String Theory Demystified

74 String Theory Demystifi ed
COMMUTATION RELATIONS FOR THE CLOSED STRING
We will derive the commutation relation [Eq. (4.3)] for the modes explicitly, and
simply state the results in the other cases. Hence we write down the left-moving
modes [Eq. (2.55)] which are functions of σ +. The left-moving modes are restated
here in the case of the closed string
µ 2
µ x µ i
XL
( σ, τ) = + p ( τ + σ)
+
2 2 2
Now since σ + τ σ = + , the derivative is
m e
µ
α
s s m − im(
τ+ σ)
∑
m≠0
2
∞
µ s µ s µ − im(
τ+ σ)
s
∂ + XL( σ, τ) = p + ∑αme
= ∑ αme
2 2 2
To get the last step, we used α 0
of Eq. (4.3). We have
m≠0
µ µ
= ( s/
2)
∞
⎡ s
[ ∂ + X ( σ, τ), ∂ + ′ X ( σ′ , τ)] = ⎢ αme
⎣ 2 m=−∞
µ ν µ − im(
τ+ σ)
=
2
m=−∞
(4.6)
µ − im(
τ+ σ)
(4.7)
p . Now let’s calculate the left-hand side
2
s
∑ , ∑
∞
m=−∞
α e
ν − im(
τ+ σ′
)
m
2
s
∞
∑
−im
( + n)
τ − im ( σ+ nσ
′ ) µ ν
e e ⎡⎣ αm, αm⎤⎦
mn , =−∞
But, this must be proportional to ( ∂∂ / σδσ ) ( − σ ′ ) , and furthermore, the term on the
right-hand side of Eq. (4.3) does not depend on τ. So, we have to remove the τ
dependence. We can do so by noting that
− im ( + n)τ
e → 1 when m=−n We will be able to enforce this condition by introducing the Kronecker delta
term δm+ n,
0 , which is 1 when m=−nand 0 otherwise. Furthermore, we take note of
the following expression for the Dirac delta function:
Notice that
1
δσ ( − σ′
) =
2π
∂
i
δσ ( − σ′
) =−
∂σ
2π
∞
− ( σ− σ′
)
∑ e
=−∞
im
m
∞
− ( σ− σ′
)
∑ me
=−∞
im
m
⎤
⎥
⎦
(4.8)