Table of contents

Attention is drawn to some special relations between 6-j and 9-j symbols. It is shown that one of the relations derived finds its application in the nuclear physics field.

The method of analytical continuation using the Pade approximant technique in the total and partial coupling constants is described and illustrated with some examples. The method can be applied to calculate the characteristics of the loose near-threshold and few-body resonance states in the case of the potential with a strong repulsive core etc.

Under the vector addition of angular momenta, each equal to s, the resulting angular momentaj occur with some multiplicity. The general formula for these multiplicities is obtained by a simple algebraic method. In the case of s=1/2 the expression proves to be a well known one. Also given are some relations connected with the multiplicities.

A generalization of a previously (see ibid., vol.9, p.18-1 (1976)) derived expansion of a determinant in terms of determinants of progressively lower order is obtained and used to discuss the eigenvalue problem in the cases when two or more elements are degenerate. The development is exact for the case of a finite determinant, and represents a perturbation series for an infinite determinant. A procedure is given for writing down the expansion to any order without using detailed algebra. An application to inhomogeneous systems of linear equations is also briefly discussed. The continued fraction methods are more straightforward to apply (particularly for the higher orders) and are more rapidly convergent than standard perturbation theory.

Current theories of the origin of inertia aer reviewed. An alternative treatment is given. This recognizes the physical extent of even the smallest masses and the interaction effects of physical measurement. It is shown that there are very good physical reasons for identifying the origin of inertia with the local system and that this simultaneously accounts for quantization and the units of length and time. Two experiments are described which illustrate some of the properties under discussion.

Attention is drawn to difficulties associated with gravitational theories based on Lagrangians formed from quadratic invariants of the Riemann tensor, and in particular with the Palatini variational method used by Yang in a gauge theory of gravitation. It is pointed out that the Yang theory is not mathematically well founded.

It is shown that an analogue of Birkhoff's theorem of general relativity exists for electromagnetic fields in a scalar-tensor theory of gravitation proposed by Sen and Dunn (1971) when the scalar field introduced in the theory is independent of time.

Linear density perturbation equation for general relativistic cosmologies using the homogeneous isotropic Robertson-Walker metric and the anisotropic Bianchi-I metric for the unperturbed model has been derived in a simple and rather general manner. The method has then been applied to consideration of the fate of density perturbation in the Brans-Dicke model of the universe. Two coupled differential equations connecting variations in the density rho and the scalar field phi of the model have been obtained. An approximate solution has been sought and a power law growth of the fluctuation density, delta rho , with time has been obtained.

Exact numerical data on self-avoiding walks are presented for the irregular network studied by Finney (1970) in connection with the Bernal model of a liquid and for a face-centred cubic lattice with randomly removed bonds. Averages for the total number of walks C_n, and polygonal closures U_n, are defined and found to be consistent with C_n approximately mu ⁿn^g, U_n approximately mu ⁿn^h where the critical parameters g and h have the same values as those obtained from the regular lattices. The mean-square end-to-end distances (R_n²) are also studied and the critical parameter theta is found to have the usually accepted value of 6/5. The results add further support to the conjecture that these exponents depend only on the dimensionality and are not affected by the irregularity of the network.

Rigorous inequalities are proved, which relate percolation probability, mean cluster size and pair connectedness respectively with magnetization, susceptibility and pair correlation function in ferromagnetic Ising models. In two dimensions the critical point is shown to be a percolation point, while in three dimensions this is not true.

A general formalism is developed to obtain series expansions of the average number of physical clusters of particles in the framework of Mayer's theory. The special case of lattice systems is investigated in more detail and some preliminary results are given on the relation between percolation (namely the formation of an infinite cluster) and condensation in fluid systems in the lowest approximation (summation of chain diagrams).

A duality theorem is proved which establishes the property of self-duality in the thermodynamic limit to be a natural property of a large class of pure three-body Ising models. A new form of low temperature expansion is obtained for triplet Ising models in zero field, which are closed polygon expansions and are highly lattice dependent. These closed polygon expansions readily establish an equivalence between pure triplet models and corresponding vertex models, and it is seen that a number of 'ice rules' are equivalent to triplet models at the dual point.

Upper and lower bounds are presented for the rate of energy loss by a high intensity Gaussian pulse of finite length propagating in a cold dense homogeneous plasma. For sufficiently high plasma conductivity these bounds coalesce yielding an explicit expression for the dissipated power.

The mathematics of reflector design under the geometrical optics approximation is treated using complex coordinates. Conformal transformations occur naturally in the new formalism and their application to reflector design appears to have great potential. Some detailed illustrations are included.

Hegerfeldt's (1974) concept of T-positivity in Euclidean random fields is generalised to non-commutative probability theory, that is, to Euclidean Fermi fields and to current algebra with possible Schwinger terms. The axioms imply the Wightman axioms. A non-Abelian form of Markovicity is introduced, and is shown to imply T-positivity if a reflection property holds. The investigation suggests a generalization of Nelson-Symanzik positivity, which might be valid in cases when the extension of the Schwinger functions to coinciding arguments is not expected to maintain both commutativity and positivity (or anti-commutativity and positivity).

The Rarita-Schwinger theory of a massless spin-³/₂ field belonging to the mixed spinor representations (1,¹/₂) and (¹/₂,1) and the unmixed spinor representations (¹/₂,0) and (0,¹/₂) is explored. The motion is indeterminate to within a group of gauge transformations. It is found that the gauge invariants transform according to the unmixed spinor representations (³/₂,0) and (0,³/₂). The gauge invariants are quantized by a new non-canonical coordinate-covariant Lagrangian procedure, and the anticommutators are found to be positive. It is shown that the energy-momentum tensor is irremediably gauge variant, thus ruling out any possibility of gravitational interaction. It is found that electromagnetic interaction is also quite impossible to achieve.

Considers aspects of the non-relativistic quantum theory of radiation that occur when the coordinates and momenta of all material particles (nuclei as well as electrons) are treated as dynamical variables. The formalism gives the complete scheme of atomic field equations for an ensemble of ions, free electrons, atoms or molecules and comprises the multipolar Hamiltonian for systems composed solely of electrically neutral atoms or molecules. It also deals with effects due specifically to motion of the nuclei, such as the Doppler effect or the recoil that occurs when photons are emitted, scattered or absorbed.

The adsorption of a charged macromolecule to a charged surface under the influence of a weak electrostatic attraction is investigated. The probability density P(x,N) to find the end point of a molecule with N monomer units at distance x from the surface is calculated analytically. In the limit N to infinity this function shows a phase transition at a critical value ( theta _c) of the adsorption energy in units if k_BT: for theta > theta _cP(x,N) is peaked near the surface and the molecule is effectively adsorbed to the surface; for theta < theta _c the heat motion is strong enough to remove the molecule from the surface and P(x,N) is effectively constant throughout the whole available volume. A possible role of such conformational phase transitions in the regulation of cellular metabolism is briefly commented on.

The change of a polymer from an extended coil form above the theta point to a dense globular form below the theta point is investigated in a self-consistent field approximation and found to be a second-order phase transition in the limit of an infinitely long polymer. Other theories of the transition are reviewed and further information obtained as to the dimensionality dependence of the transition by means of the analogy between the theta point and a tricritical point.

A Hamiltonian which shows two successive phase transitions on cooling is studied by the 1/n method correct to order 1/n². Despite the presence of Goldstone modes in the phase which orders first, the second phase transition is found to have 'ordinary' critical exponents determined solely by the components of the order parameter that go soft at the second transition.

The random-cluster formulation of the Potts model connects a number of lattice models and reveals a disagreement between series expansions published for special cases. The corrected series are given.