Definition of functional derivative

[PreposterousUniverse]

TL;DR Summary

Ambiguous definition for the functional derivative.

In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as:

$$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$

Why do they use the delta function and not some other arbitrary function?

 

[jambaugh]

The delta symbol is being used to distinguish differentials when acting on a function being treated as an independent variable rather than a dependent value.

EDIT: I got long-winded and you may want to first skip down to the end where I say "In a less trivial example..." and then come back if you like and read from the beginning.

Consider when we define differentials of variables e.g. dx, this indicates a change in the value x. Then df(x), the differential of a function of x is the change in its dependent value. It is then a function whose value depends on x and on the differential variable dx. We can of course then extend this to vector variables and functions thereof.

When looking at functional derivatives, the function itself is a variable (imagine it as an infinite dimensional vector). If we consider a functional of a function G[f] (I use brackets to distinguish this level) then this is distinct from a composition g(f(x)) as the value of G depends on the whole of the function across all of its values. For example:
$$G[f] = \int_0^1 f(x)\cdot f'(x)dx$$
When considering a functional, its value may depend not only on the function's value at x but on its derivatives and antiderivatives. It is thus important in considering functional differentials to distinguish them from variable differentials applied to functions. $\delta f$ is a differential function by which we can define the functional derivative as a limit of a difference quotient.

It was very helpful to me for understanding this to, first, fully understand the general derivative of vector-valued functions of vectors, i.e. multivariable derivatives. We often write this as the Jacobi matrix. I find it also much clearer to start with differentials, specifically the Gateaux differential. Firstly the differential of a variable is an operator mapping it to another independent variable, e.g. $d[x]=dx$. Importantly, do not think of differential variables as "infinitesimals" they are unrestricted. But they are important in the limits at 0.

Let $f$ be a function mapping vectors to vectors (possibly in different spaces). Consider then the function relation: $\boldsymbol{y} = f(\boldsymbol{x})$. We then define the Gateaux differential of the dependent value $f(\boldsymbol{x})$ as:
$$d\boldsymbol{y}= df(\boldsymbol{x}) = \lim_{h\to 0} h^{-1}\left( f(\boldsymbol{x}+hd\boldsymbol{x}) - f(\boldsymbol{x}) \right) = G_f(\boldsymbol{x},d\boldsymbol{x})$$
[edit, corrected missing h in the difference quotient above.]
Here the result is a function $G_f$ of both $\boldsymbol{x}$ and $d\boldsymbol{x}$. Provided this limit exists one then finds this function is in-point-of-fact linear in the differential variable $d\boldsymbol{x}$ and thus can be expressed as the action of a linear operator valued function $f'$ of $\boldsymbol{x}$ on the vector $d\boldsymbol{x}$:
$$d\boldsymbol{y}=df(\boldsymbol{x})=f'(\boldsymbol{x})[d\boldsymbol{x})]$$

This defines the derivative as an operator. In the vector mapping case its matrix representation is that of a matrix of partial derivatives of components of vectors, (the Jacobi matrix). Note, however this is a basis-dependent construction and we should firstly think of the derivative as an operator.

Now comes the problem of functional derivatives. Since a functional (or functional operator) acts on functions, there is already the differential and derivative defined as other operators acting on functions. To define the functional Gateau differential we must use a new symbol to distinguish it e.g. $\delta$ from conventional variable differentials.
$$\delta G[f] = \lim_{h\to 0} h^{-1}\left( G[f+h \delta f] - G[f]\right)$$
We can then use this to e.g. take the functional derivative of the conventional derivative operator. (It is trival since the derivative is linear but importantly meaningful.)

Consider the derivative (in this context) of the simple homogenous linear function $f(x)=ax$. It is not quite the constant $a$ but specifically the "operator" $a\cdot$ mapping $x\mapsto a\cdot x$.
Make $x$ a vector and $a$ a matrix and you get a clearer picture of this distinction.

Now take functional Gateaux derivative of the conventional derivative operator, $D[f] = f'$ and you get that "the derivative of the derivative is the derivative".
$$\delta D[f] = \lim_{h\to 0} h^{-1}\left(d[f+h\delta f] - D[f] \right)=\lim_{h\to 0} h^{-1}\left(h D[\delta f]\right) = D[\delta f]$$
Thus the Gateau derivative of $D[f]$ is the derivative operator $D$ (as a constant with respect to $f$ but ready to act upon $\delta f$ to express the functional differential of a linear operator.)

In a less trivial example, one can take the functional differential and derivative of the entropy functional for the probability distribution of the state vector for a classical system or of the wave function for a quantum system. In the classical case:
$$S[f] = -\kappa\int_\mathbb{R} f(x) \ln(f(x)) dx$$
Here:
$$\delta S[f] =-\kappa \int_\mathbb{R}(\ln(f(x))+1)\delta f(x) dx$$
Note that $\delta f$ is an independent function of $x$ and not the conventional differential $df(x)=f'(x)dx$ of the function value $f(x)$. If it helps, let $\delta f(x) = g(x)$ and relabel its appearence in the functional derivative of the entropy.
$$(S'[f])[g] = -\kappa \int_\mathbb{R} (\ln(f(x))+1)g(x) dx$$

 

[malawi_glenn]

Define the variation of a functional $F[\phi(x)]$ as $\displaystyle \delta F[\phi] = F[\phi + \delta \phi] - F[\phi] = \int \dfrac{\delta F[\phi]}{\delta \phi} \delta \phi \mathrm{d} x$ the $\dfrac{\delta F[\phi]}{\delta \phi}$ is the functional derivative of the functional $F[\phi(x)]$ w.r.t the field $\phi(x)$.

Now construct a localized variation of the field $\phi(x)$ at the point $x=x'$ with magnitude $\epsilon$. This means that we can write $\delta \phi(x) = \epsilon \delta(x-x')$.
Insert this in the definition of functional variation:
$\displaystyle \delta F[\phi] = F[\phi +\epsilon \delta(x-x')] - F[\phi] = \int \dfrac{\delta F[\phi]}{\delta \phi} \epsilon \delta(x-x')\mathrm{d} x = \epsilon \dfrac{\delta F[\phi]}{\delta \phi}$

This resembles the ordinary relation we have for a variation of a function $f(x + \epsilon) - f(x) = \epsilon \dfrac{\mathrm d f}{\mathrm d x}$ which is valid in the limit "small" $\epsilon$.

Your book (well I also own it so I guess it's my book too!) then motivates the defintion of the functional derivative as
$\displaystyle \dfrac{\delta F[\phi]}{\delta \phi} = \lim_{\epsilon \, \to \, 0} \dfrac{F[\phi +\epsilon \delta(x-x')] - F[\phi] }{\epsilon}$

(I used $\phi$ insteaf of $f$ here) (note: you had some errors in the way you wrote down how the book defined the functional derivative...)

Let's pray a real mathematician chimes in at some point

TIP: if you really want a good QFT book get "Field Quantization" by Greiner. It might be hard to get, but sometimes you get get it very cheap used. It covers only field quantization methods in an entire book about 350 pages. This is what most QFT books spend on 2 chapters worth of 20-30 pages.

 

[vanhees71]

Summary: Ambiguous definition for the functional derivative.

In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as:

$$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$

Why do they use the delta function and not some other arbitrary function?

The correct definition of course is
$$\frac{\delta F}{\delta f(x)}=\lim_{\epsilon \rightarrow 0} \frac{F[f(x')+\epsilon \delta(x-x')]-F[f]}{\epsilon}.$$
It's a measure for the change of the functional when you vary the function $f$ a bit only at its argument at $x=x'$.

Usually the functional is defined by some integral. E.g., if
$$F[f]=\int_{\mathbb{R}} \mathrm{d} x' f(x') g(x')$$
with some function $g$ you get
$$\frac{\delta F}{\delta f(x)} = \lim_{\epsilon \rightarrow 0} \frac{\int_{\mathbb{R}} \mathrm{d} x g(x') \epsilon \delta (x-x')}{\epsilon}=g(x).$$

 

[dextercioby]

[...]

TIP: if you really want a good QFT book get "Field Quantization" by Greiner. It might be hard to get, but sometimes you get get it very cheap used. It covers only field quantization methods in an entire book about 350 pages. This is what most QFT books spend on 2 chapters worth of 20-30 pages.

Excellent recommendation. None of the Greiner books is weak in my opinion. These are a 30-year delay alternative to most (emphasize most as opposed to all) of Landau-Lifschitz's books. Some people even called them „the German L-L”. But in Germany, there were/are other professors who published several books in a series on general theories in physics. Nolting comes to my mind. However, Springer did not translate them all into English. @vanhees71 knows better than me as he is from Germany.