Path Integral Quantization
I was reading through my notes on the path integral quantization of bosonic string theory when a general question about path integral quantization arised to me.
The widely used intuitive explanation of a path integral is that you sum over all paths from spacetime point $x$ to spacetime point $y$. The classical path has weight one (is this correct?), whereas the quantum paths are weighed by $\exp(iS)$, where $S$ is the action of the theory you are considering. In my current situation we have the <PERSON> path integral: $$ Z = \int\mathcal{D}X\mathcal{D}g_{ab}e^{iS_p[X,g]}, $$ where $S_p$ is the <PERSON> action. I have seen the derivation of the path integral by Matrix kernels in my introductory QFT course. A problem which occured to me is that if the quantum paths are really "weighted" by the $\exp(iS_p)$, it only makes sense if $\mathrm{Re}(S_p) = 0$ and $\mathrm{Im}(S_p)\neq0$. 346
If this were not the case, the integral seems to be ill-defined (not convergent) and in the case of an oscillating exponential we cannot really talk about a weight factor right? Is this reasoning correct? Is the value of the <PERSON> action purely imaginary for every field $X^{\mu}$ and $g_{ab}$?
Secondly, when one pushes through the calculation of the <PERSON> path integral one obtains the partition function $$ \hat{Z} = \int \mathcal{D}X\mathcal{D}b\mathcal{D}ce^{i(S_p[X,g] + S_g[b,c])}, $$ where we have a ghost action and the <PERSON> action. My professor now states that this a third quantized version of string theory (we have seen covariant and lightcone quantization). I am wondering where the quantization takes place. Does it happen at the beginning, when one postulates the path integral on the grounds of similarity to QFT? I am looking for a well-defined point, like the promotion to operators with commutation relations in the lightcone quantization.
Finally, in a calculation of the Weyl anomaly and the critical dimension, the professor quantizes the ghost fields. This does not make sense to me. If the path integral is a quantization of string theory, why do we have to quantize the ghost fields afterwards again? 346