Table of contents

The matrix formulation of Flaschka is used to study the integrability of nonlinear N-particle systems of exponential type (e.g. Toda lattices, electric circuits of ladder type, Volterra systems). The asymptotic, i.e., N to infinity , behaviour of the normalised eigenvalue moments ( mu _r'^(N); r=0,1,...,N) of the N-dimensional L matrices, which are constants of motion of the system, is investigated. Compact expressions of these quantities in terms of the asymptotic values of the dynamical variables (P_n, Q_n) of the Toda lattice are analytically obtained in a simple way. The applicability of these expressions is illustrated in the case (P_n to 0, Q_n-Q_n-1 to 0), which encompasses many of the motions of the Toda lattice considered in the literature.

Using a model mechanical free energy functional, the nucleation rate of superfluid ³He from the A phase to the B phase is discussed.

It is shown that, on the triangular lattice, series expansions for the Ising model and related problems in lattice statistics can be obtained by combining the contributions of finite hexagonal regions. This generalises earlier results for square lattice systems.

A Hamiltonian formalism is developed from a regular Lagrangian L^r depending on an arbitrary number of derivatives. The formalism leads to a set of Hamilton equations whose solutions are the same as those of the Euler-Lagrange equations derived from L^r. The Noether currents associated with a symmetry transformation of the Hamiltonian action are also derived.

The ground states of a quenched random Ising spin system with variable concentration of mixed nearest-neighbour exchange couplings +or-J on a square lattice (frustration model) are studied by a new method of graph theory. The search for ground states is mapped into the problem of perfect matching of minimum weight in the graph of frustrated plaquettes, a problem which can be solved by the algorithm of Edmonds. A pedestrian presentation of this elaborated algorithm is given with a discussion of the condition of validity.

One starts with a representation of a given abstract group G=H+sH (where H is its subgroup of index 2, and s is a coset representative) by a group of unitary and antiunitary operators in the state space of a quantum system. Co-representation theory maps further these operators into matrices, which are linear operators in the space Cⁿ of number columns, but it fails to preserve homomorphism. An equivalent theory in terms of unitary matrices and antimatrices (antilinear operators in Cⁿ) which is based on isomorphism with the mentioned group of operators is presented. The connection (subduction of the one side and *-induction on the other) between the set of all unitary irreducible matrix representations of H and the set of all unitary irreducible matrix-antimatrix representations of G are studied. Finally, a simple construction of the latter from the former is given.

All possible graded extensions of the Poincare Lie algebra are constructed. Massless P²=O representations of these algebras are investigated and it is found that the usual conditions on massless particle helicity states necessarily imply constraints on the spinorial generators. It is shown that these constraints are not consistent with the graded Lie algebra, except for the case of conventional supersymmetry. It follows that massless representations of such algebras are forbidden.

Fourier-transformed Fokker-Planck transport operators, T(k)=-(grad- nu . zeta nu .grad+ik. nu , have discrete spectra for most physically reasonable friction tensors zeta ( nu ). The set of eigenvalues ( lambda _alpha (k)) represents a collection of single and/or multiple-valued functions analytic in k, with no other singularities than branch points, which are also branch points for the eigen projections P_alpha (k). Corresponding analytic behaviour is found with the eigenfunctions phi _alpha (k, nu ), chosen in such a way that ( phi _alpha , phi _beta *)= delta _{alpha beta} and P_alpha =(., phi _alpha *) phi _alpha . Eigen nilpotents and generalised eigenfunctions (which are not proper eigenfunctions) only appear at branch points and possibly at some other meetings points of eigenvalues. Each eigenvalue remains real for sufficiently small real k, until it meets its first branch point. Within this region, the eigenfunctions are conveniently classified by three labels analogous to quantum numbers alpha =(n,l,m). A necessary condition for the meeting point of two eigenvalues lambda _nlm and lambda _n'l'm' to be a branch point is m=m'.

It is shown that even in the simplest case, when Yang-Mills equations with boundary conditions admit the Coulomb solution, there is a possibility for non-trivial, non-Abelian solutions. Bifurcation theory methods are used for finding these solutions.

Considers waves generated at t=0 by prescribing Cauchy data. It is shown that, under weak assumptions the waves propagate inside the forward light cone only if Z_j omega _j²(k) is a polynomial in k of degree 2.

Equivalent formulae for y_lm(r₁×r₂) are derived from boson and hyperspherical calculus. As a by-product, apparently new sum rules for 3jm symbols are obtained. Two passages from E₄ to E₃ are also discussed.

A simple and straightforward derivation of an expression recently obtained by Hage Hassan et al. (ibid., vol.13, p.2623, 1980) for y_lm(r₁×r₂) in terms of Y_l1m1( theta ₁, phi ₁) and Y_l2m2( theta ₂, phi ₂) is presented.

The Feynmann variational method is applied to a real functional integral representation of a generator for the velocity correlation functions in stationary homogeneous turbulence. The resulting approximation for the two-point correlation differs from that recently obtained by Lucke using a similar approach. It is pointed out that Lucke's approximation is based on a restricted trial function and does not correctly reproduce second-order perturbation theory in the limit of weak non-linearity. The approximation derived here does not suffer from this defect but is more complicated. Some defects of this general approach are discussed.

An iterative method which determines the low-lying eigenvalues of a Hermitian operator defined in a finite-dimensional vector space is extended to a specific type of unbounded Hermitian operator defined in a Hilbert space. As an illustrative numerical example the extended algorithm has been applied to the quantum mechanical harmonic oscillator problem.

The author describes two algorithms, one for the 'generation' of the elements of a certain class of subgroups of the symmetric group and the other for the 'generation' of the coset representatives of these subgroups in the symmetric group. Later they discuss the relevance of these algorithms in the enumeration of distinct and connected diagrams in many-body perturbation theory.

The authors discuss the high-energy behaviour of scattering phase shifts for a large class of spherically symmetric Coulomb-like potentials. Some results which have been known only for short-range potentials are extended to this larger class. In particular, by iterating appropriate Volterra integral equations they derive an asymptotic expression for the phase shifts, which is valid whenever the short-range part of the potential is integrable. When applying the results to the problem of the interference between Coulomb and short-range interactions they obtain an estimate for the high-energy behaviour of the Coulomb-interference effect in the phase shifts.

Mackey's scheme (1963) for the quantisation of classical momenta generating complete vector fields (complete momenta) is introduced, the differential operators corresponding to these momenta are introduced and discussed, and an isomorphism is shown to exist between the subclass of first-order self-adjoint differential operators, whose symmetric restrictions are essentially self-adjoint, and the complete classical momenta. Difficulties in the quantisation of incomplete momenta are discussed, and a critique given. Finally, in an attempt to relate the concept of completeness to measurability, concepts of classical and quantum global measurability are introduced, and shown to require completeness. These results afford strong physical insight into the nature of complete momenta, and lead us to suggest a quantisability condition based upon global measurability.

It is shown that Tr(J_lambda ^2p) can be developed from Tr(J_lambda ^2p-2) by means of recurrence relations. Starting from Tr(J_lambda ⁰)=Tr(I), traces of J_lambda ^2p with p<or=1 are obtained. It follows as a corollary that the Bernoulli polynomials B_n(x),n<or=2, can be generated by means of the recurrence relations developed herein.

Provides a formal derivation of a set of equations linking the S-matrix and density matrix methods. The formal manipulations involved permit a clear statement of the mathematical assumptions required in order to arrive at the associated set of time-irreversible equations. They should thus permit future investigations to put the whole theory on a more rigorous basis.

For electromagnetically as well as gravitationally bound quantum mechanical many-body systems the coefficients of absorption and induced emission of gravitational radiation are calculated in the first-order approximation. The results are extended subsequently to systems with arbitrary non-Coulomb-like two-particle interaction potentials; it is shown explicitly that in all cases the perturbation of the binding potentials of the bound systems by the incident gravitational wave field itself must be taken into account.

The authors investigate the topological effects of non simply connected space-time on supergravity with matter supermultiplets. Topological sectors are typically labelled by one or a pair of elements from the group of homomorphisms of pi ₁(M) into Z₂ and Lorentz invariant vacuum states are defined by analogy with the theta -vacua of Yang-Mills theory.

The lower and upper bounds including variation of the frequency are calculated for the partition function of oscillators of the quartic type. They are shown to be fair approximations to the partition function calculated from eigenvalues given by Hioe and Montroll (1975). The upper bound presents a very good high-temperature approximation which can be differentiated analytically giving also closed formulae for the thermodynamic functions H, E, S, C_p. Perturbation theory is discussed in terms of operator-free generating functions. As expected, it works well only for intermediate temperatures and for small anharmonicity. Using a technique by Fisher (1965) for convex functions, upper and lower bounds for the entropy are also calculated from the exact upper and lower bounds to the partition function.

The statistical mechanics of polymer molecules subject to topological constraints is formulated. In particular, the problem of two polymer loops topologically linked is studied in the limit when one of the loops is allowed to fill a macroscopic volume at a finite density. The authors show that this problem can be recast as a local gauge invariant field theory in the usual limit that the number of components goes to zero. The gauge invariance is a direct consequence of the fact that topological entanglements imposed on the system are conserved.

Starting from the high-temperature series for the susceptibility of the spin-¹/₂ ferromagnetic Ising models-square, planar triangular, SC, BCC, FCC, diamond, and HSC in four, five and six dimensions-the authors analyse the analytical structure near the critical point of the susceptibility by writing it as a Laplace transform in the variable log (1-w/w_c), where w=tanh( beta J). The interest of the method is that the coalescing singularities which sit at w=w_c are spread out and can be analysed separately, the first subdominant critical indices, appearing as stable poles of Pade approximants. They first recover the results for the two-dimensional models, with a high accuracy: gamma =1.74995 and gamma _s=0.757. The most stable results in three dimensions are obtained for the diamond lattice: gamma =1.2506+or-0.0015 and gamma _s=0.42+or-0.11. The other lattices give 0.10<_{gamma s}<0.52. However, in all three-dimensional cases the amplitude of the subdominant singularity is less than a few per cent of the leading one. Therefore the uncertainties are large, but the subdominant singularity stays closer to the value Gamma -1=0.25 than to the field theoretical predicted one gamma _s=0.75.

The Migdal-Kadanoff renormalisation group for two-dimensions is employed to obtain the global phase diagram for the site-bond correlated percolation problem. It is found that the Ising critical point (K=K_c,H=O) is a percolation point for a range of bond probability rho _B such that 1>or= rho _B>or=1-e^-2Kc. In particular, as rho _B approaches 1-e^-2Kc, the percolation clusters become less compact and coincide with the Ising critical droplets.

In a general formulation of hard-core lattice gas models the hard-square and hard-hexagon models are expressed in terms of vertex models. It is shown that the hard-square gas is a 16-vertex model on the square lattice, and in one of its representations is equivalent to a lattice ramrod model first considered by Nagle (1968). A lattice transformation of the hard-hexagon model generates a special case of the 64-vertex model on the triangular lattice. In these two reductions clear differences between these two hard-core models emerge, and it is probably not surprising that they exhibit quite different critical behaviour. It is also shown that the block-site scaling transformation can be used to provide an alternative computational scheme for obtaining accurate numerical estimates of critical point parameters for these models. The method is illustrated by calculations performed on the hard-square gas.

The wave mechanical formulation of quantum electrodynamics is investigated in an explicitly gauge invariant form and this leads to the connection between manifest charge conservation and the Power-Zienau-Woolley transformation.

A realisation of the conformal group with operator-valued dimensions is found which leaves invariant the Dirac equation for the spinor electrodynamics with massless fermions. All conformally invariant two- and three-point Wightman functions are calculated and it is shown that they are in agreement with the equation.

The behaviour of a free electron in a homogeneous but time varying external field is analysed and exact results are presented. Based on the exact wavefunction obtained, a new perturbation method for treating intense field problems is proposed. In particular, the transition amplitude of nonlinear direct and inverse bremsstrahlung is calculated.

For pt.I see ibid., vol.13, p.2817, (1980). The behaviour of a relativistic free electron in an external plane wave field is analysed and a review of the existing solutions of the corresponding Dirac equation is presented. Completeness and orthogonality of the Volkov states are also proved. Based on the exact wavefunction obtained, a relativistic generalisation of the perturbation method proposed in pt.I is elaborated as a means of treating intense field problems in a covariant manner.

It is shown that the radial motion of a charged particle moving in a helical coil may be described in terms of motion in a one-dimensional potential, the potential being related to a suitable average of the components of the magnetic field. An explicit form for this potential is given. Good agreement is found with numerical orbit calculations.

The method recently put forward, Moussiaux et at (1979) for finding the electrostatic fields around two touching spheres must be generalised to include the possibility that a small residual gap remains between the spheres. The reason is that the calculated values of some quantities are significantly affected if such a gap exists and, since a real physical system can only approximate mathematical spheres in point contact, it is important to be able to calculate deviations from ideal behaviour. The generalisation is illustrated using four different problems.

Renormalisation group equations are derived from the Ginzburg-Landau free energy for p-wave paired neutron-star matter in the presence of large magnetic fields. The effect of fluctuations on the nature of the superfluid phase transition is discussed.

It is shown that the trace of J^k_-J^l_zJ^k₊ is given by where ξ₀ is approximately of the form ξ₀ = (1 - 1/j)^l.