DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 2. BOSONIC STRINGS 20which confirms our claim for p µ . Furthermore, simply applying the definition <strong>of</strong> anaverage to X µ (σ), we see that1π∫ π0dσX µ (σ, τ) = x µ + l 2 sp µ τ, (2.38)indicating that x µ is the center <strong>of</strong> mass position at time τ = 0. Note that x µ and p µare arbitrary, i.e. if we wanted to we could explicitely fill them in, thereby specifying aninitial position and momentum for our string.2.5.2 <strong>Open</strong> string expansionA feature present in all open string expansions is that only one set <strong>of</strong> expansion coefficientsα is present. Before giving the expansions, this can already be understoodby realizing that whilst the closed strings allow oscillations in two directions, in thecase <strong>of</strong> open strings left- and right-movers need to combine into standing waves due toreflections at their endpoints, hence leaving only one set <strong>of</strong> coefficients.Dirichlet boundary conditionsThe wave equations for an open string satisfying Dirichlet boundary conditions can beexpanded as∑X µ (σ, τ) = x µ + w µ 1σ − l sn αµ ne −inτ sin(nσ), (2.39)with again µ ≠ 0. We need sin(nσ), and not cos (nσ), in order to accomodate thevanishing <strong>of</strong> the oscillators in the endpoints. The constant w µ is called the windingnumber <strong>of</strong> the string. Note thatn≠0X µ (σ = π, τ) − X µ (σ = 0, τ) = w µ π, (2.40)and hence w µ encodes the distance between both endpoints. We will come back to thiswhen we will consider T-duality and D-branes.Neumann boundary conditionsFor an open string satisfying Neumann boundary conditions, the expansion readsX µ (σ, τ) = x µ + l 2 sp µ τ + il s∑n≠01n αµ ne −inτ cos (nσ). (2.41)Note that this time we need cos (nσ), so that the ∂ σ derivatives will vanish at theendpoints. Also note that since no momentum can flow <strong>of</strong> the ends <strong>of</strong> the string, we canagain talk about a center <strong>of</strong> mass momentum p µ .This time, we can define the zero-mode asα µ 0 = l sp µ . (2.42)