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In thermodynamics and statistical mechanics, we learn that many subatomic particles, such as electrons, are indistinguishable. But are they really? I can understand from one perspective that if we swapped one electron for another, we wouldn't be able to tell the difference because of its mass, charge, size, etc.

However, I have an argument against this and I would like to obtain some clarity on the topic. My argument is simply the "identity of indiscernibles" which states that there cannot be separate entities that have all their properties in common. In other words, if multiple particles share all of their properties, then they are the exact same particle, no distinction whatsoever.

Suppose we have two electrons (one in our left palm and the other in our right palm), then they share all of their properties except one: spatial coordinates, where they are located. Thus, they are in fact two separate electrons, not one. I do agree if we swapped one for the other, there would be no distinction, but as long they occupy different positions, they are distinguishable.

So how can we say that particles, such as in a lattice, are truly indistinguishable since each one is located at a different position? Any clarity would be appreciated.

RJ Onyx Moonshadow
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I do agree if we swapped one for the other, there would be no distinction, but as long they occupy different positions, they are distinguishable.

You are simply using the word "distinguishable" in a different way than it is used in the sense of elementary particles. In its technical sense, the fact that electrons are indistinguishable means that the state of a quantum mechanical system is invariant under their interchange, or equivalently that the electron exchange operator is unitary (in fact, proportional to the identity operator).

Albatross
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This question is answered by QFT. Indeed, the position of a particle is not an observable of the theory, but rather "parameters". Therefore, it makes no sense to attach a position label to a particle, as this is something that cannot be observed.

Consider the purely relativistic effect of a particle decay and it's potential reappearance later on. Once a particle has decayed, what's its position? On the other hand, such a problem does not exist when you have enough fields that can act as "reservoirs" of energy and momentum. Particles are then just "ripples" of these fields, some of which we can actually observe.

Phoenix87
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So long as the wave functions of the particles in your hands do not overlap, you can say that they did not swap places. But this does not describe the quantum state, which describes the particles themselves as identical. While the wave functions of the particles does not overlap, there will be no change to calculations.

For the particles in a lattice, you have a tensor product consisting of wave functions for the individual particles. Suppose you detect the position of one of the particles. All you can say is "I found a particle at that position". You cannot actually say which particle it was, beyond the fact that it was at that position. This is reflected in the (anti-)symmetrising of the state. It does not say "all the particles are at all the positions", just that the state does not distinguish which particle is at which position.

Charles Francis
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