Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 306 / 1 / ii / Solution

Define the open-string endpoint momentum flux along the open string by $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Variation of the Nambu-Goto phase-space action givesThe two last equations are the Virasoro constraints. Integration of the momentum equation givesThus the necessary and sufficient condition for total momentum conservation is equality of the two endpoint fluxes, component by component. Vanishing of both fluxes is a sufficient local boundary condition.

The spatial boundary term in the action variation is $-\int dt[J_m\delta X^m{]}_0^{\pi }$. For free ends the endpoint variations are arbitrary, soIn a gauge with $u=0$ and nonzero $e$ this reduces to the usual Neumann boundary condition $X^{\prime m}=0$. The conclusion is conservation of total momentum, not that total momentum must vanish. The PDF contains the derivative in this conclusion; the TeX transcription drops it.

For endpoints confined to $x^1=0$, the Dirichlet boundary condition fixes $\delta X^1=0$. It therefore imposes no requirement that $J_1$ vanish. Preserving the fixed position requires ${\dot{X}}^1=e{P}^1+u{X}^{\prime 1}=0$ at each endpoint, but that is a different condition. For example, in $u=0$ gauge the endpoint momentum $P^1$ is zero while $e{T}^2{X}^{\prime 1}$ can be nonzero. Consequently ${\mathcal{P}}_1$ is generally not conserved: the normal endpoint forces transfer momentum to the hyperplane. A dynamical D-brane recoils and carries the missing momentum. If the hyperplane is idealized as fixed, its external support absorbs the momentum. The combined string-and-support system conserves momentum; freely varying directions tangent to the hyperplane retain their vanishing-flux condition.