Lectures on String Theory

– 38 –
Thus, we arrive at
{l i− , S j− } + {S i− , l j− } = −2i p− X 1
p + n αi n αj −n +
i X 1
p
n≠0
+ p i α j −n
n
− pj α i
−n α − n . (3.83)
n≠0
Now summing up equations (3.82) and (3.83) we obtain
{l i− , S j− } + {S i− , l j− } + {S i− , S j− } = i 4√ πT
p +
α − 0 − α− 0 + ᾱ− X
0
2
n≠0
1
n αi n αj −n . (3.84)
At first glance this expression is non-zero and it can not be compensated by the contribution of left-moving modes
{l i− , ¯S j− } + { ¯S i− , l j− } + { ¯S i− , ¯S j− } = i 4√ πT
p +
ᾱ − 0 − α− 0 + ᾱ− X
0
2
n≠0
1
n ᾱi nᾱj −n . (3.85)
However, we have to invoke the level-matching constraint which simply tells that α − 0 = ᾱ− 0
and makes both eqs.(3.84) and (3.85)
separately vanish. Thus, we have indeed shown that the most non-trivial relation {J i− , J j− } = 0 is indeed satisfied.
4. Quantization of bosonic string
4.1 Remarks on canonical quantization
According to the standard principles of quantum mechanics canonical quantization
consists in replacing the Poisson brackets of the fundamental phase space variables
by commutators
{ , } → 1
i [ , ] ,
where is the Plank constant. Thus, we consider now X(σ, τ) and P (σ, τ) as the
quantum mechanical operators which obey the following commutation relations 7
[X µ (σ, τ), X ν (σ ′ , τ)] = [P µ (σ, τ), P ν (σ ′ , τ)] = 0 ,
[X µ (σ, τ), P ν (σ ′ , τ)] = iη µν δ(σ − σ ′ ) , (4.1)
These commutation relations induce the commutation relations on the Fourier coefficients
[α µ m, α ν n] = [ᾱ µ m, ᾱ ν n] = mδ m+n η µν ,
[α µ m, ᾱ ν n] = 0 , (4.2)
[x µ , p ν ] = iη µν .
For the case of open string the modes ᾱ n are absent. In what follows we will work
in units in which = 1, so that the commutation relations read as
[α µ m, α ν n] = [ᾱ µ m, ᾱ ν n] = mδ m+n η µν ,
[α µ m, ᾱ ν n] = 0 , (4.3)
[x µ , p ν ] = iη µν .
7 Here the indices µ, ν run from 0 to d − 1.