I obviously have a problem with basics of group theory. consider an open string in flat spacetime. there are usually two common gauge to solve the classical problem and quantize the strings:
- lightcone gauge
- conformal gauge.
After quantization, you study the state space, and here I have a problem with probably the representations of the $SO(24)$ (I focus on bosonic strings). I explain my problem explicitly below:
1-in lightcone gauge: for the second excited state, $N=2$ : the mass is $M^2=1/\alpha'$ and the bases for this subspace are : $a^\mu_2 |0\rangle $ and $a^\mu_1a^\nu_1|0\rangle $
2- in the conformal gauge: for the second excited state, $N=2$ : $M^2=1/\alpha'$ and the bases for this subspace are: $a^\mu_1a^\nu_1|0\rangle$ and, not $a^\mu_2|0\rangle$. (see Polchinski problem 4.1)
Now the problem is: Do we have a spin one particle with the mass :$M^2=1/\alpha'$ ? in lightcone gauge's state space we have it but, in conformal gauge we have only spin 2 particles with this mass. Is it really a problem?! or just a paradox comes from two different representations of the $SO(24)$ ?
I hope you got my question.
thank you very much