In Greiner field quantization book, when discussing the Feynman path integral approach, the book tries to calculate the path integral
$$\tag{12.35} \int \mathcal{D}\phi \exp\bigg[\frac{i}{\hbar}\int d^4x \bigg(\frac{\hbar^2}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m\phi^2+J\phi\bigg)\bigg].$$
The book then proceeds with the standard way of how physics textbooks would do this: they treat $\mathcal{D}\phi$ as if it were just the usual Riemann integral $d\phi$, with $\phi$ treated as a single dummy variable. The result is the usual Gaussian integral.
However, at the beginning of chapter 12, we defined the precise meaning of $\mathcal{D}\phi$: first slice spacetime into a set of $M$ "elementary cells" of volume $\Delta V$, which for simplicity will be taken of equal size, centered at the coordinates $x_l, l=1,…,M$. In this way the continuous field function $\phi(x,t)$ is made into a finite dimensional veector $\phi_l(t)=\phi(x_l,t)$ with discrete index $l$ etc." (page 366 has more details on what Greiner means.)
My question is: how do we calculate 12.35 using this precise definition? I.e. how to show convergence of the prescription above to the nonrigorous result obtained by treating the Feynman path integral as a Riemann integral.
$\textbf{EDIT:}$ I now realized that Rothe's lattice gauge theory book chapter 3 has an explicit calculation using the prescription above, for the scalar field case. This appears to be exactly the kind of calculation that I was looking for.
Best Answer
Greiner is in the beginning of chapter 12 describing a heuristic discretization (rather than a precise mathematical meaning) of the Feynman path integral. A rigorous mathematical definition of path integrals is a huge topic, cf. e.g. this and this Phys.SE posts. For starters Greiner should include an $i\epsilon$ prescription for convergence. Discretization is mainly used in numerical work rather than in analytical calculations.