Suppose a particle has 50% probability of being at location $A$, and 50% probability being at location $B$ (see double slit experiment). According to QM the particle is at both $A$ and $B$ at the same time, so is there a force of gravity between the two particle superpositions? Is there self-gravity when a wave-function reaches over a finite distance?
I cannot seem to wrap my head around this. Is the gravity a proportional fraction of the entire mass based on the probabilities. How do you combine a wavefunction with Gauss' law of gravity? I have being trying to think about self-gravity for a long time now.
Answer
There is some work by Roger Penrose on the subject. The papers title is, "On Gravity's Role in Quantum State Reduction", and it discuses how the interaction of two states that have different mass distributions with spacetime can cause the wavefunction to collapse in the one state or the other. There is also a following paper that discuses the same thing in Newtonian gravity, "Spherically-symmetric solutions of the Schrödinger-Newton equations" (and there is also this that you could have a look).
There is one thing that I should point out that is also pointed out by David. In a situation as the one described in the question (double slit experiment), the particle is not at two different places at the same time and interacts with it self. It is the two states (wavefunctions) that interact to give you the interference.
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