Introduction
Hello again! In order to explain why special relativity is so important, we have to talk about symmetries. I do not want to make this extremely mathematical, but alas we still have quite a bit to cover before I can show you some nice results. To do this in a formal mathematical way, I will have to talk about Lie groups. These can get really abstract, but I will try to keep it a bit more conceptual.
Groups
Symmetries are transformations which keep some nice property invariant. A group is a mathematical object used to describe all symmetries of some object. We will use the terms symmetry and ‘element of a group’ interchangeably. We note 3 important properties:
- Symmetries can be composed, which again results in a symmetry.
- There is a neutral symmetry corresponding to doing nothing and composing with it does nothing.
- For every symmetry there is an inverse symmetry such that composing with it gives you the original element.
Note that in general the order of composition does matter. A good example of this is rotations of a cube. If you rotate 90 degrees along two different axes, then order in which you pick these axes matters. Groups where the order does not matter are called abelian, but these are far from the norm once you start looking at complicated groups.
Lie Groups
Lie groups are a special type of group. Namely, these are groups to which we can associate a smooth shape, with every point of the shape corresponding to some symmetry. In particular, we require that any smooth path through this shape gives us a smooth transition between symmetries.
The simplest example is the translation group in \(n\) dimensions to which we can associate \(\mathbf{R}^{n}\), by sending a point \(x\) to the translation sending the origin to \(x\). It is not too difficult to convince yourself that if you smoothly move \(x\) around, the corresponding translation also moves everything around smoothly.
Another good example of this is the rotation group in \(n\) dimensions. This group corresponds to all symmetries preserving angles and orientation. For \(n=2\) we get the simple case of all rotations of the plane. To these we can associate a very simple shape: again a circle! If you associate the point with angle \(\theta\) to a rotation of \(\theta\) degrees, then you will find that this gives a smooth correspondence.
Smooth shapes have a dimension. This is roughly the amount of directions you can move into, once you identify opposite directions as ‘going backwards the as moving a negative amount forwards’. Translations in dimension \(n\) have dimension \(n\) again, because you can move everything around in \(n\) directions. For the rotation group this is a bit more complicated, you can show that there are \(\frac{n(n-1)}{2}\) directions of rotation in \(n\) dimensions. So for dimensions \(1,2,3,4\) we find that there are \(0,1,3,6\) directions. In 1D you cannot rotate, so dimension 0, in 2D you can only rotate one way. In 3D you have 3 axes of rotation, so that also makes sense. However, at 4D our intuition fails and it becomes difficult to argue why there are 6 directions, so you have to trust me for now!
Generators
Suppose we have a Lie group \(G\) of dimension \(n\). Since every group has a neutral symmetry, there is a point \(1\in G\) which corresponds to this trivial transformation. Around this point, you can see what happens if you take tiny steps in one of the \(n\) directions. It turns out that if you take very small (infinitesimal) steps, everything becomes (roughly) linear. We obtain elements of the form \(1+\varepsilon A\) , where 1 is the identity transformation, \(\varepsilon\) is the size of the step and \(A\) is called the generator for transformations in this direction. Since there are \(n\) directions to go in, we obtain \(n\) different independent generators \(A_{1},\dots,A_{n}\).
Now remember that most groups aren’t abelian, implying that the order of taking tiny steps matters. If \(A,B\) are two generators, then we find that moving in the direction of \(A\) , then in the direction of \(B\), then back in A and then back in \(B\) gives us something of the form \(1+\varepsilon(AB-BA)\) (because we may ignore \(\varepsilon^{2}\) ). Since this is again a small step, there is some generator \(C\) (possibly a combination of the \(A_{1},\dots,A_{n}\) ) such that \((AB-BA)=C\). This quantity \(AB-BA\) is called the commutator of \(A\) and \(B\). From now we will denote it using the usual mathematical notation: \([A,B]\).
Groups acting on things.
If you rotate a square by 90 degrees, you again obtain a square, so this rotation is a symmetry of the square. However, if we now take a single point on the square, we find that rotating the square does not keep this point in place. Hence we get a rather interesting result: the whole can be symmetric under some transformation, even if its parts are not.
Similarly in physics, special relativity says that the laws of physics are invariant under translation. This means: translation of the entire universe. If I do an experiment by dropping a ball, then I do not get the same result if I only move the ball by 1000 kilometres while keeping the rest of my setup in place.
If \(g\) is an element of a group \(G\) (hence a symmetry) we write \(gx\) for \(g\) acting on some object \(x\). What \(G\) is, what \(x\) is and how \(G\) acts on \(x\) is not relevant for now. By linearity, we find that \((1+\varepsilon A)x = x + \varepsilon Ax\). Why this should hold can be explained as follows: if \(1+\varepsilon A\) is close to 1, which represents doing nothing, then \((1+\varepsilon A)x\) better be close to \(x\) in such a way which is linear in \(\varepsilon\).
What the above shows us, is that \([A,B]x = (AB-BA)x\) and \(Cx\) better be the same if we want any type of mathematical self-consistent notion of symmetries acting on things. This puts a lot of constraints on how these generators can act on elements and this is where the magic will occur!
A small example
The group of \(n\)-dimensional translations has dimension \(n\), as mentioned previously. Note that the order of translation does not matter (try it in your head!), hence we find that \([A,B] = 0\) for all generators \(A,B\) of this Lie group. If we want to define a notion of translation on some sort of objects, then we must have that \(ABx-BAx = [A,B]x = 0\), so \(ABx=BAx\) for any \(x\). This implies that our notion of translation must again be abelian.
For 3D we have 3 directions of rotations with generators \(X,Y,Z\) corresponding to rotation in those three axes. We find that \([X,Y]=Z, [Y,Z]=X\) and \([Z,X]=Y\) (notice the symmetry!). Hence if we want to define any notion of translation, then rotating a bit around the \(Y\)-axis then around the \(X\)-axis, compared to first \(X\) and then \(Y\) should give a difference equivalent to a small rotation around the \(Z\)-axis.
Conserved quantities
Now suppose that we have a generator \(C\) such that \([C,A]=0\) for any generator \(A\) of the Lie group. One can show that this implies that \([C,1+\varepsilon A]=0\). By combining (infinitely many) of such tiny small steps, we can (usually) reach all elements, hence \([C,g]=0\) for any element \(g\) of the Lie group. This means that no matter what symmetry transformation you perform, the action performed by \(C\) is always the same. In some sense, this action is ‘independent of perspective’ and ‘universally agreed upon’ as no amount of rotation, translation, or whatever transformations you group may contain can distort this operation. For these reason, such generators correspond to conserved quantities!
We will leave with this cliffhanger for now. Next week I will talk about the Poincare group, its generators and its conserved quantities!