Algebraic methods of quantum physics 1100-MAFK

1. Basic formalism of Quantum Mechanics, superselection sectors–why
∗-algebras in Quantum Physics?
2. Formal definition of ∗-algebras
3. Classification of finite-dimensional ∗-algebras and their representations.
4. Banach and C∗-algebras.
5. Von Neumann algebras and W∗-algebras.
6. GNS representation of a C∗-algebra
7. Tensor product of vector spaces and Hilbert spaces.
8. Tensor product of an infinite number of Hilbert spaces.
9. Spin systems in algebraic description, the corresponding C∗ and W∗-algebras.
10. Naive approach to thermal states
11. KMS states and the Tomita-Takesaki theory
12. Second quantization, bosonic and fermionic Fock spaces, creation and an-
nihilation operators
13. Exponential property of Fock spaces
14. CAR algebras and their representations
15. Stone-von Neumann Theorem
16. Canonical commutation relations (CCR) and their representations
17. Bogoliubov transformations of CCR and CAR algebras.
18. Shale and Shale-Stinespring criteron for the implementability of Bogoli-
ubov transformations.
19. Quasifree states.
20. Araki-Woods representations of CCR
21. Araki-Wyss representations of CAR.