As premise I know barely anything about physics and thus this question might be absolutely trivial or ill-posed.
I am starting to study the White noise theory and when constructing the infinite-dimensional analogue of the Schwartz space $S(\mathbb R^n)$, i.e. $(S)$ one way is to grab a random variable $\psi\in L^2(S(R),\mu)$ with chaotic representation in term of Wiener-Ito integrals:
$$\psi=\sum_n I_n(f_n).$$
At this point an operator is introduced, namely
$$\Gamma (A) \psi= \sum_n I_n(A^{\otimes n} f_n),$$ where $A$ is the Hamiltonian of a harmonic oscillator, $A=-d^2/dx^2 +x^2+1$.
The author of the notes I'm reading states that this is the "second order quantization operator", also T.Hida makes allusion to such operator.
I went to Wikipedia to read about this "second order quantization" and honestly I don't quite understand if the two things are related at all.
Do you mind giving a brief (and "for a dumb") explanation on why this operator is called like that? I suspect it has to do with the Fock space, since the Homogeneous chaoses are related via an isometry with the Hilbert space symmetric tensor power.
Thanks in advance.