I don’t wish to ignore the rest of your post, but I got caught here with something I don’t understand. You said that even if we can identify them uniquely by their position and momentum, it doesn’t follow that they are distinguishable. (If my paraphrase shows some misunderstanding, that is a good thing. Tell me about it. )
But I don’t understand this. If you’ve identified them uniquely by their position and momentum, then haven’t you ipso facto distinguished them, viz., by their position and momentum?
You can look at an individual particle (or intertwined collection of particles), but you can’t go away and come back to it later with certainty that you’re looking at the same particle; it might be another one whose position and momentum functions overlap the original one, and which is otherwise identical in all salient characteristics. This may not seem terribly significant until you realize that it causes problems in figuring out whether the system is deterministic or not; if you can’t distinguish individuals, you can’t establish whether a particle was where you found it because you predicted it to be there, or because another completely identical particle happened to be where you expected one.
To put it another way: Suppose I have two pool balls, the cue and the 8. They roll towards each other, collide in a satisfying “bonk”, and roll off in two new directions. In this collision, I can say things like “the ball that came from the north went off to the east”, since a cue ball is distinguishable from an 8 ball. But if I do the same thing with electrons, I can’t say that the electron which came from the north ended up going off to the east. Maybe it was the one from the south. In fact, both of these possible processes appear to contribute equally to the final outcome.
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