String Theory Demystified

46 String Theory Demystifi ed
This can only be true if sin kπ = 0 , which means that k must be an integer. Denoting
it by n, a simple exercise shows that Eq. (2.47) can be written as
Closed Strings
µ µ 2 µ
X = x + p τ + i
s
2
µ
s n
∑
n≠0
α
n
−inτ
e cos( nσ)
(2.53)
In the case of closed strings, the boundary condition becomes one of periodicity,
namely
µ µ
X ( τ, σ) = X ( τ, σ +2 π)
(2.54)
This condition restricts the solutions to those for which the wave number k takes on
integral values. Hence,
X
µ
L
X
x
p i
n e
µ 2
µ
s µ s αn
−in(
τ +
( τσ , ) = + ( τ+ σ)
+ ∑
2 2 2 n≠0 µ
R
µ 2
x µ
( τσ , ) = + p ( τ− σ)
+ i
2 2 2
n e
µ
α
s s n −in( τ−σ) ∑
n≠0
where n is an integer. In addition, periodicity enforces the condition that
σ )
(2.55)
(2.56)
µ µ
p = p
(2.57)
for closed strings. We saw that if this condition was satisfi ed in the case of open
strings, there was no winding of the string permitted. In the case of closed strings,
however, the situation is a little bit more involved if we allow for the possibility
where the ambient space-time includes a compact extra dimension (then p = p
µ µ
does not hold). Then, we can consider, for example, the situation where the closed
string is compactifi ed on a circle of radius R.
Using Eq. (2.57), we sum Eqs. (2.55) and (2.56) to obtain the complete solution,
focusing on the momentum term–and imposing the periodicity condition of Eq. (2.54).
This gives the total solution which can be written as
µ µ
X ( τσ , + 2πR) = X ( τσ , ) + 2 πRW
(2.58)