# Effective action as a generating functional and its derivative expansion

On page 381 of Peskin and Schroeder, equation (11.90) reads

$$\frac{\delta^2 \Gamma}{\delta \phi_{cl}(x)\delta \phi_{cl}(y)} = iD^{-1}(x,y).\tag{11.90}$$

I am having a bit of trouble interpreting this formula. On the left-hand side, we have the second derivative of the quantum effective action. However, on page 130 of Srednicki, the quantum action is written in terms of a derivative expansion, $$\Gamma[\phi_{cl}] = \int d^4x \,\left[-{\cal U}(\phi_{cl}) -\frac{1}{2}{\cal Z}(\phi_{cl})\partial^{\mu}\phi_{cl}\partial_{\mu}\phi_{cl}+\ldots\right],\tag{21.19}$$ suggesting that the quantum action is local, and hence $$\frac{\delta^2 \Gamma}{\delta \phi_{cl}(x)\delta \phi_{cl}(y)} \propto \delta(x-y).$$

However, the propagator on the right-hand side is not proportional to the delta function. So what is going on?

The classical action is local, while the quantum effective action is very much not so.

The quantum effective action is the sum over one-particle irreducible diagrams. You can easily see that these diagrams are not polynomial in the momenta. In fact, they can contain poles and logs.

• I'm confused then as to why Srednicki chooses to write the quantum action in the form I've given in the edited question. Does this not suggest that the action is local? Apr 9 at 23:27
• @awsomeguy If the derivative expansion is only local if it has a finite number of terms. If it is an infinite series, then it is generally non-local. Apr 9 at 23:34

The effective action $$\Gamma[\phi_{\rm cl}]$$ is the generator of 1PI-diagrams with bi-local propagators, and hence a non-local functional by construction.

1. Refs. 1-2 are less ambitious and do not discuss a derivative expansion per se, but instead consider $$x$$-independent field configurations $$\phi_{\rm cl}$$. From spacetime translation invariance it then follows immediately that the functional $$\Gamma[\phi_{\rm cl}] ~=~-\text{Vol}_4~{\cal V}(\phi_{\rm cl})\tag{11.50/16.2.3}$$ reduces to a function, known as the effective potential. This is of course the leading term in a derivative expansion.

2. The conceptionally simplest way to understand the derivative expansion (21.19/9-116) in Refs. 3-4 is to introduce a coupling constant $$\lambda$$ in front of each spacetime derivative $$\partial_{\mu}\to \lambda\partial_{\mu}$$ in the action $$S[\phi]~=~S_{\rm free}[\phi]+ S_{\rm int}[\phi].$$ Then the kinetic derivative terms get demoted to the interaction part $$S_{\rm int}[\phi].$$ Since the free part should remain non-degenerate, perturbation theory would then only work for massive fields. In this case the free propagators $$G_0(x\!-\!y)~\propto~ \delta^4(x\!-\!y),$$ and hence all connected Feynman diagrams become localized in position-space, so that the generator $$\Gamma[\phi_{\rm cl}]$$ of 1PI-diagrams can be written as a spacetime integral over (infinitely many) mono-local terms, i.e. the aforementioned derivative expansion.

However, this approach is in many way too$$^1$$ radical, and one would have to re-sum diagrams just to get the leading term, i.e. the effective potential.

3. A less radical approach would be to consider the tree-level formula $$\Gamma_0[\phi_{\rm cl}]~=~S[\phi_{\rm cl}]\tag{0}$$ and 1-loop formula $$\Gamma_1[\phi_{\rm cl}] ~=~\frac{i\hbar}{2}\ln {\rm Det}\left(\frac{1}{i}\frac{\delta^2 S[\phi_{\rm cl}]}{\delta \phi_{\rm cl}^k \delta \phi_{\rm cl}^{\ell}}\right),\tag{1}$$ which are manifestly local functionals, cf. e.g. my Phys.SE answer here.

To write the higher-loops $$\Gamma_{\geq 2}[\phi_{\rm cl}]$$ as a spacetime integral over (infinitely many) mono-local terms, it seems one would have to resort to methods mention under point 2. For a recursive formula, see e.g. my Phys.SE answer here.

4. By the way, it is interesting to compare the non-locality of the 1PI effective action with the non-locality of the Wilsonian effective action, see e.g. my Phys.SE answer here.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT; eq. (11.50).

2. S. Weinberg, Quantum Theory of Fields, Vol. 2, 1996; eq. (16.2.3).

3. M. Srednicki, QFT, 2007; eq. (21.19). A prepublication draft PDF file is available here.

4. Itzykson & J.-B. Zuber, QFT, 1985; eq. (9-116).

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$$^1$$ Parts of the 1-loop formula (1) would become $$\lambda$$-dependent. The functional determinant (1) can in many case be performed without expanding in $$\lambda$$,