I have this generating functional
$$ Z(J)=\int \!D\varphi(x^\mu)\, e^{ i \int\!d^4x\,\left\{\tfrac{1}{2}\left[(\partial\varphi)^2-m^2\varphi^2\right]+J\varphi \right\} } ~~.$$
I am using Zee and he uses a trick to solve this integral by imagining discretizing it (page 22.) I don't want to use the trick, I want to solve the integral up to some constant multiple associated with the path integral measure. The $\int D\varphi$ measure is such that, for discrete $q$, we have
$$\int \!Dq(t)\equiv \lim\limits_{n\to\infty} \left( \frac{-im}{2\pi\delta t} \right)^{\!\frac{n}{2}} ~\prod_{k=1}^{n-1}\int dq_k~~, $$
but Zee does not say exactly what $\int D\varphi(x^\mu)$ is in the limit where the field $\varphi$ is a continuum of an infinite number of $q$ taken in four spacetime dimensions. So... can you solve this integral? Can you tell me where this integral is worked out so I can look at it? A source would be great if you don't feel like doing the integral for me.
This was basically the same question I'm asking. The only answer is about how discretizing works. Is there a way to solve this integral without discretizing? Discretizing is automatically an approximation but Zee makes some pretty clear statements about how this is essentially the only exactly soluble generating functional in QFT. Is it exactly soluble or is it merely soluble by minimal approximation? I suppose no matter what, I'm going to have to throw in an assumption for the $i^{n/2}$ term in the path integral measure, but I'd like to leave it at that. Do I have to hand wave twice? That wouldn't seem very "exactly soluble" to me.
Here is another same version of the same question with a nice answer that still kind of avoids my question. Helpful Poster wrote:
According to Faddeev and Slavnov (Gauge fields: Introduction to Quantum Theory), "...all those properties of the Feynman integral that are used in practice in the perturbation theory are derived directly from the definition of the quasi-Gaussian integral and can be rigorously established independent on the issue of existence of Feynman integral measure. Therefore, in the framework of perturbation theory, the formalism of functional integration is a quite rigorous method, and results obtained using this method do not require additional proof." (I quote by a Russian edition).
So... you see: my question remains unanswered. Even though the proof is not required, I would still like to see it. ALSO, HOWEVER! It would be very useful for me if someone could direct me toward this non-path integral rigorous establishment of the issue. If no additional proof is required, where is the proof beyond which other proofs are not needed?
