This question relates to this link. But I still don't understand it >_<
In Polchinski's string theory vol I, p. 22, there is a divergence term (when $\epsilon \rightarrow 0$) in the zero point energy of open-string (1.3.34), $$ \frac{D-2}{2} \frac{ 2l p^+ \alpha'}{ \epsilon^2 \pi} .\tag{1.3.34}$$
It is said
The cutoff-dependent first term is propotional to the length $l$ of the string and can be canceled by a counterterm in the action proportional to $\int d^2 \sigma (-\gamma)^{1/2}$. In fact, Weyl invariance requires that it be cancelled.
I have a couple of questions related to this statement and I don't know whether it is good to split into several threads.
- About the underlying logic of regularization and renomalization. In quantum field theory, we meet a couple of divergences. One is in quantizing the scalar field. There is an infinity, we argue that is the zero-point energy. We throw it away since energy is a relative quantity. And we left the cosmological constant problem.
If the divergence in (1.3.34) is in this sense, string-theory is a quantum gravity theory. We cannot simply throw it away like QFT.
Later in QFT, we meet other divergences in the loop calculations. The reason is, the presented QFT is a low-energy theory. It has certain scale that the theory is not applicaple. Therefore we met the divergence. We regularalize it and renormalize it, where we met the terminology "counterterm".
But, string theory is regarded as a final theory(?!), if I follow the logic of QFT, why there is still divergence in (1.3.34)? What is the scale string theory is not applicable?
How the counterterm works in $\int d^2 \sigma (-\gamma)^{1/2}$?
Why "Weyl invariance requires that it be cancelled"?