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Can we born rule be used to find probability of a particle to exist in a region in QFT using the formula $\int_a^b \psi(x)\psi^*(x)dx$,where $\psi(x)$ is a fermionic field? If yes, please provide citations.

I cannot understand how can born rule be applied to a particle since a field contains multiple particle and all particles are explained by the plane wave solutions of the field.

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  • $\begingroup$ Can you be more specific why you think QFT might not be "compatible" with the Born rule? It's currently unclear what sort of answer one could give here except a simple "Yes, the Born rule applies to QFT". $\endgroup$
    – ACuriousMind ♦
    Commented Jun 6, 2022 at 13:42
  • $\begingroup$ @ACuriousMind Can we born rule be used to find probability of a particle to exist in a region in QFT using the formula $∫_b^aψ(x)ψ^*(x)dx$,where ψ(x) is a fermionic field? If yes, please provide citations. I cannot understand how can born rule be applied to a particle since a field contains multiple particle and all particles are explained by the plane wave solutions of the field. $\endgroup$
    – Ace
    Commented Jun 6, 2022 at 14:11

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The quantum fields are not wavefunctions, they are operators. The Born rule holds, but you need to apply it to the states of quantum field theory (mostly created by applying the field operators to the vacuum state), not to its operators.

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  • $\begingroup$ You mean the notation needs to be $\int_a^b <x|\phi|p><p|\phi|x>dx$. Can you please provide some citations? $\endgroup$
    – Ace
    Commented Jun 6, 2022 at 14:25
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    $\begingroup$ @Ace No, that's not what I mean (there are no position eigenstates in relativistic theories, for one). If you do not know the basics of quantum field theory necessary to understand my answer, I cannot really give you an answer here except that you should read a QFT textbook. $\endgroup$
    – ACuriousMind ♦
    Commented Jun 6, 2022 at 14:43

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