Volume 15, Issue 8, August 1974
Index of content:

Mackey‐Moore cohomology and topological extensions of Polish groups
View Description Hide DescriptionTopological group extensions with Abelian kernels are analyzed using factor sets and following the pattern of the work of Eilenberg and MacLane on extensions of groups without topology. In this analysis, the Eilenberg‐MacLane cohomology is replaced by the Mackey‐Moore one, whose cochains are Borel mappings and which is especially suitable in the case of Polish groups (Hausdorff second countable complete groups). The connection between cohomology groups of degree 2 and equivalence classes of topological group extensions with Abelian kernels is established. A fundamental sequence of cohomology groups and group homomorphisms is proven to be exact, and it is shown that, in some interesting cases, the low degree cohomology groups of topological semidirect products are determined by the corresponding cohomology groups of the factors.

Coupling of space‐time and electromagnetic gauge transformations
View Description Hide DescriptionA kind of topological extensions of a space‐time group Q by an electromagnetic gauge group J are investigated in order to determine covariance groups of electrodynamics. Here Q stands for the Poincaré group, for the Galilei group, or for their neutral components, and J is the Abelian group of all real‐valued functions of class C^{ m } (m ∈ N or m = ∞) defined in space‐time. The topological groupsJ _{φ} fQ so obtained, already important in the study of charged particles in external electromagnetic fields, are analyzed and placed in the general context of combining different symmetry groups. They are characterized by a given operation φ of Q on J and by factor sets f such that f(q,q′) is a constant gauge function for all (q,q′) ∈ Q × Q. It is shown that all these groups J _{φ} fQ are topologically isomorphic to the external topological semidirect product of Q by J relative to Φ.

Coherent states and Lie algebras
View Description Hide DescriptionThe Bloch coherent states for a spin or a system of spins and the Glauber coherent states for bosons are examined from the viewpoint of Lie algebras. It is pointed out that the Bloch coherent states are vectors in the space spanned by the basis functions for an irreducible representation of the unitary unimodular group SU(2), and that the Glauber coherent states are vectors in the space spanned by the basis functions for the infinite‐dimensional irreducible representation of a contracted group of SU(2). A deeper understanding of many of the useful properties of these coherent states is gained.

Invariants of Born reciprocity theory
View Description Hide DescriptionIn an attempt to use Born reciprocity theory as a possible scheme for explaining elementary particles we calculate all the invariants of this theory. It turned out that all the invariants are functions of the operator (x ^{2}+p ^{2}). Thus there is only one independent invariant which characterizes this theory.

Behavior of distribution functions in the thermodynamic limit
View Description Hide DescriptionRuelle has proven that the solutions of the Kirkwood‐Salsburg equation for a finite volume Λ become, in the limit as Λ → ∞, the solutions to the Kirkwood‐Salsburg equation for an infinite volume, i.e.,.The form of ε is not obtained. We show that for the first order contribution to the solution of the Kirkwood‐Salsburg equations obtained via a perturbation scheme developed in an earlier paper that,where k′_{a} is a positive real constant which can be specified and R is the minimum distance from the container walls to the particles of the system.

Proof of Zwanzig's rule of ``planar'' graphs
View Description Hide DescriptionA rigorous proof is given of Zwanzig's conjecture that only planar graphs contribute to the virial coefficients of discreet‐orientation models for a fluid of long thin rods. The proof makes use of relations between connected graphs and ``trees''—the simplest type of connected graphs.

Note on space‐times that admit constant electromagnetic fields
View Description Hide DescriptionAll space‐times that admit a covariantly constant, test, electromagnetic field are constructed. All solutions to the Einstein‐Maxwell equations with constant electromagnetic field are given.

Exact nearest neighbor degeneracy for dumbbells on a one‐dimensional lattice space
View Description Hide DescriptionRelationships are developed which describe exactly the degeneracy associated with nearest neighbor pairs of occupied sites (1–1), mixed sites (0–1), and vacant sites (0–0) for dumbbells distributed on a one‐dimensional lattice space. The first moments of these statistics are calculated, thereby permitting an evaluation of the error inherent in the use of the Bragg‐Williams approximation for this situation.

Gel'fand‐pattern technique applied to the physically important decomposition of S U(6)
View Description Hide DescriptionExplicit and easily computable formulas for the physical quantum numbers I _{3}, Y, and S _{3}, and also two additional quantum numbers for all states spanning an arbitrary irreducible representation of S U(6) are obtained by using Gel'fand‐pattern technique. This result is accomplished by establishing a correspondence between the 3 diagonal generators + 2 ``quark‐spin'' generators, and the 5 ``canonical'' generators H _{1} − H _{5}, the eigenvalues of which are given by the Gel'fand patterns of S U(6) in the ordinary way. The content of the S U(6) representations taken as examples is displayed explicitly. The possibility of doing this suggests that Gel'fand patterns may be useful even in a nonmaximal decomposition, although the patterns are intrinsically linked to the ``canonical'' chain of decomposition: . The procedure developed in the case of S U(6) is generalized to the twofold nonmaximal decomposition of .

Some observations on the operator H=−(1/2)d ^{2}/dx ^{2} +m ^{2} x ^{2}/2+g/x ^{2}
View Description Hide DescriptionIn the present work we study the differential operator H=−(1/2)d ^{2}/dx ^{2} +m ^{2} x ^{2}/2+g/x ^{2}. This operator known as the Hamiltonian of the quantal oscillator has been a matter of study since the beginning of quantum mechanics. Recently, it has become again actual after the paper of Calogero where the correspondent N body problem (developed in many works) is studied. Parisi and the author have used H as Hamiltonian, studying the anomalous dimensions in one‐dimensional quantum field theory. Finally, Klauder, using H as a simple degree of freedom example, has studied some qualitative features of quantum theories with singular interaction potentials. In the following work we are going to study H, showing that H is equivalent to ``half an harmonic oscillator'' for the odd and even eigenspaces separately.

Irreducible tensorial sets within the group algebra of a compact group
View Description Hide DescriptionFor compact topological groups (discrete or continuous) a basis of the group algebra is defined which consists of irreducible tensors only. This tensor basis is generally discussed and compared with similar constructions for finite groups and SU (2).

On analytic nonlocal potentials. II. Analyticity of the S matrix, for fixed l, its representations, and a dispersion relation for fixed t
View Description Hide DescriptionFor a class of analytic short‐ranged nonlocal potentials, we study the analyticity of the S matrix in the k plane, for fixed l, using Fredholm method for the Lippman‐Schwinger equation for the partial scattering solution, and contour deformation in the analytic continuation of the S matrix, thereby extending a representation of the S matrix in terms of Fredholm determinants. We also obtain a representation of the S matrix in terms of Jost functions, for l = 0. For a subclass of this class of potentials, we obtain a dispersion relation for the full scattering amplitude, for fixed t in the range 0 ≥ t > − 4γ^{2}, where γ is some parameter of the potential, using summation of the partial amplitude expansion of the full amplitude. Analyticity properties of the partial scattering solution, for all l, and of the regular and Jost solutions, for l = 0, are also discussed.

On analytic nonlocal potentials. III. Local correlations
View Description Hide DescriptionWe give examples of ``analytic nonlocal potentials'' which show local correlations for a finite range of the distance, bounded below and above.

The T operator and an inverse problem for nonlocal potentials
View Description Hide DescriptionFor a class of short‐ranged nonlocal potentials, the T operator T(k) is studied in the region Imk> ‐γ, where γ > 0 is some parameter of the potential. Inversion formulas are obtained which determine the potential from for any real E, for which t(E) is defined, perhaps with the exception of a countable set of points.

Representations of the Jost solutions and the S matrix for a class of analytic nonlocal potentials
View Description Hide DescriptionWe derive representations of the Jost solutions, the Jost functions, and the S matrix, for analytic nonlocal potentials belonging to a double Laplace transform class, for the s‐wave case, in terms of the spectral function for the potential.

An approximate interior solution in Brans‐Dicke theory
View Description Hide DescriptionAn approximate interior solution of the field equations of Brans‐Dicke theory is obtained for a static spherically symmetric metric which can be considered to be an analog of Schwarzschild's interior solution in Einstein's theory.

On the transport properties of van der Waals fluids. II. Explicit calculations
View Description Hide DescriptionStarting from the formal expansion of an arbitrary transport coefficient X in series of the inverse range γ of a van der Waals potential, we establish the explicit form of the first γ correction to X; we show that it can be expressed solely in terms of the Fourier transform of the long‐range interaction and in terms of the equilibrium and transport properties of the short‐range reference system. A comparison with previous work on related problems is also given.

Polynomial algebras. II. Commutation relations
View Description Hide DescriptionThe present work completes the algorithm which Bhabha had prescribed so as to set up the commutation relations of spin algebras but could implement it only for spin one algebra using a third order permutation identity. Generally this work is concerned with the setting up of the commutation relations of derived polynomialalgebras which are obtained by addition operation from a basic polynomialalgebra. To obtain these commutation relations, a set of identities called Josthna identities are introduced among the permutations of a finite set of elements. With the help of these identities it is established that commutation relations for derived algebras can be set up directly. Applications to spin and parafield algebras are considered to obtain their commutation relations which make their deduction trivial.

Lie theory and separation of variables. 4. The groups SO (2,1) and SO (3)
View Description Hide DescriptionWinternitz and coworkers have shown that the eigenfunction equation for the Laplacian on the hyperboloid separates in nine orthogonal coordinate systems, associated with nine symmetric quadratic operators L in the enveloping algebra of SO (2,1). Corresponding to each of the operators L, we employ the standard one‐variable model for the principal series of representations of SO (2,1) and compute explicitly an L basis for the Hilbert space as well as the unitary transformations relating different bases. We also compute the associated results for realizations of these representations on the hyperboloid. Three of our bases are related to well‐known subgroup reductions of SO (2,1). Of the remaining six, one is related to Bessel functions, two to Legendre functions, and three to Lamé functions. We show that there is virtually a perfect correspondence between the known theory of the Lamé functions and the representation theory of SO (2,1) and SO (3).

Composition of coherent spin states
View Description Hide DescriptionClebsch‐Gordan coefficients for the S U (2) group are computed in the coherent spin state basis introduced by Radcliffe.