Volume 20, Issue 2, February 1979
Index of content:

Group‐theoretical foundations of classical and quantum mechanics. II. Elementary systems
View Description Hide DescriptionClassical and quantum elementary systems are investigated from the point of view of invariance under a connected Lie group. The classification and characterization of elementary systems are considered in a unified way. The representation theory of symmetric and enveloping algebras is used as a tool in order to characterize the observables physically and also to analyze the analogies between classical and quantum mechanics. The results obtained are applied to the Galilei, Poincaré, and Weyl Lie groups.

Exotic spinors in superconductivity
View Description Hide DescriptionThe existence of inequivalent types of spinors in spaces which are not simply connected is mathematically investigated. The mathematical results provide a purely geometrical explanation of the charge dependence of quantized flux and Josephson current in superconductivity.

On the expansion of the propagator in power series of the coupling constant
View Description Hide DescriptionWe show by simple rearrangements of the Lippmann–Schwinger equation, that the propagator (Green’s function) of a linear physical field can be expanded in power series valid for values of the coupling constant either close to unity or very large. The new series satisfy the generalized resolvent operator equation first derived by Mockel.

Conformal flatness and the Schwarzschild interior solution
View Description Hide DescriptionThe paper gives a rigorous proof of the theorem that the only static conformally flat metric for a perfect fluid distribution is the Schwarzschild interior metric.

Invariant operators of IU(n) and IO(n) and their eigenvalues
View Description Hide DescriptionA systematic explicit evaluation of the invariant operators of IU(n) and IO(n) has been carried out. It is found that the invariant operators of IU(n) and IO(n) can be obtained from those of U(n,1) and O(n,1) by simple substitution. Similarly the eigenvalues of the invariant operators of IU(n) and IO(n) can be obtained from those of U(n,1) and O(n,1) by simple substitution. Since the invariant operators and their eigenvalues of U(n,1) and O(n,1) are closely related to those of U(n+1) and O(n+1) our results can be expressed in explicitly closed and simple form.

Automorphisms of the Bianchi model Lie groups
View Description Hide DescriptionA derivation of the group of automorphisms of the Lie group of isometries characteristic of each of the nine Bianchi types of cosmologies is presented.

Perturbative solution of the Percus–Yevick integral equation for a general class of intermolecular potential
View Description Hide DescriptionA qualitative investigation of the Percus–Yevick integral equation by perturbation method is discussed for a general class of intermolecular potential. Under some general assumptions it is proved that the Percus–Yevick integral equation has a unique solution when the particle density ρ is in the region 0<ρ<0.33, and a divergent solution when ρ is greater convergent if the supremum norms of the nth order solutions are less than or equal to n!

Classification of gauge fields
View Description Hide DescriptionIt is shown that the Lorentz invariants of an arbitrary gauge field are double valued functions of a Lorentz invariant matrix L ^{ i j } when rank L=3 and single valued functions of L ^{ i j } when rank L≠3. The question of how many Lorentz inequivalent realizations of the Lorentz invariants there are is answered. This leads naturally to a classification of an arbitrary gauge field at a space–time point, given previously by Anandan and Tod based on the rank of L. The answer to the analogous question for the gauge invariants of the SU(2) gauge field leads to a new classification of this field. Five more classifications of this field, including one which is symmetric with respect to space–time and isospin groups, are also presented.

Exceptional gauge groups and quantum theory
View Description Hide DescriptionIt is shown that a Hilbert space over the real Clifford algebraC _{7} provides a mathematical framework, consistent with the structure of the usual quantum mechanical formalism, for models for the unification of weak, electromagnetic and strong interactions utilizing the exceptional Lie groups. In particular, in case no further structure is assumed beyond that of C _{7}, the group of automorphisms leaving invariant a minimal subspace acts, in the ideal generated by that subspace, as G _{2}, and the subgroup of this group leaving one generating element (e _{7}) fixed acts, in this ideal, as the color gauge group SU(3). A generalized phase algebra A⊇C _{7} is defined by the requirement that quantum mechanical states can be consistently constructed for a theory in which the smallest linear manifolds are closed over the subalgebra C(1,e _{7}) (isomorphic to the complex field) of C _{7}. Eight solutions are found for the generalized phase algebra, corresponding (up to an overall sign), in effect, to the use of ±e _{7} as imaginary unit in each of four superselection sectors. Operators linear over these alternative forms of imanary unit provide distinct types of ’’lepton–quark’’ and ’’quark–quark’’ transitions. The subgroup in A which leaves expectation values of operators linear over A invariant is its unitary subgroup U(4), and is a realization (explicitly constructed) of the U(4) invariance of the complex scalar product. An embedding of the algebraic Hilbert space into the complex space defined over C(1,e _{7}) is shown to lead to a decomposition into ’’lepton and ’’quark’’ superselection subspaces. The color SU(3) subgroup of G _{2} coincides with the SU(3) subgroup of the generalized phase U(4) which leaves the ’’lepton’’ space invariant. The problem of constructing tensor products is studied, and some remarks are made on observability and the role of nonassociativity.

Scattering theory and polynomials orthogonal on the unit circle
View Description Hide DescriptionThe techniques of scattering theory are used to investigate polynomials orthogonal on the unit circle. The discrete analog of the Jost function, which has been shown to play an important role in the theory of polynomials orthogonal on a segment of the real line, is defined for this system and its properties are investigated. The relation between the Jost function and the weight function is discussed. The techniques of inverse scatteringtheory are developed and used to obtain new asymptotic formulas satisfied by the polynomials. A set of sum rules satisfied by the coefficients in the recurrence relaxation is exhibited. Finally, Szegö’s theorem on Toeplitz determinants is proved using the recurrence formulas and the Jost function. The techniques of inverse scatteringtheory are used to find the correction terms.

Covariant perturbed wave equations in arbitrary type‐D backgrounds
View Description Hide DescriptionWe present an approach to the fundamental tensorial quantities of general relativity which is inherently covariant and based on the irreducible representations of the Lorentz group, O(3,1). Using this technique, the wave equations appropriate to perturbing, massless, D(0,s), fields in an arbitrary curved background are studied and a relationship between the decoupling of (at least) one of the equations and the algebraic degeneracy of the spacetime is shown. It is then found that sufficient conditions for decoupling the equations determining both of the radiative components (the extremal helicities) are that the space be of type D. Using Plebański–Demiański coordinates to describe such an arbitrary vacuum spacetime (of type D), we separate the (decoupled) perturbation equations for the radiative components corresponding to spin s=0, 1/2, 1, and 2.

On the completeness of the natural modes for quantum mechanical potential scattering
View Description Hide DescriptionThe set of natural modes, associated with quantum mechanical scattering from a central potential of finite‐range is shown to be complete. The natural modes satisfy a non‐Hermitian homogeneous integral equation, or alternatively, are solutions of the time independent Schrödinger equation subject to a recently formulated nonlocal boundary condition (the quantum mechanical extinctiontheorem). An expansion theorem similar to that of Hilbert–Schmidt is formulated, valid for values of the solution of the scattering integral equation inside the range of the potential. The boundary conditionsgenerated by the quantum mechanical extinctiontheorem are shown to be closely connected with the Jost function.

Canonical Schwartz spaces and generalized operators
View Description Hide DescriptionFor a C ^{∞} multiplier ω, on R ^{ n } we define the concepts of differentiability and codifferentiability in the Von Neumann algebra generated by the regular ω representation of R ^{ n }. Analogs of the classical Schwartz space and its dual are formulated and the case where ω is fully antisymmetric is studied. Connections with the canonical Fourier transform of an earlier paper are investigated.