Table of contents

Our earlier arguments (Berry M V and Robbins J M 1997 Proc. R. Soc. Lond. A 453 1771-90) leading to the spin-statistics relation are summarized and then revisited. Constructions are described that satisfy all our previous requirements but lead to the wrong exchange sign (one such alternative is the replacement of commutators by anticommutators in the Schwinger representation of the spins of the particles); we suggest why these might be unacceptable.

We investigate the response properties of granular media within the framework of the so-called random Tetris model. We monitor, for different driving procedures, several quantities: the evolution of the density and of the density profiles, the ageing properties through the two-times correlation functions and the two-times mean-square distance between the potential energies, the response function defined in terms of the difference in the potential energies of two replicas driven in two slightly different ways. We focus, in particular, on the role played by the spatial inhomogeneities (structures) spontaneously emerging during the compaction process, the history of the sample and the driving procedure. It turns out that none of these ingredients can be neglected for the correct interpretation of the experimental or numerical data. We discuss the problem of the optimization of the compaction process and we comment on the validity of our results for the description of granular materials in a thermodynamic framework.

A cyclic thermodynamic heat engine runs most efficiently if it is reversible. Carnot constructed such a reversible heat engine by combining adiabatic and isothermal processes for a system containing an ideal gas. Here, we present an example of a cyclic engine based on a single quantum mechanical particle confined to a potential well. The efficiency of this engine is shown to equal the Carnot efficiency because quantum dynamics is reversible. The quantum heat engine has a cycle consisting of adiabatic and isothermal quantum processes that are close analogues of the corresponding classical processes.

We compute the continuum limit of the spectra for the XX model with arbitrary complex boundary fields. In the case of Hermitian boundary terms one obtains the partition functions of the free compactified boson field on a cylinder with Neumann-Neumann, Dirichlet-Neumann or Dirichlet-Dirichlet boundary conditions. This also applies for certain non-Hermitian boundaries. For special cases we also compute the free surface energy. For certain non-Hermitian boundary terms the results are more complex. Here one obtains logarithmic corrections to the free surface energy. The asymmetric version of the XX model with boundaries (this includes the Dzyaloshinsky-Moriya interaction) is also discussed.

Slow dynamics and metastability are often seen in models with quenched disorder, but rather harder to find in situations where no such disorder is present and energy or entropy barriers must be generated dynamically. Using Monte Carlo simulations we show that the 3D four-spin interaction Ising model, which possesses no quenched disorder, exhibits rather strong metastability in a broad range of temperatures around its first-order transition point, due to the shape dependence of excitations in the model and the resulting largeness of the associated critical droplets.

We present an exactly solvable model of a Gaussian (flexible) polymer chain in a quenched random medium. This is the case when the random medium obeys very long-range quadratic correlations. The model is solved in d spatial dimensions using the replica method, and practically all the physical properties of the chain can be found. In particular, the difference between the behaviour of a chain that is free to move and a chain with one end fixed is elucidated. The interesting finding is that a chain that is free to move in a quadratically correlated random potential behaves like a free chain with R²~L, where R is the end-to-end distance and L is the number of links, whereas for a chain anchored at one end R²~L⁴. The exact results are found to agree with an alternative numerical solution in d = 1 dimensions. The crossover from long-range to short-range correlations of the disorder is also explored.

Braunstein and Caves (Braunstein S L and Caves C M 1994 Phys. Rev. Lett.72 3439-43) proposed to use Helstrom's quantum information number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing other than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on n identical copies of the given quantum system.

We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two-stage adaptive measurement procedure.

When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of `distinguishability of close-by states' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.

The relativistic two-body system in (1+1)-dimensional quantum electrodynamics is studied. It is proved that the eigenvalue problem for the two-body Hamiltonian without the self-interaction terms reduces to the problem of solving a one-dimensional stationary Schrödinger-type equation with an energy-dependent effective potential which includes the δ-functional and inverted oscillator parts. The conditions determining the metastable energy spectrum are derived, and the energies and widths of the metastable levels are estimated in the limit of large particle masses. The effects of the self-interaction are discussed.

Degeneracy of resonant states and double poles in the scattering matrix of a double barrier potential are contrived by adjusting the parameters of the system. The cross section, scattering wavefunction and Gamow eigenfunction are computed at degeneracy. Some general properties of the degeneracy of resonances are exhibited and discussed in this simple quantum system.

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as `curvature/signature (in)dependent trigonometry' and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an `absolute trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, Minkowskian, Newton-Hooke and Galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are `universality' and `self-duality': every equation is meaningful for the nine spaces at once, and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of trigonometry and, in fact, any equation previously known for the three classical (Riemannian) spaces also has a version for the remaining six `spacetimes'; in most cases these equations are new.

A question of completeness of a discrete many-electron Sturmian set proposed in a series of recent publications is considered. It is shown that already in the simplest case of a two-electron system the proposed Sturmians do not form a complete set since the spectrum of a generating eigenproblem is mixed: apart from discrete eigenvalues, with which the discrete Sturmians are associated as corresponding eigenfunctions, the spectrum also contains a continuum part. A peculiar feature of the spectrum found is that infinitely many discrete eigenvalues are embedded in the continuum.

( = parity times time-reversal) symmetry of complex Hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for Hermiticity. Perhaps an equally important role is played by the related strengthened analyticity assumptions. In a constructive illustration we complexify a few potentials solvable only in s-wave. Then we continue their domain from the semi-axis to the whole axis and obtain new exactly solvable models. Their energies turn out to be real as expected. The new one-dimensional spectra themselves differ quite significantly from their s-wave predecessors.