The fermionic entanglement entropy of spatial regions in Schwarzschild and Minkowski spacetime

In this thesis we define and study the fermionic entanglement entropy for spatial subregions in Schwarzschild and Minkowski spacetime. Our starting point is always the Dirac propagator corresponding to the vacuum state (for an observer at infinity). We then introduce an ultraviolet regularization and rewrite the respective propagator as pseudo-differential operator. In Schwarzschild spacetime we consider the entanglement entropy of a black hole horizon. We use separation of variables, an integral representation of the propagator and methods from pseudo-differential operator calculus to explicitly compute the entanglement entropy of the horizon in the limiting case that the regularization tends to zero. It turns out that it equals a numerical constant times the number of angular momentum modes occupied at the horizon. A similar result is proven to hold for the Rényi entanglement entropies with Rényi index κ >$\frac{2}{3}$. In the case of Minkowski spacetime we study the entanglement entropy of bounded spatial subregions. We consider two limiting cases: one where the regularization goes to zero and one where the regularization is fixed and the size of the region tends to infinity. The corresponding limiting coefficient is obtained by applying a more general result from [F. Finster, M. Lottner, and A.V. Sobolev, The Fermionic Entanglement Entropy and Area Law for the Relativistic Dirac Vacuum State, arXiv:2310.03493 [math-ph] (2023).]. We then show the positivity of this coefficient and prove that it is proportional to the area of the considered region, giving an area law. The positivity is also proven to hold for the Rényi entanglement entropies with Rényi index 0 < κ < 2, the other main results in this part even apply for arbitrary Rényi entanglement entropies (with κ > 0).