We want to solve the wave equation. But before that I would like to first note that for the open string we have the interval,

    \[-\infty \leq \tau \leq \infty \ \ \ \ 0 \leq \sigma \leq l \]

Recall, from the introduction of this chapter, that the action S_{P} has redundancies. It is possible to compute the amount of redundancies. For example,

    \[ \tau \rightarrow \tau^{\prime}(\tau, \sigma) \]

    \[ \sigma \rightarrow \sigma^{\prime}(\tau, \sigma) \]

What we see is that we already have two arbitrary functions. But there is a third, and this third redundancy is given by way of Weyl invariance. It should be clear that we want to fix the redundancies. And, so, what this means is that we have 3 conditions to fix diff + Weyl symmetries. 

The more important question now concerns how we go about this task of fixing these invariances. The answer is that there are various ways to approach the issue. The easiest way to motivate the reasoning and logic behind the ‘best choice’ is to show the calculation, which, if indeed the ‘best choice’, will unveil relatively simple equations of motion (Polchinski, p. 17).

Recalling, analogously, our previous discussion about the flat metric, we start with:

    \[ X^{+} = \tau \]

    \[ \partial_{\sigma} \gamma_{\sigma \sigma} = 0 \]

    \[ det \gamma = -1 \]

These are gauge fixing conditions implying LC gauge fixing for the string. What we see is that in choosing the LC gauge for the time coordinate on the worldsheet we also impose two conditions on the metric (p.17). Polchinski expands more deeply on these matters, which, to save space, I will leave as a topic of future study for the enquiring reader.

As an aside, one thing we might point out here is that one can always choose LC gauge fixing locally.

In summary: as alluded previously, we are to work firstly in the LC gauge. This gauge choice is fairly conventional at this point in the study of bosonic strings. The reason, as mentioned before, has to do with how working in the LC allows us generally quick and easy access into a study of the physical characteristics of the string.


From the LC analysis of the S_{P} action we come into contact with these conditions,

    \[ X^{+} = \tau \]

    \[ \partial_{\sigma} \gamma_{\sigma \sigma} = 0 \]

    \[ \gamma = -1 \]

The Hamiltonian of the open string in the LC can also be derived (p.19),

    \[ H_{0} = \frac{l}{4\pi \alpha^{\prime} P^{+}} \int_{0}^{l} d\sigma [2\pi \alpha^{\prime} \prod_{i} \prod^{i} + \frac{1}{2\pi \alpha^{\prime}} \partial_{\sigma X^{i} \partial_{\sigma}X^{i}}] \]

From the basis of understanding that the Hamiltonian is generally a function of momentum and coordinates, we denote H = H(P, q). The equation of motion for X can be written,

    \[ \partial_{\tau} X^{-}(\tau) = \frac{\partial H}{\partial P_{-}} \]

    \[ = - \frac{\partial H}{\partial P^{+}} = \frac{H}{P^{+}} \]

For P^{+},

    \[ \partial_{\tau} P^{+} = - \partial_{\tau} P_{-} = \frac{\partial H}{\partial X^{-}} = 0 \]

This implies that P^+ is a conserved quantity (p.19).

For the remaining EoM we have to use variation, and so we must generalise H. For example,

    \[ \partial_{\tau} X^{i} = \frac{\delta H}{\delta \prod^{i}} \]

    \[= 2\pi \alpha^{\prime} c \prod^{i} \ (*) \]

    \[c = \frac{l}{2\pi \alpha^{\prime} P^{+}} \]

And, also,

    \[ \partial_{\tau} \prod^{i} = - \frac{\delta H}{\delta X^{i}} \]

    \[ \frac{c}{2} \frac{1}{2 \pi \alpha^{\prime}} \cdot 2 \partial_{\sigma}^{2} X^{i} \]

    \[ \frac{c}{2 \pi \alpha^{\prime}} \partial_{\sigma}^{2} X^{i} \ \ i = 2, ..., D-1 \]

These EoM describe D-2 free fields (p.19). But what is very nice is that, if we take the derivative of (*), we find the wave equation with c being the speed of propagation (p.20). Note,

    \[ \partial_{\sigma}^{2} X^{i} = 2\pi \alpha^{\prime} c \cdot \partial_{\tau} \prod^{i} \]

    \[ = 2\pi \alpha^{\prime} c \frac{c}{2 \pi \alpha^{\prime}} \partial_{\sigma}^{2} X^{i} \]

    \[ \implies \partial_{\tau}^{2} X^{i} - c^2 \partial_{\sigma}^{2} X^{i} = 0 \]

The solution to the wave equation is therefore as follows,

    \[ X^{i}(\tau, \sigma) = X^{i} + \frac{P^{i}}{P^{+}} \tau + i(2\alpha^{\prime})^{\frac{1}{2}} \sum_{n = - \infty}^{\infty} \frac{1}{n} \alpha_{n}^{i} e^{- \frac{\pi i n c \tau}{l}} \cos\frac{\pi n \sigma}{l} \]

To explain what is going on: when we do the Fourier mode expansion for the wave equation, the function must vanish at both ends of the string. So we must expand in terms of cosine’s. Thus, the harmonics are in terms of cosine’s and cosine expansion. There is also a zero mode.

Another way to put it: this solution is constructed from the basic idea that the wave equation, which we derived earlier, can be written in terms of a superposition of waves. Typically, we describe this superposition in terms of waves moving left and right along the string with 1-dim motion (x).

    \[ X^{\mu} = X_{L}^{\mu}(\tau + \sigma) + X_{R}^{\mu}(\tau - \sigma) \]

    \[ X_{L}^{\mu}(\tau, \sigma) = \frac{x^{\mu}}{2} + \frac{l_{s}^{2}}{2}p^{\mu}(\tau + \sigma) + i\frac{l_s}{\sqrt{2}} \sum_{k \neq 0} \frac{\alpha_{k}^{\mu}}{k}e^{-ik(\tau + \sigma)} \]

    \[ X_{R}^{\mu}(\tau, \sigma) = \frac{x^{\mu}}{2} + \frac{l_{s}^{2}}{2}\bar{p}^{\mu}(\tau - \sigma) + i\frac{l_s}{\sqrt{2}} \sum_{k \neq 0} \frac{\bar{\alpha}_{k}^{\mu}}{k}e^{-ik(\tau - \sigma)} \]

Where, as written above, x^{\mu} is the centre of mass coordinate and p^{\mu} is the momentum (p.20). The zeroth mode is written as,

    \[\alpha_{0}^{\mu} = \frac{l_s}{\sqrt{2}}p^{\mu} \ \bar{\alpha}_{0}^{\mu} = \frac{l_s}{\sqrt{2}}\bar{p}^{\mu} \]

But also recall that for the free open string we have boundary terms. These have already been discussed but may be restated for convenience, noting that they are zero at the ends,

    \[ P^{\sigma \mu} = 0\rvert_{\sigma = 0, \pi} \ \frac{\partial X^{\mu}}{\partial \sigma}\rvert_{\sigma = 0, \pi} \]

The function that satisfies these boundary conditions can be gleaned from inspection: \cos n\sigma. Hence, we obtain superpositions of this function, which is noticeable in our solution. But as we also need to solve the wave equation, \cos n\sigma is coupled to e^{in\tau}.

Secondly, the solution we’ve arrived at is built to satisfy the Virasoro constraints (\dot{X} \pm X^{\prime})^{2}. Notice, when we take the respective derivatives with respect to \tau, beginning with the last terms on the right-hand side,

    \[ \dot{X}^{\mu} = \sqrt{2\alpha^{\prime}} \sum_{n \in \mathbb{Z}} \frac{1}{n} \alpha_{n}^{\mu} e^{-in\tau}\cos n\sigma \]

    \[ X^{\prime \mu} = -i\sqrt{2\alpha^{\prime}} \sum_{m \in \mathbb{Z}} \frac{1}{m} \alpha_{m}^{\mu} e^{-in\tau}\sin n\sigma \]

From which we end up with a very nice looking result,

    \[ \dot{X}^{\mu} \pm X^{\prime \mu} = \sqrt{2\alpha^{\prime}} \sum_{n \in \mathbb{Z}} \frac{1}{n} \alpha_{n}^{\mu}e^{-in(\tau \pm \sigma)} \]

As we are still in the realm of classical physics, our investigation seeks a real solution to the wave equation, and thus we can consider the following reality condition:

    \[ X^{i}(\tau, \sigma)^{*} = X^{i}(\tau, \sigma) \]

From which the reality condition can be stated, wherein \tilde{a_{n}^{i}} = a_{-n}^{i}. This translates into a Hermiticity condition (p. 122) which we’ll use upon quantisation, a_{n}^{i}\dagger = a_{-n}^{i}.

We can also explicitly compute the centre of mass position of the string. From the conventional definition X^{i} \equiv \frac{1}{l} \int_{0}^{l} d\sigma \partial_{\tau} X^{i}(\tau, \sigma), we can compute it in terms of the total momentum of the string (p.20):

    \[ P^{i} = \int_{0}^{l} \prod^{i}(\tau, \sigma) \]

    \[ = \frac{P^{+}}{l} \int_{0}^{l} d\sigma \partial_{\tau} X^{i}(\tau, \sigma) \]

    \[ = \frac{P^{+}}{l} \cdot \frac{P^{i}}{P^{+}}l \]

Where we have arrived at the momentum of the string in transverse coordinates in the LC gauge.


Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.