Table of contents

Recent calculations on one-dimensional energy bands are improved remarkably by using a simple perturbation approach with D⁴ as the perturbation. The theoretical approach used is also relevant to the problem of including relativistic mass corrections in the Schrodinger equation.

A Monte Carlo renormalisation-group method for polymers is developed and applied to the freely jointed polymer model with excluded volume potential. Both the exponent and the amplitude of the end-to-end correlation length are obtained.

The energy conditions imply that null-homogeneous solutions of the Einstein field equations admit a non-expanding shear-free normal null congruence, and that these space-times are algebraically special.

Starting from the generalised Gibbs equation of extended irreversible thermodynamics, a thermodynamic potential which offers a suitable description of the electric current fluctuations in an isotropic rigid conductor is derived.

Ferromagnetic Ising systems with four different arrangements of antiferromagnetic impurities are considered. Using the high-temperature series expansion method, T_c is determined and compared with respective values for a system containing the same amount of impurities but distributed randomly. It is shown that T_c is much more sensitive to the concentration of antiferromagnetic bonds and their strength than to the amount of frustration present in the system.

A specific vertex model is defined on a square lattice whose point set is partitioned into two subsets (A) and (B). It is shown that if the A-B pattern of the partition does not contain elementary squares with an odd number of A points, then the vertex model is equivalent to an Ising model on a graph G. In order to construct the graph G, we make use of the properties of the A-B pattern on the square lattice. Several examples are examined and three different possibilities of behaviour are found, i.e., 'two-dimensional' Ising-type transitions, 'one-dimensional', and 'frozen in' behaviour.

The hard-hexagon model in lattice statistics (i.e. the triangular lattice gas with nearest-neighbour exclusion) has been solved exactly. It has a critical point when the activity z has the value ¹/₂(11+5 square root 5)=11.09017..., with exponents alpha =¹/₃, beta =¹/₉. More generally, a restricted class of square-lattice models with nearest-neighbour exclusion and non-zero diagonal interactions can be solved.

Temperatures where the 2nth expansion coefficient of the average free energy with respect to the external field fails to have a definite value are obtained for a random Ising model on a Cayley tree. The assumptions made for the random Ising model are that the exchange integral is a random variable, its probability distribution is independent of the pair of sites and the average value of the absolute value of the exchange integral is finite. The critical exponents kappa defined by Muller-Hartmann and Zittartz are obtained. The phase transition is of the continuous order.

The authors outline an approach to irreversible thermodynamics leading to a covariant evolution equation for the four-flux of heat in a relativistic non-viscous simple fluid. This relation results in a non-trivial generalisation of the classical case.

The discrete spectrum for a Morse oscillator is found using an SO(2, 1) algebra. Since this algebra does not prove to be appropriate to compute matrix elements for the oscillator eigenfunctions, the author construct a B₁-type algebra with the aid of an auxiliary angle variable. Matrix elements and recurrence relations are found for several useful operators using the algebra. The limit in which the anharmonicity tends to zero is studied.

The complete set of 6j symbols for the double point groups D_4d and D₄ and the complete set of 3jm factors associated with the group chain D_4d contains/implies D₄ contains/implies C₄ are calculated.

Series analysis methods are used to estimate the surface susceptibility exponent for the polymer analogue of a magnet with a free surface, on the square and simple cubic lattices. Our exponent estimates are slightly larger than the surface scaling predictions for both lattices.

The Mower sequential decay theory of quantum processes has been extended in order formally to remove certain spurious poles in the matrix elements of the resolvent operator, and to recast the results into a more symmetrical general form. A special choice of intermediate manifolds of states leads to a further simplification. An illustrative application of the theory is given.

The previously developed 'continuous temporal development and Noether's theorem' approach to the association of symmetries and conservations of the laws of nature, part of the foundation of any (classical or quantum) Lagrangian theory, is generalised significantly. Within this framework a strong result is obtained for the inverse Noether's theorem, associating symmetries with given conservations. A general symmetry-conservation formalism is proposed, into which both the previously developed 'linear temporal development and conserved eigenheit' approach, part of the foundation of any quantum theory and also applicable to certain classical ones, and the 'continuous temporal...' approach, as different as they are, fit nicely.

The study of the symmetry group of the time-dependent oscillator in N dimensions with equation of motion d²x_i/dt²+ Omega ²(t)x_i+0, i+1, ..., N, gives the full symmetry group SL(N+2, R) of N²+4N+3 operators. The Noether subgroup consisting of ¹/₂(N²+3N+6) operators and the resulting constants of motion are given. A table of the commutation relations between the operators gives the structure constants of the associated Lie algebras.

It is shown that, in the symmetric formulation of classical mechanics, the set of dynamical variables of the unconstrained systems constitutes a Jordan algebra under the plus Poisson bracket combination law defined by Droz-Vincent (1966). For the constrained systems, it is shown that the set of dynamical variables constitutes a Jordan algebra under the corresponding plus Dirac bracket combination law if some conditions are satisfied. These conditions are presented.

The motion of a conservative classical system is considered as a geodesic flow on a Riemannian manifold. Expressions for various curvature tensors are derived. It is shown that the Riemann tensor of a conservative system with N degrees of freedom is defined by ¹/₂N(N-1) curvatures c_ij(i<j), for which expressions and elementary properties are derived. The manifold has negative curvature if all c_ij<0. For closed manifolds negative curvature implies that the system is a K-system, but we show that one cannot have c_ij<0 everywhere in the configuration space of a conservative system. The results are presented of numerical calculations of the curvature of 8- and 16-particle systems interacting according to the Debye-Huckel and Lennard-Jones potential. Good agreement is found between the 8- and 16-particle curvature distribution functions. In general, more than half of the c_ij at a given point in configuration space are negative. Negative curvature turns out to be a rare phenomenon, although it is found in more than half of the configuration space of Lennard-Jones systems at low density and energy E=0.

A general theory of quantum mechanical constants of motion of arbitrary order in the momenta is developed. Because of rotational invariance the theory assumes an elegant tensorial form. The fundamental equations for the nth-order constants of motion in m variables are set up in Cartesian coordinates. The first two equations of the set do not contain the potential function and their solutions are shown to be polynomials of degree n and n-1 with coefficients satisfying certain symmetry relations. The next two equations contain the first and the second derivatives of the potential function. From these equations all unknown functions are eliminated in the special cases n=3 and n=4. The elimination gives partial differential equations for the potential function, of second and third order for n=3, and third and fourth order for n=4. The possibility of extending the results to higher values of n is indicated.

A rigorous definition is given of the one-dimensional Coulomb Hamiltonian, discuss its spectral properties, and investigate various approximations for it. For one of these approximations, which is frequently used in the literature, it is proven in particular that it converges to the semi-bounded Coulomb Hamiltonian in the strong graph limit, although its ground-state tends to minus infinity in this limit.

The design of a technically feasible experiment is discussed that would distinguish between Heisenberg's uncertainty principle and de Broglie's relation, ( Delta x)_i( Delta p)_f>or=h(cross)/2 where ( Delta x)_i is the initial localisation of the particle and ( Delta p)_f is the final scatter in the momentum of the free particle.

Two new possibilities in the axiomatic foundations of quantum mechanics are examined: first, the possibility of introducing a non-symmetric transition probability between pure states, and second, showing that the concept of orthocomplementation in the logic of events is unnecessary and of secondary importance. An axiomatic scheme is presented which does not involve the concept of orthocomplementation and yet has all the advantages of the well-known quantum logic axiomatics, because the generalised logic of events admits an extension, which is a complete orthocomplemented orthomodular lattice with the covering law holding in it. Thus, the approach to quantum axiomatics may be seen as answering both the old questions of the quantum logic approach (e.g. the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law) and the question concerning the necessity of the orthocomplementation in the logic of events, recently raised by Mielnik (1976).

Renormalisation of lambda phi ⁴ theory in curved space-time is considered in the interaction picture. A generalisation of normal ordering to curved space-time is introduced, based on the construction of adiabatic particle states in Robertson-Walker space-time. Dimensional regularisation is used to define uniquely the divergent quantities which are removed by normal ordering. It is shown that this normal ordering is sufficient to make finite all physical processes including vacuum polarisation to first order in lambda . An alternative and equivalent procedure is given which requires renormalisation of the mass and of the constant which couples the field to the Ricci scalar. The stress tensor is found to be finite to first order in lambda and it is shown that if the free-field theory in a Robertson-Walker universe predicts that particles are created by the gravitational field with a black-body spectrum then this spectrum is maintained when first-order self-interactions are taken into account. Finally, some aspects of the renormalisation of second-order physical processes are discussed.

For pt.I see ibid., vol.13, p.901 (1980). An explicit renormalisation of all second-order physical processes occurring in lambda phi ⁴ field theory in conformally flat space-time, including vacuum-to-vacuum processes, is performed. Although divergences dependent on the definition of the vacuum state appear in some Feynman diagrams, physical amplitudes obtained by summing all diagrams which contribute to a single physical process are independent of these divergences. Consequently, the theory remains renormalisable in curved space-time, at least to second order in lambda . Renormalisations of the mass m, the coupling constant lambda and the constant xi which couples the field to the Ricci scalar are required to make two- and four-particle creation amplitudes finite.

Perturbative electromagnetic fields are investigated in the Godel universe using the Debye potential (two-component Hertz potential) formalism. The Godel space-time lends itself to this approach since it is algebraically special (Petrov D). With the usual system of coordinates used in the text describing the Godel universe, the three of the basis coordinate vectors delta / delta x⁰, delta / delta x¹ and delta / delta x³ which are Killing, the field at infinity in these directions remains oscillatory, while for the remaining non-Killing coordinate x² the field decays at both extremities. The result compares dramatically well with the behaviour of the null geodesics in the Godel universe.

The authors show how it is relatively straightforward to deduce the Kruskal line element directly from the vacuum field equations, thereby avoiding the conventional approach which sets out from the analytic continuation of the Schwarzchild metric.

A class of Robinson-Trautman space-times is investigated with respect to the existence and properties of cosmic time functions and corresponding surfaces of simultaneity, partial Cauchy surfaces (PCS). The existence of an horizon-like hypersurface N generated by null curves is established. Then it is proved that in any connected PCS, intersecting N and approaching the curvature singularity in a certain way, the singularity appears as a point. Some counterexamples are given when these conditions do not hold, and the relation to the problem of representing particles by means of singularities is briefly discussed.

The Einstein-Maxwell field equations are solved completely when the line element has the form ds²=exp(2h)dt²-exp(2A)(dx²+dy²)-exp(2B)dz² where h, A and B are functions of t only, and is thus non-static. The metric admits a four-parameter group of motions. The Weyl tensor is type D and the electromagnetic field is non-null.

Particle creation due to field interactions in an expanding Robertson-Walker universe is investigated. A model in which pseudoscalar mesons and photons are created as a result of their mutual interaction is considered, and the energy density of created particles is calculated in model universes which undergo a bounce at some maximum curvature. The free-field creation of non-conformally coupled scalar particles and of gravitons is calculated in the same space-times. It is found that if the bounce occurs at a sufficiently early time the interacting particle creation will dominate. This result may be traced to the fact that the model interaction chosen introduces a length scale which is much larger than the Planck length.

The authors apply the theory of classical canonical ensembles to a one-dimensional system of 'soft rods' and obtain an exact expression for the partition function. In the model, each particle is composed of two mass points attracting or repelling one another according to a potential linearly dependent on their separation. Mass points, whether they constitute a given particle or are members of different particles, are presumed to be unable to pass one another. This is a very simple model, therefore, of a system whose internal and external degrees of freedom interact. In order to eliminate end effects without appealing to the thermodynamic limit, they employ a coordinate system with periodic boundary conditions. The methodology of evaluation is more general than the solution for their particular model which, for a finite system, involves in the partition function a modified Bessel function of half-integral order.

The behaviour of weakly charged polyelectrolytes in a poor salt-free solution is considered, using the notion of blobs. It is shown that in a very dilute solution, where different polyions do not overlap, a transition of the coil-globule type takes place, upon making poorer the thermodynamic quality of the solvent. This transition occurs in each blob separately. In the region of higher concentrations (the polyions overlap strongly, but the volume fraction occupied by the blobs in the solution is small) an additional effect is predicted: avalanche-type counter-ion condensation. This effect is due to the fact that the polyelectrolyte conformation varies self-consistently, together with the variation of the number of condensed counter ions.

Using a Mellin integral transform the authors obtain the exact bulk and finite size terms of the logarithm of the grand partition function of a spinless relativistic boson gas confined in an Einstein universe. The finite size contribution calculated by Al'taie (1978) is shown to be negligible when compared with the leading relativistic correction to bulk term unless N(kT/m)<or=1. They also verified that the transition temperature is lowered by the relativistic corrections overwhelming the shift produced by the surface effects. In the ultra-relativistic limit the finite size contribution is shown to be comparable only to higher-order mass corrections to the bulk term.

The theory of non-radiating stochastic current distributions is developed within the framework of first-order coherence theory of electromagnetic fields. Necessary and sufficient conditions for non-radiating current correlations are derived. A class of non-radiating stochastic current distributions is constructed. In contrast to these results, it can be shown that any non-trivial field correlation given on an arbitrary surface radiates.

It is shown that the fourth-field derivative, chi ₀⁽²⁾, of the free energy of the Ising model above the critical temperature can be expressed as an expansion in terms of two-connected graphs or stars. The contribution of any star graph to the expansion is expressed in terms of the weak lattice constant of the graph, and a weight, the latter being a property of the star graph only. The method is used to derive series expansions for chi ₀⁽²⁾ on the four-dimensional hyper-simple cubic, hyper-body-centred cubic and hyper-face-centred cubic lattices to order 16, 11 and 9 respectively.

The validity of the hyperscaling relation 2 Delta -dv- gamma =0 is studied for the four-dimensional spin-¹/₂ Ising model. High-temperature series expansions are derived for the fourth-field derivative chi ₀⁽²⁾ of the free energy on the four-dimensional hyper-face-centred cubic (HFCC) and hyper-body-centred cubic (HBCC) lattices to order 9 and 11 respectively. These are analysed, together with other series already available for the susceptibility chi ₀ and correlation length xi for the HFCC, HBCC and the hyper-simple cubic (HSC) lattices. All these series are found to behave consistently with the asymptotic form t^-q mod lnt mod ^p, where t is a reduced temperature variable and q is the appropriate mean-field exponent (so that hyperscaling is satisfied automatically). The best estimates for p are as follows: p=0.30+or-0.05 (HFCC), 0.32+or-0.05 (HBCC) for xi ₀ (with q=1= gamma ) and p=0.33+or-0.05 for xi ² (with q=1=2v). These estimates are in good agreement with the renormalisation group (RG) prediction of p=¹/₃. Results for chi ₀⁽²⁾ are more slowly convergent, but are still consistent with p=¹/₃ for q=4=2 Delta + gamma .

Metcalf and Yang (1978) have obtained the numerical estimate 0.3333 k for the entropy per site of hard hexagons. They therefore conjecture that the exact value is k/3. The authors have used the corner transfer matrix method to obtain a more accurate numerical estimate, namely 0.333 242 721 976 k, which contradicts the conjecture. The problem is a good example of the numerical accuracy of the method.

The two-dimensional Ising model on the square lattice with a fraction of periodically distributed frustrated cells equal to 1, ²/₃, and ¹/₂ is solved rigorously. It is shown that a phase transition, understood in the sense of a singularity in the specific heat, exists for the two latter model systems while it disappears for the completely frustrated lattice. The phase transition temperatures are lower than that of a ferromagnetic Ising model and increase with decreasing frustration. The mean-field approximation results suggest that long-range order is present in areas of nonfrustrated cells.

Using the appropriately generalised finite lattice method, series expansions of the layer ( chi ₁) and local ( chi ₁₁) susceptibilities of the square lattice Ising model have been obtained. They extend existing series by 3 and 13 terms for chi ₁ and chi ₁₁, respectively. Series analysis yields the exponent estimates gamma ₁=1.375+or-0.005 and gamma ₁₁=0.00+or-0.01, in agreement with scaling predictions. Repetition of the analysis used in the analogous self-avoiding walk problem confirms the breakdown of the renormalisation group scaling relation gamma ₁₁=v-1 for the square lattice self-avoiding walk problem found in an earlier study.

The authors consider two lattice gas models where dimers are placed on the simple cubic lattice with an attractive interaction between collinear dimers. They prove (using reflection positivity and Peierls' argument) that at low enough temperatures both models exist in an ordered state where one or two of the three orientations are preferred. No attempt is made at deciding whether the ordered state actually has dimers of two orientations or only of one orientation. Both models have earlier been considered in the two-dimensional version and proved to exhibit ordering at low temperature.

The set of possible Carnot cycles involving positive and/or negative Kelvin temperatures is analysed in terms of graphs and of transported entropy and entropy production. The formulations of the second law of thermodynamics, allowing for the existence of negative Kelvin temperatures, proposed by Ramsey (1956) and by Landsberg (1977) are shown to be logically equivalent. Some properties of the limiting temperatures T+0, +or- infinity , -0 are also investigated and a generalised third law is formulated.

A phenomenological theory is proposed of a charged, streaming N-component mixture of dissipative fluids which are polarisable and magnetisable and whose governing equations form an hyperbolic system. Non-stationary transport equations are proposed for dissipative fluxes containing new cross-effect terms, as required for compatibility with the entropy principle expressed by a new balance equation (including a new Gibbs equation). The theory formed by the set of (13N+7) equations governing the material behaviour of the system, by generalising the constitutive equations of a quasineutral media, together with Maxwell's equations, may be referred to as the electrodynamics of dissipative, streaming media. Proposed transport laws for polarisation and magnetisation generalise the well-known Debye law for relaxation, and show the possible influence of temperature and density gradients on polarisation and magnetisation. The forms of the free energy and of the Gibbs function, in the non-stationary regime, are also formulated.

The problem of acceleration and diffusion of charged particles in a stochastic magnetic field is investigated. The analysis of earlier authors has been generalised to take into account non- delta -correlated stochastic processes. Using Khasminskii's limit theorem (1966) a rigorous analysis of the acceleration and diffusion of charged particles is presented. The results are obtained under very general conditions and are free from many limitations in the earlier work. Stochastic heating of the plasma is discussed briefly.

The breaking of the replica symmetry in spin glasses is studied. It is found that the order parameter is a function on the interval 0-1. This approach is used to study the Sherrington-Kirkpatrick model. Exact results are obtained near the critical temperature. Approximated results at all the temperatures are in excellent agreement with the computer simulations at zero external magnetic field.

The physical meaning of second-order phase transitions discovered by Davies (1977) in black-hole thermodynamics is discussed. It is argued that the phenomenon has nothing to do with phase transitions occurring in rotating self-gravitating fluids in Newtonian theory. The authors show that the internal state of a black hole remains unaffected after the phase transition. They also argue that the change of sign and infinite discontinuity in the heat capacity are of purely geometric origin, i.e. are determined by the embedding of an event horizon into a black-hole space-time. This result supports the view that temperature cannot be used as a well-behaved fundamental parameter in black-hole physics.

Two new high-field polynomials L₁₄ and L₁₅ are given for the simple cubic lattice together with the complete partial generating function F₇ which determines the corresponding sub-lattice polynomials. For the simple quadratic lattice the complete partial generating function F₈ is given; this, combined with the recent work of Baxter and Enting, determines the corresponding sub-lattice polynomials to order 19. Extra high-field polynomials through L₂₁ derived by manipulation of the results of Baxter and Enting are quoted explicitly.