In QM of finitely many degrees of freedom it is well known that due to the Stone-Von Neumann theorem, the CCR
$$[Q_i,P_j]=i \delta_{ij} $$
leads to a unique representation up to unitary equivalence, on which $P_j$ acts as the derivative
$$P_j\mapsto -i\partial_j.$$
Now, in Weinberg's QFT book volume 1, chapter 9, he considers a general quantum mechanical system with coordinates $Q_a$ and momenta $P_b$ satisfying
$$[Q_a,P_b]=i\delta_{ab},\tag{9.1.1}$$
with $a,b$ allowed to have a continuous part. So that in quantum field theory we would have for instante $a\mapsto \mathbf{x}$ so that we have coordinates $\phi(\mathbf{x})$ and momenta $\pi(\mathbf{x})$ and then $\delta_{ab}$ turns into $\delta(\mathbf{x}-\mathbf{y})$.
Weinberg then considers the eigenstate basis of both operators
$$Q_a|q\rangle = q_a |q\rangle,\tag{9.1.6}$$ $$ P_b|p\rangle = p_b |p\rangle,\tag{9.1.9}$$
where $q,p$ stands for the whole collection of $q_a$ and $p_b$.
He then claims that
$$\langle q | p\rangle = \prod_a \dfrac{1}{\sqrt{2\pi}} \exp (i p_a q_a)\tag{9.1.12}$$
with a footnote on page 379 saying that it follows from the CCR that $P_b$ acts as $-i\partial/\partial q_b$ on wave functions and thus this formula follows.
Well there are quite a few objections here:
If the labels $a,b$ have infinite values, as it happens in QFT, we have a system with infinitely many degrees of freedom to which the Stone-Von Neumann theorem doesn't apply. There are infinitely many inequivalent representations of the CCR and we can't conclude that $P_b$ acts as a derivative as suggested by the author.
More alarming would be that if the labels have infinite values, the expression for $\langle q | p\rangle$ have $(2\pi)^{-1/2}$ multiplying itself infinite times, which isn't well defined.
So how to make sense of these issues? What am I missing here?
