Lectures on String Theory

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Here N is the number operator. It turns out that all eigenvalues of the number
operator N are non-negative. Indeed,
N =
∞∑ (
∑d−1
− α−nα 0 n 0 +
n=1
i=1
α i −nα i n
We see the “-” sign coming from the time-like oscillators. However, the time-like
oscillators themselves provide only non-negative contribution to N, because for any
m > 0
∞∑
∞∑
[N, a 0 −m] = − [α−nα 0 n, 0 a 0 −m] = − α−n[α 0 n, 0 a 0 −m] = mα−m
0
n=1
since commutator of two time-like oscillators contributes with the negative sign.
Thus, time-like creation operators contribute positively to N.
n=1
)
.
Virasoro primaries, descendents and physical states
Let us introduce the following useful definitions.
1. States which are annihilated by all positively moded Virasoro operators and
are eigenstates of the operator L 0 with an eigenvalue a are called Virasoro
primaries. Number a is called a weight of the Virasoro primary.
2. A Virasoro descendent of a given primary is a state that can be written as a
finite linear combination of products of negatively moded Virasoro operators
acting on the primary state.
3. A state which is both primary and descendent is called a null state.
If |Φ〉 is a primary state then L −1 |Φ〉 is its descendent. If N|Φ〉 = N Φ |Φ〉 then
NL −1 |Φ〉 = (N Φ + 1)L −1 |Φ〉.
There are two basis descendents with the number N Φ + 2, namely L −2 |Φ〉 and
L −1 L −1 |Φ〉. The counting of descendents changes at N Φ + 3. Here the candidate
descendents are
L −3 |Φ〉 , L −2 L −1 |Φ〉 , L −1 L −2 |Φ〉 , L 3 −1|Φ〉 .
The second and the third states are not identical because the Virasoro operators do
not commute. However, due to the Virasoro algebra there is one relation between
the above states
L −1 L −2 = [L −1 , L −2 ] + L −2 L −1 = L −3 + L −2 L −1 .
Thus, there are only three descendents with number N Φ + 3.