Table of contents

An extended integrity basis (EIB) of a polynomial algebra in a set of variables on which a finite group operates includes the ordinary integrity basis of invariants and linear integrity bases of covariants. The latter are defined as sets of covariants of a given type such that any other covariant of this type is expressible as a linear combination of basic ones with invariants as coefficients of this combination. A constructive method of derivation, based on successive Clebsch-Gordan reaction and elimination of redundant covariants, is described, and the 'extended Noether's theorem', which states that the EIB of a finite group in a finite set of variables is finite, is proved with its use. It is shown that EIBs in irreducible sets of variables are fundamental for a given group because overall homogeneous EIBs in any set of variables can be constructed with their use for this group. A relationship between this method and theory based on a consideration of Molien series is established. It is shown that the division of invariants into denominator and numerator invariants enables one to construct general invariant functions as well as functional covariants.

Some generating functions involving harmonic oscillator functions are obtained as Lauricella functions and Appell functions.

The correspondence between operators on a Hilbert space and phase-space functions based upon symmetric ordering, introduced by Cahill and Glauber (1969) (Weyl correspondence) is used in the present paper to define (non-linear) unitary transformations for quantum systems with the help of canonical transformations, which are bijective, i.e. one-to-one onto. These unitary transformations can be used to determine exactly the energy eigenvalues of a large class of one-dimensional quantum systems. As an example the authors calculate the exact eigenvalues of the Hamiltonian H(X,P)=¹/_2m(P²+a¹2( nu +2)/ mod X mod ^nu ), a, nu >0.

A method for deriving lower bounds to the ground states of nonrelativistic quantum Hamiltonians is developed and illustrated with examples. The bound depends on a trial function and can be made arbitrarily close to the true value, except for systems of fermions. For one-dimensional and spherically symmetric systems bounds for the excited states may also be derived. From the bound another one as found by Barnsley (1978) may be derived.

It is well known that the spectrum of a homogeneous quadratic Hamiltonian in m fermion construction-operator pairs is characterised by m mode energies. In this paper it is proved that the spectrum of a quadratic Hamiltonian with a linear part presents the same feature. Simple methods are given for the calculation of the mode energies and of the ground-state energy of the Hamiltonian. It is shown that, in contrast to the boson case, the diagonalisation cannot be carried out by a linear transformation of the fermion construction operators. Application of such a transformation can only result in a diagonalisation after the introduction of a 'ghost' particle (fermion) and a corresponding pair of Hermitian-conjugate construction operators.

For pt.I see ibid., vol.12 (1979). The quantum mechanical Coulomb and isotropic oscillator problems in an N-dimensional spherical geometry, which were shown in the previous paper to possess the dynamical symmetry groups SO(N+1) and SU(N) respectively as classical systems, are analysed by the method used by Pauli to find the energy eigenvalues of the hydrogen atom. This analysis is carried through completely for N=3 to obtain energy eigenvalues and recurrence relations among energy eigenfunctions. It is shown that Pauli's method is equivalent to Schrodinger's method of solving the radial Schrodinger equation by factorisation of the second order differential operator. The latter method is used to find the energy eigenvalues in N dimensions, and the corresponding eigenfunctions are obtained in closed form.

Conservation laws are derived using Synge's method in third approximation for a general continuous medium. These laws, together with Synge's equations of motion, are then applied to the case of a perfect fluid and a comparison is made with the results of Chandrasekhar and co-workers.

Using dimensional regularisation, a prescription is given for obtaining a finite renormalised stress tensor in curved space-time. Renormalisation is carried out by renormalising coupling constants in the n-dimensional Einstein equation generalised to include tensors which are fourth order in derivatives of the metric. Except for the special case of a massless conformal field in a conformally flat space-time, this procedure is not unique. There exists an infinite one-parameter family of renormalisation ansatze differing from each other in the finite renormalisation that takes place. Nevertheless, the renormalised stress tensor for a conformally invariant field theory acquires a nonzero trace which is independent of the renormalisation ansatz used and which has a value in agreement with that obtained by other methods. A comparison is made with some earlier work using dimensional regularisation which is shown to be in error.

The thermal flux emitted by a de Sitter due to interaction with massless fields with spin is calculated by examining the field equations of these fields. The field equations are obtained by perturbing the metric with the various fields and obtaining linearised equations. The expression for the power radiated indicates the absence of spin-half particle emission.

The authors derive the equation of evolution in the configuration space for a system of semiclassical bosons or fermions starting from the recently derived nonlinear Kramers-Chandrasekhar equation for such particles. The latter equation does not contract to the corresponding Fokker-Planck equation in the velocity space; however, it is shown that in the context of a Chapman-Enskog approximation the contraction to the physical space leads to the diffusion equation with the classical relationship between the self-diffusion coefficient and the friction coefficient maintained.

A rigorous inequality between the pair correlation function and connectivity functions is proved for the Ising model (correlated percolation). This relation shows that large correlations imply large connectivity. Such inequality becomes equality in the random percolation problem (infinite temperature). Other relations among susceptibility cluster size and perimeter are also derived which give information on the shape of the cluster for the random and correlated percolation problems.

The photon density in diluted black-body radiation is epsilon (0< epsilon <1) times that for the black-body radiation at temperature T from which it originated. If sigma is Stefan's constant and B is a geometrical factor, it is shown that the energy and entropy flux due to such radiation is Phi =B epsilon sigma T⁴/ pi Psi =⁴/₃B epsilon X( epsilon ) sigma T³/ pi (X(1)=1) where X( epsilon ) is a function calculated here for the first time. A special type of steady-state non-equilibrium situation is defined, and called effective equilibrium, for which the effective temperatures T/X( epsilon ) identical to T* of the various components of a system are equal. In this state the system cannot yield work. The maximum efficiency eta ₀ of such systems is investigated. The application to solar radiation (diffuse and direct) proves possible and involves the function lambda (x)=1-⁴/₃x+¹/₃x⁴. In order to allow for diffuse and direct radiation the calculation is somewhat more complicated than previous ones. It shows that, for a black absorber, eta ₀ approximately 0.7 (diffuse) rises to 0.93 as the radiation becomes more direct. However, for a grey absorber the efficiency might range typically from 60% to 83% for absorptivity alpha =0.9. For one pump p and a black absorber at ambient temperature T, eta ₀= lambda (T/T_p^*).

The replica method for random systems is critically examined, with particular emphasis on its application to the Sherrington-Kirkpatrick solution of a 'solvable' spin glass model. The procedure is improved and extended in several ways, including the avoidance of steepest descents and a reformulation which isolates the thermodynamic limit N to infinity . Ideas of analyticity and convexity are employed to investigate the two most dubious steps in the replica method: the extension from an integer number (n) of replicas to real n in the limit n to 0, and the reversal of the limits in n and N. The latter step is proved valid for the Sherrington-Kirkpatrick problem, while the non-uniqueness of the former is held responsible for the unphysical behaviour of the result.

Measurements of rotational diffusion coefficients by depolarised light scattering spectrometry are often disturbed by double-scattering effects. These effects may be reduced by minimising the concentration of scatterers and by using an appropriate geometry. Such an optimum geometry is proposed. Explicit expressions for the contribution of double scattering to the field correlation function are computed for the optimum geometry. The correlation functions evaluated numerically are compared with experimental data in the case of polystyrene latex spheres. The agreement is excellent. The expressions derived can be used to correct correlation functions obtained experimentally in case double scattering cannot be avoided.

This paper studies the limiting accuracy with which power spectra and the values of spectral parameters for random signals can be achieved from short batches of data. Initially, general expressions for arbitrary data are derived; these are then restricted for simplicity to wide-band signals. In addition, comparison is made with computer simulation for two well defined models, namely, photodetection of light of constant intensity and heterodyne photodetection of narrow-band Gaussian-Lorentzian light. It is shown that there is no analytical difference between operation in time or frequency space for batch data, and also that long data sets can be analysed at least as well, in terms of the accuracy with which spectral parameters can be determined by fitting, by averaging over many short batches as by processing the set as a whole.

An exact multi-soliton of the Benjamin-Ono equation is presented. The asymptotic form of the solution for large time is also given.

A generation theorem for solutions of Einstein's equations is presented. It consists mainly of algebraic steps. With its aid, one obtained from an 'old' solution (e.g. from the Minkowski space) 'new' solutions with an arbitrary number of constants. The method of repeated application of potential and coordinate transformations considered by Geroch (1972) and Kinnerley (1977) is included.

The authors take issue with some of the comments by Ashworth, Davies and Jennison (1978) on their paper (McFarlane and McGill 1978) on light ray and particle paths on a rotating disc.

The authors establish necessary and sufficient group-theoretical criteria for the existence of limit cycles in differentiable flows defined by sets of N autonomous first-order ordinary differential equations. An example is given.