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Can You Prove Two Particles Are Identical?

Mitchell, the essence of indistinguishability is NOT "that if you were to get the two particles and move them around so that they occupied the other position, that would count as the same configuration, quantum mechanically." Bob doesn't care about a version of indistinguishability that restricts the relevant properties to those important for QM. QM-indistinguishability is not indistinguishability. So if that's the notion of indistinguishability at play here, then Bob could accept that QM can determine that two electrons are QM-indistinguishable while still objecting to the in-principle possibility of determining indistinguishability simpliciter.

But if the sense of indistinguishability in which the electrons are supposed to be indistinguishable is the simpliciter sense, then I still don't see how QM responds to the underlying structure argument above.

Can You Prove Two Particles Are Identical?

I've read this post several times, as well as the previous three posts, and I still can't see how the theory guarantees the indistinguishability of any two particles. Admittedly, I'm weak on the math, so maybe there was a really clear mathematical explanation that was wasted on me (I take it that the amplitude equations were supposed to rule out distinguishability in virtue of difference in, say, spin values at the 100th decimal place, but I couldn't tell for sure).

But, much like our stubborn philosopher Bob, I still think that the theory is insufficiently comprehensive and precise to justify the positive claim of indistinguishability of any two particles. I would greatly appreciate Eliezer's critique of the following stab at an argument on Bob's behalf.

Here goes: If a particle is supposed to be an object with mind-independent properties, then it seems that an object must bear its properties either in virtue of an underlying structure, or it must bear properties basically—not in virtue of any deeper structure. If particles (electrons, in this case) bear their properties in virtue of underlying structure, then, in virtue of this structure, each will have an infinite number of positive properties. This is because the spatial relational properties borne by the sub-components to each other would admit of infinite degrees of precision. And as far as I can tell, the theory in question deals with the "macro" properties of the electron, not the properties had by an electron's structural components in relation to each other. As such, the theory wouldn't seem to tell us one way or the other about indistinguishability between electrons with respect to the degrees of difference in each's substructural spatial relationships at many hundreds of decimal places of precision.

On the other hand, perhaps particles bear their properties basically. In this case, an electron would have mass, spin, charge, or whatever other properties are detectable and relevant for experimentation and theorizing, and its having these properties would not be explicable in virtue of more fundamental structure. As such, there would be no host of substructural relational properties to distinguish particles, as there would be nothing else "down there" to bear such properties. If this is in fact the case with electrons, we would seem to be dealing with something very unlike the continuants of philosophy. Here particles are more like clusters of a small handful of properties.

But I see no way that a theory could guarantee the basicality of the properties of a particle. Therefore, I take it that the theory doesn't tell us determinately that any given particle absolutely lacks any more fundamental structure. How could it, even in principle? As such, I don't see how any theory could tell us that a particle is indistinguishable from another (due to the above argument from the ineliminable potential for infinitesimal difference in substructural relational properties).

This argument doesn't depend on the billiard ball conception, and doesn't even require sameness of identity over time for any given particle. It should apply to any two particles in any configuration space, at any given time.

Thanks for any feedback you're willing to give, and I apologize for the length of the post!