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Post# 2: A Few more Spinors and some more Minkowski Space

Last post I introduced spinors as the two-component complex vectors that are acted on by the 2x2 complex matrices that represent elements of the group SU(2). The SU(2) matrices represent rotations in ordinary 3 dimensional space. The spinors have this weird property that if you multiply one by a matrix representing a rotation of 2π you don’t get back the spinor you started with – in fact you need to rotate through 4π to do this.

We can think of the rotations in 3 space as a subset of the Lorentz transformations (with Lorentz transformations, we don’t just have rotations, but also “boosts” where one frame is moving with respect to another. The boosts mix up the time and space coordinates). In the same way, we can extend the group SU(2) to include matrices representing these boosts as well as the rotations. The “extended” group is SL(2, ℂ) and it contains SU(2) as a subgroup.

Elements of SL(2, ℂ) are still 2x2 complex matrices and they still act on the 2 component spinors.

Conventionally spinors are represented by notation such as ξA. A = 0, 1 labels the two spinor components, each a complex number. Now for a little bit of tedious, but essential, abstract algebra:

Our spinors ξA are elements in a two complex-dimensional vector space V. (Just means they each have two elements, each of which is a complex number). Now, given a vector space, what does a mathematician immediately do with it ? – well of course they define the “dual space”. This is just the space of linear maps from V to whatever field V was defined over. In this case, it’s the set of linear maps from V -> ℂ. Elements of this dual space are denoted with lower indices, for example ηA.

The complex number obtained by ηA acting on the spinor ξA is just given by ηAξA (remember repeated indices are summed over).

If you like, you can think of ξA as a column vector and ηA as a row vector, so ηAξA is just what you get when you multiply a row vector by a column vector.

Unfortunately, to define twistors, I just need one more spinor thing. If V is the vector space of spinors, ξA, then we can consider another vector space Vbar, the space of complex conjugate spinors (V should have a bar over the top, but I can’t see a Unicode way to do that). The coordinates of a conjugate spinor are denoted ξA’. Of course, there are also dual conjugate spinors ηA’.

Compactified Minkowski Space

Minkowski space M is the set of quadruples (x0, x1, x2, x3) with metric
(dx0)2-(dx1)2- (dx2)2- (dx3)2..........(1)

Of course, that’s far too simple to understand, so we’ll have to make it a bit more complicated. One of the original motivations behind twistor theory was the idea that spacetime points may not be the most fundamental objects in our models of the universe, but that the conformal structure might be more fundamental. What do I mean by the conformal structure ?

In the diagram I show a picture of a null cone in (2+1) dimensional Minkowski space. (Ideally it would be 3+1 dimensional, but not possible to draw). This is the null cone of the origin and the surface of the cone defines the points for which the expression (1) is zero. The straight lines passing through the origin which make up the surface of the cone are potential world lines of light rays passing through the origin. Anything passing through the origin at a speed slower than the speed of light would ride on a world line in the interior of the cone.

There are two halves to the cone – the blue half and the red half. The blue half defines the spacetime points which could possibly influence the origin, and the red half defines the spacetime points which could possibly be influenced by the origin. Points outside the cone cannot possibly influence, or be influenced by the origin since a signal would have to travel faster than light for that to happen.



There is a conceptual null cone such as this one at every spacetime point. From the argument above, these null cones define the causal structure of the spacetime.

The null cone is given by the expression
(dx0)2-(dx1)2- (dx2)2- (dx3)2 = 0

i.e. ηabdxadxb where

ηab = diag(1, -1, -1, -1) is the Minkowski metric.

Now here’s the useful thing – it doesn’t matter a toss if you multiply the metric by a scalar function Ω2(x). In other words if I define a new metric η^ab = Ω2(x)ηab, then η^ab defines precisely the same null cones as ηab, so it defines precisely the same causal structure. The metric defines the geometry in the sense that it allows you to compute lengths and angles. Multiplying the metric by a scalar changes the length measurements, but not the angles. Now the one case where it doesn’t screw up the lengths is if the length is zero, as it is with the worldline of a light ray.

We can use this freedom to rescale the metrics in order to embed Minkowski space inside a “slightly” larger manifold. This manifold, which rejoices in the name of “compactified Minkowski space” can be thought of as taking the bottom half of a null cone (just the cone surface, not the interior) and “sticking” it onto the top half of Minkowski space, and taking the top half of a cone and sticking it on the bottom half of Minkowski space. So you end up with a spinning top-shaped picture:



I’ve drawn an arbitrary point P in Minkowski space and a couple of photons being fired off from it in different directions. They eventually hit the top half null cone at two different points. This half cone is called “future null infinity” and is denoted by the squiggly “I” which I don’t have in Unicode, but is called “scri plus”. Correspondingly there is a past null cone at infinity, denoted by scri-minus.

There are also points representing spacelike and timelike infinity but I won’t discuss those. Although I haven’t described it here in all its gory detail, this procedure of appending the null cone at infinity is mathematically rigorous and is an enormously useful technique, and a generalisation of it played a vital part in the proof of the Hawking-Penrose singularity theorems.

Diagrams such as these, where you explicitly attach some points to represent “infinity” in the spacetime in question are generally referred to as “Penrose diagrams” in the literature, although Roger Penrose always maintained that they should be called “Carter diagrams” after Brandon Carter since the original inspiration came from him apparently.

Minkowski space with its "null-cone-at-infinity" is called compactified Minkowski space. In the next post, I'll describe twistor space and its correspondance with compactified Minkowski space. All that work on defining spinors will then have been worth while !!