Algebraic Approaches to Bose-Einstein Condensation

We review two recent applications of operator-algebraic methods to the analysis of homogeneous Bose gases at finite temperature. The first part concerns the semiclassical description of Bose–Einstein condensation in the framework of Weyl C^* C ∗ -algebras. In this setting, the semiclassical parameter is naturally linked to the density of the system, and the classical weak KMS condition emerges as the limit of quantum equilibrium states. We show in particular that the condensate structure is preserved in a suitable large-density regime. The second part discusses a joint work with Prof. Jan Dereziński which the Araki–Woods representation is used to formulate a systematic perturbative theory for interacting Bose gases at positive temperature. Within this representation, thermal effects are encoded directly in the field operators, allowing for a transparent implementation of Wick’s theorem and the computation of damping coefficients via the Fermi Golden Rule. The review synthesizes results obtained in two recent works presented at IQSA2025, emphasizing the conceptual role of algebraic methods in connecting semiclassical analysis, equilibrium states, and finite-temperature perturbation theory.