Tomita-Takesaki theory in B(H) with physical applications

Abstract

The main aim of this project is to develop a simple, yet mathematically rigorous, version of Tomita-Takesaki theory for the von Neumann algebra B(H ) with a faithful normal state. In Chapter 2 we formulate the theory in terms of tensor products. Even in this fairly general setup we can already attach physical interpretation to the modular objects and J. Namely that, the former, the modular operator induces a unique modular automorphism group t which in turn gives the time-evolution (dynamics) of some physical system. Whereas the modular conjugation implements a time-reversal. Chapter 3 presents an alternative formulation of Tomita-Takesaki theory, unitarily equivalent to the first, but with the space of Hilbert-Schmidt operators as our preferred choice of Hilbert space. To gain further insight into the theory, in Chapter 4, a certain simple physical system is explored. In particular, we look at how the system of an electron in a constant orthogonal magnetic field, together with the associated phenomenon of Landau levels, displays a modular structure in the sense of Tomita-Takesaki theory. In such a case, the algebra of observables and its commutant correspond to the two directions of the magnetic field.