Itinerant ferromagnetism is spontaneous polarization of non-localized (itinerant or 'wandering') particles, i.e. ferromagnetism in a gas. In a two-component Fermi gas it is caused by the exchange interactions, at the cost of an increased kinetic energy. That is why itinerant ferromagnetism is only expected to occur for strong repulsive interactions between the particles.

Although itinerant ferromagnetism has already been predicted in a free electron gas by F. Bloch in 1929, it has proven to be notoriously hard to find experimentally. Due to the recent successes of ultracold atomic gases as a quantum simulator for condensed matter systems and their experimental tunability, this experimental system has been suggested as a model system for the realization of pure itinerant ferromagnetism. In 2009, experimentalists from the Ketterle group at MIT have been able to reach the strongly interacting regime where itinerant ferromagnetism was predicted to occur. However, the Feshbach resonance that was used to tune the interaction strength is unstable towards molecular pairing. This molecular pairing dominated the experiment and prevented the formation of any equilibrium state (including the itinerant ferromagnetic state).

In order to understand why itinerant ferromagnetism could not yet be observed in ultracold atomic gases, I revisited the basic theory of itinerant ferromagnetism in the path-integral formalism. In particular, I studied how the direct and exchange interactions are treated in the Hartree channel of the path-integral formalism and how this treatment can be improved.

First, I identified an important stability issue in the conventional description of the interactions in the saddle-point approximation for the case of contact interactions. In order to understand this stability issue, I extended the current formalism to general interaction potentials and demonstrated that the Pauli exclusion principle is not necessarily observed in the saddle-point approximation. In order to solve this problem, I proposed to enforce the Pauli exclusion principle by using a modified interaction potential. Afterwards, I applied this new method to the case of itinerant ferromagnetism in 3D. The results suggest that dynamical stability is an important factor to take into account when studying itinerant ferromagnetism, as it greatly constrains the itinerant ferromagnetic region in the phase diagram. Finally, I demonstrated that the new method is not exact: the Pauli exclusion principle is not the same as an interaction potential, as both behave differently under a Fourier transform.