In the standard treatment of bosonic string theory the “heuristic” argument for the critical dimension goes as follows (see Ref. 1-4).
Upon quantization the mass-squared operator becomes normal ordered and an a priori unknown constant is added, just in case the normal ordered expression is not the true expression,
$$ \tag{1} M^2 = \sum_{n=1}^\infty \alpha_{-n} \cdot \alpha_n - a $$
(up to multiplicative constants).
(Note: $M^2$ has to differ only by a finite value from the normal ordered expression in order to be well defined on the Fock space vaccuum, so (1) makes sense.)
The next step is (GSW): “Let us try to calculate the normal-ordering constant $a$ directly. This normal-ordering constant arises from the formula”
$$ \tag{2.3.15} \frac{1}{2} \sum_{n=-\infty}^\infty \alpha_{-n} \cdot \alpha_n = \frac{1}{2} \sum_{n=-\infty}^\infty :\alpha_{-n} \cdot \alpha_n : + \frac{D-2}{2} \sum_{n=1}^\infty n $$
Then the LHS of (2.3.15) is suggested to be the “true“ $M^2$ so that a comparison yields $a = - \frac{D-2}{2} \sum_{n=1}^\infty n = \frac{D-2}{24}$.
Now my question: Why is the last step (equality of (1) and (2.3.15)) valid? We introduced $a$ because we don't know the true ordering, so why should the one in (2.3.15) be correct?
Even stronger: The note above implies that the LHS of (2.3.15) is no well defined operator – it really should not be the true form of (1).
References:
- Green, Schwarz, Witten: Superstring Theory, Vol. 1 (p. 96)
- D. Tong: Lectures on String Theory (p. 38 f.)
- J. Polchinski: String Theory, Vol. 1 (p. 22)
- Blumenhagen, Lüst, Theisen: Basic Concepts of String Theory (p. 44)
