String Theory Demystified

CHAPTER 4 String Quantization
Normal ordering leads to
1
2
D−2
∞
i=
1 n=−∞
D−2
∞
i i 1
i i D − 2
α−nαn= : α−nαn: +
2
2
∑∑∑∑ ∑
i=
1 n=−∞
The regularization trick can be applied to make the second term fi nite. We fi nd
D − 2
2
Again taking a = 1, one fi nds D = 26.
∞
∑
n=
1
D − 2
n =−
24
In the previous two chapters, we constructed a relativistic theory of the string, the
classical theory. In this chapter we have introduced the simplest possible quantum
extension of the classical theory. This is a theory that consists only of bosons.
While the theory is not realistic since it does not include fermionic states, it is easier
to deal with and introduces important concepts and methods that will play a role in
the full quantum theory. The classical theory was quantized using two different
methods. The fi rst method, called covariant quantization, is a straightforward
approach that imposes commutation relations on the X µ and their conjugate
momenta. This leads to negative norm states. The Virasoro constraints are imposed
to rid the theory of these states. When this is done, we fi nd that the theory must have
26 space-time dimensions. We concluded the chapter with a different approach,
known as light-cone quantization.
Quiz
µ ν
1. Explicitly calculate the commutators [ ∂ + X ( σ, τ), ∂ −′
X ( σ′ , τ )] and
µ ν
[ ∂ − X ( σ, τ), ∂ ′ + X ( σ′ , τ )] .
µ ν
2. Consider the closed string and explicitly calculate [ x , p ] .
µ ν
3. Consider the fi rst excited state of the closed string εµν ( k) α−1α−1 0 , k . Using the
condition satisfi ed by physical states ψ , in particular L1 ψ = L1
ψ = 0,
µ
fi nd ε µνk
.
∞
n=
1
n
87
Summary