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A while ago I asked a question asking what is field strength renormalization (What exactly is field strength renormalization?). I now have a better way of thinking about this, which is that it relates the one particle state of the free theory to an interacting theory. This renormalization changes the the probability of a one particle state being created after the field acts on the vacuum from 1 (in the free theory) to $1/\sqrt{Z}$ (in the interacting theory). We call $\sqrt{Z}$ the field strength renormalization. (As a side remark, this does mean that $Z \geq 1$ so that the probability $1/\sqrt{Z}$ is bounded above by 1? Why do we take the square root of $Z$?)

However I am still confused on what is the difference between field strength renormalization and wavefunction renormalization. Many sources, including answers in the question linked above, state that they are the same. Other sources say that the wavefunction renormalization is $Z$ whereas the field strength renormalization is $\sqrt{Z}$. Which one is correct? If they are the same, why do both terminologies exist but are used in different contexts? If they are different, what is the difference other than one is $\sqrt{Z}$ and the other is $Z$?

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    $\begingroup$ If each $\psi\to\sqrt Z\psi$ then $\bar\psi\psi\to Z\bar\psi\psi$, and this is easier to work with. It clearly does not matter if we call the renormalisation $\sqrt Z$ or $Z$, we all agree the symbol and how to calculate it. There is no ambiguity in the context that it appears. $\endgroup$ Dec 4, 2023 at 22:13
  • $\begingroup$ @naturallyInconsistent Thank you, I see now why we take the square root of $Z$. Does this mean that $Z \geq 1$? Also, I am still confused on what is the difference between field strength and wavefunction renormalization. $\endgroup$
    – CBBAM
    Dec 4, 2023 at 23:54
  • $\begingroup$ There is no difference. Z>1 $\endgroup$ Dec 5, 2023 at 0:32
  • $\begingroup$ @naturallyInconsistent Thank you. In that case howcome in some texts (such as Peskin & Schroeder) they use both terminology? $\endgroup$
    – CBBAM
    Dec 5, 2023 at 3:24
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    $\begingroup$ Because when you have two authors they can use different ones interchangeably? Also, if you think two things are the same thing, even as one person I would also use the terms interchangeably. $\endgroup$ Dec 5, 2023 at 4:16

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