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I'm going through the details of the path integral, and have a question about its derivation.

When we discretize the time interval and evaluate <p_n|exp(-iH*(t_n-t_n-1)|q_n-1>, a Hamiltonian of the form H(p,q)=T(p)+V(q) becomes a number T(p_n)+V(q_n-1).

However, when the Hamiltonian contains a term like 1/2*[p*f(q)+f(q)*p], it becomes f(q_n')*p_n with q_n'=(q_n+q_n-1)/2.

I understand the mathematics behind it but it seems to me that it doesn't really matter whether we use q_n-1 or q_n' as the time interval goes to positive infinitesimal.

Can anyone explain to me why this actually matters?

Thanks in advance.

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# Path integral question

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