Alternating Group S_3 January 24, 2009

Posted by keithkchan in fun stuffs, Trivia, Whatever.

As expected, after the start of the spring semester, there are many more chores or even crap to deal with, I don’t have much thoughts to post. Therefore I probably will not update this blog very often. If you check this blog regularly, feel free to reduce your frequency of visiting this blog by a factor of 2. I also find that my research productivity has dropped by at least a factor of 2 when the semester starts.

OK. In this post I would like to talk about the group, the alternating group $S_3$, another name is dihedral group $D_3$. Well, everybody in QFT II class knows it well, why do I bother talking about it on the Blackboard? First, I  want to say something, you can’t stop me from doing this. Second, I realize it may be a good analogue for the Lorentz group. This connection may turn out to be another stupid idea, as most of my “great ideas”. That’s why I want to write it down before I realize it is really stupid.

The name alternating group originates from the fact that the group elements in this group are the permutation operations of 3 (non-identical) objects. There are altogether 3!=6 ways to permute them. The other name dihedral group is from its  geometric interpretation.  We can regard the elements as rotations and reflections of an equilateral triangle. The rotations by 0 deg,  60 deg and 120 deg, and reflections about an axis passing through one of the vertex and perpendicular to the opposite side of the triangle.  Therefore there are 3 reflections.

Take a subgroup group H, which contains the 3 rotations. We denote the other 3 reflections elements by Hr.  One can check that $H^2=H$ (rotation twice is still a rotation),   $H \cdot Hr = H$ (rotation followed by a reflection is equivalent to a reflection about a rotated axis, so it is still one of the reflection), $(Hr)^2 =H$ (it is well-known that two reflections can be cast as a rotation about the intersection of axes of reflections). So H and Hr from the group $Z_2$!

Now lo and behold. I think $S_3$ is an analogue of the Lorentz group in some vague sense. The Lorentz group consists of the rotations and boosts, while the alternating group consists of rotations and reflections. Obviously I want to link the boosts to the reflections.  If you remember the commutation relations between the generators of Lorentz group J (rotations) and K (boosts). Schematically, [J,J]~J, [K,K]~J, [J,K]~K.  This looks similar to relation between H and Hr.

Of course, the alternating group is discrete and the Lorentz group is a Lie group. They are rather different. The claimed  connection may be just some bullshits. But there is carbon in bullshit, and we know that diamonds is just some crystal structure of carbon. So there exists non-vanishing chance that bullshits may become diamonds.  I will keep it in mind, we will see…