Abstract:
We calculate the ultra-relativistic Bose-Einstein condensation temperature of a complex scalar field with weak lambda Phi^4 interaction. We show that at high temperature and finite density we can use dimensional reduction to produce an effective three-dimensional theory which then requires non-perturbative analysis. For simplicity and ease of implementation we illustrate this process with the linear delta expansion.

Abstract:
This paper is a short introduction to cold atom physics and Bose-Einstein condensation. Light forces on atoms are presented, together with laser cooling, and a few atom traps: the magneto-optical trap, dipole traps and magnetic traps. A brief description of Bose-Einstein condensation is given together with some important links with condensed matter physics. The reader is referred to comprehensive reviews and to other lecture notes for further details on atom cooling, trapping and Bose-Einstein condensation.

Abstract:
This book surveys results about the quantum mechanical many-body problem of the Bose gas that have been obtained by the authors over the last seven years. These topics are relevant to current experiments on ultra-cold gases; they are also mathematically rigorous, using many analytic techniques developed over the years to handle such problems. Some of the topics treated are the ground state energy, the Gross-Pitaevskii equation, Bose-Einstein condensation, superfluidity, one-dimensional gases, and rotating gases. The book also provides a pedagogical entry into the field for graduate students and researchers.

Abstract:
We examine Bose-Einstein condensation as a form of symmetry breaking in the specific model of the Einstein static universe. We show that symmetry breaking never occursin the sense that the chemical potential $\mu$ never reaches its critical value.This leads us to some statements about spaces of finite volume in general. In an appendix we clarify the relationship between the standard statistical mechanical approaches and the field theory method using zeta functions.

Abstract:
We study the thermalization of gluons far from thermal equilibrium in relativistic kinetic theory. The initial distribution of gluons is assumed to resemble that in the early stage of ultrarelativistic heavy ion collisions. Only elastic scatterings in static, nonexpanding gluonic matter are considered. At first we show that the occurrence of condensation in the limit of vanishing particle mass requires a general constraint for the scattering matrix element. Then the thermalization of gluons with Bose-Einstein condensation is demonstrated in a transport calculation. We see a continuously increasing overpopulation of low energy gluons, followed by a decrease to the equilibrium distribution, when the condensation occurs. The times of the completion of the gluon condensation and of the entropy production are calculated. These times scale inversely with the energy density.

Abstract:
The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.

Abstract:
For a system of identical Bose particles sitting on integer energy levels, we give sharp estimates for the convergence of the sequence of occupation numbers to the Bose-Einstein distribution and for the Bose condensation effect.

Abstract:
We propose and analyze a mechanism for Bose-Einstein condensation of stationary dark-state polaritons. Dark-state polaritons (DSPs) are formed in the interaction of light with laser-driven 3-level Lambda-type atoms and are the basis of phenomena such as electromagnetically induced transparency (EIT), ultra-slow and stored light. They have long intrinsic lifetimes and in a stationary set-up with two counterpropagating control fields of equal intensity have a 3D quadratic dispersion profile with variable effective mass. Since DSPs are bosons they can undergo a Bose-Einstein condensation at a critical temperature which can be many orders of magnitude larger than that of atoms. We show that thermalization of polaritons can occur via elastic collisions mediated by a resonantly enhanced optical Kerr nonlinearity on a time scale short compared to the decay time. Finally condensation can be observed by turning stationary into propagating polaritons and monitoring the emitted light.

Abstract:
The Bose-Einstein condensation in the hard-core boson limit (HCB) of the Bose-Hubbard model with two local states and the particle hopping in the excited band only is investigated. For the purpose of considering the non-ergodicity, a single-particle spectral density is calculated in the random phase approximation by means of the temperature boson Green functions. The non-ergodic contribution to the momentum distribution function of particles (connected with the static density fluctuations) increases significantly and becomes comparable with the ergodic contribution in the superfluid phase near the tricritical point.