Table of contents

The modification of Levinson's theorem for the case of non-local potentials suggested by many authors is shown to be incorrect. The error is a result of overlooking a simple property of the inverse tangent function.

A useful coordinate system in the vicinity of a caustic point is described. The local expression of the Hamilton-Jacobi phase function of the geodesic flow and the first order WKB approximation of Einstein's equations in a neighbourhood of the caustic are given.

A non-linear integral equation is derived for the percolation probability of the correlated bond percolation problem on the Bethe lattice. Analysis of the integral equation yields exact results for the critical probability and the behaviour in the critical region.

Using exact inequalities, it is proved, for d-dimensional Ising models with ferromagnetic pair interactions, that: (i) any triplet order parameter ( sigma ₁ sigma ₂ sigma ₃) vanishes at the critical temperature with the same exponent as the conventional order parameter ( sigma ₁); and (ii) the associated susceptibilities diverge as T to T_c from above with the same exponents. The first result confirms a recent conjecture of Wood and Griffiths (see ibid., vol.9, p.407, (1976)) based on series expansions.

The Hilbert space L²(SU(2)) is used as a representation space for a (unitary) representation of the direct product group SU(2) x SU(2) and the corresponding group algebra. Three different types of operators which are closely related to the representation theory of SU(2) are used to construct convenient operator bases whose elements are irreducible tensor operators with respect to SU(2)*SU(2). A complete set of irreducible tensor operators and useful operator identities are derived.

The Hilbert space L²(SU(2)) is used as a representation space for a (unitary) representation of the semi-direct product group (SU(2)*SU(2)) s S₂ and the corresponding group algebra. Special operators are constructed which are closely related to the representation theory of the groups SU(2) and S₂ and are irreducible tensor operators with respect to (SU(2)*SU(2))s S₂. These operators are then used to define a complete set of irreducible tensor operators and to calculate two classes of Clebsch-Gordon coefficients of (SU(2)*SU(2) s S₂.

The relation between the factor systems of a group and the factor systems of an invariant subgroup is discussed both for PU and for PUA representations. The results are used to discuss the factor systems of a class of magnetic space groups.

Weight vectors and weight multiplicities are defined in terms of group characters. The characters appropriate to all the unitary irreducible representations of the unitary, orthogonal and symplectic groups are expressed in terms of S-functions. The resulting explicit formulae for weight multiplicities are used to tabulate results by making use of the definition of S-function in terms of standard Young tableaux. The results obtained give, for the first time, the k dependence of the weight multiplicities of the groups U(k), O(2k+1), Sp(2k) and O(2k). There is no limitation on the size of k nor on the dimensions of the representations.

The coordinate-free analysis developed in an earlier paper (see ibid., vol.8, p.1853 (1975)) is used to provide a simplified proof of the Bazley inequalities for establishing lower bounds to the eigenvalues at atomic Hamiltonians. It is then explained how Bazley lower bounds enable one to get other lower bounds to eigenvalues without using empirical values of higher eigenvalues.

Using Baker-Campbell-Hausdorff formulae for the exponentials of the generators of the SU₂ and SU(1,1) groups, the matrix elements of spherical and hyperbolic rotations are calculated in the angular momentum basis.

An addition theorem is derived for the regular and irregular Coulomb functions by means of the symmetry properties of the Coulomb problem in analogy to that for the spherical Bessel functions. The coefficients which enter in the addition theorem are closely related to 9j symbols of complex angular moment. For the computation of these coefficients a complete set of recurrence relations is given. In addition, some useful relations are presented for the Coulomb function in configuration space as well as in momentum space.

A formalism of time-smoothed total transition probabilities and their rates is developed which employs Laplace averages of these quantities. Under conditions pertinent to scattering processes, the Laplace-average formalism is shown to yield results equivalent to those obtained from a stationary-state formalism. Rigorous lower and upper bounds are obtained for the Laplace-averaged quantities which reduce to equalities for two-level systems. The lower bounds appear to be potentially useful estimating lower bounds for total cross sections of various processes.

For pt.I, see ibid., vol.9, p.245 (1976). A construction technique previously introduced is applied in search of new, asymptotically flat, metrics. Lengthy calculations lead to the conclusion that there can be no new non-stationary metric, approaching the Kerr metric at time-like infinity, for values of the metrical expansion parameter n smaller than two. The generality of the method is also demonstrated.

The paper considers the application of the metric ds²=dr²+r²d theta ²+dz²+2 omega r²d theta dt-(c²- omega ²r²)dt² to a rotating system. The first part of the paper considers the application of radar measurement to the rotating system. In the second part infinitesimal radar measurements are used to show that for an observer at a radius R within a rotating system, similar results are obtained, through the application of the metric, to those which would be obtained by the use of instantaneous rest frames.

The renormalization prescription of t'Hooft and Veltman (see Nucl. Phys., vol.B44, p.189 (1972)) is used to construct crossover scaling functions for the susceptibility and free energy in an isotropic n vector model. Some difficulties in interpreting the epsilon expansion in this context are discussed, and the formalism is illustrated by a calculation of the expansion factor alpha ² of a polymer in dilute solution.

The Fokker-Planck equation is constructed in the space of two variables: s, the long-range order and theta , the nearest neighbour correlation function in the quasi-chemical approximation. This generalizes the equation in the Bragg-Williams approximation. The master equation is taken as the starting point and expressions are obtained for the transition probabilities. The equations of motion for the mean values of s and theta are derived: they behave satisfactorily in an asymptotic manner and yield the thermodynamic values. The time-dependent fluctuations are not given by the approximations used; the fluctuations in equilibrium are derived from the bivariate normal distribution representing the equilibrium state.

Systematic methods are developed for calculating the partition functions of star topologies in the D-vector model. A particularly simple technique (ladder transformation) is proposed for topologies containing a 2-cycle (ladder topologies), and these constitute numerically the majority of star topologies. The fewer non-ladder topologies need individual attention and three methods are suggested (i) making a selected bond infinite, (ii) using direct averages, (iii) considering the behaviour as D to infinity . By suitable renormalization of the interaction it is shown that as D to 0 the self-avoiding walk model results.

The mathematical structures of the theories of a two-level atom interacting with radiation and an electromagnetic wave interacting with a dielectric medium in which it is propagated as a plane wave are identical. The spin structure of the Jones operator in optics representing the polarizer, is obtained in its general form in terms of Stokes parameters and identified with the optical density operator with a spin structure identical with the density operator in the quantum mechanics of the atom. While the quantum dynamical equation of the atom can be reduced to the gyroscopic form, the correspondence law giving expression to the parallelism of the two processes leads to forms of the dynamical law in topics which are identical with those of the quantum theory of the atom.

The representation of the transitional state of a two-level system interacting with the radiation field on the complex plane provides a point of comparison between polarization (in the optical sense) and atomic transition processes. This corresponds to the Poincare representation of polarization in optics. Polarization and dynamics are two aspects of an interaction process.

Using the theory of Young symmetrizers it is shown how to obtain algebra which are sufficient for causal propagation of higher-spin particles interacting with an external minimal electromagnetic field. The commutation relations of the algebra are derived already expressed in irreducible Young symmetrizer form. An example is given which is not equivalent to any known causal theory and it is shown that the algebra is infinite. Thus, the requirement of causal electromagnetic interaction is not sufficient to generate a finite algebra and non-trivial sub-algebra may exist which are causal.

Starting from the basic supersymmetry algebra of 2n-component Weyl spinors it is shown how this is enlarged to incorporate the discrete symmetries of parity and charge conjugation. The transformation properties of the superfield representations of the enlarged Dirac supersymmetry are found, and the particle content of a simple example examined.