String Theory Demystified

30 String Theory Demystifi ed
is once again the proper time τ, and the second parameter, which is related to the
length along the string, is denoted by σ :
ξ = τ ξ = σ
1 2
Coordinates on the worldsheet ( τ, σ ) are mapped onto space-time by the functions
(called the string coordinates)
X µ ( τ, σ)
(2.12)
So time and spatial position on the string are mapped onto the spatial coordinates in
(d + 1) dimensional space-time as
{ X ( , ), X ( , ), , X ( , )}
d
0 1
τσ τσ … τσ
Now, we need to write down the action for the string which will generalize
Eq. (2.8) to our new higher-dimensional world, that is, to the case of the worldsheet.
This is done in the following way. Recall that the action of a point particle is
proportional to the length of its world-line [Eq. (2.5)]. We just noted that a string
sweeps out a two-dimensional worldsheet in space-time. This tells us that if we are
going to generalize the notion of the action of a point particle, we might expect that
the action of a string is proportional to the surface area of the worldsheet. This is in
fact the case. Anticipating that the constant of proportionality will turn out to be the
string tension, we can write this action as
S =−T∫ dA
(2.13)
where dA is a differential element of area on the worldsheet. To fi nd the form of
dA, we start by considering a differential line element ds 2 and introduce coordinates
1 2
on the worldsheet as ξ = τ and ξ = σ.
Doing a little algebra we have
2
ds =−η
dX dX
=−η
µν
µν
µ ν
µ ν
∂X
∂X
dξ dξ
α β
∂ξ
∂ξ
α β
This allows us to defi ne an induced metric on the worldsheet. This is given by
γ = η
αβ µν
µ ν
∂X
∂X
α β
(2.14)
∂ξ
∂ξ