Table of contents

The authors study the ( phi ⁴)₂ theory with negative mass term and consider small variations to extremals (Morse theory), identifying the Jacobi fields (zero models). The associated topological information is recovered, interpreting the vacuum and the kink as sections in the trivial and Hopf real line bundles respectively; they consider also duality, PS limit, spinor character and statistics. They conclude with the relevance of the two other Hopf line bundles (complex and quaternionic) for vortices and instantons, respectively.

The existence, number and type of the constants of motion in non-relativistic mechanics are examined for different set-ups of Newton's equations in configuration space, the Noetherian symmetries in the Lagrangian formulation, the Hamiltonian formulation and the Schrodinger equation in quantum theory and are found to be equivalent and exhaustive, as already known. Time-dependent constants are shown to be arbitrary, but nevertheless amenable to the general symmetry methods of Katzin, Levine (1977) and Mariwalla (1979).

Deals with the following type of question. Suppose V₀(x) is so singular near 0 that the quadratic form of -d²/dx²+V₀(x) is unbounded from below, but that (-d²/dx²)_D+V₀(x) is bounded below where D refers to Dirichlet boundary conditions (BCS) at 0. Let H(a)=-d²/dx²+V_a(x) where V_a(x) to V₀(x) pointwise (AE) as a(spin down)0. Under some additional assumptions on V_a, the author proves that H(a) to H_D(0) in the norm resolvent sense. Typical for applications is the case where V_a are cut-off potentials or regularised potentials of some sort.

An approximate implicit energy level expression is obtained for a spherically symmetric anharmonic oscillator with three degrees of freedom, by using a semi-classical, WKB type Bohr-Sommerfeld quantisation rule.

The planar Ising model in an external magnetic field and with multi-spin interactions is shown to be equivalent to a vertex problem. The equivalence can be extended to higher spins and higher dimensions.

Several families of two-dimensional lattices are discussed for which the critical percolation density (p_c) can be calculated exactly. The lattices are obtained by decorating lattices for which the value of the critical density for bond percolation is already known. It is shown that finite decorations of this type do not change the value of a critical exponent. By a sequence of decorations and lattice transformations the authors obtain a set of lattices of increasing coordination number q, and calculate the limit qp_c for q to infinity for this family. The value of this limit is close to the numerical estimate of the corresponding critical density for continuum percolation.

A general five-state model, which contains the five-state Potts model and a solid-on-solid model as special cases, is studied. The authors find that the high-temperature paramagnetic and the low-temperature ordered phases are separated either by a line of first-order transitions or by an intermediate phase with algebraic decay of correlations. The phase diagram is proposed on the basis of general considerations and Monte-Carlo simulations.

The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be y_T=1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (y_T=1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of ³/₂, the difference is nevertheless substantial.

A real-space renormalisation group (RG) approach for polymers is presented and is used to calculate the exponent v for the radius of gyration of three types of molecules on a square lattice. In all three cases the excluded volume effect is taken into account. The three cases studied are linear polymers, randomly branched polymers without loops and randomly branched polymers in which branch end-points are allowed to join either to other branch ends or to polyfunctional units, thereby forming loops. The author refers to the latter two configurations as branched polymers and branched polymers with loops, respectively. The author finds that v decreases from its linear chain value as the concentration of polyfunctional units is increased and branched polymers are formed. However, v is only slightly changed as loop fugacity is increased and branched polymers with loops are produced. Next, a new two-parameter RG is developed and it is found that linear polymers and branched polymers are described by two different fixed points, indicating that they belong to different universality classes. As the fugacity for loop formation is increased, the global flow diagram remains unchanged. This result supports the hypothesis that branched polymers with and without loops, in a good solvent, belong to the same universality class.

Electrical circuits at low temperatures, T<or approximately=(h(cross) omega /k_B), require a quantum electrodynamic treatment of voltage as an operator which creates and destroys photons of frequency omega . Here, the authors consider the physical interpretation of measuring quantum Nyquist voltage noise using an amplifier with a power gain G( omega ) and Schwinger noise temperature T*( omega )<<T.

The authors show that energy dissipation is Reynolds number independent in three-dimensional turbulent flows. They obtain the small-scale Kolmogorov spectrum without any particular assumption on energy dissipation, using an approach developed in a previous paper.

An explicit expression for a massless scalar field of a point charge resting near a static black hole is obtained, and the properties of this solution are described.

Starting with an analysis of the general structure of the matrix beta ⁰ entering in first-order relativistic wave equations, the authors show that the degree of the minimal equation of beta ⁰ is determined by the size and nature of the various spin blocks of the 'skeleton matrix' of the theory. Since it is the numbers of Lorentz irreducible representations contributing to particular spins which determine the sizes of the spin blocks (and the value j_m of the maximum spin contained in psi has no direct bearing on these), the reason for the failure of the Umezawa-Visconti rule and its extrapolation by Chandrasekaran et al. (1972) becomes clear. The authors obtain some general results concerning the minimal degree in certain types of theories and on certain procedures whereby the minimal degree can be raised without altering j_m, and finally analyse a few interesting examples.

For pt.I see ibid., vol.11, p.1009 (1978). Considers a special 2n*2n matrix 1-x for which the determinant is the square of a polynomial in the x_ij. A graph G with n branches is associated to the matrix and det(1-x) and (1-x)_ij^-1 are expressed in terms of sums over subgraphs of G. The generating functions of the representation matrices D_MM'^(m)(u) of the representations (m)=(a_nb_n0...) of SU(n) are expressed in terms of the determinant and inverse of such a 2n*2n matrix.

The construction of symmetric states of N particles with angular momentum j=1 is presented. Elliott's result (1958), dealing with the angular momentum content of the symmetric subset of states belonging to the set of states of N particles with angular momentum j=1 is obtained using Racah algebra for the angular momentum coupling. The explicit functional dependence, on the total angular momentum, of the coefficients of fractional parentage (CFP) associated with one, two, three and four removed particles is found for arbitrary values of N. This dependence is expressed through the simple algebraic functions of J and N.

The SO₃-SO₂-3jm's (the '3j symbols' of angular momentum) are known to incorporate no information about the orientation of axes and may be calculated from a knowledge of character theory alone, together with some freedom of choice of phases. For many point-group embeddings, at least once in every finite-group basis of SO₃, an extra phase choice arises in the 3jm calculation. The authors show that choosing this phase corresponds to choosing the orientation of the symmetry axes of the groups.

A rigorous dimensional analysis of Bogoliubov subtracted-out type Feynman integrands is carried out, giving sufficient conditions for determining the asymptotic behaviour of the corresponding integrals in the ultraviolet region in Euclidean space.

Stokes and anti-Stokes lines are familiar in the asymptotic approximation of functions of a complex variable. The author generalises this notion and define the Stokes and anti-Stokes sets of a complex function of many (possibly complex) variables, defined by a diffraction-type integral. They are subsets of the Maxwell set of catastrophe theory, extended to complex variables. On the Stokes set the number of complex stationary points contributing to the integral changes by one, whereas on the caustic the number of real stationary points changes by two. Knowledge of the location of the Stokes set is essential to perform a full stationary phase analysis of a diffraction integral, and it imposes a constraint upon the positions of wavefront dislocations. For the canonical cusp diffraction catastrophe the author finds the explicit equation of the Stokes set, which is a broadened mirror image of the cusp caustic.

The eigenfunction method introduced by Van Kampen (1977) to solve a Fokker-Planck equation is extended and applied to a related partial differential equation. From this analysis we are able to obtain the solution of the Burgers and the damped Burgers equation in a systematic way.

In the frame of the inverse scattering problem at fixed energy, the author obtains a class of spherically symmetric potentials which are transparent in the Born approximation: all their Born phase-shifts vanish. This class consists of oscillating functions of the reduced radial variable. Amongst these functions, the Born approximation of the transparent potential of the Newton-Sabatier method is found. In the same class, the author exhibits potentials for which the Born phase-shifts vanish at and after a certain L-wave (quasi-transparent potentials). Very general features of potentials transparent in the Born approximation are then stated, and bounds are given for the exact scattering amplitudes corresponding to most of the potentials previously exhibited. These bounds, obtained at fixed energy and for large values of the angular momentum, are found to be independent of the energy.

The spectrum of the Mossbauer gamma-radiation, passed through a resonant absorber vibrated at ultrasonic frequencies, is studied in a new approach. An analytical expression is derived for the line profile, which consists of an infinite series of equidistant satellites. Their intensities depend on both the vibration amplitude and the absorber thickness. Certain ideas appearing in the investigations so far are critically discussed. Most of the phenomena considered can be explained by the classical waves dispersion theory.

Resonant/non-resonant amplitude modulation of gamma radiation occurs when it passes through a medium whose resonant/non-resonant dispersive properties are varied periodically with time. These phenomena are described from a unified point of view. Results are understood in the framework of the classical electromagnetic theory.

It is shown that Maxwell's equations can be obtained as the zero-mass limit of the Duffin-Kemmer-Petiau equation for the massive spin-one particle with appropriate identification of the components of the DKP spinor with electromagnetic field strengths and potentials. The electromagnetic Lagrangian also follows from the DKP Lagrangian in the same limit.

It is shown that Fermi's golden rule is valid for any initial state having a very narrow energy spread even if it is not an eigenstate of the free Hamiltonian. As an example, the author computes explicitly the tunnelling rate through a rectangular barrier. This method may have application in the Stark effect, autoionisation, fission, and similar problems.

Two new sets of asymptotically flat and functionally non-related electrovac solutions representing the external field of an isolated mass carrying electric charge, dipole and higher multipole moments are presented and a stationary vacuum metric is generated.

The curvature collineation equations have been solved for the two families of Petrov type-N plane-fronted gravitational wave solutions of Einstein's vacuum field equations in general relativity. Both of these solutions always have non-trivial curvature collineations, i.e. vector fields xi with respect to which the components R^mu _{nu alpha beta} of the Riemann tensor for those solutions are Lie derivable.

The second- and fourth-order scalar invariants on curved manifolds with torsion are tabulated. Relations among these invariants arising from the Bianchi, cyclic and Ricci identities are presented. The number of invariants is seen to be greatly reduced when the torsion tensor is taken to be totally anti-symmetric. The large number of fourth-order invariants indicates that, in general, the expressions normally obtained during the quantisation of a gravitational theory including torsion will be extremely cumbersome. There are certain exceptions to this statement, however. The most famous example of this is supergravity theory.

Imposing an approximate Poincare invariance on the metric tensor in the post-Newtonian scheme of general relativity the author finds a recursive method which gives restrictions to the gauge conditions at each order of approximation. Those restrictions are explicitly calculated up to the second post-Newtonian approximation (PNA) for a bounded perfect fluid. At the first PNA the harmonic gauge is chosen, which satisfies the imposed restrictions, but at the second PNA the harmonic gauge does not give a Poincare-invariant metric tensor and a modified harmonic gauge is chosen instead. Using these gauges, the gravitational interaction can be described by approximate Poincare-invariant equations of motion in the framework of predictive relativistic mechanics.

A necessary and sufficient condition for flatness of the quasi-spherical space-times of Szekeres (1975) is given. Using this, the linear approximation of these space-times is considered and it is shown that the third derivatives of the quadrupole moments vanish for matter distribution contained within a comoving surface. Hence the Einstein formula gives zero energy loss due to gravitational radiation. This agrees with other investigations, which have shown that no radiation is present in these space-times.

The authors present Monte Carlo and series results for the mean valence ( nu )_F of non-percolating clusters on the square lattice and corresponding series results for the simple cubic lattice. Mimic functions have been constructed from the series which, for the square lattice, agree well with the Monte Carlo data, for a wide range of densities. They have shown that if the percolation probability is continuous at rho _c then ( nu )_F is continuous at rho _c, and have presented a heuristic argument which indicates that ( nu )₁, the mean valence of a site in a percolating cluster, is greater than ( nu )_F, at rho _c. This allows one to infer the asymptotic behaviour of ( nu )_F in the region of rho _c.

For the two mutually reciprocal, unit, Bravais lattices ( tau ) and ( gamma ) in an m-dimensional Euclidean space, the author presents exact results for (i) a class of lattice sums J_tau (a, k, m)= Sigma '_tau exp(-a tau ²)/ tau ^2k for a>0 and k any real number and (ii) the class of complementary lattice sums S_gamma (a, l, j, m)= Sigma '_gamma gamma ^-2l( gamma ²+q²)^-1 for j>0, l any real number and (2l+2j)>m. These results may be useful in dealing with finite as well as infinite physical systems-in particular, the ones undergoing phase transitions. The asymptotic results following from the expressions are in qualitative agreement with those of Hall (1976), though quantitatively a slight discrepancy is noted in the case k<m/2 for J_tau (a, k, m) and in the case l<m/2 for S_gamma (q, l, j, m).

The renormalisation group equations for the q-state Potts model having the hierarchical interaction is investigated. A state fixed point is found to exist when q<3 to the leading order in Delta sigma = sigma -¹/₃, sigma being the interaction range parameter. Independent evidence suggest that there is no fixed point when q=3 for all sigma >¹/₃. Critical exponents associated with the new fixed point are also obtained to the first order in Delta sigma .

Asymptotic representations are obtained for the eigenfunctions of the differential equation describing scalar waves in a clad inhomogeneous planar waveguide, which has two turning points and finite boundaries. These representations are valid to all asymptotic orders in the large parameter, which is proportional to wavenumber. This is achieved, in contrast to augmented WKB theory, by finding transformations on the independent variable which map the eigenfunction exactly into known solutions of canonical differential equations. The resulting transformation equations are nonlinear but have tractable asymptotic properties.

For pt.I see ibid. vol.18, p.3057 (1980). The eigenvalue problem for the differential equation describing scalar waves in a clad inhomogeneous planar waveguide is solved using an asymptotic formulation derived in the preceding paper. The results of this earlier paper are summarised, and a method proposed for the extraction of the eigenvalue from the implicit eigenvalue equation. Although in general these calculations are conveniently carried out by numerical methods, approximate closed-form expressions can be obtained for the eigenvalue and these are given for all the asymptotic regimes of interest.

The spherical collapse of gas clouds is reduced to quadrature by specifying the mass function n(r, t). Exact solutions for both shearing and shear-free flow are constructed admitting two-parameter equations of state. A generalisation of Ritter's theorem is proved which allows the shear-free collapse of index three polytropes.

The possible phase transition to quark matter in a neutron star is considered using the neutron matter equation of state proposed recently by Canuto, Datta and Kalman (1978) which involves spin 2-meson interaction among nucleons. The corresponding equation of state for quark matter is taken from quantum chromodynamics. The Oppenheimer-Volkoff equation is then numerically solved to investigate the stability of a superdense system with quarkion cores.