Table of contents

An alternate method is used to establish the analytic expression recently given and incompletely proved by Mikhailov (see ibid., vol.10, p.147 (1977)) for the multiplicity of occurrence of any angular momentum in the vector addition of n identical angular momenta.

The spherically symmetric monopole solutions, whose existence was rigorously demonstrated by Tyupkin, Fateev and Shvarts (see Math. Theor. Phys., vol.26, p.270 (1976)) are shown to be regular everywhere.

The dielectric RPA response function is used to evaluate the thermal conductivity of a partially degenerate electron gas in the presence of classical ions, for any temperature.

Stone's Cartesian-spherical transformation formalism is shown to yield a unified, simple and concise demonstration of the properties of the regular and irregular solid harmonics.

A derivation of the line groups, which are the symmetry groups of the stereo-regular polymer molecules, is presented. Every line group is an extension of a one-dimensional translation group by a point group. The point group is either cyclic or a semi-direct product of cyclic groups. The expounded method of derivation of the line groups consists of first extending by the cyclic point groups and then obtaining the rest of the line groups by unification within the Euclidean group. To this purpose, a simple test is given to decide whether two line groups multiply into a third one. The method displays the subgroup structure of the line groups, relevant to the construction of their irreducible representations. All the line groups are derived and tabulated.

The authors have formulated a simple method, the collocation variational method, to solve the Fredholm integral equations of the second kind and have proved its convergence. As applications, the method can be usefully employed not only to solve the Fredholm equations, but also some other equations reducible to it, and, in particular, the Lippman-Schwinger equation in potential scattering.

In order to study the accuracy and properties of a certain type of arbitrary-order phase-integral approximations, especially as regards the connection formula for tracing a wavefunction from a classically forbidden region to a classically allowed region, the Schrodinger equation for the one-dimensional harmonic oscillator is solved by means of the first, third and fifth order of these approximations, and the resulting approximate wavefunctions, both when normalised and when fitted to the exact wavefunctions at + infinity , are compared with the corresponding exact wavefunctions. The numerical results show that, except for the lowest energy eigenstates, the higher-order phase-integral approximations in question are very accurate both in the classically allowed and in the classically forbidden regions.

The increasing versatility of polarisation experiments in particle and nuclear physics has brought into focus theoretical problems associated with the optimum extraction of information from experimental results. One such problem is the selection of observables whose measurement would lead to the unambiguous determination of reaction amplitudes of interest. The paper presents a solution of the problem in the form of general criteria using certain easily constructed diagrams.

Einstein's field equations are applied to infinite cylinders of fluid, and some exact solutions are obtained by a systematic method. Solutions having p varies as rho are shown to be unstable as the equation of state p= rho is approached.

The residue at the graviton ghost pole for spinor-spinor scattering of gravitating spin-¹/₂ particles is evaluated for the lowest-order graph, and proved to be non-zero. It is concluded that attempts to quantize gravity in models for which counter terms are required are unsuccessful; a possible class of models is briefly considered.

Existence with probability one of the thermodynamic limit for the free energy density of a class of classical and quantum quenched random spin systems is proved using strong laws of large numbers. When the control on the randomness is strong enough, the infinite-volume free energy so obtained is shown to be equal, with probability one, to that derived from a prescription originally due to Brout (1959). Particular mean-field models are then studied in detail. Similar arguments are pointed out concerning the corresponding thermodynamic functions, and are then applied to the existence problem of phase transitions in quenched random systems with continuous internal symmetry groups, in particular to those models recently proposed to describe the spin glass phenomenon, like the Edwards-Anderson model and its various classical and quantum extensions.

The concept of the helicity modulus Y_d(T), introduced by Fisher, Barber and Jasnow (see Phys. Rev., vol.A8, p.1111 (1973)), is applied to the ideal Bose gas in d dimensions. Above the critical temperature T_c,d, Y is found to vanish identically, while for T<T_c,d, Y_d(T)=(h(cross)² rho /m)(1-(T/T_c,d)¹2d/), where rho is the total density. The relevance of these results to more general theories of superfluidity is discussed.

Previous exact enumerations of the numbers and mean square lengths of short, first-neighbour-avoiding walks on the face-centred cubic, body-centred cubic and tetrahedral lattices have been extended to 12, 13 and 21 terms, respectively. Examination of the augmented data suggests an asymptotic expression for the mean square length of the form (R_n²) approximately An⁶5/+Bn^alpha . For the tetrahedral lattice this conjecture is supported by some new Monte Carlo data.

Necessary and sufficient conditions for the conformal invariance of a multiple integral variational problem whose Lagrangian depends upon second-order derivatives of a covariant vector field are obtained. These conditions take the form of differential identities involving the Lagrangian, its derivatives, and the infinitesimal generators of the special conformal group; they differ from the classical Noether identities in that they involve only second-order derivatives of the field, not fourth-order derivatives. The conditions are not conservation laws, but rather identities which provide a practical test for invariance which, if established, can lead to conservation laws via the Noether theorem. Finally, an application to 'generalised electrodynamics' is given.

The existence and structure of localised solutions for a non-linear spinor field are studied.

In electrodynamics the longitudinal components of all charged-particle Green functions are determined by the charged-particle propagator. Inserting these components into the Dyson-Schwinger equation leads to an integral equation for the propagator itself which may be solved and used as a basis for an iterative determination of the transverse Green functions' components. The solution for electrodynamics of scalar mesons is found in analogy with recent work on spinor electrodynamics.

It is shown that the usual (non-existing) Hamiltonian operator H' governing the interaction of a one-electron atom with transverse photons, can be written as the sum of a finite number of self-adjoint and bounded 'partial' interaction Hamiltonians L, where each L has a well defined physical meaning. The systematic 'Weisskopf-Wigner approximation scheme' consists of special sequences of partial sums of L. The system of partial sums of L is associated with a system of graphs where each graph defines uniquely a certain Weisskopf-Wigner theory and visualises its physical content in a comparable way to a Feynman graph. Finally some applications are given.

It is shown that the homotopy classes of soliton-type solutions to chiral field theories in three space dimensions are determined entirely by cohomological properties. This result is employed to construct topological currents which are identically conserved and whose integrated time component contains all the homotopical information.

Equations given by Harish-Chandra (1947) describing fields having both spins ¹/₂ and ³/₂ are minimally coupled to an external homogeneous magnetic field, and found to exhibit the usual acausality problems. One of the Harish-Chandra equations is a particular case of the Bhabha-Gupta equation and has similar acausality problems. The other is more complicated and this is reflected in additional acausal modes. The composite nature of the Harish-Chandra particles is used to discuss suggestions that high-spin problems may be resolved by regarding high-spin particles as composite.

Propagation of disturbances through relativistic Vlasov plasmas in the presence of a uniform external magnetic field has been investigated. The main purpose of the paper is to solve the dispersion relation in exact expressions for an arbitrary isotropic equilibrium distribution function. Analytic expressions in the form of single integrals have been derived for the propagations parallel and perpendicular to the external magnetic field. Expressions for the limits and asymptotic values of the frequency ratios have also been obtained. Numerical results for Maxwellian distribution are depicted graphically.

The author motes that Ninio (see ibid., vol.9, p.1281 (1976)) in his simple proof of the Perron-Frobenius theorem repeats in fact the Jentzsch proof (1912).

The new spin-³/₂ wave equation of Jayaraman (see ibid., vol.9, p.L131 (1976)) is shown to be an eight-dimensional form of the Dirac equation.