Table of contents

It is proven that the n-zone Lame equation is equivalent to a spectral problem for a quadratic element of a universal enveloping sl(2) algebra. The Riemannian surface corresponding to the eigenstates in a parametric space forms the (2n+1)-sheet surface which splits in four subsurfaces; three of them contain the same number of sheets.

The authors present the operator content of the XXZ chain in the presence of external fields. The finite-size scaling spectra can be expressed in terms of irreps of shifted U(1) Kac-Moody algebras, the shift depending on the external fields. Using an explicit Feigin-Fuchs construction (1982), they project out from these spectra the operator content of c<1 systems with free, fixed and mixed boundary conditions. They also show how this projection mechanism works for a finite number of sites.

Matrix-multipolar asymptotic modules are used for introducing and solving a hierarchy of systems of purely differential non-linear equations. Each system consists of (N+1)-dimensional evolution equations and of quite strong differential constraints. Specific constructions of matrix-multipolar asymptotic modules are given, in particular delta constructions.

The connectivity constant omega for Hamiltonian walks on Dhar's 4-simplex (1978) is found using an exact set of recursion relations. The result omega =1.399710 . . . differs from the value of omega for Hamiltonian walks on the square lattice. As a consequence, omega must depend on other lattice characteristics besides the coordination number and the fractal dimension. The first correction to the leading-order asymptotic behaviour is also obtained.

The Sutherland equation (1970), implying exact integrability of vertex models, is studied on a two-dimensional square lattice. A novel 15-vertex solution commuting with a spin-1 Hamiltonian is constructed.

The author discusses the N to 0 limit of a variety of field theories, in particular those describing spin glasses. He shows how the convergence is naturally described using a p-adic norm. Replica symmetry breaking and combinatorial calculations in these theories make sense mathematically in the p-adic regime. The p-adic order parameter is viewed naturally as a topological central extension related to holonomy (like Berry's phase) from the quenched frustration in spin glasses.

The author calculates the order parameters (local height probabilities) in the ordered phase of integrable SU(n) vertex models. The author shows that they have formally the same expression as the partition functions of the associated critical theory in a finite box with appropriate boundary conditions, once the distance to criticality in the former case is properly identified with the modular parameter in the latter. This points out a relation between off-critical integrable models and conformal theories in a finite geometry.

The Pauli algebra P, in which the usual dot and cross products of 3-space vectors are combined in an associative, invertible, but non-commutative multiplication, provides a simple but powerful approach to problems in special relativity. Even though the Pauli algebra is the Clifford algebra for Euclidean 3-space, Minkowski 4-vectors and their products in the Minkowski metric appear in a natural and covariant way as elements of P. The authors review the algebra and develop a formulation which, although closely tied to elementary vector and functional analysis, nevertheless allows a compact coordinate-free treatment of essentially all problems in special relativity. They derive a number of useful results and show how the elements are related both to traditional Minkowski-space tensors and to elements of the Dirac algebra.

The Pauli algebra is used to study the relativistic motion of charged particles in external electromagnetic fields. Plane-wave solutions to Maxwell's equations, in particular standing waves with electric and magnetic fields which are everywhere parallel, are easily represented and analysed in a gauge-independent treatment. The authors derive the 4-momentum of electromagnetic fields and show how the integrals for fields of a charged particle can be evaluated with integration limits specified in the rest frame of the moving charge. The general solution of the Lorentz force equation is shown to be a Lorentz transformation with time-dependent rotation and boost parameters which, in the case of parallel fields, are trivially related to the magnetic and electric fields, respectively. With the Pauli algebra, an earlier derivation in the Dirac algebra of the 4-velocity in constant uniform fields is greatly simplified and extended to arbitrary initial motion, and solutions are found to motion in a Coulomb fields and in the presence of arbitrarily polarised (monochromatic, directed) plane waves.

Using a vector coherent state framework and an associated vector Bargmann Hilbert space (1961), the structure of two classes of U(n): U(n-1) unit projective operators is shown to be intimately related to (6-j) and (9-j) coefficients of the U(n-1) subalgebra. Explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels.

The authors present a new method for the calculation of the rotational averages which arise in the theory of spectroscopic radiation-molecule interactions in fluid media. Based upon the principles of irreducible cartesian tensor analysis, the method presented allows them to express results either in the usual reducible form, or directly in terms of linearly independent sets of irreducible tensor products. For interactions up to and including rank 3 in the molecular response (or non-linear susceptibility) tensor, the rotational averages cast in terms of irreducible tensor products are considerably simpler in structure than the corresponding results expressed in reducible form.

An analytic continuation procedure using Taylor series is utilised to produce very accurate wavefunctions and eigenvalues for a double-minimum potential having almost degenerate eigenvalues. Comparison is made with results obtained by other means.

An explicit derivation of the propagator is given for a particle in a Dirac delta potential lambda delta (x) by making a direct application of a variational technique followed by a systematic use of Laplace transforms. Although the solution of the corresponding Schrodinger equation has been known for years, the solution of the propagator has been obtained only recently.

For an arbitrary classical gas of particles in which the interaction is given by a many-body, stable, regular and strongly regular potential the author proves the uniqueness and analyticity of the grand canonical tempered Gibbs state for every value of the chemical activity such that z^-1 not in (spectrum of the Kirkwood-Salsburg operator).

The phase diagram of the two-dimensional biaxially next-nearest-neighbour Ising (BNNNI) model is obtained using the real-space renormalisation group method with the mean-field approximation. The transitions between disordered and commensurate phases are observed to be single transitions for the ferromagnetic phase at intermediate and high temperatures and for the antiphase structure at high temperatures. However, within the approximation, successive transitions are observed between the antiphase structure and disordered phase at intermediate and low temperatures.

The local minima of the Sherrington-Kirkpatrick model (1978) are studied numerically by introducing a heuristic method. The author's approach naturally leads to their ultrametricity. For finite systems some observations on this method suggest the relation d₁-d₂ approximately d₃ among three distances of three low-energy local minima d₁>or=d₂>or=d₃. Overlap functions are studied numerically, giving consistent results with the above relation and the results of the replica symmetry breaking theory.

Phase diagrams of two inhomogeneous annealed Ising models on the triangular lattice are constructed in an exact way. The interimpurity exchange constants (bonds), of the same but arbitrary value and sign, can be located along the horizontal edges of the triangles. In the first model the remaining non-random couplings are ferromagnetic and in the second they are antiferromagnetic. The dependence of the correlation functions and magnetisation on the temperature is calculated and discussed. It is shown that, for certain values of the impurity bond couplings and a range of their concentrations, there is no net magnetic moment at T=0 and low temperatures. The magnetisation appears at some finite temperature and then vanishes at a higher one. This increase of the magnetisation with the temperature is connected with the role of entropy in determining the minimum of the free energy.

Interface phenomena, such as wetting and layering transitions of the three-state chiral clock model in three dimensions, are studied within the Bethe approximation. The Bethe approximation yields the correct sequence of first-order layering transitions. This is in contrast to the mean-field theory which is known to be qualitatively misleading for low temperatures. At higher temperatures the discontinuities of the order parameter at the layering transitions become exceedingly small so that the magnetisation profile appears to vary continuously across the interface. For finite values of the chiral field Delta the width of the interfacial layer is found to remain finite at the first-order bulk transition line. Critical wetting occurs only for Delta =0, i.e. for the symmetric three-state Potts model.

The gauge-invariance problem for a supersymmetric Schrodinger equation invariant under non-relativistic Euclidean supersymmetry is discussed. A gauge-invariant supermultiplet is considered which satisfies this equation. The equations of motion for its physical components are found to be relativistically invariant, which allows the author to conclude that Lorentz invariance is a dynamical symmetry of the non-relativistic supersymmetric gauge problem.

The central charge of the Virasoro algebra of the one-dimensional dimerised XY model with arbitrary boundary conditions is obtained at criticality. The authors obtain c=1-12 tau ² where tau (0<or= tau <or=1/2) characterises the boundary conditions. This shift of c yields relations between the XY model and both the q-state Potts model and the O(N) model. They show that the phenomenological renormalisation group equation, which is used to estimate the critical value of the parameter of the system, has no real solution for the range tau >1/26. This corresponds to the non-unitary Potts model (0<or=q<2) and the O(N) one (-2<or=N<1).

It is shown by example that the condition C_v to 0 as T to 0 is not enough to establish Nernst's theorem.

In a paper entitled 'An equivalent theorem of the Nernst theorem', Yan and Chen (1988) purport to derive the third law of thermodynamics on the basis of the second law and the conclusion that heat capacities tend to zero as the temperature approaches absolute zero. The proof is faulty and the third law is an additional postulate in thermodynamics.