under the assumption that the potential must vanish at infinity. Mathematically, the Schrödinger–Newton equation is a special case of the Hartree equation for . The equation retains most of the properties of the linear Schrödinger equation. In particular, it is invariant under constant phase shifts, leading to conservation of probability and exhibits full Galilei invariance. In addition to these symmetries, a simultaneous transformation maps solutions of the Schrödinger–Newton equation to solutions. The stationary equation, which can be obtained in the usual manner via a separation of variables, possesses an infinite family of normalisable solutions of which only the stationary ground state is stable. Relation to semiclassical and quantum gravity The Schrödinger–Newton equation can be derived under the assumption that gravity remains classical, even at the fundamental level, and that the right way to couple quantum matter to gravity is by means of the semiclassical Einstein equations. In this case, a Newtonian gravitational potential term is added to the Schrödinger equation, where the source of this gravitational potential is the expectation value of the mass density operator or mass flux-current. In this regard, if gravity is fundamentally classical, the Schrödinger–Newton equation is a fundamental one-particle equation, which can be generalised to the case of many particles (see below). If, on the other hand, the gravitational field is quantised, the fundamental Schrödinger equation remains linear. The Schrödinger–Newton equation is then only valid as an approximation for the gravitational interaction in systems of a large number of particles and has no effect on the centre of mass. It is also suggested that Schrödinger–Newton equation coupled with Unruh radiation could give rise to an interferometric experiment that could test entropic gravity. Many-body equation and centre-of-mass motion If the Schrödinger–Newton equation is considered as a fundamental equation, there is a corresponding N-body equation that was already given by Diósi and can be derived from semiclassical gravity in the same way as the one-particle equation: The potential contains all the mutual linear interactions, e.g. electrodynamical Coulomb interactions, while the gravitational-potential term is based on the assumption that all particles perceive the same gravitational potential generated by all the marginal distributions for all the particles together. In a Born–Oppenheimer-like approximation, this N-particle equation can be separated into two equations, one describing the relative motion, the other providing the dynamics of the centre-of-mass wave-function. For the relative motion, the gravitational interaction does not play a role, since it