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I have some doubts regarding equation 11.58 (see below) in the QFT book by Peskin and Schroeder. If I understand correctly, they are expanding the Lagrangian density about $\phi_{\text{cl}}$ by writing $\phi(x) = \phi_{\text{cl}}(x) + \eta(x)$, after which they do the following expansion: $$ \mathcal{L}_1[\phi] = \mathcal{L}_1[\phi_\text{cl}] + \int d^4x\; \frac{\delta\mathcal{L}_1}{\delta\phi(x)} \eta(x) + \int d^4x \; d^4y\; \frac{1}{2!} \frac{\delta^2\mathcal{L}_1}{\delta\phi(x)\delta\phi(y)} \eta(x) \eta(y) + \cdots,$$ where all the functional derivatives are evaluated at $\phi_\text{cl}$.

My question is: the Lagrangian density is simply a function of $\phi$, not a functional (cf. e.g. this Phys.SE post), why then are we taking functional derivatives instead of 'ordinary' derivatives? Also, if the formula above is correct, shouldn't there be a double integral in the term linear in $\eta(x)$ in Eq. 11.58?

Quoted from p. 371 of Peskin and Schroeder's QFT book:

P&S QFT 11.58

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OP has a point. If $$S_1~:=~\int d^4x ~\mathcal{L}_1\tag{*}$$ denotes the corresponding action functional, then the 3 last appearances of $\mathcal{L}_1$ in eq. (11.58) should strictly speaking be $S_1$ not $\mathcal{L}_1$. Be aware that such abuse of notation as in eq. (11.58) is quite common. See also e.g. this related Phys.SE post.

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  • Thank you for the clarification! That equation finally makes sense to me now. I am a bit perplexed that the authors did not explain that abuse of notation to the readers (or maybe they did and I simply skimmed over it?). Anyway, thanks! – seric Mar 03 '17 at 08:21