String Theory Demystified

CHAPTER 13 D-Branes 225
Once again we apply the quantization procedure, considering the bosonic string
theory case. The fi rst step is to write down the modal expansions. These will be
different depending on which coordinates we look at because now we have NN
coordinates and DD coordinates. The fi rst step is to write out the modal
expansions.
Now let’s recall the open string modal expansion, which can be written in the
following way:
µ
µ µ µ
αn
−inτ
X ( στ , ) = x + 2α′ p τ+ i 2 α′
∑ e cos( nσ)
(13.9)
0 0
n≠
n
Taking the derivative of this expression with respect to σ we fi nd
Clearly this expression satisfi es
∑
µ µ −inτ
∂ X ( στ , ) =− i 2α
σ
′ α e sin( nσ)
n
n≠0
µ
∂ X 0 = 0
σ
σ= , π
which are the Neumann boundary conditions. So, we take the modal expansion for
the X i to be
i
i i i αn
−inτ
X ( στ , ) = x + 2α′ p τ+ i 2 α′
∑ e cos( nσ)
(13.10)
0 0
n≠
n
For the DD coordinates, we really have two requirements that have to be satisfi ed.
We need the summation over the modes ∑ in the expansion to vanish at σ = 0, π.
n≠0 This indicates that we should use sin( nσ ) instead of cos( nσ ) which is in the
a a a
usual open string expansion. However, we also need X ( 0, τ) = X ( π, τ)
= x .
Looking at the usual modal expansion, this tells us that we must take
x → x 0
a
p → 0
a a
0
0
0
Quantization