There exists a fundamental problem in the quantum theory, which is that the collapse postulate, aimed at
describing how the wave function of a quantum system changes during a measurement, is not a Lorentz invariant process. In everyday language, the collapse is assumed to happen infinitely fast, which opens the way to possible supraluminal influences, in contradiction with the theory of special relativity. A situation “of peaceful coexistence” between special relativity and quantum mechanics is however made possible by the fact that the collapse is not a deterministic process but a stochastic one. This constitutes the so-called no-signaling condition: due to the intrinsic quantum randomness of the collapse, it is actually impossible to send a classical signal faster than light.
As was made explicit by Gisin  and others [84, 31], one of the key ingredients of this nearly miraculous situation of peaceful coexistence is the linearity of the Schroedinger equation. There exist attempts of non-linear modifications of the Schroedinger equation, like the GRW model of spontaneous localisation , in which the no-signaling condition is still respected because of the aforementioned mechanism: stochasticity and non-locality conspire in order to restore locality “on average” . Our approach is different: in our view, if one seriously considers the possibility of non-linear modifications of the linear Schroedinger evolution law, one is confronted with the possible violation of the no-signaling condition. Our goal is to study how and when such violations occur, focusing on the Schroedinger-Newton equation [76, 91, 33, 50, 29], which is a non-linear but deterministic modification of the Schroedinger equation that results from the hypothetical existence of a self-gravitational interaction at the quantum level. The project will lead to publications in renowned scientific journals, to presentations during international conferences and to the completion of a Ph.D. thesis.
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