Particle wavefunction and gravity

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.

I'm fairly sure it's not correct to say that the particle is at both A and B at the same time. If it interacts with something at A, then it's at A, not at B. And vice-versa. I believe the production of a gravitational field would be one such interaction (although perhaps we might need a quantum theory of gravity to be truly sure), so when you detect the gravitational field produced by the particle, it will appear to be "emanating" from either A or B, but not both.

This would mean that the particle can't interact with itself, since if it exists at point A to be "emitting" the gravitational field, it can't also exist at point B to be reacting to the gravitational field.

I believe the same question could apply to electromagnetic self-interaction of a charged particle. But for that case we do have a theory that should explain what happens, namely quantum electrodynamics. Perhaps someone else can explain that case in detail, or if I can figure out something, I'll edit it in here.

First a general comment - everything in the world is described by either classical or quantum fields. Point particles are a fiction, sometimes useful, sometimes not. Starting with classical field theories like Maxwell equations or general relativity, you find that you are forced to forget about point sources and repalce them by continuous charge or mass distribution, otherwise you get all kinds of nonsense (non-locality, acausality, etc. etc.). One of the reasons for that is the infinite self-force or self-energy problem that crops up already at the classical level.

We can approximate a continuous distribution of matter by a "particle" if it obeys certain conditions, roughly speaking it has to be localized and weakly interacting. By "localized" I mean that all relevant observable quantities (expectation values of operators) are localized. This is not the situation you describe - the wavefunction is approximately localized (with two centres) but it is not observable. Relevant observable quantities like the expectation value of currents will not necessarily be localized.

So, what you are asking in effect is the self-force for a particular distribution of mass (or charge). There is an answer for that, but since you are asking a question that had to do with short distance physics, the quantum mechanics of the gravitational (or electromagnetic) field comes into it. Probably not enough space-time to elaborate on this here.