String Theory Demystified

26 String Theory Demystifi ed
This allows us to write the action in the form
∫ τ η µν
S =−m d − X µ ν
X
(2.9)
This action is a nice compact form that allows us to derive the equations of motion.
As you recall from your studies of classical mechanics or quantum fi eld theory, the
quantity in the integrand is called the lagrangian:
L =−m − X µ
Xν =−m −X
2
η µν
There are two problems with the action so far developed in Eq. (2.9). First, think
about what happens in the case of a massless particle. Setting m = 0 leaves us with
S → 0 and so there is nothing to vary to obtain the equations of motion. So this
action isn’t very helpful in the case of a massless particle. Also, it turns out that
quantization is not easy when we have a square root in the action. For these reasons,
we introduce an auxiliary fi eld that we will denote a( τ ) and consider the
lagrangian
L
a X
2
1 m
= 2
− a
2 2
We can use this to defi ne an alternative expression for the action
1 ⎛ 1 2 2 ⎞
S′ = ∫ dτ
−
⎝
⎜ X m a
2 a ⎠
⎟
(2.10)
We can vary this action to fi nd an equation of motion for the auxiliary fi eld a( τ ) . We
fi nd
1 ⎛ 1 ⎞
δS′ = δ∫dτ −
⎝
⎜ X2 2
m a
2 a ⎠
⎟
1
=
2
∫
⎛ ⎞
dτ
δ
⎝
⎜
a⎠ ⎟ X
⎛ 1
2 2 ⎞
−
⎝
⎜ δ(
ma)
⎠
⎟
1 ⎛ 1 ⎞
= ∫ dτ
− −
⎝
⎜ X2 2
m
2
2 a ⎠
⎟