I think the meaning behind ‘identical particles’ is very hard to pin down without directly using mathematical definitions*. The analogy with (secretly numbered) billiard balls gives a strong intuition for non-identical particles. There are also intuitive examples that behave more like identical particles:
For example, the intuition for symbols nicely matches identical symbol/particle behaviour:
If I represent a Helium atom with the symbol “H” and no atom with “_”, the balloons interior might be described by
“H__H_H____H__H_____H_______H_H__HH____H”.
Here, it would still make sense to think ‘the Helium atom at this position’, but thinking ‘what if I wrote “the fifth H” at the position of “the third H” and vice versa?’ is not meaningful in the same way that the word “identical” remains “identical” even if I claim that I exchanged the two “i”.
Can’t we distinguish between particles through their relationships with other objects or “themselves”, including causal relationships? For example, the electrons in my body now have different (and stronger) causal effects on electrons in my body later than on electrons in your body, and by this we can distinguish them.
I think this way of distinguishing particles makes sense, but does not rely on ‘identity’ in the sense of identical particles – your example could be realized both with identical and non-identical particles, as ‘identifying’ a particle by its state remains valid in both cases.
And can’t we trace paths in spacetime for identity? Not particle-like paths, but by just relying on causality and the continuity of the wavefunction over spacetime?
The atom swap experiment would then destroy both atoms and create two atoms (possibly the same, possibly different, possibly swapped). What we could say about their identities would depend on the precise details of the view. Maybe there’s no coherent way to make this work.
A different, but consistent definition for individual particle-identity might be possible. But, as the experimental predictions** from identical particles are well-confirmed, it would still have to treat the way that two electrons have different identity in a different way than the different identity between an electron and, say, a photon. I do not see how one could get the qm-predictions without also using the identical-particle maths.
*) One (simplified) way to write it for 2 particles would be:
For two non-identical particles the wave function is defined over the space of ordered tuples of positions, e.g. (r_1, r_2). Here it makes sense to think ‘what happens if I exchanged the two particles?’ as (r_2,r_1) is generally not (r_1,_2) and ‘exchange particles’ is a meaningful term.
For identical particles instead, the wave function is defined on the space of unordered tuples, e.g. {r_1,r_2}. Here, ‘exchange particles’ is not meaningful as {r_1, r_2} and {r_2, r_1} per definition describe the same thing.
**) There are significant consequences: As the space that the wave function moves in is changed drastically, its behaviour also changes. E.g. everything solid builds on the Pauli principle, which is a consequence of identical particles
I think the meaning behind ‘identical particles’ is very hard to pin down without directly using mathematical definitions*. The analogy with (secretly numbered) billiard balls gives a strong intuition for non-identical particles. There are also intuitive examples that behave more like identical particles:
For example, the intuition for symbols nicely matches identical symbol/particle behaviour:
If I represent a Helium atom with the symbol “H” and no atom with “_”, the balloons interior might be described by
“H__H_H____H__H_____H_______H_H__HH____H”.
Here, it would still make sense to think ‘the Helium atom at this position’, but thinking ‘what if I wrote “the fifth H” at the position of “the third H” and vice versa?’ is not meaningful in the same way that the word “identical” remains “identical” even if I claim that I exchanged the two “i”.
I think this way of distinguishing particles makes sense, but does not rely on ‘identity’ in the sense of identical particles – your example could be realized both with identical and non-identical particles, as ‘identifying’ a particle by its state remains valid in both cases.
A different, but consistent definition for individual particle-identity might be possible. But, as the experimental predictions** from identical particles are well-confirmed, it would still have to treat the way that two electrons have different identity in a different way than the different identity between an electron and, say, a photon. I do not see how one could get the qm-predictions without also using the identical-particle maths.
*) One (simplified) way to write it for 2 particles would be:
For two non-identical particles the wave function is defined over the space of ordered tuples of positions, e.g. (r_1, r_2). Here it makes sense to think ‘what happens if I exchanged the two particles?’ as (r_2,r_1) is generally not (r_1,_2) and ‘exchange particles’ is a meaningful term.
For identical particles instead, the wave function is defined on the space of unordered tuples, e.g. {r_1,r_2}. Here, ‘exchange particles’ is not meaningful as {r_1, r_2} and {r_2, r_1} per definition describe the same thing.
**) There are significant consequences: As the space that the wave function moves in is changed drastically, its behaviour also changes. E.g. everything solid builds on the Pauli principle, which is a consequence of identical particles