Table of contents

The analytic structure of the Feynman amplitude for arbitrary Euclidean external momenta is discussed.

A representation on Hilbert space of canonical transformations to action and angle variables is given for a wide class of one-dimensional periodic motions. This extends the results discussed previously for the harmonic oscillator to problems not solvable in closed form. The concepts of ambiguity group and ambiguity spin continue to play a key role.

For convenience the letter opens with a summary of the accepted elementary treatment of the coupled fluxes and forces in the flow of entropy and mass according to Onsager-based irreversible thermodynamics of the steady state. It is then shown that a simple transformation entirely consistent with that treatment shows that there also exists a set of uncoupled fluxes and forces. The physical significance of the uncoupled set is clear, and it is convenient to deal with operationally. Some important questions raised by the existence of this alternative set are discussed.

The Ginzburg-Landau free energy describing p wave paired neutron star matter is used to derive renormalisation group equations. The absence of infrared stable fixed points indicates a first order phase transition.

The authors show that the massless Dirac equation and Maxwell equations are invariant under a 23-dimensional Lie algebra, which is isomorphic to the Lie algebra of the group C₄(X)U(2)(X)U(2). It is also demonstrated that any Poincare-invariant equation for a particle of zero mass and of discrete spin provide a unitary representation of the conformal group and that the conformal group generators may be expressed via the generators of the Poincare group.

The large-order behaviour of the 1/N expansion in the zero- and one-dimensional g phi ⁴ model is investigated. The one-dimensional N-vector model is also considered. The asymptotic behaviour shows oscillations due to complex instantons. The phase and the amplitude of these leading behaviours are determined theoretically and are compared with the explicit numerical calculations of higher orders. In the one-dimensional g phi ⁴ model the relevance of the large-N instanton is discussed.

A recently proposed regional variational method is used with a mixture of local and global trial functions to solve the Schrodinger equation for a discontinuous potential. Good results are obtained. In addition, it is illustrated how known features of the solution may be included in a conventional finite-element method, and the resulting equations solved efficiently.

Diffractals are waves that have encountered fractals. Fractals are geometric objects with non-integral Hasudorff-Besicovitch dimension D; they have structure down to arbitrarily fine scales. Diffractals are a new wave regime characterised by a short-wave limit in which ever finer levels of structure are explored and geometrical optics is never applicable. The diffractal studied here is the wave psi (x,z) at distance z beyond a one-dimensional random phase screen that deforms an initially plane wavefront at z=0 into a random fractal curve h(x) with power law spectrum and dimension D (between 1 and 2), after which the wave propagates freely. Some averages of psi are calculated. These are ( psi ), ( psi (x,z) phi *(x+X,z)), the spectrum of psi and the spectrum of the intensity fluctuations and, most important, the second intensity moment I₂ identical to ( mod psi mod ⁴). It is proved that the intensity fluctuations are non-Gaussian. A variety of scaling laws is derived, all involving D. For the 'Brownian' diffractal (D=1.5) all averages are expressed exactly in closed form.

The motion of relativistic particles in the field produced by two circularly polarised, electromagnetic plane waves travelling in opposite directions is studied. The Klein-Gordon equation is used, i.e. spin effects are neglected, and only motion along the beam direction is considered. If the electromagnetic beam carry opposite polarisation, the particle energy shows a band structure. The transmission of a plane particle wave through a long device is calculated. Transmission occurs only if the particle energy corresponds to an allowed energy band. For other energies the particles are totally reflected. For equal polarisation a similar structure emerges for the particle momentum along the beam direction. The observation of the band structure poses serious experimental problems.

The quantum-mechanical joint (quasi-) density in phase space (the Wigner distribution) for thermodynamic equilibrium is studied in the semiclassical limit h(cross) to 0. The approximation is of an asymptotic (WKB) type and nonperturbative in character. The equilibrium phase-space distribution can be expressed in terms of classical paths satisfying certain well defined conditions. These paths are complex-valued, i.e. classically forbidden. The resulting semiclassical expression for the equilibrium Wigner phase-space density agrees with the exact quantum result for the analytical solvable case of harmonic potentials for all temperatures.

In this work the scattering from an infinite dielectric cylinder embedded into another cylinder is considered. In the case of an eccentric circular cylinder the problem is solved using the classical separation of variables techniques combined with related translational addition theorems. When the difference in the indices of refraction of the cylinders is small, a perturbation series is developed up to second order. For non-circular arbitrary inhomogeneities, an integral equation method is developed by employing the homogeneous scatterer Green function. Numerical results are computed for circular cylindrical inhomogeneities; the convergence of the perturbation series and the properties of the scattered field are discussed.

A proof that a certain polynomial has n real zeros is presented. From this it then follows that the generalised Kerr-Tomimatsu-Sato metric has 2n distinct infinite red-shift surfaces.

A procedure for calculating the Debye potential corresponding to a stationary axisymmetric distribution of charges and currents in the Kerr metric is given. The electromagnetic potential is derived by differentiation. The potential of a point charge on the symmetry axis and the value of the charge resulting from the accretion in the case of a charged current loop situated in the equatorial plane are determined with this formalism.

A global solution of Einstein's equations for an infinite line-mass of perfect fluid is given. The interior solution depends on two arbitrary constants, and both of these are needed to describe the exterior space-time.

The solution represents the external field of an isolated mass carrying electric charge and dipole moment.

The equation for hydrostatic equilibrium has been solved for the equations of state P=K rho for finite central densities and K>or=¹/₃. The maximum of the mass for an isothermal neutron star core is at K=0.80 and not at K=1.00. Irrespective of the value of K, the maximum of the core mass occurs at the central density rho _c approximately=3*10¹⁵ g cm^-3 and the radius of the isothermal core comes out to be 12-13 km.

The variational approximations to the partition function of a particle interacting with systems of disordered scatterers, presented by Luttinger (1976) for uniformly random systems and generalised by the author to systems with arbitrary positional disorder, are used to carry out explicit calculations on one-dimensional model systems. The models consists of uniform random configurations of delta functions with a given density which are constrained by a two-particle correlation preventing a closer approach than specified for any pair. The results are used to calculate approximations to the densities of states for such systems with reasonable and interesting results.

THe authors investigate the spin-¹/₂ Ising model with nearest-neighbour interactions on the four-dimensional simple hypercubic lattice. High-temperature series expansions are studied for the zero-field susceptibility chi ₀ and the fourth-field derivative of the free energy Xi ₀⁽²⁾ up to order nu ¹⁷. The series are analysed for singularities of the form t^-1 mod 1nt mod ^p where t is the reduced temperature. For chi ₀ it is found that p=0.33+or-0.07 when q=1, in good agreement with the prediction p=¹/₃, q=1 of renormalisation group theory. The critical temperature is estimated to be nu _c^-1=6.7315+or-0.0015. Results for chi ₀⁽²⁾ are more slowly convergent but are not inconsistent with the renormalisation group prediction p=¹/₃, q=4.

For Pt.IX see ibid., vol.8, p.1461 (1975). The derivation of high-field expansions for the four-dimensional simple hypercubic lattice is described briefly. The high-field polynomials L_n are given up to L₁₅ together with the complete partial generating functions (codes) up to F₇ which determine the corresponding sub-lattice polynomials. Expansions are given for the zero-field free energy and initial susceptibility in powers of the high-temperature counting variable nu =tanh K up to nu ¹⁷, and combined with the codes these determine the susceptibility and all its field derivatives up to nu ¹⁷.

The authors have studied two variational methods of scattering, namely the Kohn method and the Harris modification of it and applied them to resonance. As a case study, they consider a coulombic acceptor impurity potential in a zero-gap semiconductor like mercury telluride in which, on physical grounds, an impurity resonance is expected and also the free particle Hamiltonian has a fairly complex structure due to the strong spin orbit coupling. The advantages and the limitations of the two methods are discussed with an emphasis on resonance. They have made a significant improvement in the algorithms of both methods over their previous applications in the literature and have emphasised the role of the nonlinear parameter which in effect has been used as a variational parameter in the theory.

The effect of an alternating electric field on the uniform superphase motion of a charge is to emit Doppler radiation along with the usual Cerenkov radiation. For relativistic velocities of the charge, general results, giving their intensities, are presented in compact form and analysed. It is found that the reduction in Cerenkov radiation is exactly balanced by the emission through Doppler radiation, i.e. the total energy radiated through the Cerenkov and Doppler radiations is always constant. Both the radiations show oscillatory behaviour.

A one-dimensional anharmonic lattice with four phonon interaction is studied. The quantum corrections to the nonlinear equation of motion are calculated and a single soliton-like solution of this equation is obtained in the coherent state representation of phonons.

Renormalisation of a field theory model for tricritical behaviour in which the operator phi ⁶ is essential for thermodynamic stability is discussed in d=4- epsilon dimensions. The crossover scaling form of the equation of state is obtained explicitly to first order in epsilon for both positive and negative values of the four-spin coupling constant. Orthodox scaling, involving Gaussian tricritical exponents is obtained, but is shown to be physically inappropriate in the ordered region of the symmetry plane. A reformulation of scaling, using classical tricritical exponents is possible, but involves an additional parameter p with its own scaling exponent phi _p=-¹/₂(1- epsilon ). This parameter, introduced by Sarbach and Fisher for the many-component, spherical model limit, is related to the coefficient of (nabla phi )² in the field-theoretic Hamiltonian.