String Theory Demystified

70 String Theory Demystifi ed
Covariant Quantization
The procedure known as covariant quantization will be familiar to you from your
studies of ordinary quantum mechanics. In a nutshell, this is the imposition of
commutation relations on position and momentum. So, using this procedure we
continue with the notion that X µ ( σ, τ)
are space-time coordinates, but we need to say
what we are taking as the momentum. This can be done in the standard way using
µ
lagrangian dynamics. Let π ( σ, τ)be
the momentum carried by the string. Given a
lagrangian density L we can calculate the momentum from the X µ ( σ, τ)
using
∂L
πµ ( σ, τ)=
∂∂X
µ ( τ )
This is easy to calculate using the Polyakov action as written in Eq. (2.31):
( σ µ )
T
S d X X X X
L T
2 µ
µ
P = ∫ σ ∂τ∂τ µ −∂σ ∂
2
µ
µ
⇒ = ( ∂τX ∂τ Xµ
−∂σσX
∂σ
Xµ
)
2
So we see that the conjugate momentum is
∂L
πµ ( σ, τ)=
∂∂X
= T∂ X
µ τ µ
( τ )
With this defi nition in hand, we are in a position to quantize the theory. To do this
we take the approach used in quantum fi eld theory, namely, impose equal time
commutation relations on the position and momenta. In ordinary quantum mechanics
the position and momentum coordinates satisfy
[ x, p ] = [ y, p ] = [ z, p ] = i
x y z
[ x, x] = [ y, y] = [ z,
z] = [ x, y] = [ x, z] = [ y, z]
= 0
[ p , p ] = [ p , p ] = [ p , p ] = [ p , p ] = [ p , p ] = [ p , p ] =
x x y y
z z x y x z y z 0
where we have set = 1. If we denote the coordinates as x where i = 1, 2, 3 then
i
these relations can be written compactly as
[ xi, pj] = iδij
[ x , x ] = [ p, p ] = 0
i j i j