Table of contents

A fully consistent realization of the quantum operators corresponding to the canonically conjugate phase and number variables is proposed, resorting to the positive discrete series of the irreducible unitary representation of the Lie algebra su(1, 1) of the double covering group of SO^↑(1, 2). The realization holds in subspace , the system Fock space minus the vacuum state. A possible way to extend it to the full space of states based on recourse to a dilated extension of Hilbert space is discussed.

The phenomena implied by the existence of quantum vacuum fluctuations, grouped under the title of the Casimir effect, are reviewed, with emphasis on new results discovered in the past four years. The Casimir force between parallel plates is rederived as the strong-coupling limit of δ-function potential planes. The role of surface divergences is clarified. A summary of effects relevant to measurements of the Casimir force between real materials is given, starting from a geometrical optics derivation of the Lifshitz formula, and including a rederivation of the Casimir–Polder forces. A great deal of attention is given to the recent controversy concerning temperature corrections to the Casimir force between real metal surfaces. A summary of new improvements to the proximity force approximation is given, followed by a synopsis of the current experimental situation. New results on Casimir self-stress are reported, again based on δ-function potentials. Progress in understanding divergences in the self-stress of dielectric bodies is described, in particular the status of a continuing calculation of the self-stress of a dielectric cylinder. Casimir effects for solitons, and the status of the so-called dynamical Casimir effect, are summarized. The possibilities of understanding dark energy, strongly constrained by both cosmological and terrestrial experiments, in terms of quantum fluctuations are discussed. Throughout, the centrality of quantum vacuum energy in fundamental physics is emphasized.

Two methods are considered for assessing the asymptotic stability of the trivial solution of linear stochastic differential equations driven by Poisson white noise, interpreted as the formal derivative of a compound Poisson process. The first method attempts to extend a result for diffusion processes satisfying linear stochastic differential equations to the case of linear equations with Poisson white noise. The developments for the method are based on Itô's formula for semimartingales and Lyapunov exponents. The second method is based on a geometric ergodic theorem for Markov chains providing a criterion for the asymptotic stability of the solution of linear stochastic differential equations with Poisson white noise. Two examples are presented to illustrate the use and evaluate the potential of the two methods. The examples demonstrate limitations of the first method and the generality of the second method.

The pressure for the imperfect (mean field) boson gas can be derived in several ways. The aim of the present paper is to provide a new method based on the approximating Hamiltonian argument which is extremely simple and very general.

We show that several well-known one-dimensional quantum systems possess a hidden non-local supersymmetry. The simplest example is the open XXZ spin chain with Δ = −1/2. We use the supersymmetry to place lower bounds on the ground-state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground-state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar spacetime supersymmetry holds for the t–J model at its integrable ferromagnetic point, where the spacetime supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed.

Some chaotic properties of a classical particle interacting with a time-dependent double-square-well potential are studied. The dynamics of the system is characterized using a two-dimensional nonlinear area-preserving map. Scaling arguments are used to study the chaotic sea in the low-energy domain. It is shown that the distributions of successive reflections and of corresponding successive reflection times obey power laws with the same exponent. If one or both wells move randomly, the particle experiences the phenomenon of Fermi acceleration in the sense that it has unlimited energy growth.

Generalizing from the case of golden mean frequency to a wider class of quadratic irrationals, we extend our renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. We give a description of all piecewise-constant periodic orbits of an additive functional recurrence generalizing that present in the golden mean case. We establish a criterion for periodic orbits to be globally bounded, and also calculate the asymptotic height of the main peaks in the correlation function.

A generalized contour deformation method (CDM), which combines complex rotation and translation in momentum space, is discussed. CDM gives accurate calculation of two-body spectral structures: bound, antibound, resonant and continuum states forming a Berggren basis. It provides a basis for full off-shell t-matrix calculations both for real and complex input energies. Results for both spectral structures and scattering amplitudes compare perfectly well with exact values for the analytically solvable separable non-local Yamaguchi potential as a testcase. Accurate calculation of antibound states in the Malfliet–Tjon and the realistic CD–Bonn nucleon–nucleon potential are presented. Calculation of antibound states in the CD–Bonn potential are not known to have been given elsewhere.

In this paper we consider a system consisting of a two-level atom, initially prepared in a coherent superposition of upper and lower levels, interacting with a radiation field prepared in generalized quantum states in the framework of multiphoton Jaynes–Cummings model. For this system, we show that there is a class of states for which the fluctuation factors can exhibit the revival-collapse phenomenon (RCP) similar to that exhibited in the corresponding atomic inversion. This is shown not only for normal fluctuations but also for amplitude-squared fluctuations. Furthermore, apart from this class of states we generally demonstrate that the fluctuation factors associated with three-photon transition can provide the RCP similar to that occurring in the atomic inversion of the one-photon transition. These are novel results and their consequence is that the RCP occurring in the atomic inversion can be measured via a homodyne detector. Furthermore, we discuss the influence of the atomic relative phases on such phenomenon.

Numerical methods are used to compute sphaleron solutions of the Skyrme model. These solutions have topological charge zero and are axially symmetric, consisting of an axial charge n Skyrmion and an axial charge −n anti-Skyrmion (with n > 1), balanced in unstable equilibrium. The energy is slightly less than twice the energy of the axially symmetric charge n Skyrmion. A similar configuration with n = 1 does not produce a sphaleron solution, and this difference is explained by considering the interaction of asymptotic pion dipole fields. For sphaleron solutions with n > 4, the positions of the Skyrmion and anti-Skyrmion merge to form a circle, rather than isolated points, and there are some features in common with Hopf solitons of the Skyrme–Faddeev model.