Lectures on String Theory

– 60 –
Here by arrow we indicated the explicit expression for the corresponding commutator. Reducing similar terms we get
[l i− , l j− ] =
4 i [pi , x j ] p−
p + + 4
i p −
p + [pi , x j ] +
2 i [x− , p − ]δ ij = 0 .
Thus, the generators of the orbital part of the total momentum have vanishing commutator. Next we compute
[l i− , S j− ] + [S i− , l j− ] = −2 p− X ∞ 1
p +
n=1 n α[i −n αj] n + (4.15)
+ 1
p +
∞ X
1
n=1 n
h
− (α j −n α− n − α− −n αj n )pi + (α i −n α− n − α− −n αi n )pj i .
Here and below we use the concise notation α [i −n αj] n = α i −n αj n − αj −n αi n .
Now we are in a position to study the most difficult commutator [S i− , S j− ]. We will do further analysis in several steps.
1. First we consider the following commutator
Therefore,
This further gives
[S i− , α − ∞ m ] = −i X 1
n n [αi −n α− n − α− −n αi n , α− m ] (4.16)
√ √
∞X 1 4πT
= −i −
n=1 n p + nαi m−n α− n + αi −n [α− n , α− m ] −[α − 4πT
−n , α− m ]αi n − p + nα− −n αi n+m .
| {z }
A
[S i− , α − m ] = −i √
4πT
p +
− i f(m)
m αi m .
X ∞ 1
− nα i
m−n
n=1 n
α− n + αi −n (n − m)α− m+n + (n + m)α− m−n αi n − nα− −n αi n+m
[S i− , α − m ] = −i √
4πT
+ i
p +
√
4πT
p +
X ∞
α i −n α− m+n − αi m−n α− n + α − m−n αi n −α −
−n αi n+m
n=1
| {z } | {z }
A
B
X ∞ m
n=1 n
α i
−n α− m+n − α− m−n αi n − i f(m)
m αi m .
The terms in the first line of the last equation can be partially cancelled upon changing the summation index and we find
that
[S i− , α − m ] = −i √
4πT
+ i
p +
√
4πT
p +
m X
n=1
X ∞ m
n=1 n
− α i m−n α− n
| {z }
A
+ α − m−n αi n
| {z }
B
α i
−n α− m+n − α− m−n αi n − i f(m)
m αi m .
Since we have
mX
α − m−n αi n =
X 0 α − m−1
k αi m−k ≡ X
α − n αi m−n
n=1
k=m−1
n=0
we see that
mX
− α i mX
m−n α− n + α− m−n αi n = − α i m−1
m−n α− n + X
α − n αi m−n
n=1
n=1
n=0
= α − m−1
0 αi n − αi 0 α− m + X
[α − n , αi m−n ]
n=1
Using the fact that P m−1
n=1 (m − n) = 1 m(m − 1) we obtain
2
√ √
[S i− , α − 4πT
4πT
m ] = i p + (αi 0 α− m − α− 0 αi m ) + i X ∞ m
p +
α i
−n
n=1 n
α− m+n − α− m−n αi n
4πT m(m − 1)
+ i
(p + ) 2 − f(m)
!
α i m
2
m
. (4.17)
Thus, under the action of the Virasoro operators α − m the spin components Si− transform in a nontrivial manner. Note that
the r.h.s. of eq.(4.17) is normal-ordered.