Volume 9, Issue 8, August 1968
Index of content:

Time‐Ordered Products in Two‐Dimensional Field Theories
View Description Hide DescriptionIt is shown that the vacuum expectation values (VEV) of are continuous functions of the time for test functions which are C ^{∞} and of rapid decrease, with λ in some neighborhood of the origin in the complex plane. The field σ(x) is the pseudopotential derived from the pseudovector current of a free two‐component massive field in two‐dimensional space‐time. A consequence of this result is the existence of Green's functions in the Federbush model. An essential technique in the proof is a theorem by Jaffe on the boundary values of limits of sequences of analytic functions.

Canonical Realizations of the Galilei Group
View Description Hide DescriptionThe general theory of the realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied to the Galilei group. Some complements to the general theory are introduced; in particular, a new kind of possible canonical realizations connected with the singularity surfaces of the functions are discussed (singular realizations). In agreement with the situation encountered in quantum mechanics, the constants d _{ρσ} appearing in the fundamental Poisson bracket relations among the infinitesimal generators cannot all be reduced to zero. There remains a single independent constant m, which, in the physically significant cases (m > 0), represents the mass of the system. No physical interpretation seems to be attachable to the realizations corresponding to m = 0. For m ≠ 0, two different kinds of irreducible realizations exist: one of a singular type which describes the free mass‐point, and another of a regular type which describes a classical particle spin. A number of physical significant examples corresponding to nonirreducible realizations are thereafter discussed and the related typical forms are constructed: specifically, the cases dealt with are the rigid rod (linear rotator), the rigid body, and a system of two interacting mass points. It is shown that the problem of the construction of the variables of the typical form is equivalent to the determination of an appropriate solution of the time‐independent Hamilton‐Jacobi equation.

Special Functions and the Complex Euclidean Group in 3‐Space. I
View Description Hide DescriptionIt is shown that the general addition theorems of Gegenbauer, relating Bessel functions and Gegenbauer polynomials, are special cases of identities for special functions obtained from a study of certain local irreducible representations of the complex Euclidean group in 3‐space. Among the physically interesting results generalized by this analysis are the expansion for a plane wave as a sum of spherical waves and the addition theorem for spherical waves. This paper is one of a series attempting to derive the special functions of mathematical physics and their basic properties from the representation theory of Lie symmetry groups.

Special Functions and the Complex Euclidean Group in 3‐Space. II
View Description Hide DescriptionThis paper is the second in a series devoted to the derivation of identities for special functions which can be obtained from a study of the local irreducible representations of the Euclidean group in 3‐space. A number of identities obeyed by Jacobi polynomials and Whittaker functions are derived and their group ‐ theoretic meaning is discussed.

Degenerate Representations of the Symplectic Groups. I. The Compact Group Sp(n)
View Description Hide DescriptionThe degenerate, irreducible, unitary representations of the compact group Sp(n), characterized by one and two invariant numbers, are considered. The explicit expressions for the basis functions spanning the corresponding representation spaces and the decomposition with respect to the maximal subgroup are given.

Korteweg‐de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation
View Description Hide DescriptionAn explicit nonlinear transformation relating solutions of the Korteweg‐de Vries equation and a similar nonlinear equation is presented. This transformation is generalized to solutions of a one‐parameter family of similar nonlinear equations. A transformation is given which relates solutions of a ``forced'' Korteweg‐de Vries equation to those of the Korteweg‐de Vries equation.

Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion
View Description Hide DescriptionWith extensive use of the nonlinear transformations presented in Paper I of the series, a variety of conservation laws and constants of motion are derived for the Korteweg‐de Vries and related equations. A striking connection with the Sturm‐Liouville eigenvalue problem is exploited.

Properties of the S Matrix for Velocity‐Dependent Potentials
View Description Hide DescriptionSolutions of the Schrödinger equation with a velocity‐dependent potential are discussed. The analytic behavior of the S matrix as a function of the momentum is investigated, and some properties of general validity are demonstrated. The configuration of the poles of the scattering matrix in the complex‐momentum plane is described in detail for the case of a spherically symmetric potential.

Irreducible Representations of the Five‐Dimensional Rotation Group. I
View Description Hide DescriptionExplicit matrix elements are found for the generators of the group R(5) in an arbitrary irreducible representation using the ``natural basis'' in which the representation of R(5) is fully reduced with respect to the subgroup . The technique used is based on the well‐known Racah algebra. The dimension formula is derived and the invariants are found. A family of identities is established which relates various polynomials of degree four in the generators and which holds in any representation of the group.

Irreducible Representations of the Five‐Dimensional Rotation Group. II
View Description Hide DescriptionA systematic study is made of the relationship between the generators of R(5) expressed in the ``natural basis,'' as discussed in I [J. Math. Phys. 9, 1224 (1968)], and the same generators in the ``physical basis'' in which representations of R(5) are fully reduced with respect to the physical three‐dimensional rotation group. In this paper, attention is confined to the traceless symmetric tensors of R(5) which are the representations appropriate to the discussion of quadrupole vibrations of the nuclear surface. For these representations, one quantum number in addition to the angular momentum and its projection is required to specify a state within a representation. The required extra label is found through the definition of ``intrinsic states'' in the natural basis, and a complete set of states in the physical basis is projected out of these intrinsic states by integrations over the physical rotation group manifold. Members of this set of physical states are not orthonormal; however, the overlap integrals are presented in two simple algebraic forms convenient for computer programming. The construction of the explicit representation matrices for the generators of R(5) is completed by giving the reduced matrix elements of the octopole generator between physical states in terms of the overlap integrals.

Approach to Scattering Problems through Interpolation Formulas and Application to Spin‐Orbit Potentials
View Description Hide DescriptionIn many problems of potential scattering, and particularly in the inverse‐scattering problem, two equations prove to be of essential interest: the Gel'fand‐Levitan equation and the Regge‐Newton equation. These and their generalizations apply, respectively, to energy‐independent and to λ‐independent potentials. In this paper, a method is devised for obtaining equations applying to more general cases. The requirement is the existence of analytic properties of the wavefunctions corresponding to special classes of potentials, enabling one to construct interpolation formulas of the Lagrange form. From these formulas, it is possible to derive integral equations which may then be generalizable to much larger classes of potentials. This method is fully developed in the case of potentials depending linearly on λ. Interpolation formulas and analytic properties of the wavefunctions in the λ plane are exhibited. Integral equations are given and proved to apply to very large classes of potentials. Existence, uniqueness, and analytic properties of their solutions are thoroughly studied. An example is given for which all the wavefunctions are calculated exactly. Application of the method to the inverse scattering problem in the presence of a spin‐orbit potential will be the object of a forthcoming paper.

Extension of the Factorization Method
View Description Hide DescriptionWe show that it is possible to extend the formalism of the factorization method for any displacement in the spectrum space of any second‐order differential equation. Following this, we show that we can extend, at least formally, the formalism for some nth‐order ordinary differential equations.

Extended Energy‐Integral Technique for Linear Differential Equations
View Description Hide DescriptionAn extended energy‐integral technique for boundary‐value problems is presented for a class of differential equations which are prevalent in mathematical physics. The extended technique provides a method for answering questions in wave propagation and stability which could not be treated by the familiar method of energy integrals.

Convergence of the Sudarshan Expansion for the Diagonal Coherent‐State Weight Functional
View Description Hide DescriptionThe mathematical properties of the original expansion derived by Sudarshan for the diagonal coherent‐state weight functional are discussed. It is shown that, for stationary fields, the expansion is a generalized function in the space Z′(R _{2}). The validity of this method of defining the weight functional in the case of arbitrary density operators and its relationship to other approaches to the problem of the diagonal representation is briefly considered.

``Lorentz Basis'' of the Poincaré Group
View Description Hide DescriptionAn explicit derivation is given for the matrix elements of the translation generators P _{μ} of the Poincaré algebra with respect to the ``Lorentz basis,'' namely, in terms of states which diagonalize the two Casimir operators of the homogeneous Lorentz group (HLG). The results are given for the cases mass μ > 0 and μ = 0 and, for the latter, for discrete and continuous spin. The transforms connecting the momentum and Lorentz bases are discussed, a detailed derivation being given for the zero‐mass discrete‐spin case. The matrix elements of G _{μ} = i[(N^{2} − M^{2}), P _{μ}] are considered and several interesting aspects of the algebras generated by N, M′, and are discussed for the cases of positive as well as zero rest mass.

Random Spin Systems: Some Rigorous Results
View Description Hide DescriptionSeveral general results are obtained for a system of spins on a lattice in which the various lattice sites are occupied at random, and the spins, if present, interact via a general Heisenberg or Ising interaction decreasing sufficiently rapidly with distance. It is shown that the free energy per site exists in the limit of an infinite system, is a continuous function of concentration, and has the usual convexity (stability) properties. For Ising systems with interactions of finite range, the free energy is an analytic function of concentration and magnetic field for a suitable range of these variables. The random Ising ferromagnet on a square lattice (or simple cubic lattice) with nearest‐neighbor interactions is shown to exhibit a spontaneous magnetization at sufficiently high concentrations and low temperatures.

Two‐Center Coulomb and Hybrid Integrals
View Description Hide DescriptionCoulomb and hybrid integrals are shown to be related to overlap integrals. They are expressed by an integral whose integrand is an overlap integral plus a finite sum of overlap integrals.

Quasibinomial Representations of Clebsch‐Gordan Coefficients. I. Square Symbol
View Description Hide DescriptionQuasibinomial representation of Clebsch‐Gordan coefficients is symbolized by a Regge‐like square. Rules for exchange of rows and columns of this square are derived. Thus our various quasibinomial forms can be read off from this square directly.

Quasibinomial Representations of Clebsch‐Gordan Coefficients. II. ``Negative'' Representations
View Description Hide DescriptionNew quasibinomial forms are derived from the quasibinomial forms given previously by making use of both positive and negative generalized powers. They turn out to be a new representation of the Wigner‐type unsymmetrical formulas of Clebsch‐Gordan coefficients for angular momenta. Consequently, formulas of Racah, Majumdar, and Shimpuku are deduced as special cases. Rules to construct a square symbol are given from which all these ``negative'' quasibinomial representations or, more precisely, expansions can be read off directly. Thus, a unified treatment of both symmetrical and unsymmetrical formulas of Clebsch‐Gordan coefficients is thereby accomplished.

Construction of Invariants for Lie Algebras of Inhomogeneous Pseudo‐Orthogonal and Pseudo‐Unitary Groups
View Description Hide DescriptionA method of constructing invariants for the Lie algebras of the inhomogeneous pseudo‐orthogonal and pseudo‐unitary groups from invariants of the (homogeneous) pseudo‐orthogonal and pseudo‐unitary groups, respectively, is presented. The method is based on the ``expansion'' (or ``deformation'') of the inhomogeneous algebras to homogeneous ones. Several examples are worked out.