Table of contents

Using arguments by E. Krahn (1925), it is shown that the n-dimensional sphere possesses the following minimum property: among all connected regions of the given volume it yields the smallest ground eigenvalue of the Laplace operator with Dirichlet's boundary condition. This result is important in establishing an adiabatic invariant for quantum mechanical systems.

By introducing the BRS transformation and Lagrangian multiplier fields, the covariant canonical quantisation is carried out for an antisymmetric tensor gauge field of third rank. The non-propagating character of the field is assured from the viewpoint of the 'quartet mechanism' of T. Kugo and I. Ojima (1979). The correct number of ghosts is counted from the effective Lagrangian.

A model is constructed for a space-time geometry that contains Einstein and Stephenson-Yang gravity together with a gravitationally charged vector field in the adjoint representation of the structure group. The model is formulated as an SL(2C) gauge theory for a metric connection and a field of orthonormal frames. A simple ansatz renders a solution to the completely coupled system which, it is argued, has the characteristic of a gravitational monopole.

Using functional methods and the automorphic formalism, it is shown that a field theory on a multiply connected background is renormalisable provided the corresponding theory on the universal covering space is renormalisable.

An exact solution of Einstein's equations that corresponds to an axisymmetric bundle of electromagnetic waves with finite cross section is presented. Outside this bundle, there is gravitational radiation parallel to the electromagnetic radiation. If no static electromagnetic fields are present, the frequency of the gravitational waves is twice the frequency of the electromagnetic waves. Einstein's energy complex vanishes identically. The covariant energy complex however yields a radial momentum.

Making use of the Muller theory of irreversible processes, some solutions of the momentum and internal angular momentum balance equations of a fluid of polyatomic molecules are studied. When one considers only dissipative phenomena described by the rotational viscosity, it is seen that even in the case in which the spin and the vorticity coincide normal mode solutions exist for the velocity and spin.

Correlation functions are calculated on the critical line of the two-dimensional planar or XY model. A crossover is found between simple scaling behaviour for 2 pi K_eff>4, and more complex scaling behaviour for 2 pi K_eff=4. At the Kosterlitz-Thouless critical point, 2 pi K_eff=4, logarithmic factors appear in the correlation functions. These corrections to simple scaling behaviour are calculated for many correlation functions, including correlation functions not previously studied.

Recent series expansion estimates of the lattice Ising model suggest strongly that at fixed temperature below T_c the perturbation expansion of the magnetisation as a power series in the external field is asymptotic. The published results are in excellent agreement with the universal features of the essential singularity predicted by field theory calculations using instantons. Further tests of the latter are discussed.

Proposes a generalisation of the method of group analysis of differential equations to the symmetry groups more general than Lie-Backlund groups, in particular, to the non-local groups of transformations.

The hermiticity and the time reversal behaviour of a tensor operator each impose restrictions on the reduced matrix elements of the operator. These restrictions are investigated for one-particle operators. The relationships between the irreducible components of an operator and those of its time reversal or Hermitian conjugate are clarified. Selection rules are determined for the non-vanishing of certain matrix elements using both time reversal symmetry and the conflicting symmetries method of B.R. Judd (1971). This generalises and corrects previous analyses which have had useful applications in solid state physics.

A constructive proof is presented to show that, for any permutation P on spin space, P+P^-1 can be expressed as a linear combination of spin-operator products Pi s_m.s_n with all the particle indices m, n in a product distinct.

The conjecture that Tr(exp(A- lambda B)) can be written as a Laplace transform with positive measure rho is considered for finite Hermitian matrices A and B by means of Bernstein's theorem. An explicit formula is given for the moments of rho in terms of divided differences of exp(A) and elements of B. For a large class of matrices A and B the moments of rho take their maximum and minimum values when A and B commute and so upper and lower bounds for the moments of rho are established; further analysis suggests that this is generally true if B is positive definite and A and B are bounded. Some inequalities for the divided differences of the exponential are derived. Also, if A and B are both positive definite, upper and lower bounds are derived for Tr(AⁿBⁿ) and Tr(AB)ⁿ in terms of the eigenvalues of A and B. Applications to problems of statistical mechanics and possibly Euclidean field theory are mentioned.

Analytic continued fraction theory is applied to study the convergence and analyticity of the infinite continued fraction representation of the Green function for a class of confinement potentials in terms of the Coulomb-like coupling constant. The possibility of a perturbative expansion in the powers (inverse powers) of the coupling constant is also investigated.

The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.

When the polarisations of two photons are correlated, the result of a measurement of a polarisation component of one of the photons is not independent of the choice of the component measured on the second photon (even if the two measuring instruments are very distant). The authors compute the probability of a change in the readings of the first instrument, associated with a given change of orientation of the second instrument. Under optimal conditions, they show that in at least 41% of cases ( square root 2-1) the readings of one of the instruments cannot be independent of a reorientation of the other one.

A set of coherent states of the semi-direct produce N(2)(X)Sp(4,R) of the Heisenberg-Wyl and symplectic groups is constructed; they are shown to be non-spreading two-dimensional wavepackets moving on closed classical trajectories. These states generalise in a natural manner the Glauber coherent states to the description, in the classical limit, of a particle in a rotating frame as well as of a charged particle in a uniform magnetic field. The properties of these states are studied.

For pt.I see ibid., vol.12, p.2415 (1979). The problem of the electromagnetic vacuum state definition in a part of Minkowski space bounded by a single spherical mirror or two concentric spherical mirrors which expand with a uniform acceleration is considered. The causal Green functions and the vacuum stress-energy tensor for these systems are obtained.

Gives a detailed description of the null trajectories in the Kerr-Newman metric. Interesting results which were not expected from what was known in the uncharged case are a considerable enhancement of the energy storage in a restricted region around the source in the naked singularity case and the singularity is completely disconnected from the asymptotic region in the sense that no photon can reach it even in the equatorial plane.

The force acting on the matter inside an arbitrary closed surface S is considered for the case of a static space-time g_{alpha beta} with a static matter tensor T^{alpha beta} satisfying T^{alpha beta} _{; beta} =0. Since the matter is in equilibrium, the body force is balanced by the surface force. The component of this force in any chosen direction is defined as the surface integral of an invariant and is transformed into a volume integral. Shrinking S to a point then yields the density of the body force. The author applies this result to special situations. For instance, it is found that in the presence of shear stress the body force can have a non-zero component perpendicular to the gradient of g₄₄.

The quantum statistical mechanics of an ideal relativistic Bose gas of massive particles is discussed. The thermodynamic functions of the system are obtained from the exact expression for the logarithm of the grand partition function. The correct procedure for carrying out the non-relativistic and ultra-relativistic limits is presented. The limitations and errors of previous calculations which relied on the binomial expansion of the density of states are pointed out. It is shown that the behaviour of the system in the limit of zero temperature is dominated by the non-relativistic effects.

Onsager's solution of the two-dimensional Ising model provides expressions for the spontaneous magnetisation and pair correlation functions. More detailed information, which provides insight into the clustering or ordering process, is obtained by expressing these correlations functions in terms of group probabilities. Each group consists of a central atom and its four nearest neighbours. Sum-rule equations for the twelve basic groups are solved to provide expressions for the group probabilities and the behaviour of the group probabilities as a function of temperature is discussed.

Generalises the author's result (see ibid., vol.12, p.L141-6, 1979) which showed that conventional equations for entropy flux and mass flux as coupled linear functions of their conjugate forces necessarily imply the existence of an uncoupled set of fluxes and forces as well. An alternative set of flux force conjugates in which each flux is collinear with its conjugate force is derived. The author's concept of disequilibrium (see Phys. Lett. A, vol.63, p.73-5, 1977) is rigorously developed to give a phenomenological model of the steady state. The theory is consistent with near-equilibrium theory but is capable of wider application. Its scope includes thermoelectric effects, flow in porous media, and biophysical as well as engineering processes. Continuity of the analysis over the whole range from reversible equilibrium to highly irreversible nonequilibrium steady states is important.

sigma models with linear realisations of U(N) and O(N) symmetries are studied without imposing a constraint on the modulus of the field vector. Exact solutions in four-dimensional Minkowski space are presented, which have the form of plane/spherical waves. The singularities of the solutions as well as those of the Lagrangian density and the energy-momentum tensor are discussed. All results hold under the assumption that space-time and internal symmetry of freedom are not mixed.

The S-matrix problem is studied for the radial potential V(r)=-( mu + lambda /r)exp(-r). The Schrodinger equation is Laplace transformed. Then results on the asymptotics of the Laplace transformation are used to express the S-matrix element S_l as a linear combination of the values of the transformed function and of its first l+1 derivatives at one point in the complex plane. By a change of variables, the evaluation of S_l and of the phase shift sigma _l is then reduced to the numerical solution of a non-singular neutral functional differential equation (on a finite interval) followed by a finite recursive procedure. A convergent numerical scheme is implemented.

A brief review of the classical aspects of the absorber theory of radiation is presented. The divergences which arise from the use of time-symmetric electrodynamics are pointed out. It is shown that the earlier difficulties can be removed by attributing differing signal velocities to be advanced and retarded interactions. This difference in signal velocities is interpreted as arising from the extended, shell-like structure of charged particles. This leads to a new calculation of the absorber response. Absorption due to the time-symmetry normalisation factor is described. It is concluded that retarded radiation is approximately consistent in the Einstein-de Sitter model, whereas in the closed Friedmann model it is likely that retarded radiation is dominant during expansion, and advanced radiation during contraction. The theory predicts that advanced radiation exists in small amounts and can be detected experimentally.

The correction to A.N. Kolmogorov's theory (1941) is estimated using the theory of stochastic differential equations. Results are in good agreement with experimental data.

A rigorous approach by E.M. Barston (1977) to stability of Lagrangian systems is used to establish both rectangle and semicircle theorems for plane parallel flow along a horizontal but otherwise arbitrary magnetic field, permeating a perfectly electrically conducting incompressible fluid under gravity. The radius of the semicircle is reduced by magnetic effects and stable stratification. A Richardson criterion for stability against constant shear flow is also derived. The analogous problem for a compressible fluid is also discussed, and for a certain class of disturbances a 'semi-dumbbell' theorem is established which is considerably stronger than the semicircle theorem. Possible astrophysical applications are discussed.

The lower-bound approximation for the real-space renormalisation group can be formulated for an n-component generalised Gaussian model in d dimensions. This approximation is studied with the aid of a 1/n expansion, which is applied to the approximate recursion equations. Critical behaviour, including the index v, thus calculated is compared with the exactly known 1/n expansion.