Is many-worlds interpretation only a philosophical matter?
It seems to me that we can't exclude a possible test for this hypothesis. I explain.
For superposition principle each world would follow the Schrodinger equation and then it seems impossible to distinguish if we collapsed in $\psi_i$ a wave function $\psi=\sum_i\psi_i$ or we ended up in a world where the state of the system is $\Psi_i=\psi_{me}(i)\otimes\psi_i$. This would be, at the multiverse level, just one state of the superposition $\Psi=\sum_i\Psi_i$ with all the other possible outcomes and their worlds. Also, if there are interferences enhancing some worlds and suppressing others, this would be testable only by someone experimenting on $\Psi$, at the multiverse level, then again invisible inside any world.
But if the interference cancels out completely some possible world, then we could be able to recognize that by statistical means. Suppose that I am doing a measure that can collapse $\psi$ in an eigenstate $\psi_i$ such that the world $\psi_{me}(i)\otimes\psi_i$ would be canceled out by interference with other worlds. Then, getting the eigenvalue of $\psi_i$ in the measure wouldn't be an evidence contrary to the many world hypothesis?
Best Answer
This question takes for granted an idea that is common among physicists, but is also false. The idea is that there are multiple interpretations of quantum mechanics that all make the same predictions. In reality, the so-called interpretations fall into three categories.
Alternatives to quantum mechanics that make different predictions, such as the pilot wave theory and spontaneous collapse theories like GRW:
Bassi et al. (2012)
Colin and Valentini (2015)
Quantum mechanics without any modifications, which implies the existence of a structure that can sometimes be approximated as a collection of parallel universes (the Everett interpretation). Experimental tests can distinguish between the Everett interpretation and alternatives like pilot wave and GRW:
Theories that are too vague to work out their implications, with the result that they are not testable, such as the Copenhagen and statistical interpretations of quantum mechanics. Such theories fudge the issue of what exists in reality and so can't be used to make testable predictions since they are basically the same as saying "quantum mechanics applies, except when it doesn't".
The interpretations that are philosophical in a bad sense, i.e. - in the sense of being useless talk that obfuscates real problems, are in category (3) not in category (2).