Volume 10, Issue 7, July 1969
Index of content:

Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction
View Description Hide DescriptionThe equilibrium thermodynamics of a one‐dimensional system of bosons with repulsive delta‐function interaction is shown to be derivable from the solution of a simple integral equation. The excitation spectrum at any temperature T is also found.

On the Growth of the Number of Bound States with Increase in Potential Strength
View Description Hide DescriptionFor a wide class of potentials, it is shown that N(λ), the number of bound states (including multiplicity) of −Δ + λV, obeys the conditionsfor λ sufficiently large. A and B are positive finite numbers. In the centrally symmetric cases, a related growth condition on l _{max}(λ), the largest l channel with bound states, is also obtained, namely,.Finally, we discuss analogous results for a larger class of central potentials and for the many‐body case.

Super Hilbert Space and the Quantum‐Mechanical Time Operators
View Description Hide DescriptionThe basic idea of super Hilbert space is to represent physical states by continuous linear functionals on a space of good functions, rather than by functions in a Hilbert space. Since L _{2} is in one‐to‐one correspondence with a subset of super Hilbert space, everything that can be done in L _{2} can be done in super Hilbert space. In addition, however, it is possible to have a time operator and thus to base relativistic quantum mechanics on covariant four‐dimensional commutation relations.

Exact Robertson‐Walker Cosmological Solutions Containing Relativistic Fluids
View Description Hide DescriptionA new derivation of the Robertson‐Walker metrics is presented which elucidates the relationship between these spatially homogeneous and isotropic models and certain spatially homogeneous but anisotropicmodels (``three‐cylinder universes''). The evolutionaryequations for Robertson‐Walker models containing as many as four distinct noninteracting relativistic fluids (each obeying a gamma‐law equation of state) are examined, and 25 exact closed‐form solutions of this type are presented explicitly.

Equations of the de Broglie Wavefield in the Case of Spherical Symmetry
View Description Hide DescriptionTwo methods of integrating the equations of the de Broglie wavefield are introduced. The first integrating method does not seem to be as suitable for the domain of the microphysics as the second one. According to the second integrating method, the equations of the de Broglie wavefield are solved for the metric with spherical symmetry. In this case, the equations of the de Broglie wavefield admit of a solution which includes both Schrödinger and Hamilton‐Jacobi methods of description as particular cases.

Analysis of Electromagnetic Radiation in the Presence of a Uniformly Moving, Uniaxially Anisotropic Medium
View Description Hide DescriptionThe change in the character of electromagnetic radiation in the presence of a nondispersive, electrically and magnetically uniaxially anisotropic medium moving uniformly in the direction of its axis of symmetry is investigated. Because of the existing symmetry, it is possible to obtain explicit time‐dependent analytic solutions for a longitudinally—and a transversely—oriented magnetic dipole current distribution density using a spectral representation in the space‐time Fourier domain. The supports of the resulting fields are found to be oblate spheroidal wavefronts which enclose the source point if v < v _{1}, where v and v _{1} signify respectively the speed of the medium and the phase speed of a wave, as measured by an observer in the rest frame of the material, and move inside circular conical regions—a phenomenon known as the Čerenkov effect—for v > v _{1}.

Generalized Bose Operators in the Fock Space of a Single Bose Operator
View Description Hide DescriptionGeneralized Bose operators b which reduce by two the number of quanta of a Bose operator a are studied in the Fock space of a. All representations of the b's as normal‐ordered (infinite degree) power series of the a's are found. The unitary operators relating the irreducible components of b to a are also exhibited. The analogous result for b ^{(k)}'s which reduce the number of a quanta by k is given and the limit k → ∞ is discussed.

Existence of Solutions for Certain Nonlinear Boundary‐Value Problems
View Description Hide DescriptionThe existence of solutions of a class of nonlinear boundary‐value problems is characterized in terms of a linear eigenvalue problem. Bounds and comparison results for solutions are derived.

Algorithm for the Calculation of the Classical Equations of Motion of an N‐Body System
View Description Hide DescriptionThe equations of motion for N‐body systems are usually integrated by means of a central‐difference algorithm. An alternative average force algorithm is described in this paper. The new method is shown to have both theoretical and practical advantages when compared to the central‐difference method.

Wave Equations, Multipole Solutions, and Characteristic Hypersurfaces for Massless Particles
View Description Hide DescriptionIn this paper, the canonical wave equation and auxiliary conditions for a massless particle with arbitrary integral or half‐integral spin s are derived from a symmetric spinor formulation. The wave equation is solved for the multipole solutions using the appropriate parity operator for each spin. Finally, the characteristic hypersurfaces for the wave equation are derived and found to agree with that of the photon. The special cases of spins one, three‐halves, and two are detailed in conclusion.

On the Width Distribution for a Complex System Whose Hamiltonian Contains a Small Interaction That Is Odd under Time‐Reversal
View Description Hide DescriptionAn approximate expression for the width distribution of a complex system with a small odd term is given. The expression is compared with available Monte‐Carlo calculations and seems to be a good approximation for large N (10 ≤ N ≤ 100).

Type D Vacuum Metrics
View Description Hide DescriptionUsing the Newman‐Penrose formalism, the vacuum field equations are solved for Petrov type D. An exhaustive set of ten metrics is obtained, including among them a new rotating solution closely related to the Ehlers‐Kundt ``C'' metric. They all possess at least two Killing vectors and depend only on a small number of arbitrary constants.

SU(6) Clebsch‐Gordan Coefficients for the Product
View Description Hide DescriptionA method is presented which makes explicit use of Young diagrams to calculate multiplet‐coupling coefficients in SU(6). The multiplet‐coupling coefficients for are given.

F Model on a Triangular Lattice
View Description Hide DescriptionThe Rys F model is formulated on a triangular lattice and solved for certain values of the vertex configuration probabilities (including those corresponding to the ``ice model''). As with the square lattice, it is found that the system undergoes a phase transition which is of infinite order.

Covariant Electromagnetic Potentials and Fields in Friedmann Universes
View Description Hide DescriptionElectromagnetic potentials and fields are found for arbitrary four‐current densities in Friedmann universes. A choice of gauge is made so that the potentials are similar to the flat‐space potentials. A formalism is developed which allows the construction of potentials and fields which are covariant with respect to spatial transformations. It is shown explicitly how these potentials are related to the flat‐space potentials through conformal and gauge transformations. Some features of the solutions in the finite models are discussed with reference to problems of interpretation raised recently by Katz.

Involutional Matrices Based on the Representation Theory of GL(2)
View Description Hide DescriptionInvolutional matrices M(a, b, c) with three arbitrary parameters are introduced, based on a matrix representation M(R) of the linear homogeneous transformation R ∈ GL(2). Symmetry properties, eigenvalues, and recursion formulas for the representation M(R) are obtained and specialized to the involutional matrices M(a, b, c). A set of special involutional matrices A(ξ), B(ξ), C(ξ), and E(ξ) with one arbitrary parameter ξ are introduced as special cases of M(a, b, c). Their relations are discussed.

Hyperplane Helicity States
View Description Hide DescriptionThe hyperplane formalism of Fleming is developed to include a discussion of the operations of the Poincaré group, as seen by an arbitrary hyperplane observer. Basis states for the m > 0 irreducible representations of the Poincaré group are re‐expressed within the framework of generalized covariance provided by the hyperplane formalism and are seen to be related to the conventional helicity states by a special Lorentz transformation.

S‐Matrix and Classical Description of Interactions
View Description Hide DescriptionIt is shown quite generally, through a correspondence between S‐matrix and classical descriptions of particle states and of their measurements, that an S‐matrix theory actually leads, under appropriate conditions, to the classical space‐time description of interactions, involving the usual classical concepts and formulas.

Partial‐Wave Expansion in the Crossed Channel for Scattering Amplitudes Invariant under the Galilei Group
View Description Hide DescriptionA partial‐wave analysis in the crossed channel is performed for a Galilean‐invariant scattering matrix using the irreducible unitary representations of the Euclidean group in two dimensions. As in the relativistic case, a formula is obtained which is useful in determining the high‐energy behavior of the scattering amplitude for fixed‐momentum transfer. In particular, the Born term is shown to correspond to a δ function in this representation. Moreover, this parametrization is related by a group contraction to the corresponding background term of the relativistic case.

Critical Point Behavior of the Ising Model with Higher‐Neighbor Interactions Present
View Description Hide DescriptionThe method of developing exact power‐series expansions for the partition function Z_{N} and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first‐, second‐, and third‐neighbor pairs. Expansions of the spontaneous magnetizationM _{0}(T) and zero field susceptibility χ_{0}(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′^{r}, b.c.c., s.c., and f.c.c. lattices, when interactions and are present between first‐ and second‐neighbor spins, respectively (second‐neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J _{1} = J _{2} = J _{3}) are present between first‐, second‐, and third‐neighbor pairs (third‐equivalent‐neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′^{r}, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M _{0}(T) and χ_{0}(T) for as characterized by the critical exponents β and γ′, respectively. For the second‐equivalent‐neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest‐neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably and hence the transition in χ_{0} is symmetrical about T_{c} (γ′ = γ). A repetition of this analysis for the third‐equivalent‐neighbor model three‐dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high‐temperature (T > T_{c} ) expansions of χ_{0}(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over‐all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second‐ and third‐equivalent neighbor models. Finally the high‐temperature expansions of Z_{N} in zero field, and of χ_{0}(T) for the second‐neighbor model are used to examine the dependence of the critical temperature T_{c} , the critical energy (E _{∞} − E_{c} )/kT_{c} , and the critical entropy (S _{∞} − S_{c} )/k on the relative strengths of J _{1} and J _{2} for values of J _{2}/J _{1} in the range 0 to 1. It is found that the variation of the critical point is well represented by,where α = J _{2}/J _{1} and lies in the range 0 ≤ α ≤ 1; and T_{c} (0) is the critical temperature of the nearest‐neighbor model. The values of m _{1} are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′^{r} lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.