Historically, have there been any experiments that ruled out an interpretation of quantum mechanics, or an experiment that could do so? I understand that interpretations are made to give the same results, but perhaps a different interpretations could give rise to a different result in some way?
There are a lot of relevant facts that can serve here as an answer. I'll give just a taste of each of them without going too deep:
- In so-called "collapse models", a wave function has a probability to spontaneously collapse. There are parameters in the theory which influence the likelihood of collapse. Various experiments can rule out parameter values in these models, and if the graph (below) ever gets filled up, then these models are completely ruled out. Below is a graph of the ruled-out parameter values from this paper written by a postdoc under Sabine Hossenfelder. The black dots are recommended parameter values by the inventors of the theories, and each colored region corresponds to an experiment. Actually the excluded area is the overlay of both graphs, they are only separate to keep the visuals more organized. GRW and Adler are two separate collapse theories.
In case you don't consider collapse models an interpretation (I would agree with you) I do have more for you:
- In the deBroglie-Bohm "interpretation" of Quantum Mechanics, there has been a recent push to measure the distribution of times when a particle arrives at a detector. Textbook Quantum Mechanics does not make predictions for this type of experiment, due to the non-existence of a good time operator (this is sometimes called Pauli's theorem due to a footnote he once wrote). However, if you allow yourself to improvise/hand-wave a solution based on textbook QM, there are a number of (non-equivalent) predictions that can be made... see this paper for a list of just a few. So, which of these distributions matches experiment? It would be great to know, and if the Bohmian approach is ruled out, that would be a huge blow for the theory. There is a working proposal for an experiment here but it has not been carried out because... money. I'm personally very excited for this one, so if you have seven figures just lying around, feel free to hit me up and I'll get you in touch with the right people.
- Also related to the dBB theory of QM, Antony Valentini has written about the possibility to measure signs of quantum non-equilibrium in the early universe. Quantum non-equilibrium in this context is the possibility that for brief moments in time, the distribution of particles may not have been $|\Psi(x)|^2$, as is usually the case in dBB. Those moments would be brief, because when the distribution is not in agreement with QM, the guidance law of dBB's particles tend to redistribute the particles into $|\Psi(x)|^2$ again.
This is not all, I do have some thoughts on MWI which for me exclude it based on having issues making predictions for repeated measurements. But, to explain that would be a long answer in itself and therefore more appropriate for a separate question. Also, @Michael Seifert's comment that Bell's Theorem ruled out local realistic hidden variable theories in general, was in my view a solid answer to this question as well, and in a literal sense the best one, since that experiment has already been done!
I do generally agree with @anna v's comment that any interpretation cannot be ruled out by experiment just by definition. However, what we in physics typically call "interpretations" of Quantum Mechanics are actually often more than that. So the lesson here is that the word "interpretation" is a misnomer for quite a few of these alternate theories of Quantum Mechanics. But sometimes it requires a deep-dive in order to appreciate that.