On pages 280-281 of "An Introduction to Quantum Field Theory" by Peskin and Schroeder, the authors discuss the path integral formulation for a general quantum system and briefly mention Weyl ordering. The presence of operator ordering issues seems clear. However, my question, details of which follow, is whether the requirement to use Weyl ordering in particular is the result of a (seemingly innocuous) choice made earlier in the calculation. Does making different choices lead to different ordering requirements?
In detail, after equation 9.9, the matrix element of a Hamiltonian term involving only the coordinates (and not the momenta) is given as $$ <q_{k+1}|f(q)|q_k> = f(q_k)\prod_i\delta(q_k^i - q_{k+1}^i) $$ which is then rewritten as $$ \tag{1} <q_{k+1}|f(q)|q_k> = f\left(\frac{q_{k+1} + q_k}{2}\right)\left(\prod_i\int\frac{dp_k^i}{2\pi}\right)\exp\left[i\sum_ip_k^i(q_{k+1}^i - q_k^i)\right]. $$ Here, the choice has been made to replace $f(q_k)$ with $f\left(\frac{q_{k+1} + q_k}{2}\right)$. I assume that one could have left $f(q_k)$ alone, or indeed replaced it with $f(q_{k+1})$.
The text then continues to show that if Hamiltonian $H$ contains only terms of the form $f(q)$ and $f(p)$ then its matrix element is given by $$ <q_{k+1}|H(q,p)|q_k> = \left(\prod_i\int\frac{dp_k^i}{2\pi}\right)H\left(\frac{q_{k+1}+q_k}{2},p_k\right)\exp\left[i\sum_ip_k^i(q_{k+1}^i-q_k^i)\right], $$ and tries to extend this to general Hamiltonians with products of $p$s and $q$s. The authors note that, due to ordering issues, this only works if the Hamiltonian is Weyl ordered, e.g. $$ <q_{k+1}|\frac{1}{4}(q^2p^2 + 2qp^2q + p^2q^2)|q_k> = \left(\frac{q_{k+1}+q_k}{2}\right)^2<q_{k+1}|p^2|q_k>. $$
Am I right to think that this particular ordering is required as a result of the choice made in (1) to replace $f(q_k)$ with $f\left(\frac{q_{k+1} + q_k}{2}\right)$? For example, would leaving $f(q_k)$ alone give a Hamiltonian matrix element (assuming only terms of the form $f(p)$ and $f(q)$) of the form $$ <q_{k+1}|H(q,p)|q_k> = \left(\prod_i\int\frac{dp_k^i}{2\pi}\right)H\left(q_k,p_k\right)\exp\left[i\sum_ip_k^i(q_{k+1}^i-q_k^i)\right], $$ and require us to order products involving $p$s and $q$s with the $q$s on the right, i.e. $$ <q_{k+1}|p^2q^2|q_k> = q_k^2<q_{k+1}|p^2|q_k>. $$
To conclude, am I right to think that Weyl ordering is, in effect, a choice, but one that was made when we decided to replace $f(q_k)$ with $f\left(\frac{q_{k+1} + q_k}{2}\right)$ in (1), and that other orderings are required if we leave $f(q_k)$ alone or replace it with $f(q_{k+1})$. If so, is there something special about Weyl ordering that makes it preferable to, say, placing all $q$s on the right?
