I´m wondering how can one get to the definition of Functional Derivative found on most Quantum Field Theory books:
$$\frac{\delta F[f(x)]}{\delta f(y) } = \lim_{\epsilon \rightarrow 0} \frac{F[f(x)+\epsilon \delta(x-y)]-F[f(x)]}{\epsilon}$$
from the definitions of Functional Derivatives used by mathematicians (I´ve seen many claims that it is, in effect, the Fréchet derivative, but no proofs). The Wikipedia article says it´s just a matter of using the delta function as “test function” but then goes on to say that it is nonsense.
Where does this $\delta(x-y)$ comes from?
Best Answer
Whenever I have troubles with functional derivative things, I just do the replacement of a continuous variable $x$ into a discrete index $i$. If I'm not mistaken this is what they call a "DeWitt notation".
The hand waiving idea is that you can think of a functional $F[f(x)]$ as of a "ordinary function" of many variables $F(f_{-N},\cdots,f_0,f_1,f_2,\cdots,f_N) = F(\vec{f})$ with $N$ going to "continuous infinity".
In that language your functional derivative transforms into partial derivative over one of the variables: $$\frac{\delta F}{\delta f(x)} \to \frac{\partial F}{\partial f_i}$$ And the delta-function is just an ordinary Kronecker delta: $$\delta(x-y) \to \delta_{ij}$$
So, gathering this up we for your expression: $$\frac{\delta F}{\delta f(x)} = \lim_{\epsilon\to\infty}\frac{F[f(x)+\epsilon\delta(x-y)]-F[f(x)]}{\epsilon} \to$$ $$\frac{\partial F}{\partial f_j} = \lim_{\epsilon\to\infty}\frac{F[f_i+\epsilon\delta_{ij}]-F[f_i]}{\epsilon} $$ Which is, to my taste, a bit redundant. But true.