Table of contents

The intelligent spin states are defined as those states which satisfy the Heisenberg equality for the spin operators: Delta J_x² Delta J_y²= mod (J_z) mod ². The 2j+1 states which behave intelligently in each angular momentum space of spin j are found explicitly. For this purpose the Radcliffe states are used, it is shown that only the real and the pure imaginary Radcliffe states are intelligent. These intelligent states also satisfy the quartic consistency condition. The result, however, does not disagree in principle with the recent claim of Kolodziejczyk and Ryter (1974) that mod mu )= mod 0) is the only state which minimizes the uncertainty product because minimum uncertainty does not necessarily imply intelligence.

The low-temperature high-field polynomials for the free energy are given for the general Blume-Emery-Griffiths (1971) model on the FCC lattice. The order parameters and susceptibilities are obtained for three values of the biquadratic interaction and a specific form of the external field Delta . One of the treated cases is the Potts model.

A simple formula is obtained by which any symmetrized power of D(j) may be expressed in terms of the symmetrized powers of D(j-¹/₂). This formula does not depend explicitly on the representation theory of the symmetric group and it gives a method of building up any symmetrized power in fully reduced form. In particular, recurrence formulae are obtained for the totally symmetrized and totally antisymmetrized powers of D(j), and a formula is given for arbitrary symmetrized powers of D(1). Finally it is proved that D(j)" contains D(j-1)ⁿ, n>or=2, as a proper subrepresentation and formulae are obtained for the symmetrized cubes of D(j).

A discussion of the number of labelling operators specifying an abstract basis for an irreducible representation is given which indicates that in the U(N) contains/implies O(N) state labelling problem, for two-rowed irreducible representations (p,q,0,...,0) of U(N), a single additional commuting labelling operator Lambda is required. An analysis of the tensor representations leads to a suitable labelling scheme involving an additional whole number label lambda . A U(N) contains/implies O(N) branching theorem, for two-rowed representations, is formulated, and the branching multiplicities written down. Combinatorial techniques developed by Green and Bracken in 1973 for the case of U(3) contains/implies SO(3), adapted to the case of U(N) contains/implies O(N), lead to a polynomial identity of the form P( Lambda , Phi ,...)=0, which implicitly defines the labelling operator Lambda (with eigenvalue lambda ) in terms of certain O(N) invariants Phi (which are functions of the U(N) generators) and other known labelling operators and invariants. The calculations also give a cubic polynomial identity, satisfied by the N*N matrix of U(N) generators in two-rowed representations. Some physical applications of the U(N) contains/implies O(N) state labelling problem are briefly mentioned.

The Fokker-Wheeler-Feynman theory of action-at-a-distance electrodynamics is not conformally invariant. It is shown how to modify it to restore conformal invariance, and it is proved that the modified theory yields Maxwell's equations and gives Dirac's expression for the radiation reaction.

For pt.II, see abstr. A66629 of 1973. The authors show that the Frechet derivative, which they have previously used to formulate a quantum-mechanical version of Hamilton's principle of stationary action, suffers from certain defects. They set up a simpler, coordinate-free, formulation of quantum mechanics based on the Lie algebra of symmetric contravariant tensor fields on the configuration manifold of the system and re-express the principle of stationary action using this formalism. They find the same relation between Hamiltonian and Lagrangian as before, but a much larger class of allowable variations, namely those associated with and C^infinity vector fields.

The theory of the phase equation for elastic scattering, with spherical symmetry, is applied to the problem of evaluating efficiently the scattering phase shifts in a partial-wave analysis. A fortran source version of the subroutine is available from the author.

Schwinger's coupled-boson representation of spherical and hyperbolic angular momentum is related to spherical harmonics and is used for the calculation of matrix elements. It is shown that radial matrix elements of the two-dimensional isotropic oscillator and the closely related hydrogenic matrix elements are proportional to 3j-symbols. Matrix elements of exponentials of hyperbolic angular momentum operators which form a formal analogy to the well known rotation matrice d_mm'^j( theta ) and which are written as t_mm^jj'( theta ) are derived. As only elementary algebra and some operator formulae are used, the algebraic method supplements analytical and group theoretic methods.

Relations between harmonic and radiating coordinates are investigated for the case of asymptotically flat space-times, and an approximation to the 'Bondi news function' in terms of the energy-momentum tensor is found.

In a preceding paper, (see abstr. A2555 of 1974), the authors obtained a solution of the field equations of general relativity for an elastic sphere of constant density. The present investigation obtains a solution of the charged elastic fluid distribution in general relativity.

It can be shown that general relativity admits structures that are strictly conserved in number for topological reasons. Such objects, called 'kinks', are associated with the particular form of the metric and it has been shown that a kink has many of the properties expected of a classical analogue of a Fermi particle. The paper examines the general form of a stationary spherically symmetric metric corresponding to a single kink centred at the origin. This is a generalization of the usual spherically symmetric Schwarzschild case. The Einstein tensor is deduced for this system, and it is shown that there is no free-field solution for this case. It is indicated how possible extensions, such as the inclusion of electric charge, may lead to the possibility of solutions.

A solution for the interior metric of a sphere has been obtained with the effective mass density as a variable quantity.

An expression is proposed for the energy-momentum density of a first-order gravitational shock wave.

An earlier result obtained by Elvey (see abstr. AA16753 of 1973) giving the Yang-Lee distribution in the beta plane for continuum systems of particles with hard cores and nearest-neighbour 'rational step potentials' is here extended to general nearest-neighbour potentials.

By using a new method, the Boltzmann transport equation is deduced from the Liouville equation. The method involves an assumption of a statistical nature, which can be relaxed so as to yield equations more general than the Boltzmann equation, or than presently known extensions of the Boltzmann equation. The method also clarifies the origin of the time irreversibility of the Boltzmann equation.

A ferroelectric ice model or KDP model, after the Slater KDP model for a square lattice, is formulated on a triangular lattice and solved for all temperatures. It is found that this system undergoes a phase transition of the first order into a frozen ferroelectric state.

The lattice gas problem with third nearest neighbour exclusion on a square lattice is shown to be equivalent to the problem of placing hard cross-shaped pentamers on the square lattice and the existence of a phase transition is proved by means of the Peierls argument.

A new method of deriving high temperature series expansions for the susceptibility of the Ising and classical vector spin models is presented. The method involves generating the expansion for the inverse of the susceptibility, in which case only contributions from star graphs need to be considered. An additional term (the thirteenth) has been added to the Ising high temperature susceptibility series for the FCC lattice. Analysis of the extended series does not indicate that revision of previous critical estimates is required. In a test of the capabilities of the method for other systems, seven terms of the susceptibility series of the FCC Heisenberg and planar vector models have been reproduced. Techniques used in generating the graph data are discussed.

The semi-classical laser equations are analysed outside the rotating-wave approximation and a new exact steady-state solution for stationary but inhomogeneously broadened atoms is obtained. Results applicable to any solid-state laser are derived and these are also relevant to a gas laser working near threshold or at very high intensities. The resonance frequency exhibits a Bloch-Siegert shift. New results in the theory of frequency pulling and pushing are obtained.

It is suggested that the conventional view of Brillouin light scattering and ultrasonic excitation in fluids in terms of phonon phase velocity and attenuation can be misleading and leads to spurious anomalies. A simple comparison is made of the two types of experiments from a fundamental point of view. The analysis of an ultrasonic experiment from first principles is given and is illustrated by an explicit example, the classical hydrodynamic fluid. It is pointed out that an important parameter is usually overlooked in the conventional ultrasonic experiment.

For pt.II, see abstr. A3682 of 1974. The theory developed in two previous papers, for studying the Weaire model in terms of returning walks, is used here to study the density of electronic states at the band edges. The relationship between topological disorder and the shape of the band edges is considered and the shape of the inner band edge is shown to depend on oscillating terms in the asymptotic expansion of the number of returning walks. The work suggests the possibility of a band gap in a fourfold coordinated structure with mod V₁/V₂ mod >¹/₂.

As distinct from conventional electrodynamics in which the advanced potential solution of Maxwell's equations is rejected on causal grounds, absorber theory allows the possibility of a mixture of advanced and retarded radiation, dependent on cosmological boundary conditions. In a recent experiment Partridge attempted to detect advanced effects by introducing a local absorber, but it was maintained by Pegg that, because a static absorber was used, only a null result was possible. The authors give the theory and a brief outline of an experiment which uses a time-asymmetric chopper absorber to alter the boundary conditions and thus the ratio of the advanced to retarded components in the mixture, provided this is non-zero initially, leading to possibly detectable effects.

A true lifetime effect on the superfluidity in neutron stars due to quasi-particle scattering processes is calculated with the Green function method and is found to lower the superfluid transition temperature by about 20%. This effect constitutes part of the impurity effects on the superfluidity in neutron stars, i.e. it partly arises from the presence of neutrons in the proton superfluid phase and protons in the neutron superfluid phase.