What we talk about when we talk about Manifest Lorentz Covariance

String Theory without the Baggage, Episode 5.

The rains have already started down low, and we’re expecting upwards of 16 inches of snow about 3000 ft. Let’s celebrate the first winter storm of of the season with a bit of a bridge post to get us into thinking about the covariant approach to the physics of mechanics.

We’ve been discussing the point particle for quite some time. I swear this is supposed to be a series on String Theory, but it’s also going to go into great detail so… thanks for your patience.

We’ve seen that studying relativistic mechanics is pretty complicated:

We’ve also hopefully shown why most folks prefer to keep their calculations as symmetric as possible for as long as possible.

Today we’re going to get into the specifics for what a manifestly Lorentz invariant, point particle theory could be. And it’s pretty nice!

The relativistic, free point particle Action is given¹ by:

\(S(\gamma) = - mc^2 \int_{\gamma} ds,\)

where 𝛾 is some path in R^3,1. It is aesthetically pleasing that the Principle of Least Action coincides with the shortest possible path through the configuration space R^3,1. But as we have seen in the main text, this particular kind of aesthetic is fleeting.

By ds we of course mean an infinitesimal line segment along 𝛾, so we can expand this:

\(\begin{equation}S(\gamma) = -mc^2 \int_{\gamma} \sqrt{\eta_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}}\,ds.\end{equation}\)

Here s is any parameterization of the wordline 𝛾 that we like. Comparing with the classical action (inclusive of some potential V),

\(S_{\rm cl}(\gamma) = \int_{\gamma}dt \, \frac{m}{2}\left( \dot{x}^{2} + \dot{y}^{2} + \dot{z}^{2} \right) - V(x,y,z).\)

the square root gives us a sense of the complications to come.

The price of maintaining Lorentz covariance in our equations is now manifest in S(𝛾). We necessarily need a dummy parameter s in our integral. Worse, with s being a dummy parameter of an integral, S(𝛾) is now invariant under any reparameterizations of s, including local ones².

At it turns out, this is similar to the problem encountered in electrodynamics, where one must typically specify a gauge in order to work in the quantum theory.

Choosing a Gauge

The simplest gauge to pick - one that will fix a representative of s - is time.

That is, s = t.

This is akin to selecting the Coloumb gauge in electrodynamics. That is, choosing a divergence-free magnetic vector potential,

\(\nabla \cdot \mathbf{A} = 0.\)

It breaks the relativistic invariance, but affords working simplifications.

Back in the world of mechanics, this gauge implies

\(S(\gamma) = -mc^2 \int dt\, \sqrt{1 - \frac{1}{c^{2}}|\dot{\mathbf{x}}|^{2}}.\)

Note the minus sign out front is required to keep the integrand convex in velocity. While complicated, we can still work with this. For example, the momentum is given by

\(p_{i} = -mc^2 \frac{d}{dx^{i}}\left(\sqrt{1 - \frac{\dot{x}^{j}\dot{x}^{j}}{c^{2}}}\,\right) = \frac{m\dot{x}_{i}}{\sqrt{1 - \frac{1}{c^{2}}|\dot{\mathbf{x}}|^{2}}}.\)

In the nonrelativistic case, where

\(|\dot{\mathbf{x}}| \ll c, \)

one can Taylor expand the integrand. Let us compute the first few terms³:

\(L(0) = -mc^{2}, \)

\(L^{\prime}(0) \propto |\dot{\mathbf{x}}|/c = 0,\)

\(L^{\prime\prime}(0) = (-mc^{2})(-1/c^{2}),\)

so

\(L = L(0) + \frac{1}{2!}L^{\prime\prime}(0)|\dot{\mathbf{x}}|^{2} + \mathcal{O}(|\dot{\mathbf{x}}|^{4}),\)

which means that

\(S(\gamma) \approx \int_{\gamma}dt\, \frac{1}{2}m|\dot{\mathbf{x}}|^{2} - mc^{2},\)

which is consistent with the standard nonrelativistic action, together with a constant potential energy of

\(V = mc^2.\)

Next time we’ll explore what it means to develop Hamiltonian mechanics of the relativistic point particle using the manifestly covariant phase space discussed last time.

In particular, we’ll see what complications arise in that formalism.

1. We typically also want to know the initial Cauchy data for solving the differential equations of motion, but right now we're more concerned with general properties.

2. That is to say, ones that vary with s.

3. Note that an extra factor of c appeared when converting the integral over ds (a distance) to one over dt (a time).