Table of contents

Studies the quantisation of certain local quantum observables defined in terms of a momentum and a region of a Riemannian configuration space. Geometric criteria are given for the quantisability of these local observables, conditions are developed under which a reconstruction of the corresponding global quantum momentum may be attempted, and explicit formulae describing such a reconstruction are given.

Phenomenological renormalisation is used to calculate the exponent nu and the connective constant of the self-avoiding walk problem on a square lattice. A transfer matrix technique is developed for the polymer problem. The results indicate that Flory's value nu =0.75 is true in two dimensions to extremely high accuracy.

By treating the field of a charge in a stationary orbit as a rotating field no radiation term arises. Stationary circular orbits of positronium are considered from this viewpoint, a new interparticle force being obtained. Bohr quantisation of canonical angular momentum leads to two sets of stationary states, one familiar and the other with properties which permit identification with the neutrino. Systems of one or two neutrinos and an electron afford models for charged pions and muons. Other systems are considered, one with properties which might permit identification with the proton.

Using a T=0 real-space renormalisation-group method, the spontaneous magnetisation is calculated on an impurity site of the quantum spin-¹/₂ Ising chain in a transverse field. Two kinds of impurities are considered: either a transverse field or a bond is different from its host value. In both cases the magnetisation vanishes at the critical field with an exponent beta ₀ different from the host exponent and varying continuously with the impurity field (or impurity bond). These results agree with recent exact investigations of equivalent classical two-dimensional models.

It is shown that group theory may be useful in relation to dimensional analysis. The group theoretical support of dimensional analysis in universes described by kinematic groups is analysed. Its close relation to the structure of the corresponding kinematic group is also displayed by means of a simple dimensionalisation hypothesis. The scheme of contractions relating the kinematic groups enables one to discuss the dimensionalisation method. The same problem is also looked at in a different way: the possibility of adding dilatation-like transformations is studied. Finally, the role of mass in both relativistic and non-relativistic theories is examined.

The simple root system of each exceptional, simple Lie algebra is explicitly constructed in a variety of forms. Each construction is based on the natural embedding in the exceptional algebra of a classical, semi-simple algebra of the same rank. The procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented Dynkin diagrams. The various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras. For each such labelling scheme the modification rules for dealing with non-standard representation labels are tabulated. The connection between the natural labels and Dynkin labels is given in detail and a comparison is made with the labels of B.G. Wybourne and M.J. Bowick (1977).

The Weyl group, W_G, of each exceptional simple Lie group G, is described in detail. Its structure is defined in terms of its coset decomposition with respect to the Weyl group, W_H, of a classical semi-simple Lie group, H, embedded naturally in G. The concepts of G-dominance and G-equivalence are defined and used to determine, from the character formula of Weyl, the branching rule associated with the restriction of group elements from G to H. The Weyl group W_G is used further to impose constraints on both the branching multiplicities for G to H and the weight multiplicities of G. These constraints are used to evaluate the weight multiplicities of F₄, E₆, E₇ and E₈ together with the branching multiplicities for E₈ to SO(16).

A technique is described for evaluating Kronecker products of irreducible representations of an exceptional Lie group, G. The method depends upon the natural embedding in G of a classical Lie subgroup H. The problem is reduced to that of finding the branching of one irreducible representation of G on restriction to H, evaluating Kronecker products in H and finally using the modification rules appropriate to G.

A non-genealogical method is proposed for the reduction of the inner product representations of the permutation group S_N. This method of determining the Clebsch-Gordan coefficients has been found to be recursive only within a given series. As such it permits the direct reduction of products of large-dimensional representations.

Direct-sum decompositions of the Euclidean space of particle momenta into collective and intrinsic subspaces are achieved using projectors on so(3) and gl(3,R) algebra spaces. The separation of the N-particle kinetic energy into the corresponding collective and intrinsic components is then simply obtained. In the case of gl(3,R) the resulting intrinsic kinetic energy is expressed in terms of an appropriate subset of the generators of the direct product group SO(N)*SO(N). A detailed comparison of these results with those of other authors is given.

An approximate analytical method that can be used for E(X) epsilon and other similar Jahn-Teller systems is discussed. It is applicable at all energies and all coupling strengths, and gives quite a good account of the energy levels and other properties of the eigenstates over the whole range of these parameters.

This formal hidden-variables representation is analysed in the context of impossibility proofs concerning hidden-variables theories. The structural analogy of this formulation of quantum theory with classical statistical mechanics is used to elucidate the difference between classical mechanics and quantum mechanics.

An upper bound for the information rate in a communication link is set by the finite number of orthogonal states available to the electromagnetic field, is the link is subject to constraints on the average power and bandwidth. This bound may not be attainable in a free-space link since the transmitter does not have complete control over what is received. Some hypothetical systems are examined to see how close they get. At high photon rates it is possible with difficulty to improve slightly on an 'x-p' system, and at low photon rates on a photon-detecting system, but it seems that these systems are nearly the best, and certainly they are not far below the upper bound, in the sense that the ratios of the information rates to the maximum are close to unity. In all these systems the normal modes (Planck oscillators) are treated as independent channels. It is suggested that a beam splitter can provide a simple model for a free-space link.

Discusses some difficulties which arise in proposals to extend the inverse scattering transform method to nonlinear quantum field theory. In particular, for the nonlinear Schrodinger equation, the author shows that the usual classical methods for obtaining the Poisson brackets of the scattering data reach an impasse if they are extended to find the commutators of the corresponding quantum operators.

Studies the quantum oscillator on the half-line defined by the Hamiltonian H=-¹/₂ delta ²/ delta x²+¹/₂x² together with vanishing boundary conditions at 0. The authors derives and discusses a new operator equation of motion of the form x+x=A for this system. This equation is used to verify J.R. Klauder's conjecture (1979) that x satisfy x(x+x)=0. It is also used to obtain systematic results on the set of n-point functions ( Omega mod x(t₁)x(t₂)...x(t_n) mod Omega ). It is suggested that these methods and results will act as a source of conjectures about the analogous pseudo-free field theories postulated by Klauder in his program for handling non-renormalisable field theories.

The diffraction of electrons by a cylindrical capacitor is formulated in terms of conventional quantum scattering theory. Using several approximations valid in the short wavelength limit, the theory is reduced to a form that allows direct comparison with formulae based on the diffraction integral, thereby clarifying the content of the latter.

Investigates conditions under which a vacuum type-D space-time admits a regular S₂ separability structure. All vacuum type-D space-times without acceleration admit such a structure. It turns out that the structure is completely determined by the Killing tensor and the metric.

For pt.I see ibid., vol.11, p.877 (1978). The energy localisation in the case of spherical symmetry is justified in reference to an exact interior solution which is a generalisation of the rho =constant Schwarzschild solution. This reveals the inadequacy of the Tolman expression for localisation which also fails to provide the correct total mass for a system which exhibits certain discontinuities. A generalised localisation expression for arbitrary static systems is proposed for further consideration.

A new two-dimensional statistical mechanics model is solved. It is a general model with 32 free parameters. The solution uses integrals over anticommuting variables.

All the correlation functions in the free-fermion 32-vertex model are calculated.

In configurational studies of polymer chains one is sometimes concerned with averages in the presence of restrictions, e.g. that the end-to-end distance of the chain is R. The standard method of calculating such averages is to assume that the separation of all pairs of elements is Gaussian. The author develops an alternative method which uses generating functions and avoids the Gaussian assumption. The results are equivalent asymptotically to those with the Gaussian assumption, but correction terms can be calculated.

A one-dimensional Ising model with a potential falling off like r^{-( sigma +1)} is considered as a limit of a set of Gaussian models. The spin-spin correlation function is derived, and the critical indices alpha , beta , nu , eta are calculated.

A perturbation calculation is given which implies that the susceptibility of the five and six-dimensional n-vector models can be written x approximately At^-1(1+Bt¹2/) and x approximately At^-1(1+Bt In t) respectively, independent of n. For n=0 and 1 it is shown that series analysis techniques can extract the correction-to-scaling' exponents, and that estimates of the critical temperatures and critical amplitudes can also be obtained. The correction-to-scaling exponents found are in agreement with those known to exist in the case of the spherical model.

A finite-size scaling formalism is outlined for quantum Hamiltonian field theory on a lattice. The scaling behaviour in the neighbourhood of a critical point is predicted. To test the theory, exact results are generated for the mass gap specific heat and susceptibility of the (1+1)-dimensional Ising model on a finite lattice. Finite-size scaling methods give results for the critical parameters which are comparable in accuracy with those obtained by standard perturbation series analysis methods.

For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.

The spectral distribution of Mossbauer radiation passed through a resonant medium, which performs additional ultrasonic vibrations, is investigated experimentally for thick absorbers. The results are in quantitative agreement with the theory.

A simple algebraic derivation of Parisi's equations (1981) for the Sherrington-Kirkpatrick model for spin glasses is given.

Perimeter polynomials are given for the bond percolation problem on the following lattices: the triangular up to D₉, simple quadratic (D₁₃), honeycomb (D₁₇), face-centred cubic (D₇), body-centred cubic (D₈), simple cubic (D₉) and diamond (D₁₃). The total number of bond clusters grouped by size is given to two further orders in each case.