String Theory, Part III: Boundary Conditions

Because we are starting with an open string,  we must define constraints on the string at the boundary points. We will start by varying the action

\displaystyle\mathit{S} = \int_{t_i}^{t_f}dt\int_0^a dx\;\mathcal{L}(\dot{y}, y^\prime) ,

where dotted denotes the derivative with respect to time, and primed denotes the derivative with respect to space. Varying the action we get

\displaystyle\delta S=\int_{t_i}^{t_f}dt\int_0^a dx\left(\frac{\partial\mathcal{L}}{\partial\dot{y}}\delta y+\frac{\partial\mathcal{L}}{\partial y^\prime}\delta y\right) \\    =\int_{t_i}^{t_f}dt\int_0^a dx\left(\mathcal{P}^t\frac{\partial}{\partial t}\delta y+\mathcal{P}^x\frac{\partial}{\partial x}\delta y\right) .

Using integration by parts, we find that this is

\displaystyle =\int_0^a dx\left.\left(\mathcal{P}^t\delta y\right)\right|_{t_i}^{t_f}

\displaystyle +\int_{t_i}^{t_f}dt\left.\left(\mathcal{P}^x\delta y\right)\right|_{0}^{a}

\displaystyle +\int dt\int dx\;\delta y\left(\frac{\partial\mathcal{P}^t}{\partial t}+\frac{\partial\mathcal{P}^x}{\partial x}\right)

We want the variation to vanish so we must analyze the three parts that I have put on separate lines. Setting the first line to zero is equivalent to knowing the state of the string at points t_i and t_f , which is not all that useful. Setting the last line equal to zero gives us our equation of motion discussed in the last section. However, setting the second line equal to zero gives us something interesting. Written out fully, the second term is

\displaystyle \int_{t_i}^{t_f}dt\left(\mathcal{P}^x(a, t)\delta y(a, t)-\mathcal{P}^x(0, t)\delta y(0, t)\right)

Let us use the notation x^*=\{0, a\} so that we make condense the above equation into

\displaystyle\mathcal{P}^x(x^*, t)\delta y(x^*, t) = 0

Notice that we are treating each end separately and not assuming that \mathcal{P}^x(a, t)\delta y(a, t) and \mathcal{P}^x(0, t)\delta y(0, t) cancel one another out. We may now say that either

\displaystyle\mathcal{P}^x(x^*, t) = 0

or that

\displaystyle\delta y(x^*, t)=0.

The first condition lets the endpoints move, but the slope remains zero at those points, or

\displaystyle\frac{\partial y}{\partial x}(x=0,t) = \frac{\partial y}{\partial x}(x=a,t) = 0

This is known as the Neumann boundary condition.

If instead we consider the second condition, we have

\displaystyle\frac{\partial y}{\partial t}(t, x=0) = \frac{\partial y}{\partial t}(t, x=a) = 0

This is equivalent to a string whose ends are attached to each of two walls, one at x=0, t=0 and one at x=a, t=0. This is called the Dirichlet boundary condition, and with it, the end points cannot move. For a long time it was believed that the Dirichlet boundary condition was an unphysical constraint, that the Neumann boundary condition was the only realistic possibility. Joseph Polchinsky was eventually the one to realize that the Dirichlet boundary condition is a result of open strings whose ends are attached to D-branes (the “D” stands for Dirichlet) and that it should necessarily be considered. It is now accepted that the ends of open strings that are attached to D-branes can never leave the surface, whereas closed strings (to be discussed) are free to break off of a surface.

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One Response to String Theory, Part III: Boundary Conditions

  1. sheabrowne says:

    Nice to see where the D in D-brane comes from. Nice exposition on boundary conditions!

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