of a suitably restricted electromagnetic or neutrino field. If the gravitational and source fields are nonradiative, the above asymptotic behavior is restricted. In this case we explicitly solve the Einstein vacuum field equations, the Einstein‐Maxwell equations and the Einstein‐Weyl (combined gravitational‐neutrino) equations. The solutions obtained are related to the known algebraically special solutions of these equations.

An action principle for superconductivity
View Description Hide DescriptionWe study a Lagrangiantheory of superconductivity which, for slowly varying field variables, is shown to be equivalent to the BCS theory; we obtain the conservation of pairs, clearly showing its relation with the second kind of gauge invariance of the Euler‐Lagrange equations. Finally, by treating the gauge function as a further field variable, we generalize the Maxwell‐type Ginzburg‐Landau equation and use it to discuss flux quantization.

Some consequences of the strengthened interpretative rules of quantum mechanics
View Description Hide DescriptionIn this paper some consequences of the strengthened interpretative rules of quantum mechanics, which were proposed in an earlier paper, are obtained. It is also seen that, in general, the usual interpretative rules are too weak to obtain these results. For example, it is proved from the strengthened rules that if f:R → R is a Borel function which is also τ‐definable, then for each observable A the procedure ``measure A and compute f(A outcome)'' is an f(A) measurement procedure. It is also shown that there exist Borel f and observables A such that the above procedure is not an f(A) measurement procedure. Two methods of measuring the sum A + B of two observables are then considered: The measurement of A and B on different systems followed by addition of the results and (if A and B commute) the simultaneous measurement of A and B on the same system followed by addition. It is proved from the strengthened rules that the first method is not a valid measurement procedure and the second is valid. Besides these, other processes such as procedures for preparing mixtures of different states, and the empirical generation of probability measures from outcome sequences are considered.

The group chain a complete solution to the ``missing label'' problem
View Description Hide DescriptionWe discuss the decomposition by constructing multiplier representations over the group manifold of S O _{n} . Explicit orthogonal and complete bases in terms of functions diagonal with respect to the canonical and noncanonical chains are provided which give a complete solution to the ``missing label'' and multiplicity problems occuring in the latter decomposition. Moreover, an integral representation for the overlap functions between the two chains is given, for which the singularity structure can be immediately ascertained. Expressions for the cases n = 3 and 4 are given.

Expansions in Breit‐Wigner amplitudes and biorthogonal functions
View Description Hide DescriptionThe problem of finding the expansion coefficients for a finite superposition of Breit‐Wigner amplitudes is discussed. Since such amplitudes are not mutually orthogonal, another set of functions orthogonal to the Breit‐Wigner functions are computed. Advantages and disadvantages of this biorthogonal set of functions are compared with conventional orthogonal functions and, in particular, the problem of reading off expansion coefficients when the modulus squared of the amplitude is known—rather than the amplitude itself—is discussed.

Spheroidal analysis of Coulomb scattering
View Description Hide DescriptionIn studying the spheroidal problem of a charged particle scattered by two charged centers, we have had to deal with a differential equation. Its solution was complicated. In this paper, we study a very similar differential equation, which appears in the Coulomb problem. The solution for this equation, however, can be put in a simple form. For completeness the spheroidal analysis of the Coulomb scattering amplitude is also discussed.

Spectral properties of phase operators
View Description Hide DescriptionThis paper studies the spectral properties of phase operators associated with the phase of the harmonic oscillator. It is shown that all such phase operators have an absolutely continuous part and that those of a certain subclass are absolutely continuous.

Brownian motion in assemblies of coupled harmonic oscillators
View Description Hide DescriptionThe generalized Langevin equation is derived for a particle of arbitrary mass in an assembly of harmonic oscillators with general interaction matrix and with an external force acting on the particle. The reduction of the equation to the ordinary Langevin equation is studied in various limits. The reduction to the Langevin equation is achieved apart from the distribution of the bath particles, and the results thus obtained are valid whether the bath is in equilibrium or not. It is found that if the Ford‐Kac‐Mazur interaction is assumed, the particle achieves Brownian motion regardless of its mass ratio to the bath particle. The weak coupling limit is effected by scaling the equation with the mass ratio. In the weak coupling limit, the explicit formula for the friction coefficient is obtained assuming a general interaction matrix. It is demonstrated that the generalized Langevin equation is a very convenient starting point for the study of Brownian motion.

A novel approach to the exact calculation of correlation functions of a one‐dimensional random Ising chain
View Description Hide DescriptionAn exact result for the partition function for a general one‐dimensional Ising chain of N spin‐1/2 particles described by the Hamiltonian is given. For an open strand, J_{N} = 0; for a closed chain, σ_{ N+1} = σ_{1}, J_{N} ≠ 0. The novelty of the trick used enables one to obtain the partition function and all the spin correlation functions for open and closed chains with equal ease. Special cases of this model have been discussed before to elucidate certain features of some biological systems. New expressions for parallel and perpendicular susceptibilities for this model are also derived. When J_{i} and H_{i} are treated as random variables, the above Hamiltonian describes a one‐dimensional Ising spin glass. In this case some simple models and formal averaging procedures are discussed.

Perpendicular susceptibility of two one‐dimensional Ising chains
View Description Hide DescriptionExact, closed expressions are obtained for the perpendicular and parallel susceptibilities of the one‐dimensional spin‐1/2 Ising model (i) with nearest neighbor interactions in a uniform magnetic field and (ii) with next‐nearest neighbor interactions but without the magnetic field, in the thermodynamic limit. Graphs of the susceptibilities as a function of temperature for various magnetic field strengths in (i), and as a function of different ratios of the interaction strengths in (ii), are discussed.

The general solution to Einstein‐Maxwell equations with plane symmetry
View Description Hide DescriptionThe general solution to Einstein‐Maxwell equations with plane symmetry is obtained. This solution has an extra Killing vector ξ^{ a } not assumed a priori. The solution may be classified in two classes depending on the sign of an integration constant. The first class depends only on the time; ξ^{ a } is spacelike. In the second class ξ^{ a } is spacelike, lightlike, and timelike in different regions like Schwarzschild's metric.

On the moments of the distribution of widths corresponding to an ensemble of random matrices
View Description Hide DescriptionIn this work, we discuss the moments of the width distribution corresponding to an ensemble of random matrices. The ensemble considered is one where the real and imaginary parts of the Hamiltonian matrices have Gaussian distributions with different widths. In particular, we give exact expressions for the moments when N = 2 (N = dimension of the Hamiltonian submatrix with a fixed set of quantum numbers) and give a method for obtaining a power series representation when N is arbitrary.

On dynamical groups: Classification of Lie algebras with Galilei subalgebras
View Description Hide DescriptionBecause of the pure group theoretical approach to the free nonrelativistic particle through an integrable irreducible representation of the quantum mechanical Galilei (Lie) algebraG, it is reasonable to construct so‐called dynamical (Lie) algebrasD which possess an integrable representation describing interacting (nonrelativistic) systems. Such dynamical algebrasD should contain the geometrical subalgebra G _{0} of G, spanned by the mass operator, the momentum, the angular‐momentum and the position operators. Furthermore, the relation between the free and the interacting system is simplified if the one‐dimensional Lie algebraT generating free time translations is a subalgebra of D. Hence preferred candidates for D are those Lie algebrasL which possess a subalgebra isomorphic to G or to G _{0}, i.e., those L for which an injective homomorphism exists. ε is called an embedding of G or of G _{0} in L. Our main result is a complete classification of (i) all nonsemisimple L with Levi decomposition with G or G _{0} embedding ε such that (ii) all complex semisimple L̄ with an embedding of the complex extension of G, (iii) all real simple L being real forms or realifications of the lowest dimensional L̄ (i.e., A _{5}, B _{3}, C _{4}) with G‐embedding.
The result gives a fairly complete list of all candidates for dynamical nonrelativistic algebras. The physical aspects of two of them, the conformal Galilei algebraG_{c} and a limitable dynamical algebraD^{t} , are discussed. A method for the construction of physically useful integrable representations for G_{c} and D^{t} is given. Some general properties of nonrelativistic and of relativistic dynamical algebras with G‐embeddings are considered.

New wave‐operator identity applied to the study of persistent currents in 1D
View Description Hide DescriptionWe show that a large class of backward‐scattering matrix elements involving Δk ∼ ± 2k_{F} vanish for fermions interacting with two‐body attractive forces in one dimension. (These same matrix elements are finite for noninteracting particles and infinite for particles interacting with two‐body repulsive forces.) Our results demonstrate the possibility of persistent currents in one dimension at T = 0, and are a strong indication of a metal‐to‐insulator transition at T = 0 for repulsive forces. They are obtained by use of a convenient representation of the wave operator in terms of density‐fluctuation operators.

On the application of quasi‐Lagrangian coordinates to random shear flows
View Description Hide DescriptionAn ensemble of homogeneous random shear flows with steady linear mean velocity profiles is considered from a purely kinematical point of view. Quasi‐Lagrangian coordinates (advected by the mean flow) are used so that a proper orthogonal decomposition of the fluctuation velocity field is possible and periodic boundary conditions can be imposed. The conditions of stationarity in time and incompressibility take on special forms when applied to wave‐vector moments. A simple application of the methodology is presented in the construction of a two‐dimensional random shear flow.

Prequantization of charge
View Description Hide DescriptionIt is shown that a necessary and sufficient condition for quantization of relativistic dynamics of a particle with charge e moving in an external electromagnetic field F is that (e/2π)F should define an integral de Rham cohomology class on the space‐time manifold.

Smoothing operators for field domains
View Description Hide DescriptionWe consider sharp‐time fields and write down, in terms of the fields, simple and explicit expressions for operators with very strong smoothing properties. When applied to any vector in Hilbert space, the resulting vector is in the domain of any power of the space‐smeared fields, and it even is entire for the fields. It is shown that in this way one obtains a common dense domain of definition on which the field operators are essentially self‐adjoint. Attention is focused on the space of rapidly decreasing C ^{∞} functions as smearing functions for the fields; here the smoothing operators are simply products of exponentials of the field smeared with Hermite functions.

Limit of the most degenerate representations of SO(p,1)
View Description Hide DescriptionThe limit of the group SO(p,1), for large Lorentz transformations, are studied. It is found that the basis functions of the most degenerate representations of SO(p,1), constructed according to the subgroups SO(p) and SO(p−1,1), asymptotically tend to the basis functions, constructed according to the corresponding contracted subgroups. This result is applied to the matrix elements of SO(p,1) to derive novel relations. The limit of the overlap functions are considered.