Philosophers talk a lot about possible worlds. I am one person who does not believe any other worlds except ours exist. But how can I know if I am right? How would one know if other possible worlds exist? What experiment could tell if other worlds exist? And also, have any philosophers talked about this?
There are a number of SEP articles about this and similar questions:
A general strategy for "knowing about" possible worlds is plenitudinous platonism:
Balaguer defines plenitudinous platonism (somewhat roughly) as the view that there exist mathematical objects of all possible kinds, or the view that all the mathematical objects that possibly could exist actually do exist. But, in general, Balaguer would define a different plenitude principle for every different kind of abstract object. Linsky & Zalta develop plenitudinous platonism by proposing a distinctive plenitude principle for each of three basic domains of abstracta: abstract individuals, relations (properties and propositions), and contingently nonconcrete individuals (1995, 554). For example, on their view, the plenitude principle for abstract individuals asserts (again, somewhat roughly) that every possible description of an object characterizes an abstract object that encodes — and, thus, in an important sense, has — the properties expressed in the description.
Balaguer and Linsky & Zalta then argue that if platonists endorse plenitudinous platonism, they can solve the epistemological problem with platonism without positing any sort of information-transferring contact between human beings and abstract objects.
Implicitly or explicitly, for referential versions of the multiverse standpoint in set theory, something like "knowledge by acknowledged consistency" is a common justifier ("as long as the axioms are consistent and have consistent implications..."). Whether this should be seen as transferring to physical multiverses is a different, if related, matter.
Most talk of possible worlds in philosophy uses that term in the sense of hypothetical worlds, or imaginary worlds, which are just supposed contexts for a particular idea. If you are a practically-minded person, that is how you will think of possible worlds. There are some people, however, who take the view that other worlds and universes exist just as our own does. There is no test we can perform today to rule-in or out such a view. Indeed, some people assert that there are other worlds which are disconnected from ours so that by definition it will always be impossible to test whether or not they are there.
To test you have to have data. Either you already have the data, that you can analyze and philosophize; Or you create your own data through experimentation.
For the data to exist there has to be some connection between the world. If the worlds do exist but there is no connection between them and us then there is no data and therefore you have nothing to work on.
There is another limitation. May be the data is there but your senses cannot be aware of them and therefore your brain has nothing to work on. We humans solve this problem by making machines that can sense what we cannot sense, such as magnetism, and translate it to us in something that we can sense, such as visuals (numbers which are when written just pictures, graphs etc), audios, vibrations etc.
About the specific test that you can do or we humans can collectively do, such as our advanced research institutes can do, there are many, too many to list. What I can do for you is give you directions to look at. Four directions.
The first one is to have lots of data. More data the better. Especially the data our senses cannot know about on their own. I gave one example already. Another is those electromagnetic waves that we cannot see.
Second direction is precision. We want more precise equipments. For example we can sense gravity but we cannot sense "gravity waves" (assuming they are real). Looking through microscope is a step in right direction for example.
Third and most important is to examine your theories. Don't take your theories for granted. See do they fit the data. If they don't then either the data is not correct (your observation or experimentation has errors) or the theory needs to be discarded or improved. Having correct theory goes a long way for you to achieve your goal.
You have to understand the basics of how this universe, our universe, our world works. This is very important because without knowing this you cannot distinguish between origin of a phenomena, that its in our world or outside.
For example, you have to understand that our universe is expanding, and not just expanding but also accelerating in that expanding. The way our universe works is that this requires energy. The source of that energy is not found in this world. It has to be coming from outside.
With a device called a Elitzur–Vaidman bomb tester.
Don't downvote just yet, I promise it's relevant to the philosophy. The other answers are not wrong that in a philosophical context, "possible worlds" generally refers to hypotheticals and imaginary scenarios. That is, a "possible" world is some state of the world that could be - it would be internally consistent and not violate any law of physics or logic - but is different from the actual state of the world.
Whether you ascribe to the Platonist position that these other possibilities have a "reality" equal to that of the world as it actually exists, or there is some character to the actual world that qualiatively differentiates it from possible-but-not-actual, the question of "how do I know" has the same answer - you need evidence, and according to Bayesian epistemology, you need evidence in the form of observations, rigorous experiment or not, that only happen in the event that possible worlds are "real." (Conversely, observations that come out the same way regardless of whether possible worlds exist or not don't help you)
An observation happening only in the event of a certain fact being true is, (reworded) what we mean by saying that the fact causes the observation. So to practically know that possible worlds exist, the possible worlds must be able to cause things in principle - their prescense must be able to impact our observations, otherwise all observations come out the same way regardless and no evidence will ever help.
Unfortunately, it's not at all clear how hypotheticals and imaginary scenarios could impact the actual world. Imagining a hypothetical can impact your future thoughts and actions, which would be observations, but whether that counts is questionable because the actual activity of imagining is done in the actual world. We can just ascribe the observations to that actual event and bypass the question. Maybe they somehow influence us from the Platonic realm? But then, if Platonic entities necessarily exist (because "a world is possible" cannot itself be a contingent fact) how can we even entertain what difference we would see in our observations and data if they didn't exist? And if we don't know what we'd see if they weren't present, we can't determine whether they exist when we see or fail to see that.
I think the bomb tester thought experiment illustrates this idea quite clearly because it's essentially the case study you're looking for - in the "multiple worlds" interpretation of quantum mechanics, they do have causal influence. Indeed, in this reading, all "quantum" behaviour is a result of multiple classical possibilities combining on top of each other. The actual mechanics of how that works aren't simple, but I will try to explain how it works briefly.
In the bomb tester (as illustrated by this diagram) you have a laser that emits single photons, and four mirrors arranged in a square. The mirrors in two opposite corners of the square are "half-silvered", which means that laser light hitting them has an equal probability to either pass through or be reflected. The laser light also has a "phase", which (briefly) works like this: if two laser pulses that are "out of phase" overlay each other, they cancel out and disappear, whereas if they are "in phase", they reinforce each other and are detected more strongly.
A photon emitted by the laser will first reach one of the half-silvered mirrors, at which point we get our "possible worlds" - in one, the photon has reflected off the mirror, and in the other, it has passed through unaffected. It is very important to emphasize that the photon has not split in two - the entire nature of photons is that they cannot be split into multiple smaller photons as we'd intuitively expect. The reflected "version" of the photon (deterministically) bounces off another mirror, and then into the second half-silvered mirror, where it branches again into two possibilities, heading to either detector C or D. It's also worth mentioning that despite the name of "multiple worlds", the whole world hasn't split, only the one photon.
What about the transmitted photon that continues to the bomb? The "bombs" in the thought experiment are either "working" or "broken", hence the need to test them. Working bombs absorb the incoming photon 100% of the time and explode. Broken bombs allow the photon to continue unaffected, also 100% of the time. It's very important to keep in mind that a given photon can only actually be observed being in one place, after which it is absorbed and ceases to be, and exploding the bomb counts as an observation.
If the bomb is working, we can just walk through the possibilities of where the photon is detected, which are, I must stress, mutually exclusive:
- the photon ends up at detector C
- the photon ends up at detector D
- the photon hits the bomb, which explodes
The wrinkle comes in what happens if the bomb is broken. In that case, the photon sails past it, bounces off the mirror and into the far half-silvered mirror like it would if it'd gone the other way. It then splits into two possibilities as before, going to either detector C or D. ...But because it has been reflected off mirrors a different number of times in each case, the phase the photon ends up with at the end of each path is different. A photon avoiding the bomb and heading to detector D has the opposite phase as one that passes by the bomb and heads to detector D, and conversely, the two "versions" heading towards detector C end up with the same phase.
This means that, if the bomb is broken (or equivalently, not there at all) the two versions of the photon arriving at detector D cancel out and experimentally, a photon is never detected there - the photon is always detected arriving at C if there is no working bomb. This means that the corresponding list of possibilities is just:
- the photon ends up at detector C, always
Now, the difference between the two lists is useful: cases 2 and 3 are what I was talking about earlier, they are observable events that can only happen when the bomb is present and working - IOW, they are caused by the bomb being present.
Now, the causality of case 3 is obvious - you can't detonate a bomb that's not there, and the bomb won't go off without the photon. So far so good. But what about case 2, detector D going off?
A photon can be detected at detector D only if the bomb is live, but the bomb is intact afterwards, because no actual photon reached its trigger to set it off - it hit detector D instead, and a photon can't be detected in two places at once. The possibility of the photon setting off the bomb has somehow influenced the final observed result even though it never "actually" happened.
I opened by calling this a thought experiment, but actual experiments have been done demonstrating the same effect, which has been termed "counterfactual measurement." While there are many ways to interpret QM, I personally find it very hard to find any explanations of this result that don't end up ascribing reality to the "possible" photon that could-have-but-didn't hit the bomb. I hope that helps answer some of your questions, and also illustrates that finding observational evidence of the normal sort of possible world (imaginary hypotheticals) is conceptually iffy because they don't work like quantum does, and need to for normal scientific methods to apply.