Berezin's correspondance of (anti-)symmetric function with functional

[Tschijnmo]

Hi all, I have recently been reading the book ``The Method of Second Quantization'' by Felix Berezin but I got trapped on just page 4, where the concept of generating functionals is introduced. It seems to be assigning each (anti-) symmetric function of N variables with a functional of a function of just the degrees of freedom of one of the particles. And in the last sentence of the page, it is commented that ``Knowing the functional \Phi(a^*) and \tilde{A}(a^*, a), one can obviously construct the vector \Phi and the operator \tilde{A}''. But even after a serious amont of thinking, I am still not able to be the obviousness here. Google search did not seem to have yielded some clear answer. Even the book seems to have been highly cited, but I really cannot find a detailed explanation to it. Could someone here give me some guidance? Thank you so much!

 

[strangerep]

Hi all, I have recently been reading the book ``The Method of Second Quantization'' by Felix Berezin but I got trapped on just page 4, where the concept of generating functionals is introduced. It seems to be assigning each (anti-) symmetric function of N variables with a functional of a function of just the degrees of freedom of one of the particles. And in the last sentence of the page, it is commented that ``Knowing the functional $\Phi(a^*)$ and $\tilde{A}(a^*, a)$, one can obviously construct the vector $\hat\Phi$ and the operator $\tilde{A}$''. But even after a serious amount of thinking, I am still not able to be the obviousness here.

That book is a difficult study, as Berezin assumes quite a lot of the reader.

The stuff on page 4 you mentioned is part of an "introduction". A more detailed explanation follows in chapter 1 (though, as I look through it, I can't help thinking there must be more helpful ways of presenting this stuff).

Given the $\Phi(a^*)$ in Berezin's eq(0.10), i.e.,
$$
\Phi(a^*) ~:=~ \sum_n \frac{1}{\sqrt{n!}} \int K_n(x_1,\cdots,x_n) \;
a^*(x_1) \cdots a^*(x_n) \; d^n x ~,
$$
we want to extract the components of the vector $\hat\Phi$ in eq(0.1) which consists of the various $K_n(\dots)$ functions in the integrand. This is usually done with the help of a functional derivative. E.g., to extract $K_1(y)$, use a single functional derivative like this:
$$
K_1(y) ~=~ \frac{\delta \Phi(a^*)}{\delta a^*(y)} \; \Big|_{a^*=0}
$$
This uses
$$
\frac{\delta a^*(x)}{\delta a^*(y)} ~=~ \delta(x-y)
$$
(i.e., a Dirac delta on the right hand side). This extracts one term from the sum of integrals, and all the others vanish after applying $a^*=0$ as the last step.

For higher order $K_n(\dots)$ we use higher order functional derivatives, apply the Leibniz product rule carefully when differentiating the integrands (which results in a factor of $n!$, iirc), and possibly introduce an extra factor of $\sqrt{n!}$ somewhere to compensate.

I hope that's enough to give you the basic idea. Such use of functional differentiation is very common when working with generating functionals.