Table of contents

The authors study the ground state energy of the s-wave hydrogen atom with the polynomial perturbation 2 lambda r+2 lambda ²r². The usual Rayleigh-Schrodinger perturbation series in powers of lambda for the ground state energy has been shown to fail for lambda <0. The authors construct explicitly a perturbation series in powers of (- lambda )^-1/2 for lambda <0 and show that it agrees well with the results of variational and Hill determinant calculations.

Three-dimensional ground-state logarithmic perturbation theory in nonrelativistic quantum mechanics is applied to the Klein-Gordon equation. The problem of pionic atoms in an external multiple field is treated in this framework.

It is pointed out that the bound and scattering states generated by a short-range two-body potential form a complete set even in the space of irregular solutions. Working with the relevant Wronskian relations an explicit eigenfunction expansion for the outgoing Jost solution is obtained. The formalism reduces to an interesting integral of spherical Bessel functions for the case of free particles.

The one-loop infinities for scalar, spinor and vector fields due to gravitational interactions are computed in the axial gravity gauge for the Einstein-Hilbert action. The results reveal an axis dependence, in addition to their expected non-renormalisability, which is expected to get progressively worse in higher orders of perturbation theory. This means that gauge equivalences between covariant and non-covariant gauges are impossible in ordinary quantum gravity (unlike renormalisable theories such as flavourdynamics or chromodynamics) unless cancellation of infinities occurs through some supersymmetric mechanism or unless a renormalisable conformal version of gravity is adopted.

Shows that the finite-size scaling assumption, and particularly a conjectured relation involving the exponent eta , are valid in the low-temperature phase of the 2D XY model, order by order in a perturbative expansion of the two-point correlation function.

Uses position-space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals (clusters of directed bonds, in which the tail of a new bond must be added to the head of existing bonds). Exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis. In addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located.

An asymptotically exact expression is given for the mean displacement (R(t)) of random walks on directed percolation clusters on lattices in arbitrary dimensions. The critical behaviour of R_infinity =lim_{t to infinity} (R(t)), the mean squared displacement and the relaxation time are discussed near the threshold probability p_c=1 in terms of critical exponents.

Discusses the ordinary transition of the semi-infinite q-state Potts model within the Migdal-Kadanoff renormalisation group scheme. Approximate surface free energies are calculated in dimension D=2. In addition, the effect of symmetry breaking surface fields is considered. The scheme predicts that only one such field is relevant.

The authors write the partition function of the double-Gaussian model, which is representative of the lambda phi ⁴ class (Hamiltonians with unbounded doubly degenerate local potentials), exactly as the product of the partition function of a Gaussian model and the partition function of a spin-¹/₂ Ising model. On the basis of this result they are able to determine the nature and location of the critical points of this model. It also follows that the block spin renormalisation group flows in the entire high-temperature region converge to a Gaussian model fixed point.

The standard massless spin-2 and spin-³/₂ equations are not conformally covariant. By varying the coefficients of various terms in these equations the authors derive conformally covariant equations. These equations are then used to construct consistent coupling terms in the original field equations representing the self-interaction and the interaction of massless spin-2 (or ³/₂) field with matter.

It is shown, that the negative susceptibility of the n=0 vector model does not imply unphysical characteristics for a related polymer system. In terms of the polymer system it corresponds to the narrowing of the distribution function of the total number of polymers in the solution with respect to the Poisson distribution.

For pt.II see ibid., vol.15, no.1, p.1-23 (1982). The massive spin-S particle equations govern spinors xi and eta of valence n=2S. The number of undotted indices of these is t and t-1, respectively, where t is an integer 1<or=t<or=n. In flat space one has various mutually equivalent theories which correspond to different choices of the value of t. On account of the symmetries possessed by xi and eta the equations become mutually inconsistent for n>or=3 when transcribed to an arbitrary Riemann space if one merely adopts minimal coupling. One therefore has the problem of constructing a separate set of non-minimally coupled equations for each relevant value of t. The case of t=n-1 is studied. The spin-³/₂ and spin-2 equations are singled out for special consideration.

Versal deformations of normal forms of elements of real classical Lie algebras o(p,q), sp(2n,R), o*(2n), sp(2p,2q) and u(p,q) are computed. Examples involving the Lie algebras o(p,q), p+q<or=6, are considered.

Tables of 6j symbols for SU₆ and SU₃ and tables of 3jm factors for SU₆ contains/implies SU₂*SU₃, SU₃ contains/implies U₁*SU₂ and SU₃ contains/implies SO₃ are presented. The tables are computer produced, using a program that implements the building up principle in a general form. These tables are useful for calculations in high energy, nuclear and solid state physics. Some other tabulations contain errors, and none uses all the symmetries available. The n independence of SU_n results is discussed by using the various symmetric group-unitary group duality relations.

A relatively simple algorithm for the decomposition of the product of two SO(2p) representations is presented. For this purpose, generalised Young tableaux are introduced and their product is defined.

Minor quasi-symmetry (P-symmetry) crystallographic point groups associated with all the 23 distinct two-dimensional and 7 distinct three-dimensional irreducible representations generated by the point groups containing degenerate irreducible representations are obtained as semi-direct products are tabulated.

The spin plethysms lambda _G(X) Delta that arise in the reduction of Delta under SO(N) to G when (1) to lambda _G are considered. It is shown that, for the simple Lie algebras of rank k, if lambda _G= phi _G, the adjoint representation of G, then phi _G(X) Delta =2^(k2)/ delta _G where delta _G is the representation of G whose highest weight is half the sum of the positive roots. Certain results for other representations are described. A remarkable series of S-functions is introduced leading to a new dimensional equality between certain representations of O(2k) and Sp(2k).

The multiplicity of the identity representation occurring in the reduction of representations of semi-simple Lie groups to their finite or continuous subgroups is given for many group-subgroup pairs.

Some properties of a linear Boltzmann collision operator acting in the L¹ space of absolutely integrable functions of the velocity are derived. The system considered consists of particles moving in a dilute equilibrium gas. The case of a constant accelerating force acting upon the particles (as encountered in electron swarm experiments) is also studied. It is found that the collision operator is dissipative operator which generates a strongly continuous contraction semi-group. It is also shown that the time evolution leaves the positivity and normalisation of the distribution function invariant.

The coagulation-fragmentation equation describes geodesic motion in an infinite-dimensional space. This space has a symmetric affine connection but no metric in general. Some advantages of the new approach are indicated.

A numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described. The method exploits Cauchy's theorem and Jordan's lemma to write the infinite integration path along different contours in the complex plane. The method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques. Application is made to Pearcey's integral P(x,y) and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem. Tables of P(x,y), delta P(x,y)/ delta x, delta P(x,y)/ delta y and the real zeros of P(x,y) are presented for the grid -8.0<or=x<or=8.0 and 0<or=y<or=8.0.

Dirac (1964) conjectured that in a constrained Hamiltonian system the surfaces of points physically equivalent to one another are generated by the totality of first-class constraints, both primary and secondary. A proof is given.

A manifestly gauge-invariant time-dependent perturbation theory is developed for a non-degenerate quantum mechanical system interacting with an arbitrary classical electromagnetic radiation field. The first- and second-order net transition rates are derived and compared with their conventional counterparts. It is found that the conventional and the gauge-invariant perturbative rates of transition agree completely.

The authors prove the existence of a class of exact eigenvalues and eigenfunctions of the Schrodinger equation for the potential x² + λx²/(1 + gx²) when certain algebraic relations between λ and g hold. Some of the properties of these solutions are discussed. It is shown that in a certain sense they may be regarded as Sturmians for the Schrodinger equation with the potential x² - λ/(g + g²x²).

The bound-state energy levels of a many-fixed-centres Coulomb potential are determined in three different ways. The three approximate methods are all based on the same technique: the Sturmian one. Numerical results are given in the case of the H₂⁺ molecular ion.

The classical Wheeler-Feynman absorber theory with a postulate of Lorentz-invariant zero-point electromagnetic radiation is proposed to explain quantum phenomena in a similar manner to that of stochastic electrodynamics. For this purpose, Cramer's model of a 'Minimum emitter-absorber transaction' is extended to the case where zero-point radiation exists, in the context of the classical Wheeler-Feynman theory. The Einstein-Podolski-Rosen paradox and the quantum levels of a charged simple harmonic oscillator are derived from the theory. It is shown that the condition for a transaction plays an essential role, such as in the radiative balance of a simple harmonic oscillator.

The soliton technique due to V. Belinsky and V. Zakharov, (1980) is applied to find stationary axisymmetric one-soliton solutions of the Einstein equations in vacuum. In order to generate (2n+1)-soliton solutions with physical signature the (unphysical) Euclidean metric is taken as the seed solution. The one-soliton solutions are a family of non-asymptotically flat metrics depending on one parameter and can be considered as being the stationary generalisations of a very simple family of static Weyl metrics. They are Petrov type I metrics except for one of its members, which is Petrov type II and can be simply related to the van Stockum class. The Ernst potential of these solutions and the use of prolate spheroidal coordinates suggest new related families of solutions which are asymptotically flat. One of them contains the Zipoy-Voorhees metric with deformation parameter delta =¹/₂ as a particular case.

The recent exact analysis of the canonical partition function and one- and two-particle distribution functions for a disc of classical one-component two-dimensional plasma of particle charge -q with a uniform neutralising charged background is extended to consider the effects of surface charge and of having the dielectric constant outside the disc, epsilon ₂, different from that inside the disc, epsilon ₁. The system is characterised by bulk density rho , the plasma parameter Gamma =q²/ epsilon ₁kT, the surface charge density sigma q and the parameter Delta =( epsilon ₁- epsilon ₂)/( epsilon ₁+ epsilon ₂). For Gamma =2, Delta =0 or 1 and any value of rho or sigma , the canonical partition function and one and two-particle distribution functions are calculated exactly. The bulk thermodynamic properties are independent of sigma and Delta . The surface properties (surface excess free energy, density profile and two-particle correlation functions) are calculated in the thermodynamic limit and shown to depend strongly on both sigma and Delta .

A general method is presented for expanding random functions in orthogonal polynomials of binomial-random variables for transport in randomly inhomogeneous media. The expansion projects the stochastic equation onto the statistically orthogonal polynomials of binomial variables. It generates an infinite set of coupled equations for the determination of kernels in the expansion where randomness is removed at the outset. The expansion in orthogonal polynomials is applied to inhomogeneous transport in bond-disordered resistor networks (bond model). The expression for the effective conductivity is obtained, to order c² (c being the fraction of broken bonds), by truncating the infinite set of coupled equations for kernels after the third term. It is found that the expression agrees with that derived from the two-bond approximation. The expansion in orthogonal polynomials is also applied to the clumped-bond model and the continuum model. Truncated equations are derived to govern the kernels in the expansions.

Describes an approach to the dynamics of particles in liquid suspension, based on Langevin equations, which allows rather direct calculation of power series expansions in time tau of such quantities as the particle velocity autocorrelation function, mean-square displacement and dynamic structure factors. The expansions are evaluated to order tau ³ if hydrodynamic interactions are neglected, but only to order tau in their presence. The longer-time diffusion coefficients are also considered, and the importance of the structural relaxation time tau _I to the theoretical development is emphasised. Further similarities between the dynamics of particle suspensions and atoms in simple fluids are pointed out.

Symmetry relations such as the star-triangle or the inverse relation are very useful in determining the partition function of two-dimensional exactly soluble models. A common construction of the three-dimensional equivalents of these symmetry relations is presented. They are used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e. the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function. The usefulness of the inverse relation is illustrated by an application to the three-dimensional Ising model.

A numerical investigation of the two-dimensional square well lattice gas and a gas of hard ramrods has been carried out using the scaling transformation. Both models appear to exhibit a single Ising-model-like second-order phase transition over the whole temperature range.

A low-frequency approximation for the scattering of a spinless charged particle by a spinless neutral target in the presence of an external electromagnetic field, originally derived for the case of a monochromatic plane wave of infinite extent, is generalised so that it applies to the more realistic case of a wave train of finite length. The case where the charged particle has spin ¹/₂ and an anomalous magnetic moment is also treated. The field is taken to be slowly varying relative to the collision time, but the spectral composition of the field is otherwise arbitrary; the static limit, corresponding to a constant crossed field, is included as a special case. The assumption that the collision is essentially instantaneous is formulated in a gauge-invariant manner, and this provides the physical basis of the derivation. As in earlier versions of the low-frequency approximation, the approximate transition amplitude is expressed in terms of the on-shell amplitude for scattering in the absence of the field.

Using the Green function technique, the authors calculate the transition probability amplitude for one atom interacting with a one-mode electromagnetic field, without assuming the rotating-wave approximation (RWA). The effects of the RWA are studied in the time and frequency domains. In the time domain, the periodicity of the typical RWA results is broken; in the frequency domain, the Bloch-Siegert shift is obtained.

The susceptibility of solutions of semiflexible macromolecules in an external orientational field of the dipole type is considered both for the isotropic and the liquid-crystalline phases. It is shown that the concentration dependence of the zero-field susceptibility in the anisotropic phase depends essentially on the character of the flexibility distribution along the chain contour. For a localised flexibility mechanism (freely jointed chain) the susceptibility in the anisotropic phase is practically independent of concentration, whereas for a persistent flexibility mechanism it increases exponentially with concentration. Similar differences exist in the concentration dependence of the mean-square end-to-end distance for freely jointed and persistent macromolecules in the anisotropic phase. In an Appendix it is shown that the problem under consideration is analogous to the well known quantum mechanical problem of the energetic spectrum of a particle in two identical wells separated by a high potential barrier.

The Brereton-Shah problem of two polymer loops topologically linked is studied by field theoretical methods in the limit that one polymer is allowed to fill a macroscopic volume at finite density rho . For L>>1 it is shown that typically for winding numbers m²<<L^alpha rho , alpha =2- epsilon nu , universality is observed with the critical exponents of the unconstrained system despite the presence of a 'dangerous' renormalisation group instability. Here L is the length of the smaller polymer loop whose mean square size (R²) approximately L^{2 nu} ; nu =¹/₂ for random flight statistics or nu approximately ³/₅ for the swelled chain case.

The dynamical renormalisation group (RG) is implemented to study the large-scale properties of incompressible conducting fluid stirred by random forces and currents. In contrast with Navier-Stokes turbulence, invariance properties and dimensional constraints do not always prescribe the renormalisation of the couplings. In dimensions d>d_c approximately=2.8, the system displays two non-trivial regimes: a kinetic regime where the renormalisation of the transport coefficients is due to the kinetic small scales, and a magnetic regime where it is due to the magnetic small scales. The results for the magnetic regime are not identical with predictions from the direct interaction approximation; this is due to vertex renormalisation of the Lorentz force. In dimensions 2<or=d<or=d_c, with sufficiently strong external currents, there is no stable fixed point: runaway of the figurative point occurs, making the RG approach self-defeating. In two dimensions, with weak forces and currents, the absolute equilibrium results of Fyfe and Montgomery (1976) are recovered.

Among the analytic studies on runaway phenomena the most sophisticated approach to solving the Fokker-Planck equation is to divide the momentum space into five distinct regions with appropriate matching between them: various expansions of the distribution function are valid in different regions. A pair of recursive relations is established to solve the Fokker-Planck equation for a relativistic plasma in the runaway region. Starting from the available asymptotic solution, a representation of the distribution function may be obtained by iteration. The application of this technique to the nonrelativistic problem is also treated.

The weakly nonlinear dispersive modulation of one-dimensional waves in a warm, collisionless, field-free, slightly non-uniform, streaming electron plasma is investigated. Equations governing the coupled slow modulations of the waves and the slow variation of the background plasma are obtained using a two-timing procedure devised by P. Gatignol (1977). Under some restrictive assumptions the complex wave amplitude is shown to vary according to a nonlinear Schrodinger equation.

Three assertions by H. Haken (1978) on master equations, normalisation of transition probability, existence and uniqueness of a stationary distribution, and approach to the stationary distribution, are not always true when the system has infinitely many states. In particular, the stochastic Lotka-Volterra model does not necessarily conclude the approach to a state in which both preys and predators are extinct.