String Theory and M-Theory

where
N =
2.4 Canonical quantization 43
∞
α−n · αn =
n=1
∞
na † n · an, (2.97)
is called the number operator, since it has integer eigenvalues. For the ground
state, which has N = 0, this gives α ′ M 2 = −a, while for the excited states
α ′ M 2 = 1 − a, 2 − a, . . .
For the closed string
1
4 α′ M 2 =
∞
α−n · αn − a =
n=1
n=1
∞
α−n · αn − a = N − a = N − a. (2.98)
n=1
Level matching
The normal-ordering constant a cancels out of the difference
(L0 − L0)|φ〉 = 0, (2.99)
which implies N = N. This is the so-called level-matching condition of the
bosonic string. It is the only constraint that relates the left- and rightmoving
modes.
Virasoro generators and physical states
In the quantum theory one cannot demand that the operator Lm annihilates
all the physical states, for all m = 0, since this is incompatible with the
Virasoro algebra. Rather, a physical state can only be annihilated by half
of the Virasoro generators, specifically
Together with the mass-shell condition
Lm|φ〉 = 0 m > 0. (2.100)
(L0 − a)|φ〉 = 0, (2.101)
this characterizes a physical state |φ〉. This is sufficient to give vanishing
matrix elements of Ln − aδn,0, between physical states, for all n. Since
L−m = L † m, (2.102)
the hermitian conjugate of Eq. (2.100) ensures that the negative-mode Virasoro
operators annihilate physical states on their left
〈φ|Lm = 0 m < 0. (2.103)