Remarks on quantum scalar field theory for a uniformly accelerated reference frame and a Schwarzschild black hole

In the present paper, the cases of a uniformly accelerated reference frame and a Schwarzschild black hole in two and four dimensions are discussed focusing on formal aspects of the corresponding quantum scalar field theories. The main emphasis is made on building consistent quantum field theories and highlighting the problems that are not considered elsewhere. In particular, some pathologies in the case of a uniformly accelerated reference frame are discussed in detail: it is shown that some quantum states that are well defined in Minkowski spacetime possess an infinite energy when considered in Rindler spacetime. In the case of a Schwarzschild black hole in four dimensions, it is shown that the standard theory providing the Unruh vacuum state turns out to be inconsistent. The problem of consistency of quantum scalar field theory in Kruskal-Szekeres spacetime is also discussed.