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In the book 'Field Quantization' (Greiner & Reinhardt, 1996), on page 37, talking about the calculus of variations, the authors introduce the notion of the chain rule for functional derivatives. Namely, if $F[\phi]$ and $G[\phi]$ are functionals, then $$\frac{\delta}{\delta \phi(y)} F[G[\phi]] = \int dx \frac{\delta F[G]}{\delta G(x)} \frac{\delta G[\phi]}{\delta \phi(y)}\,.$$ However, I am confused by the notion of functional of a functional. Earlier on page 37, the authors define a functional $F$ as

a mapping from a normed linear space of functions (a Banach space) $M = \{\phi(x): x \in \mathbb{R}\}$ to the field of real or complex numbers, $F: M \to \mathbb{R}\ or\ \mathbb{C}$.

Thus, a functional should take as input a function of a real variable. How should I think about the functional of a functional? In particular, I get stuck when I try to write even simple examples. For instance, let $F[\psi] = \int_0^1 \psi(x) dx$ and $G[\psi] = max_{x\in [0,1]} \psi(x)$; What is $F[G[\phi]]$ in this case?

Federico Marotta
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  • The word "functional" has two conflicting meanings. In abstract algebra and functional analysis, it's an element of the dual space (which is the definition you cite). In some other areas (including PDE), it is common to use the word to refer to a function (possibly non-linear) whose values are functions. It is a bit odd that the authors use both conventions in one book (even one page...) without commenting on the distinction, though. – tomasz Aug 01 '21 at 15:23

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I see no other interpretation than $G$ being a function-valued functional, e.g. $G:C^\infty(\mathbb R)\to C^\infty(\mathbb R).$ An example of such a functional is $G[\phi](x)=e^{-x^2/2}\phi'(x)^2.$

Then $F[G[\phi]]$ makes sense and $$ \frac{\delta F[G[\phi]]}{\delta\phi(x)} = \int \frac{\delta F[G[\phi]]}{\delta G[\phi](y)} \frac{\delta G[\phi](y)}{\delta\phi(x)} \, dy . $$

md2perpe
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