# Path integral quantization of bosonic string theory

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 Polyakov path integral: $$Z = \int\mathcal{D}X\mathcal{D}g_{ab}e^{iS_p[X,g]},$$ where $S_p$ is the Polyakov 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 $\mathrm{Im}(S_p)\neq0$. 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 Polyakov action purely imaginary for every field $X^{\mu}$ and $g_{ab}$?

Secondly, when one pushes through the calculation of the Polyakov 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 Polyakov action. My professor now states that this a third quantized version of string theory (we have seen covariant and lightcone quantization).

Questions:

1. 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.

2. 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?

• The first part of this question seems related to this question: physics.stackexchange.com/q/61139 @LubošMotl gave a nice answer there. Also, it can help to consider a Wick rotation so that the exponent is real and negative, like you naively expect, as mentioned in answer – innisfree May 23 '13 at 18:18

• Thanks for the clear answer! Just a small point concerning my third question: indeed it makes sense that if we consider an action of several fields, all of them have to be quantized. My confusion is that after using the path integral to calculate the partition function, only the $X$ and $g$ fields seem to be quantized, whereas the ghost fields have to be quantized separately. Why is this the case? – Funzies May 23 '13 at 17:45