Dimensional crossover of nonrelativistic bosons

In this thesis we study the dimensional crossover of a nonrelativistic Bose gas with pointlike repulsive interactions. Our focus is on the dimensional crossover from two to three dimensions, which is of particular relevance in ultracold experiments. Both the two-dimensional and the three-dimensional Bose gas feature a phase transition to a superfluid state. We investigate how the phase transition temperature changes as a function of the spatial extent of the transverse dimension, which is confined in a potential well with periodic boundary conditions. For these calculations we employ Functional Renormalization Group equations. Further, we relate the 3D and 2D s-wave scattering lengths and show how the lower-dimensional system inherits its properties from the higher-dimensional system during the renormalization group flow. In these calculations we introduce the crossover function, which is of central importance in the dimensional crossover. We also investigate other compactifications. We calculate the crossover function for a box confinement and introduce approximate flow equations for a confinement in a harmonic trap.