Wednesday, 23 November 2011

Euclidean Spinors

For my own perverse reasons, I recently felt the need to explore, in as intimate a way as possible, the Majorana-Weyl spinors in eight Euclidean dimensions (if you don't know what any of this means, I will give a quick heuristic explanation of spinors below). Not being able to find a reference in which these were described in an appropriately explicit way, I set about the task myself. Over the last couple of days I have spent some time writing up my notes on building spinor representations, starting in dimension two and working up to eight. None of this is original, of course, but it's a very pretty story, and I enjoyed doing it. In case this might be of use or interest to anybody else, I have put the notes on my website, here. They are towards the bottom, with a few other sets of notes (the new ones on spinors are probably the most complete and useful of what I have there). Any criticism/comments/suggestions are welcome in the comments here.

So for the uninitiated, what are spinors anyway? Let's start with something a bit simpler: vectors. A vector can be thought of geometrically as an arrow; it is characterised by its length and the direction in which it points. We can also represent a vector algebraically by a list of numbers, which are just its coordinates. From the geometrical picture, we know that we can rotate vectors, without changing their lengths or the angles between them. In the algebraic picture, this corresponds to mixing up the numbers in a specific way. There are two important properties of the set of all such rotations: if we do one rotation and then another, it is the same as a single rotation, and we can undo any rotation by performing the 'opposite' one. Mathematically, this means that the set of all rotations forms what we call a 'group'.

Okay, so what about spinors? They are a bit harder to visualise than vectors, because they don't live in normal space, but all that matters to us here is that they do still 'point' in a certain sense. The other important property they have is that they are closely related to vectors in a very special way: the group which rotates vectors also rotates spinors. So if we start rotating the vectors, the spinors rotate as well. Under a 360 degree rotation, we know that the vectors come back to where they started; the strange thing about spinors is that they don't. Instead, in the algebraic representation, they pick up a minus sign. Geometrically, this means that when we perform a 360 degree rotation, the spinors end up pointing in the opposite direction, albeit in spinor space… This may all sound very strange, but in fact, spinors play a crucial role in the real world: they describe real particles like electrons. So when you turn an electron completely around, it picks up a minus sign! But we really need quantum mechanics to understand what that means...

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