*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:
We are also given the variation of the action,
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 and 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 . Which path this particle takes has many possibilities – that is, there are many possible worldlines between the beginning point and endpoint. (The engaged reader who already has some knowledge of String Theory (ST) and also of the path integral formalism may begin to note a connection pertaining to how ST could come to be seen as a very special generalisation of the latter. But further elaboration on this point will be saved for a future entry). 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 – is . What we want to do, as a first step, is write down the interval in 4-dimensional spacetime:
It follows that with some clever thinking (see Zwiebach, p.80), we may take as an expression for the action
We’re not going to set 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 ). Moreover, I think as an opening entry it is worthwhile to offer the following reminder:
- In this expression, m is derived as it has the dimensions of inverse length.
- The term is kept here for pedegogical purposes to remind of Lorentz invariance.
- 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 , we are calculating the infinitesimal invariant length of the particle’s worldline. It is common that the action is written in terms of 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 along the particle’s worldline.
But what to do from here? Let’s look at the 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, and 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 by working through the following:
We can now substitute this expression for directly into our action,
Notice, we have related , our infinitesimal length along the worldline, with the time observed by a Lorentz observer, . We have also established an interval along our worldline, such that we are integrating from to .
From this it follows quite directly that the Lagranian is,
Of course, with special relativity in mind, we cannot have , 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. 81). 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 . 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 . This will take us to the expression for as found in Polchinski.
As noted in Polchinski, we’re using the metric with Minkowski signature: . 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 . Moreover, the propagation of the free point particle described by the worldline which we might denote as – this is parameterised by , such that : . Here and .
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 , where the coordinates are a function of such that the worldline is said to travel from to the end point . The interval for is .
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 is introduced, from which, if we ensure manifest reparameterisation invariance, “we can pick a different parameter on the worldline, related to by any monotonic function” such that .
The question now becomes, how do we express – 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:
Similar as before, substitute for in the integrand,
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 as written in Polchinski (1.2.2), where and . So, we can rewrite the above
Equations of motion (EoM)
With that noted, let’s go ahead and derive the EoM. We start by varying the action,
To find we note,
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 , we can substitute this in the integrand and compute the varying action.
We now rewrite the integral as a total derivative, so that the is no longer featured in the derivative.
The first term vanishes, as it concerns behaviour specifically at the boundaries of the worldline that have already been fixed. So,
But the is just , where is the relativistic momentum of the particle. Therefore,
Rearrange and we get,
As vanishes, the EoM is
On p.10 in Polchinski, he mentions that “the action can be put in another useful form by introducing an additional field on the worldline, an independent worldline metric .” The action is rewritten as,
One comment: this additional field is an auxiliary field. For Polchinski, he invokes the tetrad . This leads to deriving the alternative form of the action, . As Polchinski explains, one usefulness in this approach is that the square root of the action is difficult to quantize. This alternative form, obviously avoids that because it eliminates the square root (this subtle manoeuvre will be repeated analogously in the case of strings, both with the Nambu-Goto action and the Polyakov action). It is also useful because is of no use when trying to describe massless particles (pp. 10-11). He also points out that the path integral for will be much easier to evaluate (something we’ll explore in a future post).
I find it helpful to think about this introducing of an auxiliary field on the worldline in the following way. First, I think one can view it as a generalised Lagrange multiplier. Something of the form,
This is because, in this context, it’s not really a dynamic variable.
Keeping that in mind, for massless particles we can instead write (again in terms of Lagrangian dynamics):
Instead of , to obtain a similar expression as Polchinski we’re going to write it as . So we get,
We clean this up and we arrive at Polchinski’s expression for :
One additional comment before moving forward is that 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 . Additionally, it is noted that the and are classically equivalent. To conclude the present entry, let’s show this is true.
We can write this as,
Now, we take as the assumption that this form of the action is equivalent to (1.2.5):
As Polchinski notes, the EoM for , :
When we substitute into we arrive back at our expression for .
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 ” that “will lead to a result equivalent to the path integral”, with the latter taken “as the starting point in defining the quantum theory”. This is most certainly something we’ll revisit later.