Introduction to String Theory and D–Branes - School of Natural ...

X 1
0
X
σ
τ
X 2
X µ (τ,σ)
τ
σ
0 π
Figure 1: A string’s world–sheet. The function X µ (τ, σ) embeds the world–sheet, parameterized by (τ, σ),
into spacetime, with coordinates given by X µ .
distances on the world–sheet as an object embedded in spacetime, and hence define an action analogous to
one we would write for a particle: the total area swept out by the world–sheet:
So = −T dA = −T dτdσ (−dethab) 1/2
≡ dτdσ L( ˙ X, X ′ ; σ, τ) . (2)
So
∂X µ ∂X
= −T dτdσ
∂σ
µ 2 µ 2
2
1/2
∂X ∂Xµ
−
∂τ ∂σ ∂τ
= −T dτdσ (X ′ · X) ˙ 2 ′2
− X X˙ 2 1/2 , (3)
where X ′ means ∂X/∂σ and a dot means differentiation with respect to τ. This is the Nambu–Goto action.
T is the tension of the string, which has dimensions of inverse squared length.
Varying the action, we have generally:
δSo =
∂L
dτdσ
∂ ˙ X µδ ˙ X µ + ∂L
=
′µ
∂X ′µδX
dτdσ − ∂ ∂L
∂τ ∂ ˙ ∂ ∂L
−
X µ ∂σ ∂X ′µ
δX µ
+
Requiring this to be zero, we get:
∂ ∂L
∂τ ∂ ˙ ∂ ∂L
+ = 0 and
X µ ∂σ ∂X ′µ
which are statements about the conjugate momenta:
∂
∂τ P µ τ
+ ∂
∂σ P µ σ = 0 and P µ σ
dτ
∂L σ=π
′µ
∂X ′µδX
σ=0
. (4)
∂L
= 0 at σ = 0, π , (5)
∂X ′µ
= 0 at σ = 0, π . (6)
Here, P µ σ is the momentum running along the string (i.e., in the σ direction) while P µ τ is the momentum
running transverse to it. The total spacetime momentum is given by integrating up the infinitesimal (see
figure 2):
dP µ = P µ τ dσ + P µ σ dτ . (7)
6