String Theory Demystified

218 String Theory Demystifi ed
Now, we know that we can expand φ( x ) in a Fourier series. That is, we can
expand it in p where p is the conjugate momentum for the coordinate x. The
expansion looks like
φ( x) = ∑φ
e n
Now, of course, the exponential function is 2π periodic. Calculating φ( x+ 2 πR)
The presence of the extra term e
ip x+ R
φ( x+ 2πR)
= φe = φe
e
n
ipx
∑ n
( 2π )
∑ n
ipx ip( 2πR)
n
n
( ) means that we must take
ip R 2π
n
p =
R
So, we relearn an important rule about compactifi cation:
• Compactifi cation quantizes momentum.
For the heterotic string, we compactify each of the extra bosonic coordinates:
I I I
X = X +2π L
Here, L I I
represents the lattice spacing. If we span the lattice with basis vectors ei then
16
I 1
I
L = ne R
∑
2 i=
1
Here, the Ri are the radii of the compactifi ed dimensions. Now, we use the conjugate
momenta p I from the bosonic states, which are compactifi ed as a generator of
translations. We take 2p I
to be the generator of translations along the lattice in the
I I I
Ith direction. The periodicity condition X = X +2π L means that
e
I I
i2πp ⋅ L
=
i i
1
i