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In the context of statistical field or quantum field theory, one encounters so called generating function(al) for connected correlations, aka the following function(al):

$$ W(J) = \ln (Z(J))$$

$$ Z(J) = \int \mathcal{D} \phi e^{-S[\phi]}$$

From a probabilistic standpoint, Z(J) is just the moment generating function(al) while $\ln Z$ is the cumulant generating function(al), I think one generates $\langle \phi^2 \rangle$ for example and the other generates $\langle (\phi - \langle \phi \rangle)^2 \rangle$.

However, we further define another generating function(al), call $\Gamma(\phi)$ usually called the "Effective Action" defined via legendre transform of the function(al) $W(J)$.

  1. Why is $\Gamma[\phi]$ the effective action a useful quantity to calculate?
  2. What is the probabilistic interpretation of the 1PI diagrams it generates? (cumulant, variance etc...).
  3. If correlation functions tell me how lab measurements of $\phi(x)$ and $\phi(y)$ at space/time points x/y would correlate, what do the 1PI functions generated by $\Gamma$ physically represent?

  4. Bonus question! The convex nature of the functional W(J) (the fact the Hessian $${\delta^2 \over \delta J(x) \delta J(y)} W(J) > 0 \forall x, y$$) ensures that the legendre transform is bijective. How do we motive this fact physically? Are there cases when this is not true and the effective action is no longer a useful quantity?

Any discussion of the intuition about the quantity $\Gamma[\phi]$ would help, even if it is in the lower dimensional case where $\phi$ is not a field but a single random variable for example.

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  • $\begingroup$ If you notice I've included the term function(al) because I believe this theory can be treated both in the continuum and discrete case. In one case $\phi$ is a statistical/quantum field, and in the second case, $\phi$ as a single random variable. Please correct me if I'm mistaken. $\endgroup$ – physicsdude Jun 17 '17 at 5:36
  • $\begingroup$ Post (v2) discussed in the chat room here. $\endgroup$ – Qmechanic Jun 28 '17 at 12:26
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In probability theory $W=\log Z$ is called the generating function for cumulants. Its Legendre-Fenchel transform $\Gamma$ is the rate function appearing in theorems on large deviations such as Cramer's Theorem. I think if you want more information about the probabilistic meaning of $\Gamma$, then you should lookup the literature on the theory of large deviations. A classic book on the subject is "Large Deviations Techniques and Applications" by Dembo and Zeitouni. A nice introduction with emphasis on statistical mechanics is "A Course on Large Deviations with an Introduction to Gibbs Measures" by Rassoul-Agha and Seppäläinen.

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Replace in your formula $S,W,\Gamma$ by the same letters divided by the Boltzmann constant $k$ (the probabilistic analogue of Planck's constant $\hbar$). Then, in the limit of $k\to 0$, the effective action becomes the original action, $\Gamma=S+O( k)$. Thus the effective action is the low temperature (in statistical mechanics ) resp. semiclassical (in quantum field theory) analogue of the (zero temperature, resp. classical) action, and can be determined by an expansion in terms of $k$. As it contains all information about the probability distribution, it is a very useful quantity.

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