Many-body boson systems. Half a century later.

*(English)*Zbl 1210.82002
Theoretical and Mathematical Physics (Cham). London: Springer (ISBN 978-0-85729-108-0/hbk; 978-0-85729-109-7/ebook). x, 189 p. (2011).

Loosely speaking this book tries to explain, to replace, to simplify the KMS (Kubo-Martin-Schwinger) conditions in the Bose-Einstein condensation (BCE). The aim is mainly to revisit the mathematical framework in order to make the topic much more attractive to the large community of physicists. The author first describes Bose systems by their Hamiltonians, that is to say their energy observable. Then equilibrium states are characterized either by a variational principle, or by the energy-entropy balance criterion. Then the BCE is considered in full detail for microscopic homogeneous boson systems, and later reversible dynamics and irreversible dynamics of Boson systems are successively investigated. Lastly, the author reviews some results related to non-commutative and quantum-limit theorems with applications to quantum statistical mechanics. The potential reader needs some prerequisites in basic algebra and basic quantum mechanics.

Reviewer: Guy Jumarie (Montréal)