Following a study of the S_{NG} and S_{P} forms of the string action, we now turn our attention to the fact that the string will fluctuate. That is to say, the aim of the next few entries is to study the spectrum of string fluctuations.

To do this, we must construct our notation and establish a number of important definitions. We will start with a study of the open string and then move to closed strings. We will also want to study the quantisation of the string. In pursuing the topic of the string spectrum, we will come into contact with the critical dimension of bosonic ST: namely, D = 26, following quantisation.

Before proceeding, however, it is worth mentioning an important subtlety that leads to the discovery of the critical dimension. It remains that, in terms of a geometrical representation, the S_{NG} action is perhaps the most natural expression. But recall that with good reason, it was decided that the S_{NG} action was not fit for purpose, due the presence of the square root. So, leveraging the point particle analogue, we introduced an auxiliary metric \gamma_{ab} on the string WS. But this manoeuvring comes at a cost! We introduce a redundancy in our description. Thus, when it comes to the study of string fluctuations, we find lots of fluctuations that have nothing to do with physics. They are merely gauge transformations due to the remarkable symmetry possessed by the string.

Our study should begin with this in mind. We need to first fix the symmetries of the S_{P} action, such that it is restricted only to physical fluctuations. Following Polchinski, we will do this by employing the light-cone (LC) gauge. Moreover, and pedagogically speaking, it should be said that we must choose some gauge. We must do so because of the incredible degrees of freedom on the WS. A smart choice of gauge eliminates the redundancies and leaves only physical fluctuations. The light-cone gauge will prove incredibly useful. Although other approaches to quantisation will be utilised later, invoking the techniques of light-cone quantisation will allow us to quickly and directly explore the physical characteristic of the free open and closed string.

Notation

So far, one should recall, we have been working in Minkowski spacetime such that \eta_{\mu \nu} = (-1, 1, ... 1). In this context we have become familiar with the coordinates: X^{0}, X^{1}, ... X^{D-1}.

Now, we want to introduce light-cone (LC) coordinates as read on p.16 in Polchinski,

    \[ X^{\pm} = \frac{1}{\sqrt{2}} (X^{0} \pm X^{1}) \]

As these are our lightlike coordinates, X^{+} is a timelike coordinate and X^{-} is a spacelike coordinate.

We’ve already seen the Minkowski metric, and we have discussed the infinitesimal distances in spacetime. In the case of LC coordinates, we can take the inverse relationship and then rewrite ds^2.

    \[ ds^2 = 2dx^{+}dx^{-} - (dx^2)^2 - (dx^3)^2 \]

So, as one would expect, we can construct a light-cone Minkowski metric and therefore continue to define distances,

    \[ \hat{\eta}_{\mu \nu} \begin{bmatrix} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \]

Which means we can write, ds^2 = -\hat{\eta}_{\mu \nu}dx^{\mu}dx^{\nu}.

Now, in LC coordinates, we must understand the form contractions will take. To do so, it follows that we can write,

    \[ a^{\mu}b_{\mu} = -a^{0}b^{0} + a^{1}b^{1}+ ... + a^{D-1}b^{D-1} \]

Thus, we arrive at the expression for the dot product (p.16),

    \[ = -a^{+}b^{-} - a^{-}b^{+} + a^{i}b^{i} \ \ \ i = 2, ..., D-1 \]

Moving forward, we will also rely on the following rules to raise and lower indices (p.16):

    \[ a_{-} = a^{+}\]

    \[a_{+} = -a^{-} \]

    \[ \implies a^{+}b_{+} + a^{-}b_{-} + a^{i}b_{i} \]

Additionally, a few comments on the WS theory in the context of the light-cone. For the WS, we take the notation we’re already familiar with and rewrite it as follows,

    \[ \sigma^{+} = \sigma + \tau \ \ \ \sigma^{-} = \tau - \sigma \]

Hence, as d\sigma^{+} = d\tau + d\sigma and d\sigma^{-} = d\tau - d\sigma we find ds^2 = -d\sigma^{+}d\sigma^{-}. The induced metric takes the signature (++, +-, -+, --),

    \[\gamma_{ab} = (\begin{bmatrix} 0 & -\frac{1}{2} \\ -\frac{1}{2} & 0 \\ \end{bmatrix}) \]

From this standard manipulations apply, and I leave it to the reader to familiarise themselves with what has been established.

Finally, with regards to derivatives, in the LC we will use the below shorthand that should be familiarised (we will be using shorthand a lot),

    \[\partial_i = \frac{\partial}{\partial x^i} \]

And therefore,

    \[\partial_{+} = \frac{1}{2}(\partial_{\tau} + \partial_{\sigma}) \ \ \ \partial_{-} = \frac{1}{2}(\partial_{\tau} - \partial_{\sigma}) \]

In the next post, we’ll look at the string action in the LC.

References

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