I Mass spectrum of open bosonic strings

[snypehype46]

TL;DR Summary

This question regards some features about the excitation of an bosonic string

I'm learning string theory from the book by Zwiebach and others. I'm trying to understand the quantisation of the open string and its mass spectrum.

In light-cone gauge the mass-shell condition of an open string is given by:

$$M^2 = 2(N - 1)/l_s^2$$

where $N = \sum_{i=1}^{D-2}\sum_{n=1}^\infty \alpha^i_{-n}\alpha^i_n$ and $l_s$ is the string length scale.

Now to determine the mass spectrum of the string, we can look at the values of $N$:

- For $N=0$, there is a tachyon since $M^2$ is negative
- For $N=1$, there is a *vector boson* $\alpha^i_-1 |0;k\rangle$.
- For $N=2$, we have that $M^2$ is positive and the states are given by: $\alpha^i_{-2}|0;k\rangle$ and $\alpha^i_{-1}\alpha^j_{-1}|0;k\rangle$

Now this is what I don't understand:

- Why is the state with $N=1$ a *vector*, why is not a scalar? How does one determine if a state is a vector or scalar?

- In the material I've read, it is claimed that Lorentz invariance requires that the the state with $N=1$ is massless, but I don't understand why is this case.

- Finally, the number of states with $N=2$ is claimed to 324 because it is the number of independent components of a matrix representation of $SO(25)$, why is this? Also this state is said to have a single massive state with spin-2, why is this?

 

[haushofer]

Your first question: for N=1 you have (D-2) components labeled by i, which transform into each other if you apply a Lorentz transfo. That's some pretty weird scalar, but it makes sense for a massless vector irrep.

 

[haushofer]

Your 2nd: what do you get if you apply the momentum operator on the state and use the on-shell condition to calculate the mass (squared)? It should be zero, which is probably explained in Zwiebach.

 

[haushofer]

3d: this is basic representation theory for SO(N). A rep. for this group can always be written as the sum of an antisymmetric part (#=1/2×N(N-1)), a traceless symmetric part (#=1/2×N(N+1)-1) and a trace (#=1). See e.g. Zee's book on group theory.