In the last entry, we looked into a study of the nonrelativistic string and derived the appropriate equation of motion. In the process, a number of comments were directed toward the reader to pay closed attention to the end points of the string. It was noted that the concept of boundary conditions will become very important in the future, and I also directed the reader to review Chapter 4 in Zwiebach (2004) should more detailed engagement be necessary.

I think it is worthwhile spending a few more minutes on this before turning our attention to the relativistic point particle, which then also brings us directly into engagement with the opening pages of Polchinski.

Recall from the previous post:

    \[ S_{NR} = \int_{t_i}^{t_f} dt \int_{0}^{a} dx \mathcal{L} (\dot{y}, y \prime) \]

    \[ \implies L = \int_{t_i}^{t_f} dt \int_{0}^{a} dx (\frac{1}{2} \ \mu \ \dot{y} - \frac{1}{2}  T y\prime^2) \]

Using slightly different notation, this is the very same Lagrangian that we worked out previously. We know, too, from that entry how to work out the equation of motion. For example, when we varied the action we ended up with an expression that looks like this (after integration by parts):

    \[ \delta S = \int_{0}^{a} dx (\mathcal{P}^t \delta y)]_{t_i}^{t_f} + \int_{t_i}^{t_f } dt (\mathcal{P}^x \delta y)]_{0}^{a} - \int_{t_i}^{t_f} dt \int_{0}^{a} dx \delta y (\frac{\partial \mathcal{P}^t}{\partial t} + \frac{\partial \mathcal{P}^x}{\partial x}) \]

Now, let’s think about the behaviour of the end points of our string with more care. I’m going to rewrite the above in a slightly different way,

    \[ \delta S = \int_{t_i}^{t_f} dt \ dx[ \mu \frac{\partial}{\partial t}(\dot{y} \ \delta y) - T \frac{\partial}{\partial x}(y\prime \ \delta y) - (\mu \ \ddot{y} - T y\prime \prime) \delta y] \]

The last term on the right-hand side is our equation of motion, as before, but again written in a different way. But we’re not concerned with that here. Instead, we want to focus in on the first two terms on the right. We can expand them, as written below:

    \[ \mu \int_{0}^{a} dx [\dot{y}(t_f, x)\delta y(t_f, x) - \dot{y} \delta y(t_i, x)] + T \int_{t_i}^{t_f} dt [y \prime (t, 0) \delta y (t, 0) - y\prime (t, a) \delta y(t,a)] \]

The first-term on the right-hand side need not be considered here, as I said previously, because the initial and final configurations of the string are fixed. In other words, \delta y(t_f, x) = 0 and \delta y(t_i, x) = 0. Furthermore, this

    \[ y \prime (t, 0) = y\prime (t, a) = 0 \]

is the Neumann boundary condition for the endpoints of our string. While this,

    \[ \delta y (t, 0) = \delta y(t,a) = 0  \]

is the Dirichlet boundary condition for the endpoints of our string.

In introductory texts on boundary conditions, one will often be presented with a string such that its endpoints are pictured as attached to rings which can then only move vertically along the poles to which they are fixed. These are Neumann boundary conditions. On the other hand, Dirichlet boundary conditions are such that the endpoints of the strings are fixed, so there is no horizontal or vertical movement. (It is also possible to have mixed boundary conditions, but I should not like to risk adding too much to the picture at this time).

There is a nice way to think of all of this in relation to nonrelativistic string momentum, such as we did in the last post. But, again, I will refer the reader to Chapter 4 in Zwiebach (2004) for further review.

Now, here is the point I wanted to get to. The discerning reader may ask: since we’re working our way in to string theory and thus toward expanded physical contexts, such as when we consider the relativistic string for example, to what might the ends of the string be fixed? The answer, we will learn, is D-branes. And this is the reason why I wanted to take a few moments to engage on the topic of boundary conditions, because I think it draws a nice connection to future considerations with respect to the relativistic string.