String Theory Demystified

154 String Theory Demystifi ed
Previously a closed string was constrained by the following periodic boundary
condition:
µ µ
X ( σ, τ) = X ( σ +2 π, τ)
(8.1)
This boundary condition was stated with the implicit assumption where the string was
moving in a space-time with noncompact dimensions. Now let’s modify the situation.
As stated above, we are going to let the 25th dimension be a circle with radius R. This
changes the boundary condition in Eq. (8.1) as follows, but only for X 25 :
25 25
X ( σ + 2π, τ) = X ( σ, τ) + 2πnR
(8.2)
The interesting thing about Eq. (8.2) is that now the string will have winding states.
Simply put, the string can wind around the compactifi ed dimension any number of
times. For all other dimensions µ ≠ 25 , Eq. (8.1) still holds.
The number n in Eq. (8.2) is called the winding number. Using the winding
number we can defi ne the winding w as
w nR
=
α ′
(8.3)
We are going to see in a moment that the winding is actually a type of momentum. The
periodic boundary condition in Eq. (8.2) can be written in terms of the winding as
25 25
X ( σ + 2π, τ) = X ( σ, τ) + 2πα′
w
(8.4)
Now let’s take a closer look at the boundary condition by seeing how it affects leftmoving
and right-moving modes. This will demonstrate the fact that the winding is
µ µ µ
a kind of momentum. First recall that X ( στ , ) = XL( στ , ) + XR(
στ , ) . The left- and
right-moving modes can be written as follows:
X x p i
n e
25 1 25 α 25 α αn
L ( στ , ) = L + L ( τ σ)
2 2 2
′
′
+ + ∑
n≠0
X x p i
n e
25 1 25 α 25 α αn
R ( στ , ) = R + R ( τ σ)
2 2 2
′
′
− + ∑
n≠0
−
−
in( τ+ σ)
in( τ−σ) (8.5)
(8.6)
25 1/ 2 25 25 1/ 2 25
where we have made the identifi cations α0= ( α′ / 2) pL and α0= ( α′
/ 2)
pR.
Adding Eqs. (8.5) and (8.6) together (while ignoring the oscillator contributions) we get
25 25 α 25 25 25 25
X ( στ , ) x pL pR τ pL pR
2 2
α
= + ′
( + ) + ′
( − )σσ + modes (8.7)