String Theory Demystified

CHAPTER 6 BRST Quantization 125
The ξ 1 and ξ 2 are constants, but ς µ is a vector with 26 components. In
Chap. 4, we found that the first excited state was massless, so we expect the
state to have physical degrees of freedom in the transverse directions. That
is, it should have 24 independent components. To get rid of the extra
parameters, we create a physical state with the requirement that Q ψ = 0. It
can be shown that
2
Q ψ = 2(
p c + ( p⋅ ς) c + ξ p⋅α ) ↓,
k
0 −1 2 −1
The requirement that Q ψ = 0 enforces constraints on the parameters. With
this general prescription, there are 26 positive norm states and 2 negative norm
states.
We can eliminate the negative norm states by introducing some constraints. The
fi rst constraint is to take p⋅ ς = 0 and ξ1 = 0.
This rids the theory of the negative
norm states. Also note that p 2
= 0,
which tells us that this is a massless state. We
also have two zero-norm states:
µ
k ↓, k and c ↓,
k
µ α−1 −1
These states are orthogonal to the physical states. Eliminating them gives us a
state with 24 degrees of freedom, as expected for a massless state in 26 space-time
dimensions.
No-Ghost Theorem
The no-ghost theorem is simply a statement of the results we have seen in Chap. 4
and here, namely, that if the number of space-time dimensions is given by D = 26,
then negative norm states are eliminated from the theory.
In this chapter we introduced the BRST formalism and illustrated how it can be used
to quantize strings. This is a more sophisticated approach than covariant quantization
or light-cone quantization. It takes a middle ground, preserving manifest Lorentz
invariance while living with ghost states. The approach makes the appearance of the
critical D = 26 dimension simple to understand.
Summary