*Edited 13/02/19 for clarity of language.

Following a brief study of the relativistic point particle, this entry focuses on an introductory study of the relativistic string. More particularly, we’re going to work our way toward deriving the Nambu-Goto action. The Nambu-Goto action is first mentioned on p. 11 in Polchinski’s “String Theory“. For Polchinski, or at least this is my reading of the text, the driving focus early on is to begin with the Nambu-Goto action and arrive almost immediately at the Polyakov action before launching very directly into the subject of stringy Conformal Field Theory. The nuances of open and closed strings, light cone quantisation, the string specturm, among other introductory topics are entertained along the way. As a generalisation of the Nambu-Goto action that we’re about to derive, the Polyakov formulation that will be studied in a separate entry will prove very important when it comes to the quantum theory of strings, keeping in mind that currently we’re working in bosonic string theory.

In following Polchinski, one of the advantages of building the picture of bosonic string theory from the relativistic free point particle and its classical action to the relativistic (free and open) string, is that for a description of the relativistic string we repeat much of what we’ve already done in a more or less analogous way when studying the point particle.

It is worthwhile focusing on describing the relativistic string, the Nambu-Goto action S_{NG} and its 2-dimensional worldsheet before we ultimately set ourselves on the Polyakov action S_{P} (sometimes called the string sigma model action) because, again, we will be able to compile an extensive list of analogies to rely on for the purposes pedagogical clarity, intuition and conceptual cohesion. Or, at least in my own studies in ST, this was how I liked to build the picture. It is an approach that I think allows for one to flesh out a lot of the lovely subtleties and nuances of the theory, ultimately enabling a fairly deep and substantive understanding from the ground-up.

With the preamble out of the way, let’s start.

***

The mission here is to begin piecing together a description of a string propagating in d-dimensional spacetime. This includes defining the classical action, beginning, indeed, with what is called the Nambu-Goto action.

If a 0-dimensional point particle (0-brane) traces out a (0+1)-dimensional worldline, a 1-dimensional string (1-brane) sweeps out a (1+1)-dimensional surface that is called the string worldsheet. (Note we have yet to define the concept of branes, but we will begin to do so in the next few entries).

Additionally, just as we can parameterise the relativistic point particle’s (0+1)-dimensional worldline, we can parameterise the the (1+1)-dimensional worldsheet (WS) traced by the relativistic string. Let’s begin with this, as it will enable us to build an early picture.

Infinitesimal area element and area functional

We need an action to describe the dynamics of the string. One of the nice qualities of the Nambu-Goto action for the relativistic string is that it is constructed in such a way to be proportional to the proper area of the worldsheet. It is also standard in the literature to parameterise the position along the string by the spatial coordinate, \sigma. Generally, we say that \sigma can take the values in the range 0 \leq \sigma \leq l.

To make sense of all this, begin by invoking the concept of area in a parameter space, as illustrated above. What we’re working toward defining here is what is called the “string map”, which enables us to embed the WS in the ambient or background space. Note that the parameterised surface can be described by the coordinate functions,

    \[ \vec{x}(\xi^1 , \xi^2) = x^1 (\xi^1 , \xi^2), x^2 (\xi^1 , \xi^2), x^3 (\xi^1 , \xi^2) \]

The area to which we want to give mathematical description, as pictured, is more accurately an infinitesimal area element. Since we begin working in a parameter space, our very small square is mapped onto the surface in target space. Here spacetime is referred to as the target space, because we want to distinguish between this surface and the string worldsheet. When we map this ‘very small’ area to the surface, we achieve a parallelogram in which the sides may be denoted as d\vec{v}_1 and d\vec{v}_2.

    \[ d \vec{v}_1 = \frac{\partial \vec{x}}{\partial \xi^1} d\xi^1 \]


    \[ d \vec{v}_2 = \frac{\partial \vec{x}}{\partial \xi^2} d\xi^2 \]

Notice, again, the illustration above. One may have guessed that the main objective here is to compute the area of this parallelogram. Of course, again, by “area” I mean an infinitesimal area element. To achieve this we simply need to invoke a basic formula and perform some simple calculations,

    \[ dA = \mid d\vec{v}_1 \times d\vec{v}_2 \mid \]


    \[ dA^2 = (d\vec{v}_1 \times d\vec{v}_2) \cdot (d\vec{v}_1 \times d\vec{v}_2) = (d\vec{v}_1)^2 (d\vec{v})^2 - (d\vec{v}_1 \cdot d\vec{v}_2)^2 \]

Note, for pedagogical purposes, it should be featured that this last line comes from the identity (\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B}) = \mid A \mid^2 \mid B \mid^2 - (A \cdot B)^2. It follows that what we get is this expression for dA,

    \[ \therefore dA = \sqrt{(\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^1})(\frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^2}) - (\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^2})^2} d\xi^1 d\xi^2 \]

And so we achieve some general expression for the area element, which has exactly the same structure in Euclidean space. But this is not the area functional that we require. What we need to do is integrate,

    \[ A = \int \sqrt{(\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^1})(\frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^2}) - (\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^2})^2} \ d\xi^1 d\xi^2 \]

From this, what we have accomplished is the parameterisation of the surface by way of developing a description for the area element. One problem: this result is not very nice. Although it is reparameterisation invariant, it reparameterises in a messy way. We need to simplify, but how? Well, we can make a gauge choice, employ an induced metric, and this formula for the area will simplify greatly.

Induced metric

Suppose you have some vector d\vec{x} on the surface. It follows we can compute,

    \[ ds^2 = d\vec{x} \cdot d\vec{x} \]

    \[ d\vec{x} = \frac{\partial \vec{x}}{\partial \xi^1} d\xi^1 + \frac{\partial \vec{x}}{\partial \xi^2} d\xi^2 \implies \frac{\partial \vec{x}}{\partial \xi^i} d\xi^i \]

From this we can plug d\vec{x} back into our expression for ds^2.

    \[ ds^2 = (\frac{\partial \vec{x}}{\partial \xi^i} d\xi^i) \cdot (\frac{\partial \vec{x}}{\partial \xi^j} d\xi^j) \]


    \[ = \frac{\partial \vec{x}}{\partial \xi^i} \frac{\partial \vec{x}}{\partial \xi^j} d\xi^i d\xi^j \]


    \[ = g_{ij}(\xi)d\xi^i d\xi^j \]


    \[ \implies g_{ij}(\xi) = \frac{\partial \vec{x}}{\partial \xi^i} \frac{\partial \vec{x}}{\partial \xi^j} \]

We now arrive at our induced metric, g_{ij}(\xi), which is the pullback of the ambient space metric onto the worldsheet. It is worth highlighting in matrix form,

    \[ g_{ij} = \begin{pmatrix}\frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^1} & \frac{\partial \vec{x}}{\partial \xi^1} \cdot \frac{\partial \vec{x}}{\partial \xi^2} \\ \frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^1} & \frac{\partial \vec{x}}{\partial \xi^2} \cdot \frac{\partial \vec{x}}{\partial \xi^2} \\\end{pmatrix} \]

Now, carrying on, let’s substitute the appropriate matrix elements into our earlier expression for the infinitesimal area. Notice,

    \[ dA = \sqrt{(g_{11})g_{22} - g_{12}^2} \ d\xi^1 d\xi^2 \]


    \[ = \sqrt{det g} \ d\xi^1 d\xi^2 \]


    \[ \therefore dA = \sqrt{g} d\xi^1 d\xi^2 \]

Where, in this final result, g \equiv det g_{ij} (\xi).

This implies,

    \[ A = \int d\xi^1 d\xi^2 \sqrt{g} \]

This new way to express the area, A, is given in terms of the determinant of the induced metric. The wonderful thing about this is that we can now describe the reparameterisation invariance by way of how this metric transforms. Moreover, we can show this new form for the area is manifestly reperameterisation invariant. 

Reparameterisation invariance

We can show manifest reparameterisation invariance of the area through a number of steps.

    \[ ds^2 = g_{ij}(\xi) d\xi^i d\xi^j = \tilde{g}_{ij}(\tilde{\xi}) d\tilde{\xi}_1 d\tilde{\xi}_2 \]


    \[ = \tilde{g}_{pq}(\tilde{\xi}) \frac{\partial \tilde{\xi}^p}{\partial \xi^i} \frac{\partial \tilde{\xi}^q}{\partial \xi^j} d\xi^i d\xi^j \]


    \[ \therefore g_{ij} = g_{pq} (\tilde{\xi}) \frac{\partial \tilde{\xi}^p}{\partial \xi^i} \frac{\partial \tilde{\xi}^q}{\partial \xi^j} \]

    \[ \implies \int d\xi^1 d\xi^2 \sqrt{g} = \int d\tilde{\xi}_1 d\tilde{\xi}_2 \sqrt{\tilde{g}}\]

There is a more elegant way to put this, which I could upload in a separate entry should it be required. But for now, one should focus on how similar this is to a metric transformation in GR inasmuch that \int d\xi^1 d\xi^2 \sqrt{g} transforms via a Jacobian determinant of \xi with respect to \tilde{\xi} as \int d\tilde{\xi}_1 d\tilde{\xi}_2 \sqrt{\tilde{g}}.

To conclude these opening considerations, we want to derive the action. So we bracket our result for the infinitesimal area on the manifold for just as moment. Now we focus on the string.

String in spacetime

We have found a way to calculate the area of the surface. But since we are interested in the relativistic string action, we need to look to a description of the string (1-brane) as it moves in d-dimensional flat Minkowski spacetime.

Our concern for now is an open string (as opposed to a closed one) that, as described above, sweeps out a 2-dim surface or worldsheet. 

Again, as described above, we want to embed the string worldsheet in spacetime. Think, in other words, of spacetime surface. We will now also adopt the string theory convention of X^{\mu} coordinates such as in X^{\mu}(\tau, \sigma), where (\tau, \sigma) \equiv \xi^{a}. Oftentimes it  is also written that the WS, \sum, parameterised by \tau and \sigma defines the map \sum : (\tau, \sigma) \mapsto X^{\mu}(\tau, \sigma) \in \mathbb{R}^{1, d-1}. This is, more concisely, the string map mentioned earlier. And so what we’re doing now is parameterising the WS with one timelike coordinate, \tau, and one spacelike coordinate, \sigma. To continue in a very explicit tone, it is this mapping that enables us to embed the WS in the ambient space. Moreover, the function X^{\mu}(\tau, \sigma) will allow us to describe this embedding, while the endpoints of the string are parameterised by \tau such that \frac{\partial X^{\mu}}{\partial \tau} (\tau, \sigma) \neq 0.

We begin by writing the area as,

    \[ dA = d\tau d\sigma \sqrt{(\frac{\partial X^{\mu}}{\partial \tau} \frac{\partial X_{\mu}}{\partial \tau})(\frac{\partial X^{\nu}}{\partial \sigma} \frac{\partial X_{\nu}}{\partial \sigma}) - (\frac{\partial X^{\mu}}{\partial \tau} \frac{\partial X_{\mu}}{\partial \sigma})^2} \]

    \[ = d\tau d\sigma \sqrt{(\frac{\partial X}{\partial \tau} \cdot \frac{\partial X}{\partial \tau})(\frac{\partial X}{\partial \sigma} \cdot \frac{\partial X}{\partial \sigma}) - (\frac{\partial X}{\partial \tau} \cdot \frac{\partial X}{\partial \sigma})^2} \]

    \[ = \sqrt{(\dot{X})^2 (X\prime)^2 - (\dot{X} \cdot X\prime)^2} \]

Here we invoked the relativistic dot product. One further comment: notice the sign under the square root. I don’t always see this nuance emphasised, but it should be highlighted that for a surface with a timelike vector and a spacelike vector the square root is always positive such that Cauchy-Schwarz inequality flips (for proof of this see Zwiebach, 2004, p. 99). This means,

    \[ (\dot{X}^2 \cdot X^{\prime})^2 - (\dot{X})^2 (X^{\prime})^2 > 0 \]

It follows from this the Nambu-Goto action, whereby we integrate over the area element.

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

Where \dot{X} = \frac{\partial X^{\mu}}{\partial \tau} and X\prime^{\mu} = \frac{\partial X^{\mu}}{\partial \sigma}. Notice the \frac{T_0}{c} out front is a constant of proportionality to ensure units of action. It also turns out to be true that this T_0 term is precisely describes the tension in the string.

Further comments: NG action as manifestly reparameterisation invariant

Similar to what we did before, we can write the NG action in a manifestly reparameterisation form. First, we need our target space Minkowski metric. \alpha and \beta are from 1 and 2. \xi^1 = \tau and  \xi^2 = \sigma.

    \[ -ds^2 = dX^{\mu} dX_{\mu} = \eta_{\mu \nu} dX^{\mu} dX^{\nu} = \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \xi^{\alpha}} \frac{\partial X^{\nu}}{\partial \xi^{\beta}} \ d\xi^{\alpha} d\xi^{\beta} \]

If the string worldsheet is a curved surface embedded in spacetime, we can define an induced metric \gamma_{\alpha \beta} on the worldsheet. It was stated above that this induced metric is the pullback of the ambient space metric, \eta_{\mu \nu}, and perhaps this comment will now become more clear. Moreover, we can say that \gamma_{\alpha \beta} is the worldsheet metric induced by the target space Minkowski metric. Notice, then, for the induced metric we can compute,

    \[ \gamma_{\alpha \beta} = \eta_{\mu \nu} \frac{\partial X^{\mu}}{\partial \xi^{\alpha}} \frac{\partial X^{\nu}}{\partial \xi^{\beta}} = \frac{\partial X}{\partial \xi^{\alpha}} \cdot \frac{\partial X}{\partial \xi^{\beta}} \]

The worldsheet metric induced by the Minkowski metric can be written in matrix form,

    \[ \gamma_{\alpha \beta} = \begin{pmatrix}\dot{X}^2 & \dot{X} \cdot X^{\prime} \\ \dot{X} \cdot X^{\prime} & X^2 \prime \\ \end{pmatrix} \]

Therefore, we arrive at the NG action as the area element in spacetime:

    \[ S_{NG} = - \frac{T_0}{c} \int d\tau \int d\sigma \sqrt{\gamma_{12}^2 - \gamma_{11} \gamma_{22}} \]

    \[ = - \frac{T_0}{c} \int d\tau d\sigma \sqrt{- det \gamma} \]

    \[ = - \frac{T_0}{c} \int d\tau d\sigma \sqrt{- \gamma} \]

Where \gamma = det(\gamma_{\alpha \beta}).

In the post to immediately follow we will look at the equations of motion.