Volume 22, Issue 2, February 1981
Index of content:

Gaussian states on the algebra of the infinite classical system
View Description Hide DescriptionA class of states on the algebra of the infinite classical system is characterized by the vanishing of the higher order (n≳2) truncated correlation functions. The states are called Gaussian. It is shown how liquids and crystals can be described by Gaussian states.

Realization, extension, and classification of certain physically important groups and algebras
View Description Hide DescriptionAn associative algebra of differential forms with division has been constructed. The algebra of forms in each different space provides a practical realization of the universal Clifford algebra of that space. A classification of all such algebras is given in terms of two distinct types of algebrasN _{ k } and S _{ k }. The former include the dihedral, quaternion, and Majorana algebras; the latter include the complex, spinor, and Diracalgebras. The associative product expresses Hodge duality as multiplication by a basis element. This makes possible the realization of higher order algebras in a calculationally useful algebraic setting. The fact that the associative algebras, as well as the enveloped Lie algebras, are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws.

The Racah algebra for groups with time reversal symmetry
View Description Hide DescriptionThe elements of the Racah algebra for a general compact group with time reversal symmetry are developed. As time reversal is antilinear it is not possible to treat these groups by representation theory but we may instead use Wigner’s theory of corepresentations. The results we obtain often parallel those for linear groups but there are some important divergences. We illustrate these with the grey double point groups.

Representations of Osp(2, 1) and the metaplectic representation
View Description Hide DescriptionShift operator techniques are used to treat the irreducible representations of the superalgebra Osp(2, 1). Apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact SU(2) algebra, the case when the star conditions on the even part are those satisfied by the noncompact SU(1, 1) algebra is also treated. In this case no gradestar representations arise, and the star representations are found to consist of the direct sum of two discrete series representations of SU(1, 1). One of these representations can be realized in terms of functions of a single complex variable, and turns out to be a simple example of a metaplectic representation.

Subgroups of Lie groups and separation of variables
View Description Hide DescriptionSeparable systems of coordinates for the Helmholtz equation Δ_{ d }Ψ =EΨ in pseudo‐Riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator Δ_{ d }. They have also been characterized geometrically by the form that the metric d s ^{2}=g _{ i k }(x)d x ^{ i } d x ^{ k } can take. We complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of G, the symmetry group of Δ_{ d }. For d=3 we study all separable coordinates that can be characterized in terms of the Lie algebraL of G and show that they are of eight types, seven of which are related to the subgroup structure of G. Our method clearly generalizes to the case d≳3. Although each separable system corresponds to a pair of commuting symmetry operators, there do exist pairs of commuting symmetries S _{1},S _{2} that are not associated with separable coordinates. For subgroup related operators we show in detail just which symmetries S _{1},S _{2} fail to define separation and why this failure occurs.

Generating functions for SU(2) plethysms with fixed exchange symmetry
View Description Hide DescriptionA number of fixed‐plethysm generating are given for SU(2); a fixed ‐plethysm generating function gives the content of the component with definite exchange symmetry of the direct product of a given number of copies of an irreducible representation R _{ l }, with l running through all values. New symmetries are found relating antisymmetric to symmetric products, and relating plethysms in which the number of factors is interchanged with the factor representation label l. Expressions for two‐box plethysm generating function for SU(3) and for fixed‐plethysm generating functions for SU(2) based on reducible representations are also given.

General U(N) raising and lowering operators
View Description Hide DescriptionIt is the aim of this paper to obtain the general form of U(n) raising and lowering operators. The raising and lowering operators constructed previously by several authors are then compared. The Hermiticity properties of these operators are also investigated. The methods presented extend, with trivial modifications, to the orthogonal groups.

Expansion of a function about a displaced center
View Description Hide DescriptionAn extremely simple closed expression is obtained for the coefficients which appear in the expansion of a function of a special type about a displaced center. A conjecture about the vanishing of a certain coefficient which appear in the expansion of a Slater‐type orbital about a displaced center is also proved.

On some properties of solutions of Helmholtz equation
View Description Hide DescriptionWe give a new method to prove results of the following type. Let: (∇^{2} + k ^{2})u=0 in D _{ R }={x‖x‖?R}, k ^{2}≳0. (1) If u∈L ^{2}(D _{ R }), then u≡0 in D _{ R }. (2) If ‖x‖^{ m } u(x)→0 as ‖x‖→∞, x ^{2} _{1}+⋅⋅⋅+x ^{2} _{ N−1}?c x ^{−2p } _{ N } ,p≳0, m=1, 2, 3,..., ‖x‖[(∂u/∂‖x‖)_{−i k u }‖x‖→∞]→0, then u≡0 in D _{ R }.

Gauge equivalence of exactly integrable field theoretic models
View Description Hide DescriptionExactly integrable field theoretic models are constructed which are gauge equivalent to the n‐component or m⋅n component nonlinear Schrödinger equations and to the O(n) nonlinear σ‐model. We obtain the C P ^{ n }‐ Heisenberg model or the Grassmann–Heisenberg model and the generalized sine‐Gordon model respectively. Consequences for the conserved quantities are discussed.

Integrals over the Fermi function
View Description Hide DescriptionA method for evaluating integrals over the Fermi distribution function using results from Mellin–transform theory is presented. The connection of this approach with the operational result of Blankenbecler is explicated. The method is used to calculate the profile function for a Fermi distribution.

The set of all projective limits of a projective system of state operators
View Description Hide DescriptionLet {W _{ K }} be a projective system of state operators defined on the finite tensor products of some family {H_{ t }} of Hilbert spaces. We prove that all projective limits of {W _{ K }} on the c o m p l e t etensor product ⊗_{ t∈T } H_{ t } can be generated from every single projective limit by certain operations. In addition, we provide a necessary and sufficient condition for {W _{ K }} to have projective limits on i n c o m p l e t etensor products of the H_{ t }’s.

The domain of definition of bundle representations
View Description Hide DescriptionIt is proved that the equivalence class of bundle representations of a group G on the product bundle B_{0} with total space B _{0}=X×Y includes all representations of G on bundles B which are homeomorphic (but not necessarily naturally homeomorphic) to the product X×Y, provided the G has the same action on the fibres of B_{0} and B. The group A of the bundle B is immaterial.

An axiomatic system for Minkowski space–time
View Description Hide DescriptionMinkowski space–time is developed in terms of a set of undefined primitive elements called events, certain subsets of events called paths which correspond to the worldlines of free particles, and a temporal order relation on each path. Nine axioms describe the existence and uniqueness of paths, temporal order, connectedness, causality, collinearity, continuity, isotropy, and dimension.

The spreading of free wave packets and the entropy of position
View Description Hide DescriptionThe spreading of free wave packets is expressed by means of the entropy of position for a certain class of states. Connection between such formulation and the usual treatment is discussed.

Nonrelativistic Coulomb Green’s function in parabolic coordinates
View Description Hide DescriptionThe nonrelativistic Coulomb Green’s functionG ^{(+)}(r_{1},r_{2},k) is evaluated by explicit summation over discrete and continuum eigenstates in parabolic coordinates. This completes the derivation of Meixner, who was able to obtain only the r_{1}=0 and r_{2}→∞ limiting forms of the Green’s function. Further progress is made possible by an integral representation for a product of two Whittaker functions given by Buchholz. We obtain the closed form for the Coulomb Green’s function previously derived by Hostler, via an analogous summation in spherical polar coordinates. The Rutherford scattering limit of the Green’s function is also demonstrated, starting with an integral representation in parabolic coordinates.

On the Abel summability of partial wave amplitudes for Coulomb‐type interactions
View Description Hide DescriptionWe prove that the sum of partial wave amplitudes for Coulomb‐type potentials [e.g., V(r) =γ/r+O(r ^{−2})] is convergent in the Abel sense although it diverges in the ordinary sense. The method of Abel summation is a generalization of the ordinary summation and allows one to sum certain divergent series explicitly. It is closely connected with analytic continuation; with the help of optimal conformal mappings the convergence of the Abel sum (for long‐ and short‐range interactions) can be improved substantially. This enables us to obtain values of the scattering amplitude for each scattering angle (except forward direction). In particular, we show that the screened scattering amplitude converges in the Abel sense up to a phase factor to the unscreened one if the screening is removed.

Source equivalency based on the energy specification of order l
View Description Hide DescriptionIt is shown that the multipole expansion of electrostatic energy can be expressed in the form of energy specification S{W _{ e }} =〈[P]_{ l },∇^{(l)}φ〉, where ∇^{(l)} is a differential operator, whereas φ and [P]_{ l } represent an arbitrary test potential and equivalent reduced volume multipole density, respectively. Two electrostatic sources are l‐equivalent if their energy specifications are identical. The formalism by means of which electrostatic multipole sources can be effectively handled is developed.

On the Lorentz dipole approximation in static electrodynamics
View Description Hide DescriptionThe energy specification S{W _{ e }}=〈[P]_{ l=0}, φ〉 of electrostatic sources introduced in the preceding paper is further developed. The electrostatic potential Φ due to a source distribution is defined as a generalized function satisfying the energy specification equation S{W _{ e }} =〈−ε_{0}∇^{(n)} ⋅∇^{(n)}Φ, φ〉, where n denotes the order of a multipole approximation is reviewed. Specifically, the generalized function of an equivalent field corresponding to a classical field intensity E is introduced in terms of a given energy specification S{W _{ e }}. Besides the equivalent field, a family of generalized functions D_{ n } referred to as the characteristic fields is introduced to deal with the displacement vector D. A formal description of the so‐called polarization charges comes out of an analysis of an equivalent field. Equivalent fields of magnetostatic problems are discussed on the basis of the magnetostatic energy specification S{W _{ m }}.

An example of an H‐space
View Description Hide DescriptionThe good cut equation for a specific asymptotic shear is solved and the metric of the associated H‐space is found to be type N, asymptotically flat and positive frequency.