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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:

  1. lightcone gauge
  2. 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

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Note that MathJax is supported here. Use it to render equations. If you are not familiar with MathJax, it is essentially LaTeX code (but it must be surrounded by $ signs for inline equations and $$ signs for other equations. E.g. $$E=\sum_{j=1}mc^3$$ results in $$E=\sum_{j=1}mc^3$$ and $E=\sum_{j=1}mc^3 results in $E=\sum_{j=1}mc^3$. For a documentation of LaTeX, see LaTeX.. –  Dimensio1n0 Aug 24 '13 at 12:51
thank you, I used it ;) –  Mohsen Aug 24 '13 at 14:12
By "spin", do you mean the kind of $SO(D−2)$ representation (vector, traceless symmetric tensor), or do you mean the component of the spin, for instance $S^{12}$, for some specific state ? –  Trimok Aug 24 '13 at 14:51
I actually mean the first one. –  Mohsen Aug 24 '13 at 22:57
Are you not satisfied with the solution of the problem $4.1$ given with details in the corrected exercices p. $33-34$ ? –  Trimok Aug 25 '13 at 15:19
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