String Theory and M-Theory

4.5 Canonical quantization of the RNS string 129
Absence of negative-norm states
As in the discussion of the bosonic string in Chapter 2, there are specific
values of a and D for which additional zero-norm states appear in the spectrum.
The critical dimension turns out to be D = 10, while the result for a
depends on the sector:
aNS = 1
2
and aR = 0. (4.89)
As before, the theory is only Lorentz invariant in the light-cone gauge if aNS,
aR and D take these values.
Let us consider a few simple examples of zero-norm spurious states. Recall
that these are states that are orthogonal to physical states and decouple from
the theory even though they satisfy the physical state conditions.
• Example 1: Consider NS-sector states of the form
|ψ〉 = G −1/2|χ〉, (4.90)
with |χ〉 satisfying the conditions
G1/2|χ〉 = G3/2|χ〉 = L0 − aNS + 1
|χ〉 = 0. (4.91)
2
The last of these conditions is equivalent to (L0 − aNS)|ψ〉 = 0. To ensure
that |ψ〉 is physical, it is therefore sufficient to require that G 1/2|ψ〉 =
G 3/2|ψ〉 = 0. The G 3/2 condition is an immediate consequence of the
corresponding conditions for |χ〉. So only the G 1/2 condition needs to be
checked:
G 1/2|ψ〉 = G 1/2G −1/2|χ〉 = (2L0−G −1/2G 1/2)|χ〉 = (2aNS−1)|χ〉. (4.92)
Requiring this to vanish gives aNS = 1/2. This choice gives a family of
zero-norm spurious states |ψ〉. Such a state satisfies the conditions for a
physical state with aNS = 1/2. Moreover, |ψ〉 is orthogonal to all physical
states, including itself, since
〈α|ψ〉 = 〈α|G −1/2|χ〉 = 〈χ|G 1/2|α〉 ⋆ = 0, (4.93)
for any physical state |α〉. Therefore, for aNS = 1/2 these are zero-norm
spurious states.
• Example 2: Now let us construct a second class of NS-sector zero-norm
spurious states. Consider states of the form
|ψ〉 =
G−3/2 + λG−1/2L−1 |χ〉. (4.94)