Lectures on String Theory

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As for the case of classical string the first term here vanishes due to the level matching
condition α 0 = ᾱ 0 , while the second sum is an anomaly which appears due to noncommutativity
of the oscillators in quantum theory. Thus, for general values of a
and d the theory is not Lorentz invariant: the quantum effects destroy the Lorentz
invariance which was present at the classical level. However, for special values
d = 26 , a = 1
the anomaly term vanishes and the Lorentz invariance is restored!
4.2.2 The spectrum
In the light-cone gauge the spectrum is generated by acting with transversal oscillators
on the vacuum state. We first discuss the spectrum of open strings.
The mass operator is the light-cone gauge for open strings is
M 2 = 1 α ′
∞
∑
n=1
( )
α−nα i n i − a ,
where, as was discussed in the previous chapter, the normal-ordering constant a
should be equal to 1 in order to guarantee the Lorentz invariance of the light-cone
theory.
The ground state |p i 〉 carries no oscillators and it has a mass
α ′ M 2 |p i 〉 = −|p i 〉 =⇒ α ′ M 2 = −1.
This is a tachyon.
The first excited state is α i −1|p j 〉. It is a d − 2 component vector which transforms
irreducibly under the transverse group SO(24). We see that
α ′ M 2 (α i −1|p i 〉) = (1 − a)α i −1|p i 〉 = 0 ,
i.e. this vector is massless.