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When we perform a Legendre transform on the connected generate functional $W[J]$ we get the quantum action (or 1PI action)

$$ \Gamma[\phi] = W[J(\phi)] - \int\mathrm{d}^4x\,\phi J,\quad\phi(J)=\frac{\delta W}{\delta J}. $$

Then it can be shown that

$$ \Gamma[\phi] = S[\phi] \mp\frac{1}{2}\log\det\left(\frac{\delta^2S}{\delta\Phi(x)\delta\Phi(y)}\right)_{\Phi=\phi} +\ldots, $$

where $S[\phi]$ is the classical action and the dots represent higher corrections. It is said that the lowest quantum correction (ie, the term involving $\log\det$) is the result of a resummation of one loop diagrams.

  1. Why is the $\log\det$ term identified with a one loop correction? I took a look at the proof, but it seems to me that there is no connection to the one loop diagrams at all.

  2. Why is $\frac{\delta^2S}{\delta\Phi(x)\delta\Phi(y)}$ identified with the propagator? Is it the free propagator or the exact propagator (including interactions)?

  3. Is it possible to get the same one loop correction to the action using the Wilson effective action instead?

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closed as too broad by Qmechanic Jun 10 at 5:39

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