String Theory Demystified

CHAPTER 2 Equations of Motion 27
Now we set δ S′ = 0. Since this means that the integrand must be 0, we obtain the
equation
1 2 2
− X− m = 0
2
a
2 2 2
⇒ X+ m a = 0
This is the equation of motion for the auxiliary fi eld. From this we fi nd an expression
for the auxiliary fi eld given by
2
X
a = −
m
2 (2.11)
Using Eq. (2.11), we can show that the action in Eq. (2.10) is equivalent to Eq. (2.8),
which we do in the next example.
EXAMPLE 2.2
Show that if a= ( −X
/ m ) / 2 2 1 2 2 2
, the action S′ = 12 / dτ[( 1/
a) X −m
a]
can be recast
∫ η µν
in the form S =−m − dX dX
µ ν .
SOLUTION
Let’s start by recalling that 2
X X µ
= ηµν Xν
. So we can rewrite the action S ′ in the
following way:
1 ⎛ 1 2 2 ⎞
S′ = ∫ dτ
−
⎝
⎜ X m a
2 a ⎠
⎟
1
=
2
1
=
2
∫
⎛
dτ
⎜
⎝
⎛
dτ
⎜
⎝
∫
2
−m
X
X2 2
−m −
2
X m
2
1 ⎛ −m
⎞
2 2
= d X −m −X
2
2 ∫ ⎜
⎟
⎝ X ⎠
τ
∫
2
2
⎞
⎟
⎠
2
−m
X 2
−m −η
X X
2
X µν
µ ν
⎞
⎟
⎠