String Theory Demystified

CHAPTER 4 String Quantization
Using Eqs. (4.3) and (4.8) together with our previous result, we have
2 ∞ s − i( m+ n)
τ − i mσ+ nσ
′ µ ν
∑ e e αm αm
2 mn , =−∞
( ) iη
[ , ] =
2T
µν
µν
iη
⎛ i
= −
2T⎝ ⎜
2π
µν
= η
2
∂
δσ ( − σ′
)
∂σ
1
2πT me im
∞
− ( σ− σ′
)
∑
m=−∞
∞
− σ− σ′
me im(
)
∑
m=−∞
We have also used Eq. (2.50) to relate s and the string tension T. This gives us
the commutation relation for the modes
µ ν µν
⎡⎣ αm, α ⎤ n⎦ = η mδm+
n,
0
µ ν
Equation (4.5) can be used to show that the αm and α commute. We can write all
n
of the commutation relations for the modes of the closed string as
µ ν µν µ ν µν
⎡⎣ α , α ⎤⎦ = mη δ ⎡⎣ α , α ⎤⎦ = mη
δ
µ ν
m n m+ n, 0 m n m+
n, 0
m n
⎞
⎠
⎟
75
⎡⎣ α , α ⎤⎦ = 0 (4.9)
In the chapter quiz you will also derive a commutation relation for the center-ofmass
position and momentum of the string
[ x , p ] = iη
µ ν µν
COMMUTATION RELATIONS FOR THE OPEN STRING
µ ν µν
In the case of the open string, it can be shown that together with [ x , p ] = iη,
the
commutation relations are
THE OPEN STRING SPECTRUM
µ ν µν
⎡⎣ α α ⎤⎦ = mη
δ + 0 (4.10)
m, n m n,
With the commutation relations in hand, we can proceed to fi nd the states of the
string. Because the open string case is simpler, we consider this fi rst. Notice that in
our quantization procedure where we have imposed commutation relations on the