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Closed Strings
The spectrum of the closed string can be obtained from the combination of the leftmoving
states and the rightmoving ones. The ground state () is given by
. That means that the ground state includes a tachyon. The first
excited state () is massless and it is given by
. This state can be naturally
decomposed into irreducible representations of the little group as follows

(9) 
The first term of the rhs is interpreted as a spin 2 massless particle
(graviton). The second term is a range 2 antisymmetric tensor
. While the last term is an scalar field
(dilaton). Higher excited massive states are combinations of
irreducible representations of the corresponding little group SO
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Open Strings
For the open string, the ground state includes once again a tachyon since
. The first exited state
is given by a massless vector field in 26 dimensions. The second excitation level
is given by the massive states
and
which are in irreducible
representations of the little group SO.
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