The Relativistic Point Particle
We want to calculate the action of a point particle under a boost. It is natural to begin with
where is the path of the worldline, and the Lagrangian,
which retrieves our classical interpretation at low order approximation. Using the Euler-Lagrange equation, we then find that
which is precisely what we expect as a cursory approach.
The relativistic point particle requires more than simple boosts, but we will still remain in flat spacetime. As such, we have the metric , however, for the action we want this in terms of some timelike value, , so we can instead say that , where Our action will then be
If we add in the electromagnetic field, it can be shown that the relativistic action of a charged particle is
Next, it must be noted that
which gives us reparameterization invariance. A good choice for that allows us to solve several equations is the equality . This is known as static gauge because, physically, we are examining slices of space at static moments.
The Relativistic String
If we construct the matrix
for arbitrary parameters , we find that is the induced metric on , where
Furthermore, we find that for , the area
is also invariant under reparameterization.
We therefore define our two-dimensional worldsheet for a string by the parameters and , where is timelike, and is spacelike.
Furthermore, we can define proper area on the worldsheet to be
Here, the dot is the relativistic dot product. Now the action has units of (the same as , coincidentally), so by multiplying the proper area by we obtain the correct units and thus our action becomes
This is called the Nambu-Goto action of a relativistic string.
Our reparameterization invariance allows us to construct the matrix
such that the Nambu-Goto action becomes