String Theory Demystified

124 String Theory Demystifi ed
We can identify the structure constants as
k
fmn = ( m− n)
δ + ,
To see how the physical spectrum can be constructed in string theory, we consider
the open string case. The states are built up from the ghost vacuum state. Let’s call
the ghost vacuum state χ . This state is annihilated by all positive ghost modes.
Let n > 0, then
m n k
b χ = c χ =0
n n
The zero modes of the ghost fi elds are a special case. They can be used to build
the physical states of the theory. Using the anticommutation relations [Eq. (6.3)],
the zero modes satisfy
{ b0, c0}
= 1
2 2
Using Eq. (6.3) it should also be obvious that b0= c0=
0.
We also require that
b0 ψ = 0 for physical states ψ . Now we can construct a two-state system from
the zero modes of the ghost states. The basis states are denoted by ↑ , ↓ . The
ghost states act as
b0 ↓ = c0
↑ = 0
b ↑ = ↓ c ↓ = ↑
0 0
We choose ↓ as the ghost vacuum state. To get the total state of the system, we take
the tensor product of this state with the momentum state k to give ↓,k . To
generate a physical state, we act on it with the BRST charge Q. It can be shown that
Q ↓ , k = ( L −1) c ↓,
k
0 0
The requirement that Q ↓ , k = 0 gives the mass-shell condition L0 − 1= 0,
which
describes the same Tachyon state we found in Chap. 4. Higher states can be
generated. We will have mode operators for each of the three fi elds: the X µ ( σ, τ)
plus the two ghost fi elds. To get the fi rst excited state, we act with α−1, c−1, and b−1
as follows:
ψ = ( ς⋅ α + ξc + ξ b ) ↓,
k
−1 1 −1 2 −1