In my last post I described some thoughts on the relationship between invariance principles and symmetry groups. Here I would like to describe the particular example of special relativity. After all, we have all heard that the speed of light is constant, but how do we go from this statement to Lorentz invariance? The path is not entirely straightforward.
First, we need to figure out what the “the speed of light is constant” should actually mean to us. Suppose the position of a particle moving at the speed of light is moving in a straight line so its 3-velocity is \(ct\), say aligned with one of our coordinate axes. Then it’s 4-vector position is given by \(x^\mu=(t,t,0,0)\) in units where \(c=1\). Strictly speaking, we should do as mentioned in my last post and make the distinction between the particle’s coordinates in Minkowski space and its velocity, which is an element of the tangent space. However, the distinction doesn’t change anything that will be important for this discussion since Minkowski space is flat, so we will just use the position vector to simplify things. This does introduce a finicky point about translations, but we will ignore them here.
The main characteristic of such vectors is that \(x^\mu x_\mu=x^\mu\eta_{\mu\nu}x^\nu=0\) where \(\eta_{\mu\nu}=\text{diag}(-1,1,1,1)\) is the metric on Minkowski space in rectangular coordinates and the summation convention is enforced. Since this characterizes vectors corresponding to the trajectories of light, they are often referred to as being light-like.
So then, when Einstein said that the speed of light is a constant in different reference frames, he could have said equivalently that all transformations which leave our physics invariant also map light-like vectors to light-like vectors. That is, if the matrix \(\Lambda^\mu_\nu\) is to be a symmetry of our physics, then it is necessary that for all light-like vectors \(x^\mu\), \(\Lambda^\mu_\nu x^\nu\) should also be light-like.
If we write this requirement out, it tells us that
$$(\Lambda^\mu_\check{\mu}x^\check{\mu})\eta_{\mu\nu}(\Lambda^\nu_\check{\nu}x^\check{\nu})=x^\mu\eta_{\mu\nu}x^\nu=0$$
for all light-like vectors \(x^\mu\). The largest group of such transformations is known as the conformal group.
If it weren’t for our discussion last time, we might think that this is a pretty strange result. After all, we know that group used in special realtivity is not the conformal group, but is the Lorentz group which is the group of transformations leaving \(x^\mu y_\mu\) invariant for any \(x^\mu,y^\mu\), whether light-like or not.
But we now see that any transformation which leaves \(x^\mu y_\mu\) invariant for any two vectors will also leave the inner product of a light-like vector with itself invariant. As a result, the Lorentz group must be a subgroup of the conformal group.
Proceeding as we did for the case of the orthogonal group, we should find a transformation in the conformal group which is not in the Lorentz group, and then investigate whether it is a symmetry or now. For the sake of example, consider dilations. These are transformations which map the metric to a scalar multiple of itself: \(\eta_{\mu\nu}\rightarrow \omega\eta_{\mu\nu}\) for some \(\omega\in\mathbb{R}_+\). This will clearly leave the square of a light-like vector invariant since we would just be multiplying zero by \(\omega\). Hence, such transformations are elements of the conformal group. These transformations certainly wouldn’t leave other inner products invariant though, since they would map \(x^\mu y_\mu\rightarrow \omega x^\mu y_\mu\). So we know that dilations also aren’t elements of the Lorentz group.
If we look at the Maxwell Lagrangian, we would see that the \(F_{\mu\nu}F^{\mu\nu}\) term actually turns out to be invariant under dilations. The issue comes in as soon as matter exists at the same time — it can be shown that the matter coupling \(J^\mu A_\mu\) is not invariant under dilations classically, though there are very special cases in quantum field theories in which invariance under the conformal group is recovered.
So, as soon as we accept that things exist and some of them are magnets, we have to abandon conformal invariance in favor of some subgroup thereof, the next largest Lie subgroup being the Lorentz group itself. This is how the statement “the speed of light is a constant” translates into Lorentz invariance.
We could, of course, go further with this and think about subgroups of the Lorentz group, and this is what people have done. In particular, an extraordinary amount of effort between the 40’s and 80’s was dedicated to this question, phrased somewhat differently, in the particle physics community. Both in terms of theory and experiment.
In four space-time dimensions, the Lorentz group can be shown to have four connected components as a Lie group. The component connected to the identity, known as the proper orthochronous Lorentz group, is connected to the other two components by the operations of parity, \(P\), and time-reversal, \(T\), which negate the space and time components, respectively. They work very much like the \(P_1\) operator described in the \(O(n)\) example last time.
The result of all these investigations by the particle physics community was to find that, in fact, neither \(P\) nor \(T\) are actual symmetries of nature. So, at the end of the day, we are left with only the proper orthochronous Lorentz group (referred to as just the Lorentz group from here on) as a symmetry of nature.
We could, of course, worry that the true symmetry of nature is a subgroup of the Lorentz group, but while possible, this seems unlikely. The group of rotations in space is a subgroup of the Lorentz group, and this group seems to be a good symmetry. As a matter of fact, the observation that all particles can be ascribed a definite value of spin follows only when the group of rotations is a good symmetry of nature.
If we accept that rotations should be a symmetry of nature, then we are forced to accept Lorentz symmetry or abandon it for rotations alone since there is no subgroup of the Lorentz group which contains rotations as a subgroup besides the Lorentz group and the group of rotations themselves. Since the part of the Lorentz group which is not the rotations is generated by the boosts, we would need to abandon the idea that boosts are good symmetries.
However, this idea was explored, though their reasoning was somewhat different, over 130 years ago by Michelson and Moreley. If we abandoned boosts, there would exist a preferred reference frame and their experiment showed that this was not the case. It might be possible that there exist some loophole which allows a yet-unknown sector of nature which violates boost invariance but which didn’t alter the results of the Michelson and Moreley experiment, but such a result would be so antithetical to everything we know about how nature works that it would be an extreme shock to the system.
