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Volume 8, Issue 1, January 1967

Certain Bootstraps and Related Equations
View Description Hide DescriptionMethods of deriving existence and nonexistence theorems for certain classes of nonlinear integral and other equations are discussed. The methods are illustrated by examples from high‐energy physics of equations of the bootstrap type.

Two‐Meson Solution of the Charged Scalar Static Model with Bound States
View Description Hide DescriptionA two‐meson solution of the charged scalar static model is constructed for the case in which the coupling is sufficient to produce bound states. This is done by taking the solution given previously for the case in which bound states are absent and increasing the coupling. As usual, bound states appear as poles in scattering amplitudes. However, since one of the mesons in the three‐body states can bind to the source, bound‐state meson scattering channels also appear. The new amplitudes relating to bound‐state meson scattering are obtained, and their unphysical singularities are examined. The enlarged scattering matrix is unitary.

Common Interpretation of Phase Transitions in Various Models
View Description Hide DescriptionA discussion is presented of the formal analogy which exists between the Bardeen‐Cooper‐Schrieffer (BCS)model for superconductivity and the several variations of the model suggested by Kac for the exact study of phase transitions. Some of the algebraic techniques, pioneered by Haag for his formulation of axiomatic field theory and his application of this theory to the BCSmodel, are shown to extend to all these cases. Some arguments are given in the beginning of the paper to point out the necessity of a new interpretation of the existence of various phases in the same physical system. The results obtained so far for equilibrium situations confirm the consistency of the proposed tentative description of the phenomena considered.

van der Waals Wiggles, Maxwell Rule, and Temperature‐Dependent Excitations
View Description Hide DescriptionStarting from a temperature‐independent Hamiltonian, we discuss for some models the coexistence of the van der Waals wiggles and the Maxwell‐van der Waals isotherms. The main tool of this paper is provided by the temperature‐dependent excitations techniques developed in an earlier paper. Some ferromagnetic models and lattice‐gas models are treated along similar lines. The van der Waals wiggles and the Maxwell‐van der Waals isotherms appear below the critical temperature as two solutions of a system of self‐consistent equations. The choice between the two solutions is then dictated by energy considerations.

High‐Energy Behavior of Feynman Integrals with Spin. II
View Description Hide DescriptionThe analysis of a previous paper is applied to determine the leading behavior and coefficients of specific lower‐order terms of planar graphs in the nucleon‐meson scattering process. The results of Cheng and Wu in verifying the Reggeization hypothesis in sixth order are rederived and a method of extending the analysis to nth order is indicated.

Quasi‐Classical Analysis of Coupled Oscillators
View Description Hide DescriptionA set of harmonic oscillators is coupled for a certain time, after which the coupling is removed. The initial and final states of the set are expressed as superpositions of the coherent states of the uncoupled oscillators, as in Glauber's formalism. It is shown that the expansion kernel for the final state can be obtained approximately by substituting into the kernel for the initial state the classical equations of motion of the amplitudes of the oscillators. The evolution of a density operator for the system can be similarly calculated. The approximation is the more exact, the longer the time of interaction compared with the periods of the uncoupled oscillators.

Calculation of Exchange Second Virial Coefficient of a Hard‐Sphere Gas by Path Integrals
View Description Hide DescriptionBy direct examination of the path (Wiener)‐integral representation of the diffusionGreen's function in the presence of an opaque sphere, we are able to obtain upper and lower bounds for that Green's function. These bounds are asymptotically correct for short‐time, even in the shadow region. Essentially, we have succeeded in showing that diffusion probabilities for short‐time intervals are concentrated mainly on the optical path. By integrating the Green's function, we obtain upper‐ and lower‐bound estimates for the exchange part of the second virial coefficient of a hard‐sphere gas. We can show that, for high temperature, it is asymptotically very small compared to the corresponding quantity for an ideal gas, viz.,,where Λ is the thermal wavelength and a is the hard‐sphere radius. While it was known before that B _{ex ch}/B ^{0} _{ex ch} is exponentially small for high temperatures, this is the first time that a precise asymptotic formula is both proposed and proved to be correct.

Generalization of the Variational Method of Kahan, Rideau, and Roussopoulos. II. A Variational Principle for Linear Operators and its Application to Neutron‐Transport Theory
View Description Hide DescriptionAdditional applications of a generalized form of the variational method of Kahan, Rideau, and Roussopoulos are presented. Equations used in neutron transporttheory, such as the spherical harmonics operator form of the Boltzmann equation, are derived from the generalized variational functional and an interpretation of these operator equations in terms of flux‐ and source‐generating operators is suggested. A relationship between this variational method and the variational method of Lippmann and Schwinger is established, and it is shown that the least‐squares variational functional of Becker for linear equations can be derived from a generalized variational functional.

Representations of Inhomogeneous SU(n) Groups with Mixture of Homogeneous and Inhomogeneous Operators Diagonal
View Description Hide DescriptionExplicit representations of some inhomogeneous compact SU(n) groups and their algebras are constructed with certain mixtures of diagonal homogeneous and inhomogeneous operators.

General Coupling Coefficients for the Group SU(3)
View Description Hide DescriptionA Hilbert space method, previously applied to the group SU(2), is employed to examine the representations D^{λμ} and the reduction of the direct‐product representation of the group SU(3). The base vectors λμ;α>, an orthogonal Hilbert space of homogeneous polynomials, are transformed to the base vectors λμ;α>_{c}, and are associated with the complex conjugate representation by an explicit R‐ conjugation operation. For the general direct‐product representation D^{λ1μ1 } ⊗ D^{λ2μ2 }, explicit expressions are derived for the vector coupling coefficients and the number of times the irreducible representation D^{λ3μ3 } is contained in the direct product. Two methods of labeling the degenerate states are given, the reduction of the direct product is shown to be complete, and the symmetry relations of the 3(λμ) coefficients are discussed.

Recoupling Coefficients for the Group SU(3)
View Description Hide DescriptionThe Hilbert space method, employed in the previous article to obtain the coupling coefficients of SU(3), is used here to obtain the recoupling, or 6(λμ), coefficients of SU(3). The coefficients are formulated in terms of a generating function involving an integral, and an explicit expression is integrated out for the general nondegenerate case. The symmetries of the 6(λμ) coefficients are discussed.

Surface‐Energy Tensors
View Description Hide DescriptionIn this paper the surface‐energy tensors are defined and general expressions for their first variations are obtained. These quantities are appropriate for an investigation of the equilibrium and the stability of a charged liquiddrop (held together by a constant surface tension) using the method of the tensor virial, which was developed by Chandrasekhar and Lebovitz for the theory of self‐gravitating masses. The definitions are shown to be consistent with the conservation laws.

Surface‐Energy Tensors for Ellipsoids
View Description Hide DescriptionThe surface‐energy tensors are evaluated for ellipsoidal surfaces in terms of particular types of elliptic integrals which have simple algebraic recursion relations that are useful for numerical evaluation. The final expressions are given in a form appropriate for an investigation of stability using the method of the tensor virial.

Stability of Axisymmetric Figures of Equilibrium of a Rotating Charged Liquid Drop
View Description Hide DescriptionThe method of the tensor virial is used to investigate the equilibrium and stability of a rotating, uniformly charged (with a total charge Q), incompressible liquiddrop (of volume V) held together by a constant surface tensionT. The tensor virial theorems provide relations which define the sequences of equilibrium figures and perturbation equations which govern the oscillations. Since spheroids satisfy the virial theorems of orders one, two, and three, the stability of exact axisymmetric equilibrium figures with respect to second‐ and third‐harmonic deformations can be inferred from the nature of the characteristic oscillation frequencies of spheroids. As the rotation increases for a given fission‐ability parameter x = Q ^{2}/10TV (0 ≤ x < 1), a sequence of oblate spheroids representing initially stable rotating ground states exhibits a neutral point (where an ellipsoidal sequence bifurcates), but remains stable (in the absence of dissipation) until one of the three second‐harmonic modes of vibration (the toroidal or γ mode) becomes overstable. Later, for third harmonics, it exhibits a second neutral point (where a sequence of asymmetric figures bifurcates), but again remains stable until the onset of the associated overstability. A third neutral point does not indicate the bifurcation of pear shapes. The error in using oblate spheroids is less than 5% up to the first overstable point. One class of saddle shapes can be represented by prolate spheroids (which must rotate about the symmetry axis) in the neighborhood of x = 1. They are initially unstable with respect to the other two second‐harmonic modes of vibration (the pulsation or β mode and the transverse‐shear mode).

Expanding Universe in Conformally Flat Coordinates
View Description Hide DescriptionEinstein's field equations for a homogeneous isotropic universe are written in terms of a conformally flat coordinate system for both the closed and the open cases. It is shown that these can be readily integrated for equations of state of the form p = aρ in terms of algebraic functions. Explicit solutions and expressions for the radius and Hubble factor are given for a = 1, ⅓, and for p = 0.

One‐Dimensional Ising Model with General Spin
View Description Hide DescriptionThe one‐dimensional Ising model with general spin S has been formulated as an eigenvalue problem of order 2S + 1. Two methods to reduce the order to [S + 1] have been developed for calculating the energy and the susceptibility at zero external field.Exact solutions for S = and S = 1 have been obtained. Numerical calculations of S = , 1, and ½ have been compared.

Canonical Root Vectors of SU_{n}
View Description Hide DescriptionThe present paper extends the study of SU_{n} to the determination of particular root vectors and fundamental weights. The root vectors are chosen so as to show explicitly the canonical nesting of subgroups SU_{n} SU _{ n−1} × U _{1}, SU _{ n−1} SU _{ n−2} × U _{1}, etc. Explicit matrix representations of the fundamental representations are given. The results are all dependent upon the use of n vectors which define the fundamental region of SU_{n} . Eigenvalues of fundamental invariant operators are also given in terms of these n vectors.

Representation Functions of the Group of Motions of Clifford Space
View Description Hide DescriptionWe find the representation functions of the group of motions of the three‐dimensional Clifford space and of the three‐dimensional Einstein space. These functions are generalizations of spherical and cylindrical waves of three‐dimensional Euclidian space and reduce to these familiar functions in the Euclidian limit. The generalization of the plane wave is also found.

Role of the Integral‐Operator Method in the Theory of Potential Scattering
View Description Hide DescriptionA recently developed method in the theory of potential scattering is investigated. The authors supply analytic computations for determining the location and the type of singularities, which occur in several scattering amplitudes that are constructed from physically relevant potentials.

Variational Principle for Eigenvalue Equations
View Description Hide DescriptionA variational principle is developed to provide an estimate of an arbitrary functional of the eigen‐functions of a set of eigenvalue equations. It is shown that the variational formalism is equivalent to a functional Taylor series expansion of the desired functional about the trial functions. The relationship of this work to perturbation theory is considered, and it is shown that the formalism can be used to construct higher‐order variational principles, i.e., those for which first‐order errors in the trial functions leads to an nth‐order error in the desired functional. Finally, it is shown that the variational principle of Borowitz and Vassell for estimating off‐diagonal matrix elements, as well as the usual Rayleigh quotient, are special cases of the principle presented here.