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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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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 at 21:57

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