In the last entry we explored the relativistic string and arrived at the Nambu-Goto action,

    \[ S_{NG} = - \frac{T_0}{c} \int_{\tau_i}^{\tau_f} d \tau \int_{0}^{\sigma_1} d\sigma \sqrt{(\dot{X} \cdot X^{\prime^2}) - \dot{X}^2 X^{\prime^{2}}} \]

    \[ \implies \mathcal{L} (\dot{X}^{\mu}, X^{\prime \mu}) = - \frac{T_0}{c} \sqrt{(\dot{X} \cdot X^{\prime^2}) - \dot{X}^2 X^{\prime^2}} \]

Equations of motion

We can obtain the EoM by setting the variation of the action equal to 0. So, we vary the action

    \[ \delta S = \int_{\tau_i}^{\tau_f} d \tau \int_{0}^{\sigma_1} d\sigma [\frac{\partial \mathcal{L}}{\partial \dot{X}^{\mu}} \frac{\partial (\delta X^{\mu})}{\partial \tau} + \frac{\partial \mathcal{L}}{\partial X^{\mu \prime}} \frac{\partial (\delta X^{\mu})}{\partial \sigma}] \]

Where \delta \dot{X}^{\mu} = \delta (\frac{\partial X^{\mu}}{\partial \tau} = \frac{\partial \delta X^{\mu}}{\partial \tau}) and likewise in terms of the analogue for X^{\prime \mu}.

However, notice or think about how complicated things become when we look to compute \frac{\partial \mathcal{L}}{\partial \dot{X}^{\mu}} and \frac{\partial \mathcal{L}}{\partial X^{\prime \mu}}. The result we obtain for both terms is below,

    \[ \frac{\partial \mathcal{L}}{\partial \dot{X}^{\mu}} = \frac{(\dot{X} \cdot X^{\prime}) X_{\mu}^{\prime} - (X^{\prime^{2}} \dot{X}_{\mu})}{\sqrt{(\dot{X} \cdot X^{\prime^{2}}) - \dot{X}^2 X^{\prime^{2}}}} \]

    \[ \frac{\partial \mathcal{L}}{\partial X^{\prime \mu}} = \frac{(\dot{X} \cdot X^{\prime}) X_{\mu}^{\prime} - (\dot{X})^{2} X_{\mu}^{\prime}}{\sqrt{(\dot{X} \cdot X^{\prime^{2}} - \dot{X}^2 X^{\prime^{2}}}} \]

So, to put it succinctly, we want to try to simplify things a bit. To do so, let’s set \frac{\partial \mathcal{L}}{\partial \dot{X}^{\mu}} = \mathcal{P}_{\mu}^{\tau} and \frac{\partial \mathcal{L}}{\partial X^{\mu}\prime} = \mathcal{P}_{\mu}^{\sigma}. Now we can perform a very direct and explicit substitution,

    \[ \delta S = \int_{\tau_i}^{\tau_f} d \tau \int_{0}^{\sigma_1} d\sigma [ \mathcal{P}_{\mu}^{\tau} \frac{\partial (\delta X^{\mu})}{\partial \tau} +  \mathcal{P}_{\mu}^{\sigma} \frac{\partial (\delta X^{\mu}}{\partial \sigma} - \delta X^{\mu} (\frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau} +  \frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma})] \]

Tidying things up,

    \[ \delta S = \int_{\tau_i}^{\tau_f} d \tau \int_{0}^{\sigma_1} d\sigma [ \frac{\partial}{\partial \tau}(\delta X^{\mu} \mathcal{P}_{\mu}^{\tau}) + \frac{\partial}{\partial \sigma}(\delta X^{\mu} \mathcal{P}_{\mu}^{\sigma}) -  \delta X^{\mu} (\frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau} +  \frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma})] \]

At this point, the main thing is that we’re going to restrict the variation such that \delta X^{\mu}(\tau_{f}, 0) = \delta X^{\mu}(\tau_{i}, \sigma) = 0.

    \[ \implies \delta S = \int_{\tau_i}^{\tau_f} d\tau [\delta X^{\mu}  \mathcal{P}_{\mu}^{\sigma} ]_{0}^{\sigma_1} - \int_{\tau_i}^{\tau_f} d\tau  \int_{0}^{\sigma_1} d\sigma \ \delta X^{\mu} (\frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau} +  \frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma}) \]

We know the second term on the right-hand side must vanish for all variations \delta X^{\mu} of the motion. Therefore,

    \[ \frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau} +  \frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma} = 0 \]

Thus, we arrive at the EoM for a free relativistic string. This holds for open or closed strings. But the main thing to notice is that, again, it is extremely complicated. For example, consider taking the second derivative of \mathcal{P}_{\mu}^{\tau} with respect to \tau. It’s just not very nice.

To simplify matters further, emphasis is placed on the choice of (\tau , \sigma) coordinates. In a very direct way, we need to put constraints on the solutions to the above equation.

Boundary conditions, enter Dp-branes

Let’s reconsider the following result,

    \[ \delta S_{NG} = \int_{\tau_i}^{\tau_f} d\tau [\delta X^{\mu} \mathcal{P}_{\mu}^{\sigma} ]_{0}^{\sigma_1} - \int_{\tau_i}^{\tau_f} d\tau \int_{0}^{\sigma_1} d\sigma \ \delta X^{\mu} (\frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau} + \frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma}) \]

The first term on the the right-hand side concerns the string endpoints. If one were to expand this out they would arrive at a collection of terms for each index, \mu. Ultimately, we need boundary conditions for each term. With that goal in mind, we can impose two sorts of boundary conditions at the endpoints of the string: Dirichlet boundary conditions or Neumann boundary conditions. We actually have quite a bit of freedom when it comes to our choice, thanks to the construction of the action.

One way to think of this is by denoting \sigma_{*} = \{ 0, \sigma_{1} \} \rightarrow \mathcal{P}_{\mu}^{\sigma} (\tau, \sigma_{*})\delta X^{\mu}(\tau, \sigma_{*}). Here, \sigma_{*} represents some \sigma-coordinate at the endpoints, which, as we’ve already established, will equal either 0 or \sigma_{1}.

For Neumann boundary conditions,

    \[ \frac{\partial \mathcal{L}}{\partial X^{\prime \mu}} = \mathcal{P}_{\mu}^{\sigma} (\tau, \sigma_{*}) \rightarrow \mathcal{P}_{\mu}^{\sigma} \rvert_{\sigma_{1}} = (0, \pi) = 0 \]

Neumann boundaries, otherwise known as free boundaries, mean that for open strings the ends can move freely. The physics of the endpoints is interesting and worth review if one is not familiar (see Zwiebach, 2004), as the endpoints of an open string always move with the speed of light. This also means their worldlines are lightlike. Additionally, it can also be shown that the momentum is conserved at the end of the string.

(For periodic boundary conditions in the case of closed strings, where the string does not have timelike boundaries: X(\sigma_{1}) = X(\sigma_1 + \pi).)

Fixing \sigma_{*}, the Dirichlet boundary condition \frac{\partial X^{\mu}}{\partial \tau}(\tau, \sigma_{*}) = 0, where \mu \neq 0. Here the string endpoint is fixed in time, and so the \tau derivative vanishes.

The EoM can then be written as,

    \[ \partial_0 \frac{\partial \mathcal{L}}{\partial \dot{X}^{\mu}} + \partial_1 \frac{\partial \mathcal{L}}{\partial X^{\partial \mu}} \]

Or, equivalently, \partial^{\alpha} \mathcal{P}_{\alpha}^{\mu}.

What is nice about this discussion is that we arrive at an intuitive introduction to the concept of the spacelike surfaces of D-branes or, more concisely, Dp-branes, with p-dimensionality. (For example, a D0-brane is a particle like object. A D2-brane is like a hyperplane).

To approach it differently: from the case of classical mechanics, we know that if a string has Dirichlet boundary conditions then the ends of the string are fixed. But this raises the obvious question: to what, in this case, might the ends be fixed? The objects that constrain the motion of the endpoints are D-branes. More elaborately, with Neumann boundary conditions of p timelike and spacelike conditions and D-p Dirichlet boundary conditions, we can say that the ends of the string are fixed on some p-dimensional D-brane.

As this post only serves as a brief introduction, D-branes will be discussed in more detail another time. A few comments in the meantime: 1) I think that while limited the above description some intuition about branes which are quite complex objects in ST. 2) Dp-branes do not break Lorentz invariance on the grounds that they are space filling objects. However many higher-dimensions are theorised – say, 10-dimensions for example – the D-brane would fill 3D space and also some of the extra dimensions. 3) Momentum is not conserved at the ends of the string in the Dirichlet directions (translation invariance is broken).

Generalising p-brane action

As an aside, recall the action for a point particle,

    \[ S_{PP} = -m \int d\tau (- \dot{X}^{\mu} \dot{X}_{\mu})^{\frac{1}{2}} \]

Though a slight distraction from the early pages of Polchinski, it should be noted that this action can be generalised, as we have seen, not only to the case of a string sweeping out a (1+1)-dim worldsheet, but also to a p-brane sweeping out a (p+1)-dimensional world-volume. We can parameterise the brane by, again, invoking timelike and spacelike coordinates. In that we’re considering D-dimensional spacetime, p<D, we can picture a D2-brane sweeping a world-volume in higher dimensional spacetime.

The generalised action is,

    \[ S_{p} = -T_{p} \int d{\mu}_p \]

Where T_p is the brane tension. As for d{\mu}_p, this is the volume element. It is (p+1)-dimensional and looks like this,

    \[ d{\mu}_p = \sqrt{- det G_{\alpha \beta}} \ d^{p+1} \sigma \]

Here, the induced metric is given by,

    \[ G_{\alpha beta} = g_{\mu \nu} (X) \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} \]

Slope parameter \alpha^{\prime}

There is one last comment to be made here, this time in reference to p. 11 in Polchinski. Recall the string tension in the NG action.

An alternative parameter to the tension is \alpha^{\prime}. This parameter has been used since the early days in string theory. It is a proportionality constant, and, if one is already familiar with the Regge trajectories, they will understand \alpha^{\prime} in terms of the relation between the angular momentum, J, of a rotating string and the square of the energy E. (For a bit of history and intuition, see this lecture by Leonard Susskind). In that \alpha^{\prime} is a famous constant in string theory, as Polchinski notes, and in that it has units of spacetime-length-squared, which is the Regge slope, we observe:

    \[ T = \frac{1}{2 \pi \alpha \prime} \]

Where \hbar = c = 1. From this another famous result can be arrived at regarding the string length, \mathcal{l}_s. It is written as follows,

    \[ \mathcal{l}_s = \sqrt{\alpha \prime} \]

A full treatment of this can be found in Chapter 8 of Zwiebach, 2004.

In closing, the general convention then is write the Nambu-Goto action in this form,

    \[ S_{NG} = - \frac{1}{2 \pi \alpha \prime} \int_{\sum} d\tau d\sigma \mathcal{L}_NG \]

Where \sum is the worldsheet that we’ve already considered, and \mathcal{L} is the Lagrangian.