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I understand that two fermions cannot simultaneously have the same <momentum, spin> state. I know this is also true of two anti-fermions. But is it possible for one fermion and one anti-fermion to occupy the same state?

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As you stated, no two fermions or anti-fermions can occupy the same state. And here by "state" we mean all quantum numbers defining the properties of our particle. So since electric charge for e.g. electron and positron are different, they can be in the same state.

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  • $\begingroup$ Thanks very much for this. I'm still confused because it seems to me that what can use the commutator relations to argue for a different conclusion. I've posted a follow up here: physics.stackexchange.com/questions/605831/… . $\endgroup$ Jan 6, 2021 at 22:17
  • $\begingroup$ Isn't that if they do they annihilate? $\endgroup$
    – Alchimista
    Jan 7, 2021 at 8:40
  • $\begingroup$ they will annihilate very fast. even the positronium does en.wikipedia.org/wiki/Positronium $\endgroup$
    – anna v
    Jan 7, 2021 at 9:31
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Pauli exclusion principle holds for identical particles. With that in mind, one fermion and one anti-fermion can be in the same state (if that particular fermion is not its own antiparticle).

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The small singlet triple splitting of positronium proves that there is no exchange interaction in this system, hence that its state is not antisymmetric, nor symmetric, under the exchange of p and e.

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    $\begingroup$ p is usually a proton, not a positron. If you want to abbreviate, you should probably use $e^-$ and $e^+$. $\endgroup$
    – PM 2Ring
    Jan 6, 2021 at 21:57

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