Volume 11, Issue 2, February 1970
Index of content:

On the Accuracy of the Adiabatic Separation Method
View Description Hide DescriptionNumerical experiments performed for a model of two strongly coupled oscillators indicate that the adiabatic separation method yields accurate results even where the condition of adiabaticity is violated to a very high degree, except in those cases where two levels are degenerate in the adiabatic approximation. An accurate solution for those cases can be obtained by diagonalizing the 2 × 2 Hamiltonian submatrix built on the two degenerate adiabatic states. It is conjectured that the adiabatic separation method can be expected quite generally to yield highly accurate results, at least for states belonging to the discrete spectrum.

Perturbation Theory for Large Coupling Constants Applied to the Gauss Potential
View Description Hide DescriptionAs an example of a perturbation technique for large coupling constants g ^{2}, we investigate the solutions and eigenvalues of the Schrödinger equation for a Gauss potential. In particular, we obtain the regular solution, valid for r ^{2} < 1/g, in terms of confluent hypergeometric functions by expanding the potential in the neighborhood of the origin. The Jost solution is obtained in an analogous manner in terms of a certain integral and is valid for r ^{2} > 1/4g. Both solutions are eigensolutions belonging to the same eigenenergy E = k ^{2}. These eigenvalues are derived in the form of large‐g asymptotic expansions which are useful and valid over a wide range of g. A noteworthy aspect of the investigation is the close analogy of the underlying mathematics with that of well‐known periodic equations.

Parastochastics
View Description Hide DescriptionThe generalized commutation relations A_{i}B_{j} − λB_{j}A_{i} = Γ_{ ij } I are introduced, where A_{i} and B_{i} are adjoint of each other, I is the identity, and Γ_{ ij } is a real covariance. For λ = +1 (−1, or 0, respectively) the parastochastic function A_{i} + B_{i} is given an interpretation in terms of Gaussian random functions (a two‐valued stationary Markov process, or infinite symmetric random matrices of the type considered by Wigner in connection with energy levels of heavy nuclei, respectively). In the Fock‐space realization, A_{i} and B_{i} appear as destruction and creation operators for bosons (fermions or boltzmannons, i.e., distinguishable particles, respectively). A few purely algebraic theorems are proved, which are applied to linear stochastic equations(equations with random coefficients). Existence and uniqueness being presupposed, mean Green's functions are shown to satisfy closed master equations. A linear functional differential master equation is obtained for equations with Gaussian coefficients. It is shown that the often‐used first cumulant‐discard closure assumption, which leads to a very simple master equation, is exact for differential equations with a two‐valued stationary Markovian coefficient. A parastochastic reformulation of the theory of Kraichnan is given, and his nonlinear master equations are, for the first time, rigorously derived without any recourse to perturbation theory or to diagrams. Kraichnan's random‐coupling model is obtained by replacing scalar stochastic quantities by Wigner matrices or, equivalently, bosons by boltzmannons (i.e., changing λ from +1 to 0). Finally, the nonlinear Kraichnan equation,,is reduced to a linear parastochastic equation in the Fock space; existence, uniqueness, boundedness, and asymptotic behavior are obtained.

Unique Hamiltonian Operators via Feynman Path Integrals
View Description Hide DescriptionThe old problem of how to represent uniquely a prescribed classical Hamiltonian H as a well‐defined quantal operator Ĥ is shown to have a clear answer within Feynman's path‐integral scheme (as expanded by Garrod) for quantum mechanics. The computation of Ĥ involves the momentum Fourier transform of a coordinate average of H. A differential equation for a reduced form of the Feynman propagator giving Ĥ from H is found; and the example of polynomialH worked out to give the Born‐Jordan ordering rule for Ĥ in this case.

Bethe‐Salpeter Equation and the Goldstein Problem
View Description Hide DescriptionThe Goldstein problem for the ladder‐approximation Bethe‐Salpeter equation for a spin‐½ fermion‐antifermion system bound to zero total mass by particle exchange is re‐examined. It is suggested that the problem arises because attention has previously been focused on the wrong Dirac‐space sectors of the equation. Criteria limiting the acceptable behavior of solutions are investigated and it is shown that discrete spectra for the values of the coupling constant are allowed to exist for the T‐A sector and S‐V sector solutions. Continuous spectra are always excluded.

Approximations for the Frequency Spectrum of a Simple Lattice
View Description Hide DescriptionThe simple cubic lattice with harmonic forces between nearest neighbors only is considered. Starting with the expression for the spectrum of squared frequency G(x) as a Fourier transform of a product of Bessel functions, the asymptotic expansions about singular points are studied. For a range of values of the ratio of noncentral to central force constants σ, simple approximations are obtained which give quantitative information on the spectrum, not only very near the singular points, but over a substantial part of the frequency range. For σ = ½, it has been found that a cusp at x = ½ has been overlooked in previous work. In this case, essentially the entire spectrum is dominated by five singularities (at x = 0, ¼, ½, ¾, 1), and can be represented accurately by simple expressions. The approximations developed break down for very small σ, but appear to work well for ¼ ≤ σ ≤ 1.

Calculation of High‐Field Distribution Functions in Semiconductors
View Description Hide DescriptionThe relation between various methods proposed recently for calculating the hot‐carrier distribution function in semiconductors is discussed. In particular, it is shown that the method deduced heuristically by Rees from considerations of the stability of the steady state is an iterative prescription for solving a suitably chosen integral form of the Boltzmann equation. It is shown that this method is essentially an adaptation of Kellogg's method with additional sufficiency conditions imposed to guarantee the existence of a positive solution and the convergence of the iterative process. The essential ingredient of these conditions is that the kernel of the integral equation be positive. It is further pointed out that the self‐scattering process introduced by Rees belongs to a larger class of operators that ensure the required positivity of the kernel.

Ising‐Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice
View Description Hide DescriptionThe asymptotic behavior of the pair correlation ω_{2}(r) = 〈σ_{ 0 }σ_{ r }〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero‐point behavior of the correlation is of the form ω_{2}(r) ∼ ε_{0} r ^{−½} cos ⅔πr, where r = r is the spin separation and being the decay amplitude of the pair correlation at the Curie point(critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth‐order correlations ω_{4}(r) = 〈σ_{ 0 }σ_{ δ }σ_{ r } σ_{ r+δ }〉 − 〈σ_{ 0 }σ_{ δ }〉 〈σ_{ r }σ_{ r+δ }〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = r is obtained for all fixed temperatures.

Ising‐Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices
View Description Hide DescriptionA detailed discussion of pair correlations ω_{2}(r) = 〈σ_{ 0 }σ_{ r }〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given. The asymptotic behavior of ω_{2}(r) for large spin separation is obtained for ferromagnetic and antiferromagnetic lattices. The axial pair correlation for the ferromagnetic triangular lattice has the same qualitative behavior as that for the ferromagnetic rectangular lattice: There is long‐range order below the Curie pointT _{C} and short‐range order above. It is shown that correlations on the anisotropicantiferromagnetic triangular lattice must be given separate treatment in three different temperature ranges. Below the Néel point T _{N}(antiferromagneticcritical point), the completely anisotropic lattice exhibits antiferromagnetic long‐range order along the two lattice axes with the strongest interactions. Spins along the third axis with the weakest interaction are ordered ferromagnetically. Between T _{N} and a uniquely located temperature T _{D}, there is antiferromagnetic short‐range order along the two axes with the strongest interactions, and ferromagnetic short‐range order along the other axis. T _{D} is named the disorder temperature because it divides the short‐range‐order region T _{N} < T < T _{D} from the region T _{D} < T < ∞, in which the axial pair correlations have exponential decay with temperature‐dependent oscillatory envelope. There is no singularity in the partition function at T _{D}, so there are only two thermodynamic phases: ordered below the Néel point, and disordered above. Correlations at T _{D} decay exponentially. Finally, special consideration is given to the anisotropicantiferromagnetic lattice when the two weakest interactions are equal, and T _{N} = T _{D} = 0. The single disordered phase exhibits exponential correlation decay with oscillatory envelope for T > 0. The exact values of the axial pair correlations at T = 0 are calculated. For large spin separation r along the strong interaction axis, ω_{2} = (−1)^{ r }, and along the weak (equal) interaction axes ,where,and E is a decay constant relating to pair correlations at the Curie point of a square lattice.

On Hidden‐Variable Theories
View Description Hide DescriptionAn abstract definition of a general hidden‐variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense. It is noted that the Bohm‐Bub hidden‐variables example is contained in this theory and an attempt is made to clarify the position of this theory with respect to hidden‐variable impossibility proofs. The general definition is used in the consideration of quantum‐mechanical ordering and the measurement process.

Domain of Dependence
View Description Hide DescriptionThe various properties of the domain of dependence (Cauchy development) which have been found particularly useful in the study of gravitational fields are reviewed. The basic techniques for constructing proofs and counterexamples are described. A new tool—the past and future volume functions—for treating certain global properties of space‐times is introduced. These functions are used to establish two new theorems: (1) a necessary and sufficient condition that a space‐time have a Cauchy surface is that it be globally hyperbolic; and (2) the existence of a Cauchy surface is a stable property of space‐times.

Mathematical Description of a System Consisting of Identical Quantum‐Mechanical Particles
View Description Hide DescriptionIn this paper a rigorous description of a system consisting of identical particles is given for which the particle number is a superselection rule. A state of such a system is described by a sequence of density operators D = (D ^{0}, D ^{1}, …, D^{n} , …), where D^{n} acts in n‐particle space and the asymptotic behavior is determined by the requirement,where   denotes the trace norm. The (bounded) observables in turn are described by sequences of bounded self‐adjoint operators: B = (B ^{0}, B ^{1}, …, B^{n} , …) such that.The expectation value of the observable B in the state D can be expressed as,whereΓ denotes the expansion operator, whereas L stands for its adjoint, the contraction operator. The analog to the n‐representability problem, i.e., the so‐called representability problem, is put forward and its solution is connected with the solutions of the n‐representability problem for different n. Finally, a possible mathematical foundation of the BCS theory is given.

Evaluation of a Unitary Integral. II. Mandelstam Iteration
View Description Hide DescriptionThe methods of a previous communication are used to derive a general expression for the first Mandelstam iteration of inputs of the form,where the sum is over terms representing t‐channel resonances in narrow‐width approximation. The coefficients g_{j} (s, t) can be either analytic in s, as for an expansion in t‐channel partial waves, or meromorphic in s, as in a Padé‐ or Schlessinger‐type expansion.

Evaluation of Some Fermi‐Dirac Integrals
View Description Hide DescriptionThe evaluation of Fermi‐Dirac integrals is discussed for cases in which the Sommerfeld method fails. Such cases occur when the integrand has a singularity at the Fermi surface and when the integrand is a rapidly oscillating function. As examples, the first‐order exchange integral for electrons and the free‐energy integral of the noninteracting electron gas in a magnetic field are evaluated. The method uses a contour‐integral representation of the Fermi function (previously mentioned by Dingle), supplemented by Mittag‐Leffler type expansions.

Parametrization of Crossing Symmetric Amplitudes
View Description Hide DescriptionThe crossing equations of Balachandran and Nuyts are solved to give the most general expansion of the π^{0}‐π^{0} amplitude consistent with crossing symmetry and in which the partial wave expansion is manifest. We find that crossing symmetry imposes 2L + 1 constraints on the Lth partial wave if the lower waves are known. The simplest such constraint (in the units were ) is,where f _{0}(s) is the s‐wave π^{0}‐π^{0} amplitude.

Random Walks on Lattices with Traps
View Description Hide DescriptionWe consider random walks on simple cubic lattices containing two kinds of sites: ordinary ones and ``traps'' which, when stepped on, absorb the walker. We study two related problems: (a) the probability of returning to the origin and (b) the situation in which the particle can meet its end, not only by absorption at a trap, but also by a process, called spontaneous emission, which has a constant probability per step. In problem (b), we ask for the probability that emission, rather than absorption, occurs. The solution to (a) is known for 1 dimension, and given here for the 3‐, 4‐, … dimensional cases; the 2‐dimensional case remains unsolved. The solution to (b) is known for the 1‐, 3‐, 4‐, … dimensional cases; we give it for 2‐dimensional case.

Polariton‐Phonon Interaction in Molecular Crystals
View Description Hide DescriptionA general theory is presented concerning the interaction between the polaritons and the acoustic phonons in molecular crystals. Using a one‐phonon approximation to truncate the hierarchy of the Green's functions involved and disregarding mixing of different polariton bands, we derive an expression for the dielectric permeability of the crystal. The absorption coefficient for the coupled polariton‐phonon spectrum is found to have an asymmetric Lorentzian lineshape even if the frequency dependence of the energy shift and spectral width is neglected. The damping function causing the asymmetry of the spectral line depends entirely on the coupling between the dressed electron subsystem and the phonon field. Expressions are developed for the energy shift and spectral width of the resonance line, and their temperature dependence is discussed. Far from the resonance peak, the frequency and temperature dependence of the absorption coefficient is established. In the transparent region of frequencies of the crystal, the expression for the absorption coefficient consists of two terms that have delta‐function distributions and are peaked at different frequency regions, depending on whether or not the polariton and the phonon fields are coupled. In the limiting case where retardation can be ignored, the bare exciton‐phonon interaction is discussed. The average energy of the crystal resulting from the polariton‐phonon interaction at finite temperatures has been derived in a closed form.

On Some Relations between the Solutions and the Parameters of Second‐Order Linear Differential Equations
View Description Hide DescriptionThe calculation of the Green's function for a point charge near an anisotropic planar diffuse layer leads to the differential equationU″ − k ^{2} D ^{2}(u)U = −2kδ(u − u′). Solutions for small and large k are obtained readily by a procedure equivalent to the principle of invariant imbedding. It is shown that the additional terms introduced in the solution for large k, due to the lack of smoothness in D(u) specifying the dielectric properties of the layer, may lead to singularities in the image potential. Similar considerations apply in the case of 1‐dimensional wave propagation through a stratified medium.

Construction of Weight Spaces for Irreducible Representations of A_{n}, B_{n}, C_{n}, D_{n}
View Description Hide DescriptionAn algebraic technique is presented by means of which the weight space for the irreducible representations (λ) of A_{n}, B_{n}, C_{n}, D_{n} can be constructed from the weight spaces associated with the representations (λ′) of the subalgebras A _{ n−1}, B _{ n−1}, C _{ n−1}, D _{ n−1}. Since each chain ends with A _{1}, all weight spaces of the classical simple Lie algebras associated with (λ) can be constructed, ultimately, from the well‐known representations of A _{1}.

Born Approximations in Statistical Mechanics
View Description Hide DescriptionThere are various possibilities in defining Born approximations for the evolution of density matrices. They differ, however, from the ordinary Born approximation for pure states by the fact that they do not, in general, conserve the fundamental properties of physical states. We shall describe these possibilities geometrically in the so‐called Liouville space of Hilbert‐Schmidt operators and investigate their significance in nonequilibrium dynamics. The results admit of an analogous interpretation in classical statistical mechanics.