About field operators

I put the question right now, writing some definitions later.

I want to calculate $ ||hatPsi^+(x)|0rangle||^2$, where $$hatPsi^+(x) = sum_{k = 1}^{infty}u^*_k(x) hat a^+_k
$$
is a field operator. So I do
$$ ||hatPsi^+(x)|0rangle||^2 = langle 0|hatPsi(x)hatPsi^+(x)|0rangle$$
Using the field operator’s commutation relation, this is equal to
$$ langle 0|(delta(x-y) pm hatPsi^+(x)hatPsi(x))|0rangle$$
Thus due to the fact that every annihilation operator destroys the vacuum state, only the delta term survives,
$$ langle 0|delta(x-y)|0rangle$$
But this results ends up being infinite, thus losing any physical sense. I can understand this but I’m not able to mathematically demonstrate it. Can someone enlighten me?

EDIT

For the sake of clarity, I’ll add more information which I actually understand. In the course of statistical mechanics I’m following, we’re facing the second quantization formulation. In particular, the ladder operators $Psi$ and $Psi^+$ have been introduced, and they’ve been defined as
begin{equation}
hatPsi(f) = sum_{k = 1}^{infty} f_k hat a_k, ; qquad hatPsi^+(f) = sum_{k = 1}^{infty} f^*_k hat a^+_k
end{equation}
They act by creating-destroying a particle in the state
$$ |frangle = hatPsi^+|0rangle$$
This should be equivalent of saying that a particle is in the state $f(x)=sum_{k = 1}^{infty}f_k u_k(x)$, in the coordinates representation. In what I just wrote $k$ denotes the $k$-th state, $u_k(x)$, $u^*_k(x)$ are the $k$-th element (and its complex coniugate) of the orthonormal basis of the Foch space, $f_k$, $f^+_k$ are scalars, and $hat a_k$, $hat a^+_k$ are the annihilation and creation operators of a particle in a state $k$.
After that, the quantum field operators $hatPsi(x)$ and $hatPsi^+(x)$ have been introduced, and they’ve been defined as
begin{equation}
hatPsi(x) = sum_{k = 1}^{infty} u_k(x) hat a_k, ; qquad hatPsi^+(x) = sum_{k = 1}^{infty}u^*_k(x) hat a^+_k
end{equation}
It’s been said that in a sense they are not true operatores, since every ladder operator is weighted by a function, and not by a scalar, like the creation-annihilation operators $hatPsi^+(f)$ and $hatPsi(f)$ defined before. Nevertheless they are useful since every $hatPsi^+(f)$ can be written as
$$ hatPsi^+(f) = int d^3x , hatPsi^+(x) f(x)$$