Virasoro operators in bosonic String Theory

[Orbb]

In a recent lecture on String Theory, we encountered the divergent sum 1+2+3+.... when calculating the zero mode Virasoro operator in bosonic String Theory. This divergent sum is then set equal to a finite negative constant - the argument for doing so was a comparison with the definition of the Zeta function. However, is still have trouble with this argument, and my question is, wether there are any further justifications for taking this step. Thank you for your answers!

[confinement]

Here is an article about this topic, although it does not treat the particular example that you mention:

http://en.wikipedia.org/wiki/Zeta_function_regularization

[Thomas Larsson]

A safe way to compute the central charge is to check how the Virasoro generators act on the vacuum. The normal-ordered expression

L_m = sum_{n=-infty}^infty : a_{m-n} a_n :

is equivalent to the following relations (perhaps modulo some signs that I don't have the energy to check) (and m > 0 in the first three lines):

L_m |0> = sum_{n=0}^m a_{m-n} a_n |0>

L_0 |0> = h |0>

L_-m |0> = 0

[L_m, a_n] = (m-n) a_{m+n}

The point is that in the second formulation, all sums are finite, so you don't have to worry about regularization.