JLA

@Noiralef Yes, that's exactly what I mean. The idea is that, for any $\epsilon>0$ any path in phase space can be approximated by line segments connecting $N+1$ points in phase space, for large enough $N\,,$ where successive points differ by less than $\epsilon\,,$ eg. $|q_{n}-q_{n-1}|<\epsilon\,.$ From this point of view, the ordering of limits is natural.

@Qmechanic I mean that if you let $\epsilon\to\infty$ and integrate over $p_N$ you get the phase space path integral in quantum mechanics, with the hamiltonian being zero. Note that, what I'm calling $\epsilon$ is unrelated to what they call $\varepsilon\,.$

Let us continue this discussion in chat.

@hft it just happens that we are both using $\epsilon$ for notation, I've now changed my notation to $\varepsilon.$ I'm not sending anything to $\infty,$ I just observed that if we do let $\varepsilon=\infty$ then we get the usual phase space path integral with zero Hamiltonian. What I wrote is a mathematically well-defined way of defining the integral. Edit: I changed the notation back because I realized we were already using different notations. $\epsilon$ and $\varepsilon$ have nothing to do with each other.

@hft This isn't really coming from anywhere, but I've now included a reference to the phase space path integral on this site.

@hft Oh, of course. Well not quite, if it were that simple I could compute it. What I wrote defines the domain of integration though, I'm not sure how to clarify it further.

@hft What is $p_{12}?$

$q_0,p_0,q_N,p_N$ are are fixed and the domain of integration is as indicated: $|q_n-q_{n-1}|<\epsilon\,,|p_n-p_{n-1}|<\epsilon\,.$

I just edited the domain of integration again, do you find this more clear?

@hft it just happens that we are both using $\epsilon$ for notation, I've now changed my notation to $\varepsilon.$ I'm not sending anything to $\infty,$ I just observed that if we do let $\varepsilon=\infty$ then we get the usual phase space path integral with zero Hamiltonian. What I wrote is a mathematically well-defined way of defining the integral. Edit: I changed the notation back because I realized we were already using different notations. $\epsilon$ and $\varepsilon$ have nothing to do with each other.

@hft This isn't really coming from anywhere, but I've now included a reference to the phase space path integral on this site.

@hft I meant two votes to close. I'd like to know where the question needs clarification is what I'm saying.

I've gotten two downvotes to close without any explanation. Can I have some?

@hft Oh, of course. Well not quite, if it were that simple I could compute it. What I wrote defines the domain of integration though, I'm not sure how to clarify it further.

@hft What is $p_{12}?$

$q_0,p_0,q_N,p_N$ are are fixed and the domain of integration is as indicated: $|q_n-q_{n-1}|<\epsilon\,,|p_n-p_{n-1}|<\epsilon\,.$