Table of contents

An allegation of a flaw in the theory developed by Dirac for the Hamiltonian formation of singular dynamical systems is denied. It is shown that the Dirac theory is perfectly correct, and adequate as a Hamiltonian theory-but also there is a gain to be had by looking at the structural features involved in the original Lagrangian picture, including those 'canonical' features necessary for the purpose of quantisation; it is indicated how the controversy about first-class secondary constraints may be examined in this way.

The connection between topological singularities of dispersion relations and analytic singularities with respect to frequency of spectral densities and wavefunctions in dispersive media due to time harmonic sources is demonstrated. The topological singularities are discussed within the framework of imperfect bifurcation theory, regarding the frequency as a distinguished bifurcation parameter and the wavenumbers as bifurcation variables, and using a recently obtained classification of topological singularities up to codimension four which is given in terms of a list of normal forms. To each normal form corresponds an analytic singularity, governed by characteristic exponents which are tabulated.

The Lie-Backlund transformation is calculated for a KdV equation with explicit space dependence, for which the Lax equation has a moving eigenvalue in the t plane. The corresponding similarity variable and the Painleve equation are deduced. The role of the Lie-Backlund generators in relation to the conservation laws is also discussed.

Relations B_a=S^-1(a)BS(a) and B_a^-1=S(-1)B_aS(-1), connecting the scale transformation (ST) and the Backlund transformation (BT) for the cylindrical Korteweg-de-Vries equation, u_t+6uu_x+u_xxx+u/2t=0, are obtained. One analogous relation between the BT and translation in x is also considered.

The conjecture that successive time derivatives of the Boltzmann entropy alternate in sign during free thermal relaxation governed by the nonlinear Boltzmann equation is shown to be false: the Bobylev-Krook-Wu solution of this equation is an explicit counter-example.

A two-dimensional model of the dynamics of Z(N) spins on a triangular lattice with three-body interactions is analysed. The properties of the model under duality are shown to be identical to those of a wide class of models in two, three and four dimensions. Monte Carlo simulations reveal the existence of three phases for N>4, with an intermediate phase which is most likely soft.

The transfer-matrix method is used to find the exact phase diagrams and the correlation length exponents v for the one-dimensional site, bond and site-bond percolation problems with bonds connecting the Lth-nearest neighbours (L up to 3). For the site percolation, the authors' results agree with the exact result obtained from the generating function method, v=L, while for the bond percolation they found much richer critical phenomena. If all the L bonds have equal occupation probability, their results predict v=L(L+1)/2.

The combined electrostriction and magnetostriction Casimir force on a compact sphere placed in a vacuum is calculated, provided that the medium satisfies the relationship epsilon mu =1, epsilon being the permittivity and mu the permeability. This special class of media is known to possess attractive properties (Brevik and Kolbenstvedt, 1982): cut-off terms that otherwise represent a difficulty in Casimir calculations for ordinary non-magnetic media (Milton 1980, Brevik 1982) simply cancel out in the surface force expression. This note examines the question whether similar cancellations occur in the striction force when the magnetostriction part is included. The answer is negative.

The fields near a slowly accelerating 't Hooft-Polyakov monopole in the Prasad-Sommerfield limit are obtained.

A position-space renormalisation group (PSRG) approach has been developed for general site percolation problems in which the site occupancies are correlated, rather than independent, random variables. PSRG parameters are systematically introduced to describe multiple-site correlations. At the two- and four-site levels of approximation in the square lattice the authors find only two physical fixed points. One characterises random, or pure, percolation and has a universality class which is shown to encompass a variety of locally correlated site problems, including unfrustrated plaquettes and four-coordinated sites in a random-bond model. The second describes 'Ising-correlated' percolation and at the two-site level yields an excellent estimate (to within 2%) for the nearest-neighbour spin correlation function of the Ising model.

A listing is proposed for the steps in the logic of dimensional analysis. Starting with the definition of a physical concept, the arguments proceed through the principles of measurement, by the specification of dimensional equality and the subsequent limitations of functional operations up to a derivation of the Pi-theorem of this analysis.

The authors extend an idea due to Woodhouse (1975) to give a coordinate-free characterisation of the orthogonal separation of one coordinate for the Hamilton-Jacobi equation on a pseudo-Riemannian manifold, in terms of an involutive family of Killing tensors. The coordinates can be computed from the Killing tensors.

In a physically significant case, closed formulae for the transformation coefficients between the standard and non-standard representations of the symmetric group S_N are obtained and the results are explicitly exhibited.

The general form of nonlinear evolution equations and their Backlund transformations connected with the quadratic in the spectral parameter, Z₂-graded, arbitrary-order linear matrix spectral problem is found. The Hamiltonian structure of the integrable equations is discussed. The infinite family of Poisson brackets which corresponds to the class of equations under consideration is given. Relativistic-invariant integrable equations are considered. The explicit forms of elementary and soliton Backlund transformations are found. A nonlinear superposition principle is obtained.

Some quasi-free irreducible representations of the CAR algebra of a non-relativistic free Fermi field are constructed, which are covariant under certain subgroups of the extended Galilei group. The construction of fully covariant (but reducible) representations as direct integrals of partially covariant ones is described. This method is also applicable to arbitrary C*-algebras and other covariance groups.

The authors develop the stochastic mechanics of a non-relativistic quantum particle with spin 1/2 in a possibly inhomogeneous magnetic field. They do not make any assumption about the inner structure of the particle, and treat spin components as discrete random variables.

Develops the stochastic models (SM) method which is an approximate computer simulation technique for treating many-body systems in thermodynamic equilibrium, suggested by Alexandrowicz (1971, 1972). The method is applied as a test to the critical region of the square Ising lattice. First the authors define the exact transition probabilities (TP), which are parameter independent; since their calculation is impractical for a large lattice they also define approximate TP (based on two parameters) which can be improved systematically. The saving in computer time, compared with previous work, (where 10 parameters have been used) is significant, and the accuracy of the free energy is increased by a factor of 4-100. The results for chi , the magnetic susceptibility, are also improved, in particular for T<T_c. Using the finite size scaling theory and assuming v=1, the authors estimate at T_c, the exponent gamma to be 1.80+or-0.12.

The authors use Noether's theorem within a restricted class of dynamical transformations involving velocity dependence to obtain first integral constants of the motion not available as Noether point transformation constants.

The authors study a general type of equation of motion for two point particles in flat space-time involving radiation reaction terms and previous states of motion of the particles at retarded and multiply retarded times: the type includes the Lorentz-Dirac electromagnetic equation and approximate gravitational equations, and may be applicable to approximations to other nonlinear theories. Existence and uniqueness are proved subject to given asymptotic conditions in the infinite past and regularity conditions at all times, provided the coupling constant is sufficiently small (i.e. small-angle scattering).

A proof is given, on the basis of a theorem due to Garding (1944), for a lower bound on the degree (l+2) of the minimal equation of matrices beta ^mu in first-order unique-mass relativistic wave equations. It is not necessary, for the applicability of the theorem, that a hermitising operator should exist or that the equation be irreducible; and the generalisation of the bound to multimass equations is also straightforward. The bound is not, in general, linked to the physical spin or spins s allowed by the wave equation or the maximum spin j_m contained in the wavefunction. However, in the physically important case of irreducible equations which admit a hermitising operator, the bound becomes (l+2)>or= (2j_m+1), which is stronger than the bound (2s+1) suggested in the recent literature.

Two approaches to semiclassical quantisation of integrable systems using periodic classical orbits are considered. They both lead to approximate formulae for the density of states function (a delta function at each energy level). The first, due to Gutzwiller (1971), involves a sum over isolated stable periodic orbits of the system, and leads to the harmonic approximation to the eigenvalues. The second, due to Berry and Tabor (1976), involves a sum over families of periodic orbits, and leads to the EBK ('torus') approximation to the eigenvalues. Here, the author extracts a modified form of the Gutzwiller series from the Berry-Tabor series by using a uniform approximation, and hence show that the complete spectrum involves both these series. The analysis demonstrates that genuine semiclassical quantisation rules for generic systems, using periodic orbits, will involve uniform approximation, which more closely reflects the underlying classical structure than do the existing stationary phase approximations.

The author shows how the exact polynomial *exp(polynomial) bound-state solutions to the radial Schrodinger equation may be constructed for any potential of the form V(r)= Sigma _{delta in D} G_delta r^delta , where D is a finite set of arbitrary rational numbers and some of the couplings G_delta are not independent.

A new method based on hypervirial relationships is used to calculate transition moments for linear potentials of bounded systems, generalising previous results. The deduced equations are applied to the model of an electron in a finite unidimensional crystal under the influence of an external uniform electric field.

The authors obtain the exact solution of the Schrodinger equation for a particle confined to (i) an equilateral triangle, (ii) a tetrahedral box with corners (- pi / square root 2,- pi / square root 2,- pi / square root 2), ( pi / square root 2,- pi / square root 2, pi /2); (- pi / square root 2, pi / square root 2, pi /2) and ( pi square root 2, pi / square root 2,- pi /2). The energies are: for (i) E_nl=(8h(cross)² pi ²/9mL²)(n²+l²-ln) where L is the side of the triangle and l, n are distinct non-zero integers and for (ii), E_lmn=(h(cross)²/8m)*(3(l²+m²+n²)-2lm-2mn-2nl) where l, m and n are distinct non-zero integers. The wavefunctions have been classified according to the irreducible representation of the corresponding symmetry groups.

Using very recent results on ergodic theorems for superadditive processes on R^d, the authors prove the existence of the density of states for a wide class of random Schrodinger operators. In particular, new non-asymptotic estimates on the density of states are obtained and examples are discussed.

The theory of sequential quantum processes has been extended to Liouville space via the use of non-Hermitian projection operators in order to treat the evolution of the quantum density operator and to enable physically important matrix elements of the density operator to be calculated. The formal relationship of master equation methods to the theory of sequential quantum processes is established, and a new set of coupled master equations is derived. Special choices of projection operators lead to further simplification of the results. The Markoff approximation is also examined.

For pt.I see ibid., vol.15, p.2157 (1982). A system consisting of discrete states and continuum states (which form a so-called internal reservoir) is treated, illustrating the theory of sequential quantum processes in Liouville space developed in the preceding paper. The populations and coherences associated with the discrete states satisfy Markovian master equations when the interaction matrix elements between discrete and continuum states are significant over a broad band of continuum states. The population of a single discrete state decays exponentially with time, whilst the population of two coupled discrete states (one only coupled to the continuum states) may exhibit Rabi oscillations. For the latter case of two coupled discrete levels, the population of particular continuum states approaches a two-peak form for long times (Autler-Townes splitting).

A Hermitian, quaternionic metric is constructed in which the real part is the usual metric of a curved space-time and the purely quaternionic part is an (anti) self-dual SU(2) Yang-Mills field. Demanding that a potential exists for the SU(2) field implies that Einstein's field equations with a cosmological constant are satisfied.

In the literature, various systems of liner eigenvalue equations from which the Einstein-Maxwell equations for stationary axisymmetric exterior fields follow as the integrability conditions were derived. In the present paper, these linear systems are shown to be equivalent; the explicit transformations mapping one form to another are given.

For a quantum field coupled to a classical background g_{mu nu} -field the author proposes a recursive technique which relates the diagonal matrix element ( psi mod T_{mu nu} mod psi ) to its value at t=- infinity . The lowest non-trivial order is then employed to renormalise the semiclassical theory of gravity. The existence of two important classes of solutions of the linearised theory is briefly discussed.

The thermodynamic behaviour of a massive relativistic ideal Bose gas in an (even) d-dimensional space is studied. The thermodynamic functions are expressed as polynomials (or rational functions) in k_BT/mc² whose coefficients are proportional to the Bose-Einstein functions. Their closed forms permit a complete analytic discussion of the crossover from the non-relativistic limit to the ultra-relativistic limit. This transition is equivalent to the dimensional crossover from a d-dimensional to a 2d-dimensional nonrelativistic gas behaviour.

Rigorous results are presented for d-dimensional lattices decorated with nu -dimensional classical spins, whose sites are occupied by Ising spins. It is shown that for suitable interactions the effect of such decorations is to generate temperature-dependent interactions between the Ising spins. It is also shown that for a system composed of two Ising-classical spins systems, which interact via a single bond, the canonical average of an arbitrary function of the spins (Ising and/or classical) in one of the systems does not depend on the interaction parameters contained in the other.

The finite-temperature behaviour of a relativistic field with a renormalisable phi ⁶ self-interaction exhibiting spontaneous symmetry breaking is studied in one space-one time dimension. Using functional diagrammatic methods, the temperature-dependent effective potential and the critical temperature up to two loops are calculated. The nature of the phase transition is also investigated and is clarified to be one of first order.

The two-dimensional anisotropic Potts model at all temperatures satisfies a local inverse relation leading to an inverse functional equation on the transfer matrix. The pertinence of the related functional equation for the partition function is discussed and verified perturbatively, by introducing a diagrammatic expansion corresponding to a partial resummation on one parameter of the model. Possible exploitations of the inverse relation in order to determine the partition functions are sketched. This indicates that the inverse relation can be used without referring to a star-triangle relation.

The authors derive low-density series expansions for the mean size of finite clusters on an anisotropic triangular lattice. By varying a bond density parameter this model includes site percolation on the square and triangular lattices and interpolates smoothly between the two. They identify the critical line and their results are consistent with the critical exponent gamma being constant along this line.

The authors consider the problem of bond-site percolation on a triangular lattice in which bonds are separated into two classes with different occupation probabilities. Pure site and pure bond percolation on the triangular and square lattices are special cases of this system. They use an approximate real space renormalisation group treatment to identify the critical surface and present evidence for the universality of critical exponents for site, bond and mixed bond-site percolation, except at a single point in the space which corresponds to percolation in one dimension.

The generators of the full quasispin group SO(8) and of its subgroup SO(7) are constructed from coupled products of annihilation and creation operators for nucleons. The mutually commuting generators for each of these groups are used to identify the irreducible representations occurring in the nuclear l shell for which l<or=3. The embedding of SO(8)*SO(2l+1) in SO(16l+8) is examined, and the apparent confluence of spinor and non-spinor representations when l=0 is shown to correspond to the well known automorphism for D₄. A quasispin group complementary to G₂ is sought in the nuclear f shell without success. The extension of isospin and spin to an additional spin space leads to the introduction of a unitary symplectic group USp(8) within the new quasispin scheme.

The problem of determining the spectrum of isotropic turbulence can be thought of as one of finding the most appropriate joint probability distribution for the flow taken as a whole. From the point of view of information theory, what one means by the most appropriate distribution is clearly defined and easily justified; it is the probability distribution that maximises the information theory entropy, subject to whatever constraints one can impose on the flow. In this work, the relevant constraints are taken to be the Reynolds number and energy dissipation rate of the flow, energy balance (on average) at every point in wavenumber space, and adherence to the Navier-Stokes equations. Using these constraints, it is shown that the maximum entropy formalism leads to a pair of coupled equations describing the distribution of energy in the turbulent spectrum, and the correlations between the amplitudes of velocity components with nearly identical wavenumbers. Although solutions to these equations are not presented, it develops that if a power-law solution exists, it can only be the Kolmogorov law E(k) varies as k^-53/. In arriving at this result, a useful concept is that of the 'turbulent temperature', defined as the reciprocal of the derivative of the entropy with respect to the local energy dissipation rate. This quantity plays a role directly analogous to the thermodynamic temperature, governing the rate of energy exchange between different wavenumbers. It is found that, within the spectrum's inertial subrange, the turbulent temperature is virtually constant, with only a minute temperature gradient required to drive the energy cascade.

A special case of the spin glass model is considered in which the number of spin components m becomes infinitely large. The authors derive a field theory Hamiltonian for this model and show that its upper critical dimensionality is eight. The critical exponents for this theory are calculated in an expansion in epsilon =8-d to second order. They notice that this second-order expansion is identical to that for the p to infinity limit of the Q³ model in epsilon =6-d.

An algebraic method due to Ihrig (1975) has been recently developed further for four-dimensional space-times by McIntosh and Halford (1981) to determine the metric tensor from the components of the Riemann tensor. This method has been extended to electrovac fields in this paper.

The perturbed ground-state eigenvalues for the harmonic oscillator with negative quartic and with cubic energy distortion are calculated by numerical integration. These agree with the author's results using an asymptotic series.