Table of contents

The symmetrised versions of the Carr (1978) and Post (1977) lower bounds for N-fermion systems are examined. It is shown that both formulae, known also as the SHIP and SHRIMP models, are incorrectly derived.

A detailed derivation of the energy lower bound model (SHRIMP model) for a translation-invariant many-fermion system has recently been proposed (Carr 1978). The author asserts that an N-particle shell model retaining antisymmetry in all N particles is obtained. The present author wishes to point out the error in this derivation which makes the SHRIMP model wrong leaving correct the related RIP and HIP models.

Derrick's theorem (1964) concerning the existence of soliton-like solutions of non-linear scalar wave equations in Minkowski space is extended to the curved background space exterior to a charged, non-rotating black hole.

A simple physical argument restricts the value of the excluded-volume exponent nu of a polymer. The result is nu <²/₃ in three dimensions.

The analogy between the branching model of electrical conductivity and the graph theoretical treatment of f-functional polycondensation in chemistry is quantified by an application of the concept of elastically active network chains in rubber theory to random resistor networks.

An approximate solution for equations for non-linear electron plasma waves in an unmagnetised plasma has been given recently by Shukla and Tagare (1977). It is shown that their approximation is not quite valid and a more rigorous study is made.

It is demonstrated that a compact spinor space-time manifold need not be parallelisable. Necessary and sufficient conditions are derived for a parallelisation and counter examples are constructed.

An integro-differential equation of motion for the radiating electron is proposed. In some special cases the stability of the solutions is investigated.

A Lagrangian formulation for a relativistic classical spinning point particle interacting with the electromagnetic field is given. Although it meets all criteria usually imposed for this system, it is nevertheless unrealistic, in that the centres of mass and charge get separated in electromagnetic fields. It is suggested that unless special care is taken, the same can happen also in different treatments of the problem.

It is shown that Schrodinger's equation for a translation-invariant system consisting of N particles with arbitrary masses interacting via Hooke's law pair potentials with the same coupling constant can be solved exactly; explicit solutions are found for the case N=3. Exact solutions are also found explicitly for the translation-invariant problem in which a particle with mass m₀ interacts with N identical particles of mass m₁ via a Hooke's law pair potential with coupling constant k₀², and the identical particles interact with each other via Hooke's law pair potentials with coupling constant k₁². The latter solution provides a basis problem for an energy lower-bound method for translation-invariant atom-like systems.

Continuation of previous work (ibid., vol.11, no.7, p.1227 (1978)). Earlier exact solutions of Schrodinger's equation for translation-invariant systems of particles interacting by Hooke's law pair potentials are augmented to include systems consisting of an arbitrary number S of groups of identical particles. Exact solutions can always be found whenever the number of distinct masses plus the number of distinct coupling constants does not exceed (2S+1). The case S=2 is solved in detail and is applied to show that harmonic matter is never stable.

New sum rules involving particular products of many 3-j symbols are established. These sum rules are useful in multiphoton calculations.

The author studies the motion of N-dimensional wave packets in quadratic potentials without friction and also under the influence of two different quantum frictional potentials. Differential equations for the coordinate and momentum uncertainties and for the correlations are presented and solutions for oscillatory motion and barrier penetration are given where possible. The barrier penetration probabilities are compared with each other.

A method of studying the scattering of a high-energy electron from a two-centre Coulomb potential is presented. The Dirac equation for the problem is solved by a generalised form of the Sommerfeld-Maue approximation, using a spheroidal phase shift analysis at the final stage. For a screened potential, the total cross section has been calculated and averaged over a random orientation of the axes of the two-centre systems. For a long-range potential, the differential cross section can be obtained after the averaging is done numerically. The spheroidal phase shifts are obtained by comparing the asymptotic behaviour of the radial equation with that of the central Coulomb radial equation and using a generalised JWKB method. The method is studied by considering the simple case of two fixed point charges. Some results, for this case, are also presented.

From two examples, one with an exponentially screened and the other with a smoothly cut-off Coulomb potential, arguments and partial proofs are given for the following conjecture. The absolute values of screened-Coulomb T matrices may converge in the limit of vanishing screening to the results of standard (i.e. short-range) theory. The condition is that the values of the momenta should be chosen before the limiting process.

A set of recursion relations is derived by the aid of which analytical expressions for Tr(J_-^kJ_z^lJ₊^k) can be built up in a chain procedure.

The author considers solutions of the Einstein-Maxwell field equations satisfying the conditions F_ij;k⁺l^k=fF_ij⁺, F_ij;k⁺n^k=gF_ij⁺ where F_ij⁺ is the self-dual Maxwell bivector and lⁱ and nⁱ are the principal null vectors of F_ij⁺. The fields are algebraically general and the most general solution is found in closed form when lⁱ and nⁱ have non-zero twist. Several twist-free solutions are also obtained. Vacuum space-times admitting a test Maxwell field are also considered. In all cases the metric admits a three-parameter group of motions acting transitively on three-dimensional orbits. Fields satisfying the dual conditions with lⁱ and nⁱ replaced by mⁱ and mⁱ are also investigated and similar results are obtained.

The authors consider a number of explicitly soluble models of a massless, non-conformally-coupled scalar quantum field propagating in certain special Robertson-Walker background space-times. Regularisation is effected using covariant point-splitting, and renormalised stress tensors are constructed according to the ansatz, described in a companion paper (ibid., vol.11, p.603 (1978)) based on the DeWitt-Schwinger expansion. The results are all finite, conserved and free of regularisation ambiguities. From these special cases a great deal is deduced about the structure of renormalised stress tensors in more general models in which computation-approximation techniques will be necessary.

The expression for the density matrix describing particles of one sort (electrons or positrons) created by an external electromagnetic field from the vacuum is obtained. The explicit form of the density matrix is found for the case of constant and uniform electric field. Arguments are given for the presence of a connection between the thermal nature of the density matrix describing particles created by the gravitational field of a black hole and the equivalence principle.

The author investigates the possibility of a Hawking-type effect due to the cosmological event horizon. He considers the reduced two-dimensional de Sitter metric. There appears to be radiation from the horizon towards the origin at temperature T defined by kT= square root Lambda '/2 pi where 3 Gamma ' is the cosmological constant. The observer dependence of this radiation is discussed.

The author discusses a simple model in which the vacuum expectation value of a complex scalar field phi is non-zero in the flat-space vacuum. Generalising to curved space and choosing the conformal vacuum in Friedmann models it is found that in hot big bang models symmetry is restored at early times whereas in cold models two new effects arise-a classical curvature term and a zero-point fluctuation term. These have opposite effects and lead to restoration only in closed universes.

A special case of the 20-vertex model on a triangular lattice is solved exactly for all temperatures by the method of Bethe ansatz. The model exhibits no phase transition.

The average end-to-end distance R_N of a flexible polymer chain of N segments in a good solvent varies as N^nu , where, according to Flory, the size exponent nu =3/(2+d) in d dimensions (1<or=d<or=4). The original derivation of this formula for the exponent nu has been shown by des Cloizeaux (1976) to be without foundation. The authors demonstrate that the Flory formula is valid in a model for which only a finite (but large) number of spherical harmonics are retained in the harmonic (partial wave) expansion of the propagator.

The helicity modulus (superfluid density) is evaluated for spin systems in the magnon regime and for an interacting Bose fluid in the Bogoliubov approximation. The quantal XY model and the Bose system each yield the same temperature dependence for the helicity modulus; the behaviour is distinct from that of the square of the order parameter.

The authors study bond percolation processes on a d-dimensional simple hypercubic lattice. Exact expansions for the mean number of clusters, K(p), and the mean cluster size, S(p), in powers of 1/ sigma , where sigma =2d-1 and p<p_c, are derived through fifth and fourth order, respectively. The zeroth-order terms are the Bethe approximations. The critical probability p_c is found to have the expansion, probably asymptotic, p_c= sigma ^-1(1+2¹/₂ sigma ^-2+7¹/₂ sigma ^-3+57 sigma ^-4+...), while the cluster growth parameter lambda can be expanded as lambda = lambda _B(1-2 sigma ^-2-...) where lambda _B is the Bethe approximation for lambda . They also present series data for the mean cluster size and the cluster growth function for d=4 to 7. Numerical analysis suggests that the critical dimension, d_c, for bond percolation is d_c=6, as it seems to be for the site problem. The evidence also supports the conjecture that the value of a particular critical exponent in a given dimension is the same for both bond and site processes.

A natural generalisation of lattice-lattice scaling is suggested which holds exactly for the second most singular amplitudes for the Ising model on the triangular, honeycomb, square and Kagome lattices. In general, this scaling theory requires three scaling parameters g, n and m, but includes extended lattice-lattice scaling as the special case of m=n. The generalised form of lattice-lattice scaling does not seem to be applicable to the three-dimensional Ising model. The connections between lattice-lattice scaling, its extension and generalisation, and the critical equation of state including correction terms are established and discussed.

Many of the standard two-dimensional problems of statistical physics can be transformed into 'Coulomb gas' problems in which there are two kinds of 'charges' represented by integers n and m. Such a transformation works for the Ising model, the three- and four-state Potts models, the Ashkin-Teller model, any many others. In general the n-n and m-m interactions have the Coulomb character in which the interaction is, for large separations, proportional to the logarithm of the distance. On the other hand, the n(r)-m(R) interaction is for large distances proportional to i times the angle Phi (r-R) which measures the angular position of R relative to r. This latter interaction is akin to that between a magnetic monopole and an electric charge.

A matrix interpretation of Tits-Vinberg construction for the Lie algebra of exceptional groups is proposed. The authors use 3*3 matrices over Rosenfeld algebra completed with its automorphisms which are induced via external multiplication of imaginary units. This allows one to express exceptional gauge theories in a compact and comprehensible form suitable for the purposes of model building.

A theory of photodetection is presented which takes fully into account the relaxation of the photosensitive atoms representing the detector. Atoms, electromagnetic field, and atom-field interaction are described in terms of the Dicke Hamiltonian with additional coupling of the atoms to a reservoir. The time evolution of the joint detector and field states is derived using master equation techniques. Simple relations between the dynamics of field and atomic moments are obtained in the case of zero detector temperature. The attenuation of the field moments as well as the photocounting probability for any initial field state are found to involve not only the atom-field, but also the atom-reservoir coupling parameters.

Patches of reverse tilt in a twisted nematic liquid crystal display device may be removed by inducing the same small angle of pretilt at both solid surfaces. Continuum theory is used to investigate the effect of this initial tilt upon a Freedericksz transition in a twisted nematic in an electric field. In particular, the author describes the various initial alignments that can occur and verify that the field has little influence until it approaches the threshold value corresponding to a similar cell without tilt.

Rigorous lower bounds on the connective constants for site and bond animals have been derived for d-dimensional simple hypercubic lattices. These bounds establish for arbitrary d that the connective constants for site and bond animals are strictly greater than the connective constant for self-avoiding walks.