AG Mathematische Physik, Ricardo Correa da Silva (FAU): Modular Structures on a von Neumann Algebra on Hilbert-Schmidt Operators and Applications to Thermodynamical Equilibrium States of Infinite Degenerated Systems

Ricardo Correa da Silva (FAU)

Modular Structures on a von Neumann Algebra on Hilbert-Schmidt Operators and Applications to Thermodynamical Equilibrium States of Infinite Degenerated Systems

After an introduction to Tomita-Takesaki Modular Theory and KMS states, we will characterize all cyclic and separating vectors for a well-known von Neumann algebra acting on the Hilbert-Schmidt operators aiming to study the thermal equilibrium states (KMS states) for this algebra. Then, we discuss the description of infinitely degenerate Hamiltonians in this algebra, in particular, the example of the Landau levels showing that there is no cyclic and separating vector such that the modular Hamiltonian corresponds to the Landau Hamiltonian, as argued in other work. Finally, we try to reproduce the thermodynamical limit of the system confined in a finite box, for physical reasons, to better understand how the degeneracy grows as the box radius goes to infinity.