What is a Spinor?
Laboratoire de Physique Fondamentale, FRANCE
16 May 2023
A pedagogical and simplified presentation of the notion of spinor is given, so that an intuitive understanding of its properties is possible.
Latest version: https://phy.clmasse.com/spinor.html
Introduction
Cartan introduced the spinor while studying the representations of the group $SO(3).$ Since then, it has been used extensively, especially in the theory of fermions and of the weak nuclear force. Though, its properties remain non intuitive, even if the mathematical theory is fully developed. In particular, or rotation of $2π$ results in the spinor changing sign, $4π$ is then necessary to recover the initial situation. This is embodied in the so-called belt trick.
§1. Spinor and stereographic projection
We present a simplified version of a spinor, but containing all the core features and easily generalisable, mainly in the sake of clear graphical representation. So let us start with the real column vector $$ζ=\pmatrix{x\\y}$$ with the condition $$ζ^tζ=xx+yy=1.$$ It is the common way of writing a (unit) spinor, and from it we define the projector $${\cal P}=ζζ^t=\pmatrix{xx& xy\\yx& yy},$$ whose trace is $1.$ A third representation is the so called fundamental field $$w=y/x.$$ Both ${\cal P}$ and $w$ do not change if the sign of $ζ$ is changed. It is the most curious property, corresponding to the fact that for a $SU(2)$ spinor, the sign changes for a $2π$ rotation around any fix axis. But virtually always only quadratic functions are used.
Now we turn to the stereographic projection, and all the gist is contained in the connection of these two seemingly disparate objects, see fig. 1. The stereographic projection associates to a point $M$ on a circle of center $O$ and radius $1$ another point $W$ projected on a straight line passing by the center, called the $w$ axis, in the following way. A pole $N$ on the circle is chosen as an intersection of the circle with a straight line perpendicular to the $w$ axis and passing by $O.$ Then a straight line is drawn between the considered point and the pole. The distance from $O$ to the intersection of this line with the $w$ axis is the stereographic projection $w.$ Finally we call $θ$ the angle $\angle SOM.$
In order to perform the calculation, we consider the circle as a trigonometric circle, whose $0$ is $S,$ the opposite pole to $N.$ By drawing a line through $M$ and parallel to the $w$ axis, defining a point $H$ on the line $(ON),$ we get two similar triangles: $NMH$ and $NWO.$ It is then obvious, since the radius is $1,$ that $$w=\overline{HM}/\overline{NH}={\sin(θ)\over1+\cos(θ)}.$$ It is the known formula.
The traditional stereographic projection from a two dimensional sphere is obtained by rotating the figure around the line $(ON).$ an angular variable is then adjoined to $w$ which then become a complex number or a double of real numbers.
§2. Trigonometry, topology, and more…
It seems to be the end of the story, however, something important is lacking, a further construction that would fulfil the promise of the title. Indeed, we can add a second trigonometric circle centered at $N,$ see fig. 2. Now a theorem of Euclidean geometry says that the angle $\angle HNM$ is just half the angle $\angle HOM$ (because the triangle $NOM$ is isosceles,) that is, $θ.$ It is then not less obvious that $$w=\tan(θ/2)={\sin(θ/2)\over\cos(θ/2)},$$ as can be verified using the trigonometric formulæ of the double arc.
But we have forgotten our spinor. Owing to the above formula, we assign $$x=\cos(θ/2),\quad y=\sin(θ/2).$$ These are the rectangular coordinates with respect to $N$ of the point $Z$ on the second trigonometric circle. The projector then becomes $${\cal P}=\frac12\pmatrix{1+\cos(θ)& \sin(θ)\\\sin(θ)& 1-\cos(θ)}$$ It projects on the line $(NM)$ a point with coordinate reckoned on the axis $x$ and $y.$ The set of all ${\cal P}$ has the topology of $S^1.$
Now, let us observe the behavior of the directed lines $(\vec{OM})$ and $(\vec{NM})$ when the point $M$ runs along the circle, starting from $S.$ When $M$ gets at $N,$ the line $(NM)$ is no longer defined, but by continuity it is the straight line tangent to the circle at $N.$ Then it rotates in the same direction while its own direction is reversed. At full circle, it recovers its initial position but with its direction reversed. In few words, it behaves like the Möbius strip with respect to the line $(OM),$ realising the projective space $P^1,$ or the non trivial zeroth Hopf bundle
with base space $S^1$ and fiber $S^0,$ or structure group $O(1)=\{-1,1\}$. The point of reversal is where the transition function is $-1.$ The curious behavior of a spinor in a $2π$ rotation is thus similar to the one of the Möbius strip for a full turn. In term of $ζ,$ this corresponds to a change of sign since the variation of $θ/2$ is only $π.$ The point $Z$ runs only a half turn, since it is the intersection of the line $(NM)$ and the second trigonometric circle, and by the way there are two of them. The projector behaves normally, in conformity with its representation of the line $(OM).$ The fundamental field $w$ is immune to all that, the point $W$ runs along the compactified line and $w$ is infinite at the point of reversal, with only one infinity.
Concluding remarks
The stereographic projection is know from Antiquity, it was a very efficient trick to perform astronomical calculations on the celestial sphere. As a conformal transformation, it is still much used in various mathematical applications. In this minimal demonstration, a crossroad between different tools of mathematical physics has been shown. It turns out that it can be extended in much more complex situations, providing an efficient method to deal with all that is spherical in all dimensions, almost as much as for rectangular case, and can even be generalised to hyperbolic problems. For that reason, the author calls it the magic circle.