String Theory Demystified

28 String Theory Demystifi ed
Now, let’s use a simple algebraic trick to rewrite the fi rst term. Remember from
complex variables that i 2
=− 1.
This means that
−m
X X
2
2
2
= ( −1)( −1)
=− −m
X iX
2
2
=− −m
X iX
2
2
−m
X X
2
2
2 2
=− −miX
=−m −i
X
4 4
2 4 2
4 2
But i 4
4
=+ 1,
and so −m − i X 2
=−m − X 2
=−m −η
X X
µν
action is
⎛ 2
1 −m
2
S′ = ∫ dτ⎜X
−m −η
X X
2
µν
2 ⎝ X 1
µ ν
= ∫ dτm −ηη
X X −m −η
X X
µν
µν
2
∫
=−m dτ −η
X X
2
µ ν
⎞
⎟
⎠
( µ ν )
( µν
µ ν ) = S
This demonstrates that the two actions are equivalent.
Strings in Space-Time
µ ν . Therefore the
At this point we have reviewed some basic techniques that help us calculate the
equations of motion for a free relativistic point particle. We are going to extend this
work to the case of a string moving in space-time. A point particle has no extent
whatsoever, so can be described as a zero-dimensional object. We have seen that its