Lectures on String Theory

– 110 –
The oscillator ground state is defined in both sectors as
α µ m|0〉 = b µ r |0〉 = 0 , m, r > 0 .
Here the dependence on the center of mass momentum is suppressed. In the Ramond
sector one also has the zero mode b µ 0. The level number operator is
where
N = N (α) + N (b) ,
N (α) =
N (b) =
∞∑
α −m α m ,
m=1
∞∑
r∈Z+θ>0
rb −r b r .
Note that the zero mode in the Ramond sector does not contribute to the number
operator! This leads to the fact the mass operator commutes with b µ 0: [b µ 0, M 2 ] = 0,
i.e. the states |0〉 and b µ 0|0〉 have the same mass. These states are degenerate. On the
other hand, all other oscillators α µ n, b µ r with n, r < 0 increase α ′ M 2 by 2n and 2r units
respectively. This means that in the NS-sector the ground state is unique and it has
Lorentz spin zero. In the R-sector the ground state is degenerate and since b µ 0 form
the Clifford algebra the ground state is a spinor of the Lorentz group SO(d − 1, 1).
This explains why in the NS-sector all the states are space-time bosons, while in the
R-sector they are all fermions. Indeed, all creation operators have vector Lorentz
index and by this reason they cannot convert a space-time boson into a space-time
fermion or vice versa. If we will write the Ramond ground state as |a〉, where a is a
SO(d − 1, d) spinor index, the b µ 0 act on it as the usual Γ-matrices
b µ 0|a〉 = 1 √
2
(Γ µ ) a b|b〉 .
Here Γ µ are the usual Γ-matrices of the d-dimensional Minkowski space and they
satisfy the Clifford algebra {Γ µ , Γ ν } = 2η µν .
We will not go into discussion of the covariant quantization but will just state
that consistency of the quantum theory will impose the following restrictions on the
constant a of the normal ordering ambiguity (for the Ramond and Neveu-Schwarz
sectors) and the dimension d of the target space-time:
a NS = 1 2 , a R = 0 , d = 10 .
The same result follows from the condition of non-anomalous Lorentz algebra in the
light-cone gauge. Instead of d = 26 found for bosonic string, quantum fermionic
string chooses to live in a ten-dimensional world.