Stringy Things

# Notes on String Theory: Relativistic Free Point Particle

1. Introduction

String theory (ST) can be considered a special, if not unique, generalisation of point particle theory. As such, most contemporary textbooks and courses on ST begin with a review of the kinematics and dynamics of the relativistic point particle. This series of notes shall be no different.

As a sort of preface to Polchinski’s textbook, we will entertain a cursory review of point particle theory so as to emphasise the analogue picture that may be constructed between the study of the point particle worldline and the future construction of the 2-dimensional worldsheet theory of the string (pp. 9-10). This will also enable us to extract some important lessons and insights for when we begin to examine the logic behind the construction of the first principle string action, namely the Nambu-Goto action. It will also aid in deepening intuition when we go on to study the kinematics and dynamics of the string. But the analogue can certainly be extended to more advanced topics, as it will become evident for example when we study the path integral formalism in string theory and even when we expand our ideas in bosonic string theory to superstring theory.

Having said that, as we are to partake in a cursory review, this post begins under the assumption that the reader is already deeply familiar with the classical theory of the point particle. Should one require further review, the reader is directed to Barton Zwiebach’s introductory textbook, `A First Course in String Theory‘.

2. Point Particle Action

In this section we will quickly review the free relativistic point particle action and how it is constructed.

The first goal is to formulate the action describing a relativistic particle of mass ${m}$. To construct the basic first-principle Lagrangian, we take as our starting point the image of a particle moving in spacetime, beginning at the origin and ending at some point ${(ct_f, \vec{x}_{f})}$. To describe the motion of this particle, we use spacetime coordinates ${x^{\mu}}$ where ${x^{0}}$ is the timelike coordinate. We may also denote ${x^{i}}$ as the spacelike coordinate, where ${i \neq 0}$. From the principle of least action, we know there are many possible worldlines between the starting and end points, as illustrated above representing only one spatial dimension.

Now, we must ensure that we invoke a Lorentz-invariant theory. That is to say, our choice of integral should not depend on our choice of reference system, such that, for any worldline, all Lorentz observers should compute the same value for the action. For instance, consider a particle worldline ${\mathcal{P}}$, as illustrated above. All Lorentz observers agree on the proper time of a particular worldline, and thus all Lorentz observers should agree on the proper time of ${\mathcal{P}}$. One logical approach would therefore be to formulate the action of the worldline ${\mathcal{P}}$ as being proportional to the proper time.

In anticipation of string theory and the fact that we will eventually be working in a higher-dimensional space, we consider the trajectory of the particle in D-dimensional Minkowski space ${\mathbb{R}^{1, D-1}}$. So, instead of one time dimension and three spatial dimensions, we have one time dimension and ${d = D - 1}$ spatial dimensions. In the simple picture that we’re currently constructing, we may quantitatively formulate the basic first-principle point particle action by recalling the interval in 4-dimensional spacetime,

$\displaystyle - ds^2 = -c^2 dt^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2 \ \ (1)$

The infinitesimal proper time is equal to ${ds/c}$, so the integral of ${ds/c}$ over ${\mathcal{P}}$ gives the proper time along this worldline. In other words, in taking an integral over ${ds/c}$, we are calculating the infinitesimal invariant length of the particle’s worldline. But in order to obtain units of action we must also include an additional Lorentz invariant multiplicative factor with units of energy or units of mass times velocity-squared, ${mc^{2}}$, which is the rest energy of the particle. Here ${m}$ is of course the rest mass and ${c}$ is the fundamental velocity in relativity. From this, we may state the point particle action as follows,

$\displaystyle S = -mc \int_{\mathcal{P}} ds \ \ (2)$

One can think of this integral similarly to many other instances, where we are taking the sum over many small increments ${ds}$ along the particle’s trajectory. Moreover, it is common that the action is written in terms of ${ds}$ for the reason that it is a Lorentz scalar. But what we want to do is look a bit more deeply at the associated Lagrangian. To do this, we should further inspect the ${ds}$ in the integrand.

If the goal is to study the Lagrangian and, indeed, an integral of that Lagrangian over time – say, ${t_i}$ and ${t_f}$ which are world-events that we’ll take to define our interval – this is because it will enable use to establish a more satisfactory expression that includes the values of time at the initial and final points of our particle’s path. More pointedly, if we fix a frame – which is to say if we choose the frame of a particular Lorentz observer – we may express the action (2) as the integral of the Lagrangian over time. But, first, in order to arrive at such a Lagrangian, we must return to our interval (1) and relate ${ds}$ to ${dt}$,

$\displaystyle -ds^2 = -c^2 dt^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2$

$\displaystyle ds^2 = c^2 dt^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2$

$\displaystyle ds^2 = [c^2 - \frac{(dx^1)^2}{dt} - \frac{(dx^2)^2}{dt} - \frac{)dx^3)^2}{dt}] dt^2$

$\displaystyle \implies ds^2 = (c^2 - v^2) dt^2$

$\displaystyle ds = \sqrt{c^2 - v^2} dt \ \ (3)$

We have now related ${ds}$ to ${dt}$. In the fixed frame, it also follows that ${x^{\mu} = (t, \vec{x})}$. The action is then found to be,

$\displaystyle S = -mc^{2} \int_{t_{i}}^{t_{f}} dt \sqrt{1 - \frac{v^{2}}{c^{2}}} \ \ (4)$

Notice, we have arrived at an integral over time between the initial and final points of the particle’s worldline ${\mathcal{P}}$. The Lagrangian for this action describing the relativistic free point particle is,

$\displaystyle L = -mc^{2} \sqrt{1 - \frac{v^{2}}{c^{2}}} \ \ (5)$

This Lagrangian gives us a hint that it is correct, because its logic breaks down when ${v > c}$, which confirms the definition of the proper time from elementary considerations in special relativity (i.e., the velocity should not exceed the speed of light for the proper time to be a valid concept). We can also see the Lagrangian (5) is correct by computing the canonical momentum ${\vec{p}}$ and the Hamiltonian.

For the canonical momentum, we take the derivative of the Lagrangian with respect to the velocity,

$\displaystyle \vec{p} = \frac{\partial L}{\partial \vec{v}} = -mc^{2}(-\frac{\vec{v}}{c^{2}})\frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{m\vec{v}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \ \ (6)$

Now that we have an expression for the relativistic momentum of the particle, let us consider the Hamiltonian. Given the Hamiltonian may be written generally as ${H = \vec{p} \cdot \vec{v} - L}$, the computation follows

$\displaystyle H = \frac{m\vec{v}^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} + mc^{2}\sqrt{1 - \frac{v^{2}}{c^{2}}} = \frac{mc^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \ \ (7)$

This expression for the Hamiltonian should make sense on first look. If we instead write the result in terms of the particle’s momentum (instead of its velocity) by inverting (6), we find an expression in terms of the relativistic energy ${\frac{E^{2}}{c^{2}} - \vec{p} \cdot \vec{p} = m^{2}c^{2}}$. This tells us we’re on the right track, and that we’ve recovered basic relativistic physics for a point particle.

The issue that we have, however, is that this Lagrangian (5) is not reparameterisation invariant. In other words, although the action (4) is correct, notice how it is based on the time ${t}$ parameterising the position. Under Lorentz transformation, there should be a symmetry between ${t}$ and ${\vec{x}}$, with the requirement that the value of the action be independent of the parameterisation chosen to calculate it. This means we need a new Lagrangian.

3. Reparameterisation invariance

There are a few different ways we may approach the construction of this new Lagrangian. In the literature, it is convention to use ${\tau}$ parameterisation. In a sense, time is promoted to a dynamical degree of freedom without actually being a dynamical degree of freedom (Tong, p.10). This is done by leveraging the power of gauge symmetry. This means we parameterise the worldline ${\mathcal{P}}$ by ${\tau}$. The ${\tau}$ parameter necessarily increases as the worldline evolves from the initial point ${x_{i}^{\mu}}$ to ${x_{f}^{\mu}}$ and ranges in the interval ${[\tau_{i}, \tau_{f}]}$. With this parameterisation, there is complete ${\tau}$ labelling for the position along the worldline. This means that the trajectory of the particle is now described by the coordinates as functions of ${\tau}$,

$\displaystyle x^{\mu} = x^{\mu}(\tau) \ \ (8)$

In this way, through the power of gauge symmetry, ${\tau}$ parameterisation enables us to treat space and time coordinates on the same footing (i.e., there is sufficient mixing), which means even the time coordinate ${x^{0}}$ is parameterised. And, so, we may update the space of our theory such that ${x^{\mu}(\tau) \in \mathbb{R}^{1, D-1}}$ and ${\mu = 0,...,D-1}$.

What we now want to do is return to the integrand ${ds}$ in (2) and update it using the parameterised worldline. More pointedly, to ensure the length of our spacelike curve is reparameterisation invariant we note that,

$\displaystyle ds^2 = - \eta_{\mu \nu} dx^{\mu} dx^{\nu} = - \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} (d \tau)^2 \ \ (9)$

Where $\eta_{\mu \nu}(x)$ is the flat Minkowski metric and where $\mu, \nu = 0, 1, ... D-1$, describing the background spacetime geometry. In ST, the metric will have the signature $(-, +, +, +)$ unless specified to be otherwise. This is because, in ST, it will be convenient to place a negative with the time component.

Similar as before, we substitute for ${ds}$ in the integrand and the action takes the form,

$\displaystyle S = -mc \int_{\tau_i}^{\tau_f} \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} d\tau \ \ (10)$

How do we interpret this form of the action? It is quite simply the proper time along the worldline. The value of this action is still the same in any frame, as its value does not depend on the choice of parameter. This means the action (10) is reparameterisation invariant. A fixed observer may calculate the action using some parameter ${\tau}$. But a different parameter ${\tilde{\tau}}$ on the worldline could just as easily be chosen and related to ${\tau}$ by any monotonic function of the general form ${\tilde{\tau}=\tilde{\tau}(\tau)}$. This is what leads our theory to be described as manifestly reparameterisation invariant.

This can be more simply described in how if we change the parameter ${\tau}$ to ${\tilde{\tau}}$ we have by the chain rule,

$\displaystyle \frac{dx^{\mu}}{d\tau} = \frac{dx^{\mu}}{d\tilde{\tau}}\frac{d\tilde{\tau}}{d\tau} \ \ (11)$

And if we substitute this into the action (10) we get,

$\displaystyle S = -mc \int_{\tau_i}^{\tau_f} \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tilde{\tau}} \frac{dx^{\nu}}{d \tilde{\tau}}} \frac{d\tilde{\tau}}{d\tau} d\tau$

$\displaystyle = -mc \int_{\tilde{\tau}_i}^{\tilde{\tau}_f} \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tilde{\tau}} \frac{dx^{\nu}}{d \tilde{\tau}}} d\tilde{\tau} \ \ (12)$

Which is of the exact same form as (10) and shows reparameterisation invariance of the point particle action. The upshot is that reparameterisation invariance is a gauge symmetry, which is not even an honest symmetry and which means that we’ve introduced a redundancy in our description as not all D degrees of freedom ${x^{\mu}}$ are physically meaningful. This means that, although our present system has D degrees of freedom, one of the degrees of freedom is a sort of counterfeit. This counterfeit degree of freedom shows up when we study the momenta, in which we find that not all of the momenta are independent satisfying the constraint ${p_{\mu}p^{\mu} + m^{2}= 0}$. As a note, this constraint should be recognised as just the mass-shell condition for the relativistic point particle. (However, as this discussion is only meant to serve as a basic review of the point particle case, I do not wish to proceed further with consideration of these topics. The reader is instead directed to the surrounding literature for more information).

Another key message at this point, as it relates to string theory, is that this new form of the action (10) is now also manifestly Poincaré invariant from the worldline perspective. Manifest Poincaré invariance is something we will discuss in much more detail in the context of the string.

It should also be noted that the action (10) is equivalent to ${S_{pp}}$ action read in the first pages of Polchinski (p.10). The only difference is notational, where Polchinski is using the string theory convention of uppercase letters, such that the spacetime coordinates are denoted as ${X^{\mu}}$. This implies that ${X^{0}}$ is the timelike coordinate and now ${X^{i}}$ run over the spacelike coordinates. And so, in Polchinski we read ${\frac{dx^{\mu}}{d \tau} = \dot{X}^{\mu}}$ and ${\frac{dx^{\nu}}{d \tau} = \dot{X}_{\mu}}$, where the dot denotes ${\tau}$-derivative.

From this point forward, we will use conventional notation in ST. In these notes we will also use the convention ${\hbar = c = 1}$. The flat Minkowski metric will be denoted as ${\eta_{\mu \nu}}$ with signature ${(-, +, +, +)}$. This is because, in ST, it will be convenient to place a negative with the time component.

So, to align with Polchinski now using conventional notation, we can rewrite (10) as the following,

$\displaystyle S_{pp} = -m \int d\tau (-\dot{X}^{\mu} \dot{X}_{\mu})^{\frac{1}{2}} \ \ (13)$

4. Deriving ${S_{pp}^{\prime}}$

There is another problem with the action (10) and equivalently (13). When we go to quantise this theory, the square root in the integrand proves troublesome because it is non-linear. It is also the case that the ${S_{pp}}$ action is of no use when trying to describe massless particles.

What we want to do is rewrite the ${S_{PP}}$ action in yet another equivalent form. That is to say, the action (13) can be put in another useful form by introducing an additional field on the worldline, an independent worldline metric ${\gamma_{\tau \tau}(\tau)}$. This new form for the action is given in Polchinski (p.10), where he invokes the tetrad ${\eta (\tau) = (- \gamma_{\tau \tau} (\tau))^{\frac{1}{2}}}$

$\displaystyle S_{pp}^{\prime} = \frac{1}{2} \int d \tau (\eta^{-1} \dot{X}^{\mu} \dot{X}_{\mu} - \eta m^2) \ \ (14)$

It will prove beneficial to dedicate a moment of time to discuss this new form of the action. The main idea to focus on is the introducing of the independent worldline metric ${\gamma_{\tau \tau}(\tau)}$ as an additional field on the worldline. We can think of it simply in terms of an einbein (a 1-dimensional version of a vielbein). Moreover, this additional field on the worldline – the einbein field – is not a dynamical field. Instead, it is an auxiliary field. One can view it as a generalised Lagrange multiplier. If, for simplicity of example, we denote this additional field as ${e(\tau)}$, we get something of the form (again in terms of Lagrangian dynamics),

$\displaystyle S_{pp}^{\prime} = \frac{1}{2} \int d\tau (e^{-1} \dot{X}^{2} - em^{2}) \ \ (15)$

Where the introduction of the einbein means our Lagrangian becomes ${L = \frac{\dot{X}^2}{2e} - \frac{em^2}{2}}$, and where we have simplified the notation by setting ${\dot{X}^{2} = \dot{X}^{\mu}\dot{X}_{\nu}\eta_{\mu \nu}}$. This approach may perhaps look for familiar. Notice, (15) very much looks as though the worldline theory has been coupled to 1-dimensional gravity. This observation refers again to viewing ${e(\tau)}$ as an einbein. That is why, in following Polchinski, if we change the notation and write the action in a more indicative way, we arrive at (14) where the worldline metric is ${\gamma_{\tau \tau} = (\gamma^{\tau \tau})^{-1}}$ and where ${e = \sqrt{-g_{\tau \tau}}}$.

As Polchinski explains (p.11), one usefulness in this approach is that we eliminate the square root from the integrand. This alternative form of the action (14) is therefore easier to quantise. Finally, it will prove to be true that the path integral for ${S_{pp}^{\prime}}$ will be much easier to evaluate.

A few additional comments before moving forward. Note, ${S_{pp}^{\prime}}$ is classically equivalent (on-shell) to ${S_{pp}}$ (we will prove this in a moment), and one will notice that it is polynomial in the fields ${X^{\mu}(\tau)}$. Hence, (14) can be described as a polynomial action. It can also be shown that, just as with the ${S_{pp}}$ action, ${S_{pp}^{\prime}}$ is reparameterisation invariant. In other words, (14) has the same symmetries as the original form of the action that we derived.

5. Classical Equivalence

To show that ${S_{pp}}$ and ${S_{pp}^{\prime}}$ are classically equivalent, we consider the variation of ${S_{pp}^{\prime}}$ with respect to ${\eta (\tau)}$. Beginning with the action (2.14), the computation is as follows,

$\displaystyle \delta S_{pp}^{\prime} = \frac{1}{2}\delta \int d\tau (\eta^{-1} \dot{X}^{2} - m^2\eta)$

$\displaystyle = \frac{1}{2} \int d\tau (- \delta (\frac{1}{\eta})\dot{X}^{2} - \delta (m^{2} \eta))$

$\displaystyle = \frac{1}{2} \int d\tau (- \frac{1}{\eta^{2}}\dot{X}^{2} - m^{2}) \ \ (16)$

From the calculus of variations, we set ${\delta S^{\prime}_{pp} = 0}$. That is to say, we set the integrand to be zero. In doing so, we find the field equations for ${\eta (\tau)}$,

$\displaystyle \eta^{2} = \frac{\dot{X}^{2}}{m^{2}}$

$\displaystyle \implies \eta = \sqrt{\frac{-\dot{X}^{2}}{m^{2}}} \ \ (17)$

Proposition 1 If we substitute (17) for ${\eta}$ back into the ${S_{pp}^{\prime}}$ action (14), we recover the original ${S_{pp}}$ action (13).

Proof:

$\displaystyle S_{pp}^{\prime} = \frac{1}{2} \int d\tau [(-\frac{\dot{X}}{m^{2}})^{-1/2} \dot{X}^{2} - m^{2}(\frac{\dot{X}^{2}}{m^{2}})^{1/2}]$

$\displaystyle = \frac{1}{2} \int d\tau [(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2} - m^{2}(\frac{\dot{X}^{2}}{m^{2}})^{1/2}]$

$\displaystyle = \frac{1}{2} \int d\tau [(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2} - m (- \dot{X}^{2})^{1/2}] \ \ (18)$

Recall, ${\dot{X}^{2} = \eta_{\mu \nu} \dot{X}^{\mu}\dot{X}^{\nu}}$. Substituting for ${\dot{X}}$ in the square root,

$\displaystyle = \frac{1}{2} \int d\tau [(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2} - m (- \eta_{\mu \nu} \dot{X}^{\mu}\dot{X}^{\nu})^{1/2} \ \ (19)$

Now, for the first term on the right-hand side of the equality, we employ some subtle algebra. From complex variables recall that ${i^{2} = -1}$.

$\displaystyle (-\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2} = (-1)(-1) -(\frac{m^{2}}{\dot{X}^{2}})^{1/2} \dot{X}^{2}$

$\displaystyle = -(-\frac{m^{2}}{\dot{X}^{2}})^{1/2} i^{2} \dot{X}^{2}$

$\displaystyle = -(-\frac{m^{2}}{\dot{X}^{2}} i^{4} \dot{X}^{2})^{1/2}$

$\displaystyle = -(-m^{2}i^{4}\dot{X}^{2})^{1/2}$

$\displaystyle -m (-i^{4}\dot{X}^{4})^{1/2} \ \ (20)$

As ${i^{4} = 1}$, it follows ${-m(i^{4}\dot{X}^{2})^{1/2} = -m (-\dot{X}^{2})^{1/2}}$. Now, substitute for ${\dot{X}^{2}}$ and we find ${-m (-\eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu})^{1/2}}$. We substitute this result into (2.19) and we get,

$\displaystyle S_{pp}^{\prime} = \frac{1}{2} \int d\tau [m(- \eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu})^{1/2} - m (- \eta_{\mu \nu} \dot{X}^{\mu}\dot{X}^{\nu})^{1/2}$

$\displaystyle = -m \int d\tau (- \eta_{\mu \nu}\dot{X}^{\mu}\dot{X}^{\nu})^{1/2} = S_{pp} \ \ (21)$

$\Box$

This ends the proof, demonstrating that ${S_{pp}}$ and ${S_{pp}^{\prime}}$ are classically equivalent.

There is much more that can be studied for the case of the relativistic free point particle from which further insights may be extracted for the case of the string, including the quantum theory and things like the geodesic equations for arbitrary background geometry. But for our purposes this ends the discussion. In the next post, we will discuss the Nambu-Goto action for the free relativistic string.

6. Summary

To summarise, one may recall how in classical (non-relativistic) theory the evolution of a system is described by its field equations. One can generalise many of the concepts of the classical non-relativistic theory of a point particle to the case of the relativistic point particle. Indeed, one should be familiar with how in the non-relativistic case the path of the particle may be characterised as a path through space. This path is then parameterised by time. On the other hand, in the case of the relativistic point particle, we have briefly reviewed how the path may instead be characterised by a worldline through spacetime. This worldline is parameterised not by time, but by the proper time. And, in relativity, we learn in very succinct terms how freely falling relativistic particles move along geodesics.

It should be understood that the equations of motion for the relativistic point particle are given by the geodesics on the spacetime. This means that one must remain cognisant that whichever path the particle takes also has many possibilities. That is, there are many possible worldlines between the beginning point and endpoint. This useful fact will be explicated more thoroughly later on, where, in the case of the string, we will discuss the requirement to sum over all possible worldsheets. Other lessons related to the point particle will also be extended to the string, and will help guide how we construct the elementary string action.

References

Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “String Theory and M-Theory: A Modern Introduction”.

David Tong. (2009). “String Theory” [lecture notes].

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

Barton Zwiebach. (2009). “A first course in String Theory,” 2nd edition.

*Edited on 18/01/19 for minor typos. Edited on 14/02/19 to include the addition of further substantiating comments on a number of points, and to ensure clarity of language.

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