In string theory, If an open string obeys the Neumann boundary condition, then in the static gauge, one can show that the end points move at the speed of light. The derivation is straightforward, but how can this apply to the massive string?
Another question, in what sense are the end points of open strings fixed in spacetime? I know nothing about D-branes but I'm interested to know how can this be done.
Have a look at these lecture notes by David Tong. When you vary the Polyakov action to obtain the equations of motion for the open string, you get two boundary terms. As usual, you want these to be zero so that you can invoke the principle of least action. You can do this by requiring 1) Neumann boundary conditions, 2) Dirichlet boundary conditions or 3) mixed Neumann/Dirichlet boundary conditions. The latter case means that one end point of the string is fixed on a so-called D$p$-brane and the other end point is free to move in space.
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