1. On classical aspects of Bose-Einstein condensation
- Author
-
Pettinari, Lorenzo
- Subjects
Mathematical Physics - Abstract
Berezin and Weyl quantizations are renown procedures for mapping classical, commutative Poisson algebras of observables to their non-commutative, quantum counterparts. The latter map is famous for its use on Weyl algebras, while the former is more appropriate for continuous functions decaying at infinity. In this work we define a variant of Berezin quantization map, which operates on classical Weyl algebras $\mathcal{W}(E,0)$ and constitutes a positive \textit{strict deformation quantization}. We use this map to compare classical and quantum thermal equilibrium states for a boson gas and to compute its rigorous classical limit. For this scope, we first define a purely algebraic notion of KMS states for the classical Weyl algebra and verify that in the finite volume setting there is only one possible KMS state, which can be interpreted as the Fourier transform of a Gibbs measure on some Hilbert space. Then, we show how the classical KMS states have thermodynamic limits which can manifest condensation in the zero mode, similarly to what happens in the standard Bose-Einstein condensation. Lastly, we prove that there exist sequences of quantum KMS states for the infinite volume Bose gas, that converge weakly$^*$ to classical KMS states. Moreover, since the different thermal phases are preserved by this limit, it is shown that a quantum condensate is mapped to a classical one., Comment: 46 pages
- Published
- 2024