String Theory and M-Theory

46 The bosonic string
Determination of the space-time dimension
The number of zero-norm spurious states increases dramatically if, in addition
to a = 1, the space-time dimension is chosen appropriately. To see this,
let us construct zero-norm spurious states of the form
|ψ〉 = L−2 + γL 2
−1 |χ〉. (2.116)
This has zero norm for a certain γ, which is determined below. Here |ψ〉 is
spurious if |χ〉 is a state that satisfies
(L0 + 1)|χ〉 = Lm|χ〉 = 0 for m = 1, 2, . . . (2.117)
Now impose the condition that |ψ〉 is a physical state, that is, L1|ψ〉 = 0 and
L2|ψ〉 = 0, since the rest of the constraints Lm|ψ〉 = 0 for m ≥ 3 are then
also satisfied as a consequence of the Virasoro algebra. Let us first evaluate
the condition L1|ψ〉 = 0 using the relation
This leads to
L1, L−2 + γL 2
−1 = 3L−1 + 2γL0L−1 + 2γL−1L0
= (3 − 2γ)L−1 + 4γL0L−1. (2.118)
L1|ψ〉 = L1 L−2 + γL 2
−1 |χ〉 = [(3 − 2γ) L−1 + 4γL0L−1] |χ〉. (2.119)
The first term vanishes for γ = 3/2 while the second one vanishes in general,
because
L0L−1|χ〉 = L−1(L0 + 1)|χ〉 = 0. (2.120)
Therefore, the result of evaluating the L1|ψ〉 = 0 constraint is γ = 3/2. Let
us next consider the L2|ψ〉 = 0 condition. Using
L2, L−2 + 3
2 L2
−1 = 13L0 + 9L−1L1 + D
(2.121)
2
gives
L2|ψ〉 = L2
L−2 + 3
2 L2 −1
|χ〉 =
−13 + D
|χ〉. (2.122)
2
Thus the space-time dimension D = 26 gives additional zero-norm spurious
states.