Table of contents

The Dirac quantisation of a magnetic monopole is readily derived from the Lie algebra of the Poincare group (taken as dynamical group). The representation can be reinterpreted in order to define a position operator for massless particles.

The Voigt functions, which are employed in such areas as neutron physics and spectroscopy, are shown to be capable of representation in terms of circular functions and confluent hypergeometric functions.

Points out some subtleties in the existence problem of quasi-periodic states for quantum Hamiltonians periodic in time.

Computes the critical indices of percolation, using a method in which the integrals are computed at fixed dimension, at the order of two loops. Following this method the authors obtain the series, giving the critical exponents, expanded with respect to the coupling constant, while in the epsilon -expansion method the expansion of the series is performed with respect to the coupling constant and to the number of dimensions D. Thus the authors think that they have obtained better control of the position of the fixed point at a dimension not too near to size: they are also able to improve the numerical values of the exponents at D=3.

The first- and second-order phase transitions of the q-state Potts models are obtained in arbitrary dimension d. Critical and tricritical behaviours merge and annihilate at q_c(d), clearing the way to first-order transitions at q>q_c(d) by the condensation of effective vacancies. The value of q_c(d) decreases with increasing d, from diverging as exp(2/(d-1)) at d to 1⁺, to q_c(2)=3.81 (cf exact value of 4), to lower values at d>2. For given d, a changeover in critical behaviour occurs at q₁(d), as the critical fixed points merge from the Potts-lattice-gas region to the undiluted Potts limit. It is suggested that the power law singularities of the percolation problem (q to 1⁺) have logarithmic corrections.

In a conformal symmetric theory the author proposes a conformal covariant energy-momentum tensor, in terms of which other conformal currents are expressed.

The critical region in the disordered phase near the bicritical and tetracritical points is obtained. For this purpose the condition for the validity of the Ornstein-Zernike approximation for the correlation functions is used. The consideration holds for 2<d<4 dimensions of space. The effect of the interaction between the orderings near the multicritical points is discussed.

A closed expression is obtained for the n-dimensional Wigner oscillator function, and a diagrammatic technique is developed for finding the explicit dependence of the function on Euler angles at any dimension n.

For pt.I see ibid., vol.11, p.2133-47 (1978). The line groups are the symmetry groups of stereoregular polymer molecules. For quantum-mechanical applications one needs their unitary irreducible representations (reps). All reps of the line groups whose isogonal point groups are D_nd and D_nh(n=1, 2, 3,...) are constructed. These reps are derived (directly or by induction) from those of the corresponding invariant subgroups of index two, which are the line groups with isogonal point groups C_nv.

Complementary variational principles for the solution of certain linear equations are developed. It is shown that these may be used iteratively for the solution of nonlinear equations. Examples are presented with applications in particle theory, electromagnetic theory, communication theory and the Thomas-Fermi statistical theory for atoms.

Compares Turschner's approximation to the eigenvalues of Hamiltonians -(del)²+c mod x mod ^nu with previously published numerical results. The authors examine evidence that Turschner's approximation yields a lower bound to the ground state energy both for boson and fermion systems. They discuss the convergence of Turschner's approximation to the exact energies E_n^{( nu )} as n to infinity , and derive a simple expression for Turschner's approximate eigenvalues in terms of the hypergeometric function.

By scattering in a central potential V(r) the state function is split into irregular parts, Psi = Phi +X, where Phi satisfies the Schrodinger equation with a modified potential V+W. The non-Hermitian term W vanishes in r space except on the incident (z) axis, where it is singular and non-local. In r space Phi and X become logarithmically singular on the z axis, and the asymptotic difference between Phi and a plane wave is assumed to be of O(V). The scattering amplitude can be expressed by an integral containing Psi , Phi and W. Half-shell F-matrices are defined, which are closely related to the T-matrix. The formalism is valid also for Coulomb scattering, where Phi and X become equal to the usual irregular solutions. The classical limits of the phases of Coulomb's Phi and X are found, and they coincide with the incoming and scattering part of the action function respectively. The theory is applied to the Yukawa case, were Xi and Phi are given to all orders. General orthonormality relations for Phi are established by means of reciprocal functions.

In some recent experiments, it has been established that the Newtonian gravitational potential can influence the phase interference between alternative virtual paths taken by a quantum test particle. The authors consider theoretically the influence of general relativistic fields on the phase interference of superfluid flows.

Outlines the procedure for the complexification of the tangent bundle over a four-dimensional space-time manifold. By introducing a connection and metric compatible with the complex structure, the authors form the geometrical basis for a new (complexified) theory of gravitation whose fundamental gauge group is U(3,1). They further prove that the Lagrangian for the theory is necessarily real when the connection is compatible with the metric.

Treats a percolation model with anisotropic bond occupation probabilities in order to study the crossover behaviour of the effective spatial dimension of the system. Previous position-space renormalisation group (PSRG) studies of this problem on the square lattice show a crossover from two-dimensional to pseudo-one-dimensional critical behaviour. To investigate the possible reasons for this surprising result, which is the opposite of the behaviour observed in thermal critical phenomena, the authors first develop alternative PSRG schemes. They study the predictions of these groups for asymptotically large rescaling parameters where the approximations in the PSRG are thought to become negligible. From a consideration of the approximations involved, they are led to a decimation transformation that uses an anisotropic cluster. The results of this method indicate that anisotropic percolation is in the same universality class as isotropic percolation, in complete analogy with thermal critical phenomena.

Kikuchi's approximation (1951) for describing phase transitions in magnetic systems is reformulated so as to reduce the number of self-consistent equations to be solved and emphasise the analogy with mean field theory. The method is used to calculate the threshold for bond percolation by exploiting the analogy with Potts model.

Static correlation functions for the sine-Gordon chain and related one-dimensional models are described in terms of the transfer operator formalism. The characteristic signatures of kink-solitons within this formalism are discussed and classes of 'kink-sensitive' and 'kink-insensitive' correlations are emphasised. Special attention is given to the integrated structure factor integral S(q, omega )d Omega and the consistency of several calculations and models of S(q, omega is addressed. Possible applications to the interpretation of scattering experiments in quasi-one-dimensional ferro- and antiferromagnetic materials such as CsNiF₃ and (CD₃)₄NMnCl₃ are discussed.

Minus the density of the effective action evaluated at the lowest eigenfunction of the (space-time) derivative part of the second (functional) derivative of the classical action, is proposed as a generalised definition of the effective potential, applicable to twisted as well as untwisted sectors of a field theory. The proposal is corroborated by several specific calculations in the twisted sector, namely phi ⁴ theory (real and complex) and wrong-sign-Gordon theory, in an Einstein cylinder, where the exact integrability of the static solutions confirms the effective potential predictions. Both models exhibit a phase transition, which the effective potential locates, and the one-loop quantum shift in the critical radius is computed for the real phi ⁴ model, being a universal result. Topological mass generation at the classical level is pointed out, and the exactness of the classical effective potential approximation for complex phi ⁴ is discussed.

Starting from a finite QED in the Johnson-Baker-Willey formulation (1964), structures of UV divergences of (0 mod S mod 0) in QED are analysed. To obtain the finite QED in the vacuum region several eigenvalue conditions for the bare electron charge are introduced.

It is established that the spectral anasatze for longitudinal Green functions in first approximation of the gauge technique fulfil the covariance properties expected on general grounds in ultraviolet and infrared regimes, but possibly not at intermediate momenta. Explicit calculations in quantum electrodynamics conform these statements.

By using second-order perturbation theory in the small parameter epsilon =4-d to 0, the author determines a specific value of the excluded volume parameter u equivalent to the fixed point value given by renormalisation group theory. For this value of the excluded volume parameter each expansion series in epsilon can be summed to an exponential function. The author thus studies the total number of configurations, C, the number of configurations returning to the origin, U, and the mean square end-to-end distance, (R²), of the polymer coil. An interdimensional relationship developed by Kosmas and Freed (1978) is used to extrapolate the authors results to lower dimensions. Finally, the author compares his results with those of previous theories and lattice enumerations, discussing possible differences between the Gaussian excluded volume model used by him and the self-avoiding walk model, close to dimensionality d=1.

The method used by Lebedev and Skalskaya (1962) for solving the electrostatic problem of a grounded sphere lying on a plane in an electric field, is generalised to the case of a portion of sphere protruding from a plane electrode. Expressions are developed for the electrical field on the surface and along the axis of the sphere. Results are checked in elementary cases for which the solution is already known. Numerical values of charge and force coefficients are tabulated.

Correlation function expressions are derived for the contributions of the frequency-dependent relaxation effect to the conductivity and dielectric permittivity of the electrolyte solution. The evolution equation for the time-dependent irreducible ionic pair correlation function is derived in the mean field approximation by the application of the generating functional method.

Extends the author's work (1979) on solution of the Bethe-Salpeter equation for the ladder approximation, Gamma _L, to the effective interaction in a Fermi liquid. This permits treatment of the case of bare potentials having finite range, with a spatial dependence which is either exponential or is representable as the Laplace transform of another function, and an exponential decay in time. The integral equation is transformed to real space and the kernel replaced by a differential operator, in a generalisation of an approach developed by Hahne, Heiss, and Engelbrecht (1979) for interactions depending only on time. This procedure simplifies calculation of the terms in the iterative solution of the integral equation, as is demonstrated by explicit calculation, for several cases, of the first two iterative terms. It is found, in agreement with earlier results, that going from a zero-range or Dirac delta interaction to one of finite range can change markedly the analytic character of Gamma _L and its calculation.

Studies the interface between two coexisting phases in an anisotropic, (n+1)-component Landau-Ginzburg model, with a single easy axis and O(n) symmetry in the transverse components, which may serve to represent a d-dimensional uniaxial ferromagnet below its Curie point, T_c. An exact solution of the mean-field equations of motion due to Sarker et al. (1976) indicates the existence of a bifurcation temperature, T_B<T_c. For T<T_B, the interface profile is consistent with the usual Bloch picture, with a non-zero density of transverse magnetisation within the wall. For T>T_B, the profile is Ising-like, with no transverse magnetisation, and approaches the known universal profile as T to T_c. Analyses of fluctuations about this solution shows that the bifurcation is quite analogous to a second-order phase transition. The amplitude of transverse magnetisation vanishes as T to T_B^- and an associated susceptibility diverges with the exponents of the (d-1)-dimensional n-vector model.

Boundary layer theory is applied to the micro-magnetic equations describing the distribution of magnetisation in small spherical ferromagnetic particles. A solution corresponding to a single domain wall naturally emerges. The spatial variation of magnetisation so obtained compares well with previous numerical treatments.