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Is it the case that chirality is an intrinsic property of elementary fermions? I mean, is it the case that all electrons have left handed chirality while all positrons have right handed chirality? And is it possible to measure the chirality of a particle?

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No, massive particles necessarily have components corresponding to both left-handed and right-handed chirality. "Only left-handed fermions and right-handed antifermions interact with the weak interaction", but right-handed Fermions do exist and chirality can change dynamically.

Regarding measurability, see this question.

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