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Volume 10, Issue 6, June 1969

Degenerate Representation Functions for SO(v), SU(v), and and Their Analytical Reductions
View Description Hide DescriptionWe have studied the problem of obtaining the rotation matrix elements d^{N} (θ) = 〈Ne ^{−iθJ 2 }N〉, where N〉 refer to a particular degenerate class of basis vectors of a symmetry group G which embraces the rotations SU(2)_{ J } as a subgroup. For G = SO(v), SU(v), and we prove that these particular representation functions are proportional to the Gegenbauer polynomials and respectively. The reduction of such functions into one another according to the formula has been solved in generality for complex values of N and corresponds to the reduction of Regge poles of G into Regge poles of one of its subgroups. The reduction formula for functions of the second type has also been derived; here one simply meets an infinite series.

Comments on Kinetic Equation for Autocorrelation Functions
View Description Hide DescriptionThe non‐Markoffian kinetic equation for the one‐particle momentum autocorrelation function, derived by Zwanzig and studied in great detail recently by Berne, Boon, and Rice, is analyzed in the weak coupling limit. It is shown that, in this limit, this kinetic equation remains non‐Markoffian because the kernel which determines the memory effects only decays very slowly. More precisely, it tends to zero over times of the order of the relaxation time itself and not, as could be expected, over the much shorter collision time. The comparison with the more traditional approach, based on the solution of a transport equation, is also discussed.

Gel'fand‐Kirillov Conjecture on the Lie Field of an Algebraic Lie Algebra
View Description Hide DescriptionThis is an article written to review with sufficient detail the so‐called Gel'fand‐Kirillov conjecture concerning the isomorphisms between the quotient fields of the algebras generated by canonical variables [p_{i}, q_{j} ] = δ_{ ij }1 and the quotient fields of the universal enveloping algebras of algebraic Lie algebras. This conjecture sheds new light on the relation between the universal enveloping algebra of an algebraic Lie algebra and the Lie algebras of dynamic groups in quantum mechanics.

Broken Symmetry and Generalized Bose Condensation in Restricted Geometries
View Description Hide DescriptionIt is suggested, on the basis of soluble models, that generalized Bose condensation results from a broken symmetry associated with a nonvanishing pairing amplitude with very large low‐momentum components, leading to a nonzero ``pairing density'' p_{c} as well as a nonzero generalized condensate density ρ_{ c }. For systems of interacting bosons it is proved that (1) nonzero p_{c} implies nonzero ρ_{ c }, and (2) p_{c} = 0 in one or two dimensions and, more generally, for geometries finite in one dimension and infinite in the other two (films) or finite in two dimensions and infinite in one (pores). It is pointed out that this does not exclude superfluidity in such geometries, but does show the need of a new mechanism to explain it.

Nonlinear Perturbations
View Description Hide DescriptionThe perturbation theory of Bogoliubov and Mitropolsky for systems having a single rapid phase is generalized to systems having several rapid phases. It is shown that one can avoid the classic problem of small divisors to all orders in the perturbation theory. The method has the advantage of providing a single approach to many problems conventionally treated by a variety of specialized techniques.

Phase‐Integral Approximation in Momentum Space and the Bound States of an Atom. II
View Description Hide DescriptionThe phase‐integral approximation of the Green's function in momentum space is investigated for a particle of negative energy (bound state) which moves in a spherically symmetric potential. If this potential has a Coulomb‐like singularity at the origin, it is shown that any two momenta can be connected by an infinity of classical trajectories with a fixed energy. The summation of the usual phase and amplitude factors over these trajectories is the approximate Green's function. If there is orbital precession, there are not only poles along the negative energy axis, but also weaker singularities which are not examined in detail. The poles are found at the energies which are given by the semiclassical quantum conditions: angular momentum = (l + ½)ℏ and action integral for the radial motion = (n + ½)2πℏ, where l and n are integers ≥ 0. The residues at these poles give the approximate bound‐state wavefunctions as a product of the asymptotic formula for Legendre polynomials with the asymptotic solution of the radial Schrödinger equation. It is conjectured that the occurrence of poles in the approximate Green's function is a direct consequence of the periodic character of the classical motion.

Dynamics near Equilibrium of Systems Described in Thermal Hartree‐Fock Approximation
View Description Hide DescriptionThe dynamics of systems described in Hartree‐Fock approximation is studied near a stationary point of the free energy. It is shown that the second‐order free‐energy functional is a constant for the linearized self‐consistent equation of motion. This leads to the stability criterion derived by Mermin. A simple collision model is constructed and is shown to satisfy the Htheorem. It exhibits the effect of critical slowing down. The formalism is shown to be applicable to superconductors.

Next‐Nearest‐Neighbor Ising Model
View Description Hide DescriptionThe next‐nearest‐neighbor two‐dimensional Ising model is cast into a form which resembles a one‐dimensional interacting many‐fermion system. An approximation, which has previously been shown to be successful in giving the critical properties of the two‐dimensional ferroelectric problem, is used. From the approximate expression obtained, the critical indices are found to be α = α′ = 0, β = ⅛, which agrees with the results obtained from series expansions and plausible physical arguments. The critical temperature obtained agrees to within 6% of the series expansion results.

Geometrization of the Brans‐Dicke Scalar Field
View Description Hide DescriptionThe Brans‐Dicke gravitational scalar field is geometrized in the spirit of the Rainich‐Misner‐Wheeler geometrization of electromagnetism. Geometric equations are derived which imply that the Brans‐Dicke field is present and an explicit expression is given for this field in terms of geometrical quantities.

Expansion of the Density Matrix of an N‐Fermion System in Terms of the Correlation Densities of Fermion Clusters
View Description Hide DescriptionThe N‐fermion density operator D = Ψ〈〉Ψ is decomposed into the densities of clusters of correlating fermions. The m‐body cluster densities are ``orthogonal'' to the one‐body ``Fock‐Dirac''‐type densities ρ_{1}, i.e., The reduced‐correlation‐density matrices obtained differ from the conventional reduced‐density‐matrices and are particularly convenient for the treatment of fermion‐fermion correlations.

Algebraic Matrices for the Configurations (d + s)^{ n } p
View Description Hide DescriptionThe construction of the algebraic matrices for the configurations (d + s)^{ n } p in L–S coupling is discussed. Particular emphasis is given to configurations with a half‐filled d shell. From a total of 42 parameters specifying the various interactions for the configurations (d + s)^{ n } p only the matrices of 8 parameters need to be calculated explicitly for the complementary configurations (d + s)^{ n } p, n > 6. The matrices of the other 34 parameters can be obtained from the corresponding algebraic matrices of (d + s)^{ n } p, n ≤ 6, either directly or by simple changes in sign.

General Interaction Picture from Action Principle for Mechanics
View Description Hide DescriptionIn this paper we consider a general action principle for mechanics written by means of the elements of a Lie algebra. We study the physical reasons why we have to choose precisely a Lie algebra to write the action principle. By means of such an action principle we work out the equations of motion and a technique to evaluate perturbations in a general mechanics that is equivalent to a general interaction picture. Classical or quantum mechanics come out as particular cases when we make realizations of the Lie algebra by derivations into the algebra of products of functions or operators, respectively. Later on we develop in particular the applications of the action principle to classical and quantum mechanics, seeing that in this last case it agrees with Schwinger's action principle. The main contribution of this paper is to introduce a perturbation theory and an interaction picture of classical mechanics on the same footing as in quantum mechanics.

Quadratic Action Principle of Relativity
View Description Hide DescriptionBy an improved mathematical technique, the field equations derivable from a Lagrangian which is quadratic in the curvature components can now be studied in greater detail. The intercalation of a high‐frequency metrical plateau between the flat Minkowskian metric and the macroscopic physical world has the consequence that the resulting perturbation equations are no longer of fourth but only of second order, thus making a comparison with Einstein's equations possible. The principal difference is that the free vector of Einstein's theory is now restricted by a divergence condition, with the result that the equations of the electromagnetic field, expressed in terms of the vector potential, become solutions of the macroscopic field equations. The cases of fourfold symmetry with imaginary time and 3 + 1 symmetry with real time are discussed. The gravitational phenomena and the second‐order interaction terms, needed for the construction of material particles, remain outside the limits of the present investigation.

Representations of the Orthogonal Group. II. Polynomial Bases for the Irreducible Representations of the Orthogonal Group
View Description Hide DescriptionPolynomial bases for the irreducible representations of the orthogonal group, which are characterized by the Gel'fand pattern, have been obtained. The method used is very similar to Moshinsky's and is a generalization from the unitary group to the orthogonal group. The Wigner coefficients of O(3), commonly called the Clebsch‐Gordan coefficients of R(3), are rederived by means of the polynomial bases obtained in this paper.

Codiagonal Perturbations
View Description Hide DescriptionMatrix methods are developed for calculating the eigenvalues and eigenvectors of a large class of quantum‐mechanical operators which may be regarded as perturbed forms of special‐function operators. Specific representations are obtained for the latter, including all of the important cases treated by Infeld and Hull. To these are added representations for terms sufficient to generate forms corresponding to the Mathieu equation, the Lamé equation, and others. A rapidly convergent computational scheme applicable to asymmetric matrices, which retains its stability even when the perturbing terms become large, is described; and its use is illustrated by application to the operator p(1 − q ^{2})p − α^{2} q ^{2}, corresponding to the Legendre‐like form (d/dx)(1 − x ^{2})(d/dx) + α^{2} x ^{2}. Though group‐theoretic considerations are stressed, appropriate correlations with differential and integral equations are presented throughout.

Lattice Dynamics of Simple Cubic Lattices with Long‐Range Interactions
View Description Hide DescriptionAn analytic study is made of the dispersion relations and frequency spectra of simple cubic lattices in which there exist long‐range potentials of the form 1/r^{p} . The rigid‐ion approximation is used. An expansion of the dispersion relations about the maximum propagation vector in the first Brillouin zone is obtained for 1 ≤ p ≤ 3 and the contribution of the region about this point to the vibrational frequency spectrum of the lattice is studied.

Sufficient Conditions for Stability of the Faddeev Equations
View Description Hide DescriptionFunctional analysis techniques are used to obtain stability theorems for the Faddeev equations in the quantum‐mechanical three‐particle nonrelativistic scattering theory. Sufficient conditions are obtained in order that the solutions of these equations not be sensitive to small variations of the off‐shell two‐particle amplitudes. These conditions provide criteria for the validity of some of the previous formal investigations of the Faddeev equations.

Evaluation of ``Kondo'' Integrals
View Description Hide DescriptionExact and approximate expressions are given for the integrals.

Magnetization of Ising Model in Nonzero Magnetic Field
View Description Hide DescriptionKnowing only the zero‐field magnetization (e.g., Yang's result) of the Ising model in any number of dimensions, one can construct a lower bound on m(h), the magnetization in finite field. Knowledge of u, the internal energy per bond, enables a more efficient lower bound to be constructed. Both are applications of the Griffiths inequality, as recently generalized by Kelly and Sherman, and should prove useful in the lattice gas problem where it is essential to know m(h).

Group Representation in a Continuous Basis: An Example
View Description Hide DescriptionGiven an irreducible unitary representation of a noncompact group, what happens if one tries to diagonalize one of the noncompact generators? We study some aspects of this question on an example, chosen to be a representation of the discrete series with j = −½ of the special real linear group in two dimensions.