We have the formula: \begin{equation} \langle F \rangle = \frac{\int Dx \times F[\phi] exp\{i/\hbar S[\phi]\}}{\int Dx \times exp\{i/\hbar S[\phi]\}} \end{equation}
Now, I am wondering how a change from the mid point prescription to any other prescription will affect the expectation values of the operators/functionals.
To do this, I wish to calculate the expectation value using the mid point prescription itself, which I am unable to do(due to novelty of concept, and thus, lack of familiarity). This is where i'd like someone to just give me an idea on how to go about it.
Examples of operators whose expectation values I'd like to calculate are $\hat p,\hat x \hat p,(\hat x \hat p)^2$ etc. to name a few.
Note: Mid point prescription is the convention where the value of a quantity over a range of data is the value at the midpoint of the range of data. For taking the integral over all possible paths, we break the path down in small intervals of $\epsilon$. The problem here is, suppose for a quantity like velocity, what is the value we assign to it on any of the small intervals? Here we assign it the value at the midpoint of these intervals, i.e:$v_i=(x_i-x_{i-1})/\epsilon$. Refer to chapter 2, equation 2.24 of Feynman and Hibbs.
