# Srednicki's explanation of the 1PI quantum action

In chapter 21 (p.127-129) of Srednicki's book the quantum action $$\Gamma(\phi)$$ is defined in formula (21.1) I won't repeat here (it's quite long). Then he considers the following path integral:

$$Z_\Gamma(J) = \int \mathcal{ D} \phi \exp\left[ i\Gamma(\phi) + i \int d^dx J\phi\right] = \exp[iW_\Gamma(J)].\tag{21.4}$$

$$W_\Gamma(J)$$ is given by the sum of connected diagrams (with sources) in which each line represents the exact propagator, and each $$n$$-point vertex represents the exact 1PI vertex $$\bf{V}_n$$.$$W_\Gamma(J)$$ would be equal $$W(J)$$ if we included only tree diagrams in $$W_\Gamma(J)$$.

Isolating the tree-level contribution to the path integral by means of introduction of the dimensionless parameter called $$\hbar$$ we have the following path integral:

$$Z_{\Gamma,\hbar}(J) = \int \mathcal{ D} \phi \exp\left[ \frac{i}{\hbar}\left(\Gamma(\phi) + i \int d^dx J\phi\right)\right] = \exp[iW_{\Gamma,\hbar}(J)]\tag{21.6}$$

It is followed by the following text:

In a given connected diagram with sources, every propagator (including those connected to sources) is multiplied by $$\hbar$$, every source by $$1/\hbar$$, and every vertex by $$1/\hbar$$.

1. I don't get this assertion. There is no hint neither why it should be like this.

Later on $$W_{\Gamma,\hbar}(J)$$ is developed in a series of loop occurrence in diagrams, or in different words a series in orders of $$\hbar$$:

$$W_{\Gamma,\hbar}(J)= \sum_{L=0}^{\infty} \hbar^{L-1} W_{\Gamma,L}(J).\tag{21.8}$$

I can understand the formula under the assumption that I take the above cited assertion for correct/granted. Srednicki's keeps on saying:

If we take the formal limit of $$\hbar\rightarrow 0$$, the dominant term is the one with $$L=0$$, with is given by the sum of tree diagrams only. This is just what we want. We conclude that:

$$W(J) = W_{\Gamma, L=0}(J).\tag{21.9}$$

1. This assertion is also curious (I don't understand it ). I thought that $$W(J)$$ would be the sum of all connected diagrams whereas here it seems to be only a subset, i.e. all connected diagrams without loops. I assume that it is related with what was said above (the reason I included the part above).

Actually, it seems that Srednicki's explanation of the quantum action is elegant, so I would really appreciate if somebody could explain the mentioned assertions to me what they actually mean.

1. The $$\hbar$$-weights are a direct consequence of eq. (B9) in my Phys.SE answer here:
• A vertex has $$\hbar$$-weight$$=-1$$.
• An internal propagator comes with $$\hbar$$-weight$$=+1$$ (rather than $$-1$$) because an internal propagator is accompanied by 2 source differentiations (which each carry $$\hbar$$-weight$$=+1$$).
• By the same token, an external leg (=source+propagator) has $$\hbar$$-weight$$=0$$ because it is accompanied by 1 source differentiation. Hence a source has $$\hbar$$-weight$$=-1$$.
2. $$W(J)$$ is by definition a sum of all possible connected diagrams made out of bare propagators and (amputated) bare vertices. The statement is that it is also the sum $$W_{\Gamma, L=0}(J)$$ of all possible trees made out of full propagators and (amputated) 1PI vertices. This is also explained in my Phys.SE answer here.