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I will be referring to the following document (page 16-17): http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf

I would like to understand the expression of the generating function Z0[J] on page 17.

The document mentioned that I need to make a Wick rotation by letting τ=it and, as a result, I also need to let p0 --> ip0.

I observed that the expression for Z0[J] on page 17 is obtained by substituting p0=ip0 in the expression of Z0[J] on the bottom of page 16, which is the expression of the generating function in Euclidean space.

However, I'm wondering why didn't the document let τ=it in the expression of Z0[J] on page 17?

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  • $\begingroup$ This aspect of QFT (duality between quantum field theory in Minkowski spacetime on one hand and statistical physics on Euclidean space on the other) is infamous for how poorly it is covered in introductory textbooks. It is still not completely understood, actually. Look for the relation between Wightman and Osterwalder-Schroeder axiomatics for the modern treatment. $\endgroup$ – Prof. Legolasov Aug 31 '18 at 10:51
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    $\begingroup$ Related post by OP: physics.stackexchange.com/q/425674/2451 $\endgroup$ – Qmechanic Aug 31 '18 at 11:07
  • $\begingroup$ $Z^M_0[J] =Z^E_0[J]$. $\endgroup$ – Qmechanic Sep 1 '18 at 15:30

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