# How does the sum of natural numbers arise in the derivation of critical string dimensions?

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:

1. Green, Schwarz, Witten: Superstring Theory, Vol. 1 (p. 96)
2. D. Tong: Lectures on String Theory (p. 38 f.)
3. J. Polchinski: String Theory, Vol. 1 (p. 22)
4. Blumenhagen, Lüst, Theisen: Basic Concepts of String Theory (p. 44)
• Equality (2.3.15) is just a consequence of the commutation relations between the oscillators. Now from Lorentz invariance, the mass of a string with excitation number 1 should be 0 (in light-cone gauge, for instance, you have only the transverse oscillators, and only a massless particle has $D-2$ degrees of freedom), so $a=1$. But if you apply the relation (2.3.15) on such a state, you see that the second sum is precisely $a$, because the first sum is 1 thanks to normal ordering. Hence the second sum equals 1 as well. – Antoine Mar 12 '15 at 13:56
• This question (v1) seems to essentially asking Why is it legitimate to regularize the sum $\sum_{n=1}^\infty n=-\frac{1}{12}$? This particular sum is also discussed here, and on Math.SE here. See also this, this and this Phys.SE posts and links therein. – Qmechanic Mar 12 '15 at 14:14
• @user40085: One cannot apply (2.3.15) on any state with finite excitation level because the LHS is not well defined (doesn't give a finite result). – Florian Oppermann Mar 12 '15 at 14:53
• @Qmechanic: No, that's not my question. My question is, why it is justifiable to claim that the LHS of (2.3.15) equals $M^2$. Any way I can clarify my OP? – Florian Oppermann Mar 12 '15 at 14:55