On Peskin & Schroeder's QFT book, page 281, in the first paragraph, the book considered Path Integral of following form: $$\langle q_{k+1}|H(q,p)|q_k\rangle\\=\left(\prod_i \int \frac{d p_k^i}{2 \pi}\right) H\left(\frac{q_{k+1}+q_k}{2}, p_k\right) \exp \left[i \sum_i p_k^i\left(q_{k+1}^i-q_k^i\right)\right]. \tag{9.10} $$
And the book want to cope $H$ contains products of $p$'s and $q$'s. I am troubled for following description in book
In general this formula must be false, since the order of a product $pq$ matters on the left-hand side (where $H$ is an operator) but not on the right-hand side (where $H$ is just a function of the numbers $p_k$ and $q_k$).
Could you please explain this for me?
