*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.

In the opening pages of volume one of Polchinski’s “String Theory“, we are immediately introduced to an expression for the action of a relativistic point particle. That the relativistic point particle should first come into focus makes sense, as it requires variational techniques to be formulated and a fairly nice analogy can be drawn between the parameterised worldline of the particle and the parameterised worldsheet of an open string (which we’ll begin to think about in a following entry).

As found in Polchinski, another terrific textbook by Becker, Becker and Schwarz (2006) also introduces a study of bosonic string theory with the relativistic point particle. I think it serves good pedagogical insight to emphasise why this approach makes sense: preceding a study of the relativistic string and thus also of its classical dynamics, there are many concepts that can and will be generalised from a study of the relativistic point particle. Perhaps this is something that will become increasingly clear in hindsight, but I will try to highlight some of the concepts as we go.

To begin, in Polchinski (p. 10) we are given the Poincaré-invariant action of the form:

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

We are also given the variation of the action,

    \[ \delta S_{pp} = -m \int d \tau \dot{u}_{\mu} \delta X^{\mu} \]

Where u^{\mu} = \dot{X}^{\mu}(-\dot{X}^{\nu} \dot{X}_{\nu})^{\frac{1}{2}}.

Let’s unpack this.

First, a comment on the Poincaré-invariance of the action. One will likely be familiar with Lorentz invariance (boosts and rotations), as they are a common symmetry typically introduced to the student early in the study of Special Relativity. But there is a larger set of symmetries that includes the addition of translations, which is defined as the Poincaré group and thus follows the notion of Poincaré-invariance. So, one might say, Poincaré symmetry is the complete symmetry of Special Relativity. (If one requires additional reading, see this Wikipedia article or section 2.3 in Becker, Becker, Schwarz).

The key point here is that in referencing Poincaré invariance of the action, we are saying that the action is reparameterisation invariant.  That is, the action does not depend on parameterisation such that it “would be proportional to the proper time along the worldline” (p.10).

With that noted, let’s derive the above classical action for the relativistic point particle and also the equations of motion. If there is space, we’re also going to want to show the classical equivalence between S_{pp} and S_{pp} \prime since this is something Polchinski mentions (p.11). (In a following post I may divert slightly from Polchinski and show the expansion of the action for the relativistic particle in terms of the generalisation to the p-brane action).

The relativistic particle

We want to 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}. Which path this particle takes has many possibilities – that is, there are many possible worldlines between the beginning point and endpoint. It is also noted that, since our relativistic particle is moving through spacetime, when considering its classical motion we must be cognisant that this motion can no longer be assumed as a straight line in a more familiar Euclidean space. Rather, we must now take into account motion given by geodesics on the spacetime.

The action along the worldline is going to be proportional to the proper time. In other words, the action is going to be proportional to the invariant length of the worldline. The invariant length – sometimes called the “line element” of the metric \eta_{\mu \nu} – is ds. What we want to do, as a first step, is write down the interval in 4-dimensional spacetime:

    \[ - ds^2 = -c^2 dt^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2 \]

It follows that with some clever thinking (Zwiebach, 2009, p.91), we may take as an expression for the action

    \[ S = -mc \int ds \]

We’re not going to set c = 1 for pedagogical reasons (although throughout the remainder of my notes, and certainly throughout much of the literature, it is common to work in the units \hbar = c = 1). Moreover, I think as an opening entry it is worthwhile to offer the following reminder:

  1. In this expression, m is derived as it has the dimensions of inverse length.
  2. The c term is kept here for pedegogical purposes to remind of Lorentz invariance.
  3. The action asserted above gives the proper time along the worldline multiplied by minus the rest energy of the particle.

A few more insightful notes would include that, as is common, we invoke a Lorentz-invariant theory. Moreover, our choice of integral should not depend on our choice of reference system. It must be invariant under Lorenz transformations. Additionally, and for this reason, one sees that, in taking an integral over ds, we are calculating the infinitesimal invariant length of the particle’s worldline. It is common that the action is written in terms of ds for the reason that it is a Lorentz scalar. With that box checked, one can also think of this integral similarly to many other instances where we are taking the sum over many small increments ds along the particle’s worldline.

But what to do from here? Let’s look at the ds in the integrand. If the name of the game here is to arrive at a more familiar 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 initial and final points of our particle’s path.

So, from our interval we obtain an expression for ds by working through the following:

    \[ -ds^2 = -c^2 dt^2 + (dx^1)^2 + (dx^2)^2 + (dx^3)^2 \]

    \[ ds^2 = c^2 dt^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2 \]

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

    \[ \implies ds^2 = (c^2 - v^2) dt^2 \]

    \[ ds = \sqrt{c^2 - v^2} dt \]

    \[ ds = c \sqrt{1 - \frac{v^2}{c^2}} dt \]

We can now substitute this expression for ds directly into our action,

    \[ S = -mc^2 \int_{t_i}^{t_f} \sqrt{1 - \frac{v^2}{c^2}} dt \]

Notice, we have related ds, our infinitesimal length along the worldline, with the time observed by a Lorentz observer, dt. We have also established an interval along our worldline, such that we are integrating from t_i to t_f.

From this it follows quite directly that the Lagrangian is,

    \[ L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}} \]

Of course, with special relativity in mind, we cannot have v > c, so there is a maximal velocity contained in this expression. As Zwiebach writes, “This could have been anticipated: proper time is only defined for motion where velocity does not exceed the speed of light” (p. 92). For pedagogical purposes, it should also be noted that a lot of physics from special relativity can be recovered from this expression, including an expression for the relativistic momentum and Hamiltonian. From this it also follows that when varying the action above, we arrive at the EoM in which \frac{d\mathcal{P}_\mu}{d\tau} = 0. I once again refer to the reader to Zwiebach’s notes, should they wish to spend more time considering these topics.

Classical action and reparameterisation invariance

Now, the engaged reader may have already guessed or foreseen a problem: we need to ensure reparameterisation invariance \tau \rightarrow \tilde{\tau}(\tau). This will take us to the expression for S_{pp} as found in Polchinski.

As noted in Polchinski, we’re using the metric with Minkowski signature: \eta_{\mu \nu} = (-, +, +, +). The particle’s worldline, as it moves through spacetime, may also be described as its trajectory. The parameterisation of this worldline is arbitrary, with the convention being to use a real parameter \tau. Moreover, the propagation of the free point particle described by the worldline which we might denote as \gamma – this is parameterised by \tau, such that \gamma: \tau \rightarrow X^{\mu}(\tau). Here X^{\mu}(\tau) \in \mathbb{R}^{1, d-1} and \mu = 0, i.

To put it differently: the key idea is that the worldline of our relativistic point particle is described by parameterisation necessary to compute the action. We invoke the notation X^{\mu}(\tau), where the coordinates X^{\mu} are a function of \tau such that the worldline is said to travel from X_{i}^{\mu} to the end point X_{f}^{\mu}. The interval for \tau is [\tau_{i}, \tau_{f}].

As David Tong (p.9) notes, even the time component of the description of our particle’s motion is parameterised inasmuch that it is promoted to a dynamical degree of freedom, via gauge symmetry, without it really being a dynamical degree of freedom. One is likely familiar with having position described as a function of time, but recall that our particle is travelling through spacetime and we need to account for this. So we have it that the previously mentioned parameter \tau is introduced, from which, if we ensure manifest reparameterisation invariance, “we can pick a different parameter \tilde{\tau} on the worldline, related to \tau by any monotonic function” such that \tilde{\tau} = \tilde{\tau}(\tau).

The question now becomes, how do we express ds – the infinitesimal invariant length – in terms of a parameterised worldline? Moreover, how might we proceed with this requirement such that we satisfy the conditions of our action being reparameterisation invariant? Well, to make the length of our spacelike curve reparameterisation invariant we note that:

    \[ ds^2 = - \eta_{\mu \nu} dx^{\mu} dx^{\nu}= - \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} (d \tau)^2 \]

Similar as before, substitute for ds in the integrand,

    \[ S = -mc \int_{\tau_i}^{\tau_f} \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} d\tau \]

One can see that this expression for the action is reparameterisation invariant, which, in a very real sense, is a gauge symmetry of the total system. If it is not clear, one can most certainly check that the action is invariant under the transformations described above (to save space, I refer the reader to p.10 of Tong’s lecture notes from the reference list).

With confidence in the reparameterisation invariance of the action, notice, also, that the above action is equivalent to S_{pp} as written in Polchinski (1.2.2), where \frac{dx^{\mu}}{d \tau} = \dot{X}^{\mu} and \frac{dx^{\nu}}{d \tau} = \dot{X}_{\mu}. So, we can rewrite the above

    \[ S = -mc \int_{\tau_i}^{\tau_f} d\tau (-\dot{X}^{\mu} \dot{X}_{\mu})^{\frac{1}{2}} \]

Canonical momentum

The canonical momentum associated with X^{\mu} can be found by taking the derivative of the Lagrangian with respect to the velocity,

    \[\mathcal{P}^{\mu} = \frac{\partial \mathcal{L}}{\partial \dot{X}_{\mu}} = m \frac{\dot{X}^{\mu}}{\sqrt{-\dot{X}^2}} \]

Equations of motion (EoM)

With that noted, let’s go ahead and derive the EoM. We start by varying the action,

    \[ \delta S = -mc \int_{\mathcal{P}} \delta (ds) \]

To find \delta (ds) we note,

    \[ (ds)^2 = - \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} (d \tau)^2 \]

    \[ 2(ds)\delta(ds) = \delta (- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} (d \tau)^2) = -2 \eta_{\mu \nu} \delta (\frac{dx^{\mu}}{d \tau}) \frac{dx^{\nu}}{d \tau} \]

    \[ \implies \delta (dS) = - \eta_{\mu \nu} \frac{d (\delta x^{\mu})}{d \tau} \frac{dx^{\nu}}{d s} d \tau \]

From this we can also arrive at an equivalent expression for the variation as read in Polchinski (1.2.3).

Since we have an expression for \delta S, we can substitute this in the integrand and compute the varying action.

    \[ \delta S = mc \int_{\tau_i}^{\tau_f} \eta_{\mu \nu} \frac{d (\delta x^{\mu})}{d \tau} \frac{dx^{\nu}}{d s} d \tau \]

We now rewrite the integral as a total derivative, so that the \delta x^{\mu} is no longer featured in the derivative.

    \[ \delta S = mc \int_{\tau_i}^{\tau_f} d \tau \frac{d}{d \tau} (\eta_{\mu \nu} \delta x^{\mu}(\tau) \frac{dx^{\nu}}{d s}) - \int_{\tau_i}^{\tau_f} d \tau \delta x^{\mu}(\tau) (mc \ \eta_{\mu \nu} \frac{d}{d \tau} \frac{d x^{\nu}}{ds}) \]

The first term vanishes, as it concerns behaviour specifically at the boundaries of the worldline that have already been fixed. So,

    \[ \delta S = - \int_{\tau_i}^{\tau_f} d \tau \ \delta x^{\mu}(\tau) \ \eta_{\mu \nu} \ \frac{d}{d \tau} (mc \frac{d x^{\nu}}{d s}) \]

But the mc \frac{d x^{\nu}}{d s}) is just mu^{\nu} = \mathcal{P}^{\nu}, where \mathcal{P} is the relativistic momentum of the particle. Therefore,

    \[ \delta S = - \int_{\tau_i}^{\tau_f} d \tau (\frac{d}{d \tau} \delta x^{\mu}) \mathcal{P}_{\mu} \]

Rearrange and we get,

    \[ \delta S = - \int_{\tau_i}^{\tau_f} d \tau \ \delta x^{\mu} (\tau) \eta_{\mu \nu} \frac{d \mathcal{P}^{\nu}}{d s} \]


    \[ = - \int_{\tau_i}^{\tau_f} d \tau \ \delta x^{\mu}(\tau)\frac{d \mathcal{P}_{\mu}}{d s} \]

As \delta x^{\mu}(\tau) vanishes, the EoM is

    \[ \frac{d \mathcal{P}_{\mu}}{d \tau} = 0 \]

Classical equivalence

Polchinski notes (p.10) that the action S_{pp} 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 as,

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

It will prove beneficial to dedicate a moment of time to discussing this new form of the action, including the introduction of an einbein (a 1-dim version of a vielbein), because in the future we will construct an analogue in the case of the classical string.

For Polchinski, he invokes the tetrad \eta (\tau) = (- \gamma_{\tau \tau} (\tau))^{\frac{1}{2}}. This leads to deriving the alternative form of the action, S_{pp}^{\prime}. As Polchinski explains, one usefulness in this approach is that the square root of the S_{pp} action is difficult to quantise. This alternative form, S_{pp}^{\prime} obviously avoids that because it eliminates the square root. It is also useful because S_{pp} is of no use when trying to describe massless particles (pp. 10-11). Finally, it will prove to be true that the path integral for S_{pp}^{\prime} will be much easier to evaluate.

One can think of this introduction of an auxiliary field on the worldline in the following way. Note, firstly, that S_{pp}^{\prime} is classically equivalent (on-shell) to S_{pp}, and one will notice that it is polynomial in the fields X^{\mu}(\tau). The einbein field itself is not a dynamical field. Instead, it is an auxiliary field. One can view it as a generalised Lagrange multiplier. And so, for massless particles, we get something of the form (again in terms of Lagrangian dynamics),

    \[ S = -m \int d\tau = -m \int d \lambda \sqrt{-g_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}}  \implies S = \frac{1}{2} \int d \lambda (\frac{1}{e(\lambda)} g_{\mu \nu} \frac{dx^{\mu}}{d \lambda} \frac{dx^{\nu}}{d\lambda} - m^2 e(\lambda)) \]

The introduction of the einbein means our Lagrangian becomes L = \frac{\dot{x}^2}{2e} - \frac{em^2}{2}, where \dot{x}^2 = g_{\mu \nu} dx^{\mu} dx^{\nu} < 0.

Instead of e(\lambda), to obtain a similar expression as Polchinski we’re going to write it as \eta(\tau). So we have,

    \[ S_{pp}^{\prime} = \frac{1}{2} \int d \tau (\frac{1}{\eta (\tau)} g_{\mu \nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} - m^2 \eta(\tau)) \implies S_{pp}^{\prime} =   \frac{1}{2} \int d \tau (\eta^{-1} g_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau} - \eta m^2) \]

We clean this up and we arrive at Polchinski’s expression for S_{pp}^{\prime}:

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

One additional comment before moving forward is that S_{pp}^{\prime} can be shown to also be reparameterisation invariant. In other words, as Polchinski notes, this action has the same symmetries as the original form that we derived.

To conclude the present entry, let’s show S_{pp} and S_{pp}^{\prime} are classically equivalent.

    \[ S = -mc \int_{\tau_i}^{\tau_f} \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} d\tau \]

We can write this as,

    \[ S = -mc \int_{\tau_i}^{\tau_f} d\tau (-\dot{X}^{\mu} \dot{X}_{\mu})^{\frac{1}{2}} \]

Now, we take as the assumption that this form of the action is equivalent to (1.2.5):

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

As Polchinski notes, the EoM for \eta, \frac{\delta s}{\delta \eta} :

    \[ \eta^2 = \frac{- \dot{X}^{\mu} \dot{X}_{\mu}}{m^2} \]

When we substitute \eta into S_{pp}^{\prime} we arrive back at our expression for S_{pp}.

As an aside, my professor who is offering me guidance in my self-studies has pointed to a question over the presumption of “the quantum theory for S_{pp}” that “will lead to a result equivalent to the S_{pp} \prime path integral”, with the latter taken “as the starting point in defining the quantum theory”. This is most certainly something we’ll revisit later.

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.