String Theory Demystified

72 String Theory Demystifi ed
µ ν µν
Now we compute the derivative of [ X ( σ, τ), π ( σ′ , τ)] = iη
δ( σ − σ′
) with
respect to s. Since X ν ( σ′ , τ)
is not a function of s, only X µ ( σ, τ)
is affected.
Proceeding and again using Eq. (4.2) we fi nd
µν ∂
µ ν
iηδσ ( − σ′ ) = T[ ∂ σ X ( σ, τ), ∂ + ′ X ( σ′ , τ)]
∂σ
µ ν
+ T[ ∂ X ( σ, τ), ∂ ′ X ( σ′ , τ)]
σ
µ
ν
= T[( ∂ −∂ ) X ( σσ, τ), ∂ ′ X ( σ′ , τ)]
+ −
µ
ν
+ T[( ∂ −∂ ) X ( σ, τ),
∂ ′ X ( σ′ , τ)]
+ −
−
+
−
µ ν
µ ν
= T[ ∂ X ( σ, τ), ∂ ′ X ( σ′ , τ)] −T[ ∂ X ( σ, τ), ∂ ′ X ( σ′ , τ)]
+ +
− +
µ
ν µ ν
+ T[ ∂ X ( σ, τ ), ∂ ′ X ( σ′ , τ)] −T[ ∂ X ( σ, τ), ∂ ′ X ( σ′ , τ)]
+
− − −
Now we can utilize the commutation relation for the conjugate momenta in
Eq. (4.1). We have
µ ν
µ
ν
0 = [ π ( σ, τ), π ( σ′ , τ)] = [ T∂ X ( σ, τ),
T∂ X ( σ′ , τ)]
τ
2 µ
ν
= T [ ∂τX ( σ, τ), ∂ τ X ( σ′ , τ)]
µµ
ν
= T[ ∂τX
( σ, τ), ∂ τ X ( σ′ , τ)]
(Since it equals zero, we divide
by T for later convenience)
µ ν
= T[(∂
+∂ ) X ( σ, τ),( ∂ ′ + ∂ ′ ) X ( σ′ , τ)]
= T[ ∂ X
+
+ − + −
µ
ν µ ν
( σ, τ), ∂ ′ X ( σ′ , τ)] + T[ ∂ X ( σ, τ),
∂′
X ( σ′ , τ)]
+ + −
µ ν
µ ν
+ T[ ∂ X ( σ, τ), ∂ ′ X ( σ′ , τ)]
+ T[ ∂−X
( σ, τ), ∂ −′ X ( σ′ , τ)]
− +
µ ν
Now since [ π ( σ, τ), π ( σ′ , τ)]
= 0, let’s form the sum [p m (s, t), p n (s ′, t)] +
ih mn (/ s )d (s - s′). We obtain
µ ν µν ∂
[ π ( σ, τ), π ( σ′ , τ)] + iη
δσ ( − σ′
)
∂σ
µ
ν µ ν
= T[ ∂ + X ( σ, τ), ∂ + ′ X ( σ′ , τ)] + T[ ∂ + X ( σ, τ),
∂−′
X ( σ′ , τ)]
µ ν
+ [ ∂ ( σ, τ), ∂ ′ ( σ′ , τ)]
+ [ ∂ X ( σ, τ), ∂ X ( σ′ , τ)]
µ ν
T − X + X T − −′
µ
ν µ ν
+ T[ ∂ X ( σ, τ),
∂ ′ X ( σ′ , τ)] −T[ ∂ X ( σ, τ), ∂ ′ X ( σ′ , τ)]
+
+ − +
µ ν µ
+ T[ ∂ + X ( σ, τ), ∂ −′ X ( σ′ , τ)] −T[ ∂−X
( σ, τ), ∂ −′ ( ′ , )] X ν σ τ
τ