Electrons are indistinguishable particles, however, when I set up two independent experiments (at two positions), I can talk about "the electrons in Experiment x". What's going on here?
I can model each electron by $|x, s \rangle$ such a state, so one electron can be in a 4-dimensional space spanned by: $|position1, +1/2 \rangle$, $|position2, +1/2 \rangle$, $|position1, -1/2 \rangle$, $|position2, -1/2 \rangle$.
The system of two electrons in there is 16 dimensional, but because of the anti symmetry we only have 6 basis vectors:
$|p1,+1/2, p1 -1/2 \rangle$, $|p2,+1/2, p2 -1/2 \rangle$, $|p1,-1/2, p2 -1/2 \rangle$, $|p1,-1/2, p2 +1/2 \rangle$, $|p1,+1/2, p2 -1/2 \rangle$ and $|p1,+1/2, p2 +1/2 \rangle$
In what way can we now talk about "the electron in experiment 1"? Or do we just mean the electron that happens to have position 1? What is the right way to think about this?
It seems that the question is being missunderstood: We already know that Electrons (ALL the electrons) are not distinguishable. The standard model describes electrons as excitations of an anticommuting quantum field, and this is a established model for electons. I don't ask wether this is true, and I don't ask why this is true.
YET when I set up an Ion trap or a quantum-computer that realizes q-bits via quantum states of electrons, it is an experimental fact that I talk about the "quantum states of the experiment". I don't talk about "all the electrons in the universe", I'm talking about the electrons in the specific machine that I have build.
And when I build another machine beneath the first one, then those two machines will (hopefully) work independently from one another.
This means although electrons are not distinguishable, we at least have a way of talking about electrons whose state will have a measurable effect in either Machine A or Machine B.
This two facts are seemingly contradictory. My question is how this contradiction can be resolved.
