I haven’t been keeping up with this as much as I would like, mainly because I have been busy. But I am committed to continuing to reupload many of my notes on Polchinski’s textbooks. It is fun for me to go through it all again in my spare time, and I’ve noticed that since the time of first working through the textbooks there is more I can add to many topics.
It is worth remembering that, in the last note, we reviewed the classical worldline and polynomial action for the relativistic point particle. We also discussed reparameterisation invariance and calculated the equations of motion. In this note, the focus is to construct the first-principle Nambu-Goto action for the relativistic string as given in equations (1.2.9a-1.2.9b) in Polchinski’s textbook.
Often in popular literature and discourse I read descriptions of the string that almost shroud it in mystery. How could the fundamental constituents of matter be described by a bunch of strings? Other times, caricatures of string theory can leave the impression that to view all elementary particles as vibrating strings is somewhat arbitrary. Why not some other type of objects? It is suggestive of a certain arbitrariness to the idea of modelling fundamental particles as strings; but the development of string theory is, in fact, well motivated. Ultimately, all that we’re doing is extending the concept of point particles that we all know and love, and this is first and foremost evidenced in the Nambu-Goto action. But, in terms of the bigger picture, what we see is that in studying the string and its dynamics an entire universe of implications emerge. It reminds me of a great line in David Tong’s lecture notes that is worth paraphrasing: we find that the requirements demanded by the tiny string are so stringent that we are led naturally to a description of how the entire universe moves. On many occasions it is, indeed, like “the tail is wagging the dog”.
As we’re following Polchinski’s textbook, which only covers the Nambu-Goto action in a few words, if the interested reader would like to spend more time studying this action I would recommend ‘String theory and M-theory’ by Katrin Becker, Melanie Becker, and John H. Schwarz, especially the exercises, or for an even more gentle introduction see Barton Zwiebach’s textbook ‘A first course in string theory’.
2. Area functional
To arrive at the Nambu-Goto action, let us first recall from the last note that a p-brane may be described as a p-dimensional object moving through D-dimensional flat spacetime with . If a 0-dimensional point particle (0-brane) traces out a (0+1)-dimensional worldline, it follows that a 1-dimensional string (1-brane) sweeps out a (1+1)-dimensional surface that we call the string worldsheet. And just as we can parameterise the relativistic point particle’s (0+1)-dimensional worldline, we can parameterise the (1+1)-dimensional worldsheet traced by the string. Coming to grips with this idea is the first task.
The main idea is that the worldline of a particle is replaced by the worldsheet , which is a surface embedded into D-dimensional Minkowksi spacetime. Given that the path of a point particle can be described by a single parameter, the proper time , which multiplied by c, is the Lorentz invariant proper length of the worldline; for strings, we will define the Lorentz invariant proper area of the worldsheet in a completely analogous way. As we’ll see, the first-principle string action is indeed proportional to this proper area.
To start, we see that because the string worldsheet is a (1+1)-dimensional surface, its requires two parameters which we will denote as and . We will also limit our present considerations to the case of an open string (we will talk about closed strings in a later note). In order to define the appropriate area functional, we want to sketch a grid on the spacial surface of the string worldsheet with lines of constant and ; then we want to embed this spatial surface in the background target space.
The target space is the world where the 2-dimensional surface lives. Ultimately, we want to distinguish between the area we parameterise and the actual physical string worldsheet. In order to accomplish this, we define a one-to-one map, which we may call the string map. The purpose of the string map is therefore to take us from the parameter space that we have constructed to the target space in which the physical surface propagates. Indeed, as we’ll see, the string action is in this precise sense defined as a functional of smooth maps.
To construct the string map, we first formalise the notion of area in parameter space, with this parameter space itself defined by the range of the parameters and . One can, in principle, view the parameters we have selected as local coordinates on the surface. And so, as emphasised above, we can think of the worldsheet as a physical surface, which is in fact the image of the parameter space under the one-to-one string map written as . The parameterised surface can therefore be described by the coordinate functions
The area to which we want to give mathematical description is more accurately an infinitesimal area element. Since we begin working in a parameter space, and since our very small square is mapped onto the surface in target space, when we map this very small area from the parameter space to the surface we achieve a parallelogram the sides of which may be denoted as and . We can express this as follows:
in which we have defined the rate of variation of the coordinates with respect to the parameters . If we multiply this rate by the length of the horizontal side of the infinitesimal parallelogram, we get the vector that represents this side in the target space.
The main objective is to now compute the area of this parallelogram.
Since we have already labelled the sides of the infinitesimal area in the parameter space, we simply need to invoke the formula for the area of a parallelogram:
Here denotes the angle between the vectors and . Written in terms of dot products in which such that
From this result, notice that we can now substitute for and using (2). Doing so gives
We have now obtained a general expression for the area element of the parameterised spatial surface. Written as the full area functional we have
where the integral extends over the ranges of the parameters and . This functional is reparameterisation invariant, which can be quickly verified by reparameterising the surface with tilde parameters that then gives back (6) when and .
The main issue is that the area functional (6) is not very nice, and reparameterisation invariance is not completely general. We want reparameterisation invariance to be manifest.
3. Induced Metric
Suppose we have some vector on the surface that we have so far pencilled into the target space. We know that we can describe this surface through the string mapping functions . What if we then consider tangent to the surface ? We could then let denote the length of this tangent vector, and hence we could invoke some early idea of a metric on .
Given the vector tangent to the surface, with its length, we can write
But what is in terms of the parameter space coordinates that we constructed? In other words, can we relate with ? This is precisely what our mapping accomplishes such that we can express in terms of partial derivatives and derivatives of :
with the summation convention assumed for the repeated indices over possible values 1 and 2. If we now return to (7) and plug back into our equation for we see that we can now write
But notice something interesting. If we set , this means we can write a more simplified equation of the form
in which the quantity is called the induced metric. It is a metric on the target space surface precisely in the sense that, as play the role of coordinates on , we see that (10) determines distances on this surface. It is said to be induced because it uses the metric on the ambient space in which lives to determine distances on . More technically, we say that the induced metric is the pullback of the target space metric onto the worldsheet.
A question we can now ask is whether, upon constructing a metric on the target space surface, does this then lead us to an equivalent expression for (6)? Observe that, in matrix form, we have for the induced metric
What is this telling us? Notice that if you compute the determinant of the matrix , you find the same quantity that resides under the square root in (6). This is a massive hint that the construction is on the right track. So, let’s substitute the appropriate matrix elements into our earlier expression for the infinitesimal area. This is what we find,
where . This implies,
This new way to express the area, , is now given in terms of the determinant of the induced metric. And although we are not yet done constructing the Nambu-Goto action, we see quite clearly from (13) why Polchinski says that the action in equations (1.2.9a-1.2.9b) in his textbook is proportional to the area of the worldsheet.
4. Reparameterisation invariance
The wonderful thing about this last result (13) is that we can now show manifest reparameterisation invariance in a much simpler way, as it may now be described by way of how the induced metric transforms.
To do this, we invoke a different set of parameters and therefore also a different metric, and then we show that the original vector does not depend on our original parameterisation.
We begin with
We then use the chain rule
Next, recall that the change of variable theorem tells us how the integration measure transforms
where is the matrix defined by and similarly
where is defined by . Using this and returning to (15) we can rewrite this equation for and such that
If we denote , and if take the determinant of the right-hand side of (18) we find
Clearly, then, if we take the square root we obtain
which is the transformation property for the square root of the determinant of the metric.
Finally, we conclude using (16) and (20) with the fact that we can show that (13) is reparameterisation invariant
There is perhaps a much more elegant way to show this proof. But for now, one should focus on how (21) is just a standard metric transformation inasmuch that transforms via a Jacobian determinant of with respect to as .
5. String propagating in spacetime
Let us now work toward constructing the Nambu-Goto action as it appears in equations (1.2.9a-1.2.9b). Up to this point we have taken the approach of mapping from a parameter space to a target space in which the surface lives. But we are interested in the case of surfaces in spacetime. These surfaces are obtained by representing in spacetime the history of the string as it propagates, in the same way the worldline of the point particle is described by representing its history.
Spacetime surfaces, such as string worldsheets, are not all that different from the spatial surfaces we considered in the previous sections. Instead of the coordinates and , for a relativistic string we should parameterise the string worldsheet in such a way that we account for both the proper time and the string’s spatial extension. Another way to put this is that, if our interest is to consider surfaces in spacetime (the worldsheet traced by the string), we now use to denote the proper time and to denote the spacial extension of the surface. Given usual spacetime coordinates, which we write following string theory conventions , the surface is then described by the mapping functions
Hence, we come to the point emphasised at the outset of this note. The string worldsheet action formally defines the map . If it is still not clear, remember that what we’re working toward is a description of the string worldsheet as a curved surface embedded in spacetime. This embedding is given by the fields , in which the parameters and can be viewed (locally) as coordinates on the worldsheet. So the string map tells us that given some fixed point in the parameter space, we are performing a direct mapping to a fixed point in spacetime coordinates. Typically we drop the arguments and leave them implicit, with the inverse of the map taking the worldsheet to the parameter space.
It is also worth noting that the functions describe how the string propagates and oscillates through spacetime, while the endpoints of the string are parameterised by such that . In our present case, we are considering an open string; but if is periodic then we’d be working with a closed string embedded in the background spacetime.
Getting back to the task at hand: to find the area element we proceed in similar fashion as before, except now we must use relativistic notation. So for the area element we have and describing the sides of an infinitesimal parallelogram in parameter space. In spacetime, this becomes a quadrilateral area element. We therefore set-up a direct analogue with our expression for in (4) where we consider the vectors and spanning the quadrilateral,
Notice that we may substitute for and into (4),
We now invoke relativistic dot product notation so that we ensure that what we are working with is the proper area. The object under the square root turns out to be negative, but we can switch the sign without violation of any rules. The basic idea is that, for a surface with a timelike vector and a spacelike vector the square root is always positive such that Cauchy-Schwarz inequality flips. This means, . We also want to integrate (24). So, putting everything together, we have
We can still simplify our expression for the area using the more compact notation, and . This means we can write,
Now comes the important part. From (26) there are a few ways we can approach the Nambu-Goto action. The most direct approach is to remember how, inasmuch that we are generalising the point particle action, we may anticipate the existence of some constant of proportionality. Indeed, it is completely reasonable to anticipate an action of the form general form . And this proves to be the case, because it follows that we may write the Nambu-Goto action for the string as
where is a constant of proportionality to ensure units of action. To explain this, consider the following. Given that the string action is proportional to the proper area of the worldsheet, the area functional has units of length squared. We see this because has unites of length, i.e., , and there are four under the square root. Each term in the square root also has two derivatives and two derivatives, with their units cancelling against the derivatives. Since must have the units of action with having units , the total proper area must be multiplied by the value . We know that the string will have a tension, , which has units of force. We also know that force divided by velocity has the units ; so to ensure units of action the proper area is multiplied by .
6. Manifest Reparameterisation Invariance of the Nambu-Goto Action
We still shouldn’t be completely satisfied with this early form of the Nambu-Goto action (28). How do we know, for instance, that what we have ended up with is manifestly reparameterisation invariant? It is crucial that the action be dependent only on the embedding in spacetime and not the choice of parameterisation.
To explore the action (28) in a deeper way, we first need to invoke the target space Minkowski metric, , and we should consider a differential line element of the form
We may now expand the derivatives acting on ,
where and run from 1 and 2. Similar as before for the spatial surface, we can define an induced metric. In this case, the induced metric on the string worldsheet is given as . It is simply the pullback of the target space Minkowski metric, . This allows us to define the induced metric as,
This means we can write the more compact equation for the line element
because, while the induced metric describes distances on the string worldsheet, it also includes the metric of the background spacetime in its definition. But, to ensure clarity of knowledge, let’s think about this induced metric a bit more. In matrix form, it is a matrix with components
And so, the induced metric may be written in matrix form as
Therefore, as we showed in the case of the spatial surface, we extend the logic of the previous examples and manifest reparameterisation invariance is seen to be featured with the help of the induced metric
The final observation is that, as Polchinski notes (p.11), for the string tension an alternative parameter is . This proportionality constant has been used since the early days of string theory; one may recognise it as the Regge slope, which has to do with the relation between the angular momentum, , of a rotating string and the square of the energy . In that has units of spacetime-length-squared, we therefore observe
where we’ve set . This is equation (1.2.10) in Polchinski. Hence, we may now rewrite in its more conventional form as read in equations (1.2.9a-1.29b):
The answer in Exercise 1.1b reveals more explicitly how the string tension is related to the Regge slope. It just requires that we write things in terms of the transverse velocity. To keep these notes focused and organised, at the conclusion of each chapter we’ll go over the solutions to the exercises and so we’ll return to this question then.
6.1. Equations of motion
Before we get to the symmetries of the action (36), let’s quickly look at its equations of motion. To simplify matters, we can write the Lagrangian as with the Euler-Lagrange equations reading as
From the chain rule, it is therefore clear that we need to calculate
For the first term in brackets, we use the identity for the variation of the determinant , which can be easily verified. Hence,
For the next term we find
Putting everything together
As the metric contains the embedding , these equations are highly non-linear. But this is not unexpected given the fact that the action (35) is non-linear. One way to interrept these equations is that, as a minimal surface area is being demanded by the stationary action, in Zwiebach’s textbook one is motivated to think analogously of the image of static soap film in some Lorentz frame. In this case, we think of the film as a spatial surface in which every point is a saddle point.
Finally, the last topic covered concerns the symmetries of the action (36).
Poincare group: As the Nambu-Goto action is completely and directly analogous to the action for a relativistic point particle, one might rightly anticipate that the action for a string is invariant under the isometry group of flat spacetime, which is the D-dimensional Poincare group
This symmetry group consists of consists of Lorentz transformations satisfying algebra and transformations. This symmetry is manifest and can be read-off from (36) since the Lorentz indices are contracted in the correct way to obtain a Lorentz scalar. But to see it explicitly just note that are flat spacetime vectors. Under the transformation we see that . Hence
where we used .
From the perspective of the worldsheet theory, the Nambu-Goto action is a 2-dimensional field theory of scalar fields , and Poincare invariance is in fact an internal symmetry.
Diffeomorphism invariance The Nambu-Goto action is also invariant under diffeomorphism transformations, or reparamterisation of the coordinates, which we’ve already observed by the very nature of how we construct (36) such that .
6.3. Concluding remarks
The main issue with the action (36) is the presence of the square root, which complicates matters when we attempt to quantise the theory or take the massless limit . That is why, analogous to the case of the relativistic point particle, we’ll want to get rid of this square root and construct a classically equivalent action. This is known as the Polyakov action and, following the progression in Polchinski, it is the topic of the next note.
In the meantime, I want to point out that there is still much more to be learned about the Nambu-Goto action and its dynamics. There are some quite famous and important results, which are not covered in Polchinski’s textbook. It is notable, for instance, that from an analysis of the worldsheet momentum densities
we can evaluate the components of the canonical momenta explicitly
From this analysis, we can obtain an equation that we can interpret as the generalised momentum flow of the particle worldline. This helps give a bit more insight and intuition into the direct analogue we’ve established between point particle theory and the theory of strings. Furthermore, by imposing the appropriate boundary conditions we can show for the equations of motion that
which, given the equations for the worldsheet momentum, we find the 2-dimensional wave equation given the choice of coordinates , , and . In the same analysis, we can find very important conditions such as the Virasaro constraints that govern the dynamics of the string.
There is also much more that can be studied: boundary conditions and motion of the string endpoints, which provides a first introduction to D-branes; tension and energy of the stretched string; transverse velocity; among other interesting topics. All of this of course comes up also in our study of the Polyakov action; but for the interested reader, Zwiebach’s textbook referenced at the outset covers all of these topics in detail in the context of the Nambu-Goto action.