Table of contents

The presence of counter-rotating terms in the extended Dicke Hamiltonian alters the temperature dependence of the soft-mode frequencies near the critical temperature. At resonance and in the long-wave limit, frequency doubling does not occur. The zero-temperature limit of excitation energies close to equilibrium is satisfactory.

The authors obtain an expression for ensemble-averaged eigenvalue spectrum for large symmetric matrices, where each matrix element is described by a Gaussian probability density function with same mean (non-zero) and variance. The eigenvalue density function is a sum of two semicircle distributions.

A model of two photon absorption in a cavity with an external driving field is presented. Phase and amplitude fluctuations in the driving field are taken into account to model a partially coherent multimode source. The photon statistics of the light in the cavity are investigated through a calculation of the second-order correlation function of the light field in the steady state. For a coherent driving field the field inside the cavity will exhibit photon antibunching for low intensities. As the fluctuations in the driving field are increased the photon antibunching is lost. For a particular coherent field, g⁽²⁾(0) is reduced with increasing laser intensity. This reduction with laser intensity is slower with increasing laser amplitude fluctuations. This could provide a possible explanation for the recent experimental observations of Kransinski and Dinev (1976).

Arnol'd showed that simple semiclassical methods applied to potentials with discrete symmetries yield 'quasimodes' which do not have the same degeneracy properties as true eigenstates; here it is shown that the true semiclassical states are simple linear combinations of the quasimodes.

Using gamma -matrix representations of the SO(5) algebra, a special ansatz for the solutions of the Yang-Mills equations is tested. Non-trivial solutions are found, and their charges are computed.

It is proved that the 'local mean-field theory' in a two-component quantum plasma never satisfies the perfect screening sum rule and the third sum rule simultaneously unless the 'local field corrections' depend on frequency.

The percolation problem is solved exactly in one dimension. The functions obtained bear a strong resemblance to those of the n-vector model on the same lattice. Further, a ghost field is included exactly in all dimensions d, thereby treating the 'thermodynamics' of percolation without appealing to the Potts model. In particular, it is shown that for d=1 that the nature of the singularities near the critical percolation probability, p_c=1, is described by alpha _p= gamma _p=1, beta _p=0, and delta _p= infinity . The pair connectedness and correlation length are calculated explicitly, and eta _p= nu _p=1, in agreement with the hyperscaling relation d nu _p=2- alpha _p. Finally, scaling is demonstrated for both the cluster size distribution and the percolation function analogous to the Gibbs free energy, and the scaling powers are explicitly evaluated; in particular, the exponents sigma =1 and tau =2, are found.

An effective dimensionality d_p is introduced for the purpose of providing a quantitative characterisation of the degree of ramification of the clusters that occur at the percolation threshold. It is found that d_p is directly related to percolation critical exponents, and that 1<or=d_p<or=d, which in turn places bounds on certain scaling powers and critical exponents. The exponents-when renormalised according to Suzuki's 'extended universality' prescription-have an appealingly simple form in terms of d_p; in particular, the renormalised mean cluster size exponent is d_p, while both the order parameter and 'decay of correlation' exponents are given by the co-dimension d-d_p.

An umbilic point U on a surface Sigma is a place where the two principal curvatures of Sigma are equal. U is a singularity of Sigma in three different senses: (i) it is the source of elliptic (E) or hyperbolic (H) umbilic catastrophes in the envelope of normals ('focal surface') of Sigma ; (ii) it has index +or-¹/₂ depending on whether the principal curvature directions of Sigma (defining the lines of curvature) rotate by +or- pi during a circuit of U; (iii) it has a pattern of the 'star' (S), 'lemon' (L) or 'monstar' (M) type depending on the configuration of lines of curvature near U. We calculate the average density of umbilic points, and the fractions alpha of umbilics of the different sorts, for the case where Sigma is a surface whose (small) deviation from a plane is specified by a Gaussian random function. It is always the case that alpha _S= alpha _+or-12/=¹/₂. The other fractions depend on the degree of isotropy of the statistics of Sigma . In the isotropic case the elliptic umbilic fraction is alpha _E=1- alpha _H=0.268, and the monstar fraction is alpha _M=¹/₂- alpha _L=0.053.

For pt.III see ibid., vol.9, p.1801 (1976). Using Poisson's summation formula of dimensionality one or two, it is shown how a class of two-dimensional lattice sums involving modified Bessel functions can be reduced to rapidly convergent one-dimensional sums. The resulting formulae provide a simplification, and in many cases generalisation, of the ones obtained previously by Fetter (1975) and by Hautot (1975).

Expressions are found for the coefficients in the relations which exhibit spheroidal harmonics as linear combinations of spherical harmonics and vice versa. In particular it is shown that in the case of prolate spheroidal harmonics the non-vanishing coefficients k_LMl in the relation P_L^M(u)P_L^M( nu )= Sigma _ik_LMlr^lP_l^M(cos theta ) are given by k_LMl=(-1)^(L-l)2/2^-La^-l(L+M)!(L+l)!/(L-M)!(l+M)!(1/2L+1/2l)!(1/2l-1/2l)!, where l=L, L-2, L-4,...,M' with M'=M or M+1 according as L-M is even or odd, respectively.

The author shows that the measure of angular momentum uncertainty is the invariant ( Delta J)² identical to (J²)-(J)², rather than sums of products like Delta J₁ Delta J₂, and that the critical states which minimise Delta J are eigenstates of maximum weight of J.n. They also determine the critical states for the associated groups E(2), O(2,1), O(4) and E(3), and construct coherent O(3) spin states mod z) which are superpositions of normal spin eigenvectors and which tend to the classical limit as mod z mod to infinity .

A phenomenological, general relativistic theory of dissipative elastic solids whose equations form a hyperbolic system is proposed. The non-stationary transport equations for dissipative fluxes containing new cross-effect terms, as required by compatibility with irreversible thermodynamics, have been adopted. As opposed to some conventional theories which are parabolic and predict instantaneous propagation of wavefronts, the theory formulated, consisting of 14 partial differential equations (in the case of special relativity), of total order 17, is hyperbolic and predicts, for all existing propagation modes, finite front speeds. The complete system of special relativistic propagation modes of an elastic solid is determined from the linearised equations. There are four mutually distinct non-trivial propagation modes, two for longitudinal waves and two for transverse waves. If the rigidity modulus decreases to zero one obtains as a special case the normal modes for fluid according to the theory of Muller (1972) and Israel (1976). Weber's equation is also briefly discussed.

Necessary and sufficient conditions for the separation of the Hamilton-Jacobi equation for the geodesics in an n-dimensional Riemannian or pseudo-Riemannian manifold are obtained. The integrability of these conditions is investigated and several results are found which are of particular interest in the four-dimensional space-times of general relativity.

Jaffe and Shapiro (1972) have recently shown that material particles (tardyons) in a spherically symmetric Schwarzschild field move in a light-like manner when their velocities at infinity are beyond a certain limit. The authors preent here a more general discussion of the problem for different gravitational fields and extend the study to faster-than-light particles (tachyons).

The classical equations for the SO(3) gauge field coupled with the strong gravity metric are considered in the f-g theory of Salam et al. The asymptotics of the regular static spherically symmetric solutions are obtained.

It is sometimes impossible to construct a relativistic star cluster with the same structure as a given relativistic fluid sphere. The reason is the unphysical nature of the distribution function which is sometimes negative in part of phase-space or sometimes contains non-integrable singularities. A simple test is derived that can be applied to a given fluid sphere to provide a set of sufficient conditions for the distribution function to be unphysical. The test is applied to some fluid spheres and clusters that have been studied in the literature.

Two distinct procedures have been followed in the literature for determining D, the self-diffusion coefficient for a dilute macromolecular solution. One is a direct statistical mechanical calculation based on the fundamental Einstein relationship; the other consists of averaging the diagonal part of the diffusion tensor which describes the hydrodynamic interactions. Subject to an approximation common to both types of calculation, that the macromolecules remain essentially fixed on the time scale of momentum relaxation, it is shown that both procedures lead to identical results for D. The result is valid for any order of approximation to the low-concentration hydrodynamic interaction term. It is briefly argued that relaxing the fixed-particle approximation should not lead to appreciably different results.

Extended lattice-lattice scaling (that is, universality of scaling corrections) is investigated for the two- and three-dimensional Ising models. It is shown to break down for non-Archimedean lattices in two dimensions, and for all lattices in three dimensions. In the course of the investigation several new higher order amplitudes for the two- and three-dimensional Ising models are obtained. These include an exact evaluation of the amplitude of the second most singular term of the susceptibility of the Kagome lattice, for which we find C₁⁺=0.086936625, C₁^-=-0.0023063996.

A low density expansion for the pair connectedness in a gas is obtained by methods similar to those used for a solid. In the case of an ideal lattice gas a change of variable enables the usual expansion for a solid to be re-derived. The new formulation allows the use of renormalisation techniques and k-space diagrams. The possibility of interactions is also considered.

The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T_c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .

Monte Carlo calculations on off-lattice chains with various ratios lambda of bead diameter to bond length show that gamma and gamma ' in mean square h=aN^gamma or mean square r=a'N^gamma ', respectively, are not 6/5 throughout but depend on lambda . It is generally accepted that the theoretical predictions of the two-parameter theory do not depend on the kind of interaction potential between non-bonded beads. This assumption, however, proves to be wrong. The hard-sphere potential employed in this work yields results inconsistent with those of the perturbation theory using a pseudopotential with a delta function.

The applications of the Domb-Joyce model to polymer chains of finite length is discussed. Techniques are developed which permit the computation of the mean square end-to-end length (R_N²( omega )) of a lattice walk with an interaction omega . Numerical estimates of the first three coefficients of the series alpha ²=1+k₁ Omega ²+... are obtained for various N, and these estimates are extrapolated to N infinite, thus yielding the two-parameter value. The results are in excellent agreement with theoretical predictions, and confirm the universality hypothesis of Domb. Some consideration is given to the possible form of corrections to the two-parameter function.

A generalised Dicke model including: (i) the spatial variation of the electromagnetic field; and (ii) quantum Debye vibrations of atoms around their equilibrium positions is proposed. It is proved that the free energy per particle for such a model can be calculated in the thermodynamic limit exactly and rigorously.

A very simple treatment is presented to describe a phase transition present in Dicke's model for super-radiance, in the off-resonance case and for a single radiation mode. The results agree with previous authors, although the use of mod n( states provides a transparent physical interpretation. The results show that if the A² term is kept in the Hamiltonian, the phase transition persists, thus clarifying some recent controversy on the subject. The thermodynamic variables are calculated above and below T_c. The phase transition is shown to be a consequence of an instability of the ground state. The behaviour of (a^Dagger a) and (S_Z) is explored as a function of temperature. (a^Dagger a) goes to zero as (T_c-T) near the critical temperature.

Dyson's perturbation theory analogue for quantum electrodynamical processes with arbitrary initial and final states in an external field creating pairs has been discussed. The interaction with the field is taken into account exactly. The possibility of using Feynman diagrams, together with modified correspondence rules for the representation of the above mentioned processes has been demonstrated.

A general method for deriving the superfield equations of motion is proposed. These equations contain the supplementary conditions of irreducibility with respect to the supersymmetry group. The method uses projection operators which single out the irreducible representations and especially the algebraic roots of these operators. It is found that the standard equations of motion for spin-vector and symmetric tensor fields can in fact be obtained by extracting the square roots of the projection operators for spin-³/₂ and spin-2, respectively. The spinor superfield equation is deduced and discussed in detail.

The Einstein-Maxwell field equations of gauge supersymmetry are solved. It is found that certain auxiliary gauge fields contribute as much to the energy-momentum tensor as the electromagnetic field itself and give rise to negative energy densities, unless they grow a superheavy mass by spontaneous symmetry breaking.

The author uses Schwartz's method to study a relativistic problem whose mathematical structure is similar to Goldstein's eigenvalue problem, i.e. the problem of a zero energy Klein-Gordon particle in an external Coulomb field. It is shown that Schwartz's method fails to give a correct answer. The result suggests that additional information about the eigenvalue spectrum for energies different from zero is needed before deciding whether Goldstein's cut-off independent solution discussed recently by Delbourgo and Prasad is physically meaningful.