Index of content:
Volume 16, Issue 10, October 1975

Exact recursive evaluation of 3j‐ and 6j‐coefficients for quantum‐mechanical coupling of angular momenta
View Description Hide DescriptionAlgorithms are developed for the exact evaluation of the 3j‐coefficients of Wigner and the 6j‐coefficients of Racah. These coefficients arise in the quantum theory of coupling of angular momenta. The method is based on the exact solution of recursion relations in a particular order designed to guarantee numerical stability even for large quantum numbers. The algorithm is more efficient and accurate than those based on explicit summations, particularly in the commonly arising case in which a whole set of related coefficients is needed.

Semiclassical approximations to 3j‐ and 6j‐coefficients for quantum‐mechanical coupling of angular momenta
View Description Hide DescriptionThe coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.

An inverse problem in statistical mechanics
View Description Hide DescriptionWe study some one‐dimensional systems specified by their nearest neighbor distribution. It is found that systems exhibiting second order phase transitions can be constructed. This paradoxical situation is resolved by a consideration of the peculiarities of the underlying potential. It is shown that the usual compressibility equation is not satisfied for these systems.

Direct determination of the Iwasawa decomposition for noncompact semisimple Lie algebras
View Description Hide DescriptionA direct method for the determination of the Iwasawa decomposition of any noncompact semisimple real Lie algebra is described in detail. It is based on the canonical form of the Lie algebra. The physically important Lie algebrass o (3,1), s o (4,1), s o (3,2), and s o (4,2) are treated as illustrative examples.

Two‐point characteristic function for the Kepler–Coulomb problem
View Description Hide DescriptionHamilton’s two‐point characteristic function S (q _{2} t _{2},q _{1} t _{1}) designates the extremum value of the action integral between two space–time points. It is thus a solution of the Hamilton–Jacobi equation in two sets of variables which fulfils the interchange condition S (q _{1} t _{1},q _{2} t _{2}) =−S (q _{2} t _{2},q _{1} t _{1}). Such functions can be used in the construction of quantum‐mechanical Green’s functions. For the Kepler–Coulomb problem, rotational invariance implies that the characteristic function depends on three configuration variables, say r _{1},r _{2},r _{12}. The existence of an extra constant of the motion, the Runge–Lenz vector, allows a reduction to two independent variables: x≡r _{1}+r _{2}+r _{12} and y≡r _{1}+r _{2}−r _{12}. A further reduction is made possible by virtue of a scale symmetry connected with Kepler’s third law. The resulting equations are solved by a double Legendre transformation to yield the Kepler–Coulomb characteristic function in implicit functional form. The periodicity of the characteristic function for elliptical orbits can be applied in a novel derivation of Lambert’s theorem.

On the quantum mechanical treatment of dissipative systems
View Description Hide DescriptionTwo types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time‐dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.

The renormalized projection operator technique for nonlinear stochastic equations. III
View Description Hide DescriptionThe solution of the nonlinear stochastic equationL (x,t,ω) ψ (x,t,ω) =g (x,t)+f [ψ (x,t,ω)] is found via the renormalized projection operator technique and is approximated to be 〈ψ (x,t) 〉=FFd x′d t′ 〈G _{ p }(x,t ‖x′,t′) 〉 {ψ_{int}(x′,0)+g (x′,t′) } −FFd x′d t′〈P (x′,t′) 〉 = o e ing−brace)=FFd x ^{″} d t ^{″}〈G _{ p }(x,t ‖x′,t ′) G _{0}(x′,t′‖x ^{″},t ^{″}〉 [ψ_{int}(x ^{″},0)+g (x ^{″},t ^{″})]}+ FFd x′d t′〈G _{ p }(x,t ‖x′,t′〉 〈f [ψ^{ H }(x′,t′)]〉. The terms 〈G _{ p }(x,t ‖x′,t′) 〉 and 〈G _{ p }(x,t ‖x′,t′) G _{0}(x′,t′‖x ^{″},t ^{″}) 〉 are the stochastic one‐ and two‐point Green’s functions. Also three conditions are shown that the projection operator must have in order to insure convergence.

Nonlinear canonical transformations and their representations in quantum mechanics
View Description Hide DescriptionIn the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions f _{α} (α=1,...,2n) of x _{ i }, p _{ i } (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions f _{α} have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher‐dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. We discuss briefly, in the concluding section, the possibility of extending our analysis to canonical tranformations that do not conserve the spectrum of the relevant operators.

Higher‐dimensional unifications of gravitation and gauge theories
View Description Hide DescriptionWe give a comprehensive geometric treatment of Kaluza–Klein type unifications of non‐Abelian gauge theories with gravitation. The appearance of a cosmological term is noted.

Exact quantization of the nonlinear Schrödinger equation
View Description Hide DescriptionBy means of an ’’inverse scattering transform,’’ we can exactly quantize the one‐dimensional nonlinear Schrödinger equation ih/Ψ_{ t }=−(h/^{2}/2m) Ψ_{ x x }−ε^{2}(Ψ*Ψ) Ψ for any value of ε^{2}=real. When ε^{2}<0, the eigenvalues of the number operator, field momentum operator, and the Hamiltonian are found to be e x a c t l y the same as the linear case. In other words, by quantizing the exact theory, no effects corresponding to ’’renormalization’’ are found, and the zero point energy is independent of ε^{2}. When ε^{2}<0, the Hamiltonian is unbounded from below, and, in addition to the above spectra of eigenvalues,bound states can occur. Each bound state can be interpreted to be a bound state of n ’’excitations,’’ moving in a coherent fashion and with a binding energy proportional to the cube of the number of excitations. This problem is also formally equivalent to the N‐body problem with a delta‐function interaction solved by Bethe, with which we shall contrast our results, and we shall conclude by making certain remarks concerning ordinary field quantization versus ’’scattering space’’ quantization.

Perturbative solution to order βε of the Percus–Yevick equation for the square‐well potential
View Description Hide DescriptionThe radial distribution function (RDF) of a fluid is considered for the case of the square‐well potential. If the RDF is expanded in powers of the depth of the square‐well, ε, the first two terms are, in most applications, the most important. The Percus–Yevick (PY) integral equation for the RDF is examined and the resulting integral equations for these terms obtained. The first set of equations are just the PY equations for hard spheres which have been solved analytically. In this paper, the remaining equations for the terms of order βε, where β=1/k _{ B } T, are solved analytically and the results examined. We have speculated in the past that the PY theory could be used to obtain estimates of higher‐order terms in a perturbation expansion of the RDF. We find that the PY theory cannot give reliable estimates of these higher‐order terms for the square‐well potential at high densities.

Internal‐labeling operators
View Description Hide DescriptionA simple method is described for finding all possible ’’missing label’’ operators when a semisimple group is reduced according to a maximal semisimple subgroup. The operators may be chosen to be Hermitian and hence lead to an orthonormal basis. The solution is worked out for all seven cases of one missing label. In each case two independent subgroup scalars are found in the enveloping algebra of the group; either of them can be used as the missing label.

An asymptotic formula for the twisted product of distributions
View Description Hide DescriptionThe expansion theorem for the twisted product associated with the Weyl form of CCR for n degrees of freedom is generalized to involve tempered distributions.

Master equations in quantum stochastic processes
View Description Hide DescriptionPauli master equation is derived rigorously in the framework of quantum stochastic processes.

Semisimple graded Lie algebras
View Description Hide DescriptionThe concept of metric is introduced for graded Lie algebras. Semisimple graded Lie algebras are defined in terms of metric conditions of nonsingularity. It is shown that for this class of algebras the metric tensor generates a quadratic Casimir operator. Also for this class, the grading representation is irreducible and its weights are related to the roots of the Lie algebra (’’root‐weight theorem’’). The problem is solved to find all semisimple graded Lie algebras. For SU(N), N≳2, for O(N), N≳5, and for all exceptional groups there are none. For all other semisimple Lie algebras there is one and only one. These are explicity constructed in terms of a convenient realization of Sp(2N) matrices. SU(2) is discussed in some detail and a new group [GSU(2)] is found which leaves a mixed c‐number/q‐number quadratic form invariant. We also define irreducible tensor operators for this group. SU(N), N≳2, provides examples of nonsemisimple gradings.

Transport equations for the Stokes parameters from Maxwell’s equations in a random medium
View Description Hide DescriptionBeginning with Maxwell’sequations in a random medium and following a perturbation procedure, we obtain transport equations for the Stokes parameters. We compare our equation with Chandrasekhar’s transport equation and find that they agree when the random medium is specialized appropriately. We also examine the role of degeneracy in the perturbation analysis.

Gauge transformations and normal states of the CCR
View Description Hide DescriptionLet ω be any normal state on the CCR‐algebra, and τ_{χ} the gauge transformation corresponding to the continuous character χ on the test function space as an Abelian group; denote by K the set of these characters; then we prove that the linear hull of {ω_{0}τ_{χ}‖χ‐K} is norm dense in all normal states. It is also proved that the theorem is in general false if we take the convex hull.

New class of exact solutions of the Dirac equation
View Description Hide DescriptionA new class of exact solutions of the Dirac equation with external electromagnetic fields is derived by assuming a set of field‐dependent solution matrices which obey an algebra isomorphic to the Pauli matrices. The method of exact solution may be applied to any field having a four‐vector potential A ^{μ} depending only on k ^{μ} x _{μ}, but for which the field tensor and initial electron momentum are such that A ^{μ} A _{μ}, A ^{μ} p _{μ}, (σ^{μ} ^{ν} F _{μ} _{ν})^{2}, and (σ^{μ}νF′_{μ} _{ν})^{2} are independent of k ^{μ} x _{μ}. Exact solutions for a circularly polarized propagating electromagnetic wave in an isotropic medium, for a screw symmetric static magnetic field, and for a rotating uniform electric field are given in terms of the roots of a quartic equation. A class of solutions is given explicitly in the weak field limit to lowest order in e A/m c ^{2}. The vacuum limit of the solution of a wave propagating in a medium is shown to be the Volkov solution.

On the determination of the relativistic wave equations associated with a given representation of S L (2,C)
View Description Hide DescriptionStraightforward algebraic techniques are presented and used to determine the structure of wave equations whose relativistic covariance is governed by two representations of S L (2,C), S _{0}(Λ) = (1,1/2) ⊕ (1/2,1) ⊕ (1/2,0) ⊕ (0,1/2) and S _{1}(Λ) = S _{0}(Λ) ⊕ (1/2,0) ⊕ (0,1/2), subject to the requirements that the equations should be parity preserving, admit an invariant Hermitian bilinear form realized by a numerical matrix η, and that they should describe a particle with a unique mass and spin. It is shown that S _{0}(Λ) leads to a unique algebraic structure, that of the Rarita–Schwinger equation, whereas S _{1}(Λ) leads either to a trivial extension of the former case or to a family of equations whose matrices have a minimal algebra with degree one higher than that of the former case. One such example reproduces the equation presented by Glass. When, contrary to custom, a singular η matrix is considered, it is shown that S _{1}(Λ) allows for equations whose coefficient matrices are reducible but indecomposable. These equations are completely equivalent to the Rarita–Schwinger equation in the free case, but the added components may enter the dynamics in the presence of certain interactions. The present examples serve to illustrate techniques which may be applied in the study of any relativistic wave equation.

On some geometrical aspects of classical nonconservative mechanics
View Description Hide DescriptionIt is shown that a scleronomous, holonomic dynamical system with nonconservative forces moves in such a way that the differential equations of motion are geodesic lines in a linear connected space L _{ n }. The space L _{ n } is semimetric and semisymmetric. The geodesic line on which the tangent at a point remains tangent if it is parallel displaced along the curve is simultaneously the curve of stationary length between two points in the space L _{ n }. A necessary condition for the stationary length is derived by making use of the noncommutation rule for the differential of variation and the variation of differential. The noncommutation rule is obtained from a quadrilateral, which is called the fundamental quadrilateral of variational calculus. By using the noncommutation rule, the variational principles of Maupertius and Hamiltonian type for nonconservative mechanical systems are presented.