Tests of Fundamental Quantum Mechanics and Dark Interactions with Low Energy Neutrons - Extended Version

Among the known particles, the neutron takes a special position, as it provides experimental access to all four fundamental forces and a wide range of hypothetical interactions. Despite being unstable, free neutrons live long enough to be used as test particles in interferometric, spectroscopic, and scattering experiments probing low-energy scales. As was already recognized in the 1970s, fundamental concepts of quantum mechanics can be tested in neutron interferometry using silicon perfect-single-crystals. Besides allowing for tests of uncertainty relations, Bell inequalities and alike, neutrons offer the opportunity to observe the effects of gravity and hypothetical dark forces acting on extended matter wave functions. Such tests gain importance in the light of recent discoveries of inconsistencies in our understanding of cosmology as well as the incompatibility between quantum mechanics and general relativity. Experiments with low-energy neutrons are thus indispensable tools for probing fundamental physics and represent a complementary approach to colliders. In this review we discuss the history and experimental methods used at this low-energy frontier of physics and collect bounds and limits on quantum mechanical relations and dark energy interactions.