From a QM (not QFT) context, one particle, we start with a hamiltonian H(q,p) and develop something like(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\langle q'',T|q',0\rangle \approx \int e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\prod_{j=1}^N{dq_j}\prod_{j=0}^N{\frac{dp_k}{2\pi}}[/tex]

where [tex]\delta t = T/(N+1)[/tex] and [tex]\dot{q}_j \equiv (q_{j+1}-q_j)/\delta t[/tex] and where the approximation becomes equality for small delta-t or, equivalently, large N. In that case we write

[tex]e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\rightarrow e^{-i\int_0^T{[H(q(t),p(t))-p(t)\dot{q}(t)}]dt}[/tex]

Now generally when we look at a Riemann sum, we are dealing with agivenfunction [tex]f(x)[/tex] and looking at a sum over [tex]f(x_i)\delta x[/tex] for finer and finer slices [tex]\delta x[/tex].

However, here the individual [tex]q_l[/tex]'s are not part of a given function [tex]q(t)[/tex] nor do I see any reason to expect them to approach anything like an integrable function [tex]q(t)[/tex]. Indeed, since each [tex]q_l[/tex] is a variable of integration over the entire real line, I can't see them settling down to anything like a piece-wise continuous function for large N.

For any discrete set of time slices for a free particle, given that a particle at [tex]q_j[/tex] at time [tex]t_j[/tex], then there is a non-zero probability that it be found atanyother value of [tex]q[/tex] at time [tex]t_j + \delta t[/tex]. The smaller we make [tex]\delta t[/tex], the more jagged most of the potential paths become. I don't see how they approach something we can integrate over.

All of my QFT books make no mention of this. Can someone recommend a link or text which covers this limit with more rigor?

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# Is the path integral well defined

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