String Theory Demystified

CHAPTER 8 Compactifi cation and T-Duality 155
The total center of mass momentum of the string is
25 25 25
p = p + p
(8.8)
L R
Along the compactifi ed dimension, the string acts like a particle moving on a circle.
The momentum is quantized according to
25 K
p = (8.9)
R
where K is an integer called the Kaluza-Klein excitation number. This is an important
result—without the compactifi ed dimension, the center of mass momentum of the
string is continuous. Compactifying a dimension quantizes the center of mass
momentum along that dimension.
Looking at Eq. (8.7) then, the fi rst term involving the momenta is the total center
of mass momentum of the string. We call this the momentum mode. The second
term, however, also involves momentum. In fact this term is the winding mode of
the string, which satisfi es
α ′ 25 25 ( p − p )= nR
2
L R (8.10)
Looking at Eq. (8.3), we see that the winding w can be defi ned in terms of the
momentum of the left- and right-moving modes as
w nR
= p p L R
′ =
1
( − )
α 2α′
25 25 (8.11)
Modifi ed Mass Spectrum
Compactifying a dimension will lead to a modifi ed mass spectrum. To obtain the
mass spectrum for the state with a compactifi ed dimension, let us begin with the
Virasoro operators. Recall that
L = p p
0
R R n n
4
′
∞
α µ
+ ∑α
⋅α
µ −
(8.12)
Note the repeated index which is an upper and lower index on the fi rst term in
the right—so we have an implied sum. Here the index µ ranges over the entire
n=
1