The Spin Chronicles (Part 3): Spin frames

What’s a Spin Axis, Anyway?

This post was inspired by a question from Bjab, who asked: "What do you mean by ‘spin axis’?" Well, buckle up because I’m about to explain that! I’m not entirely thrilled with the answer I have right now, but hey—it works (kind of). Hopefully, in the near future, I'll come up with something that satisfies both Bjab and my restless curiosity. For now we have to be happy with the standard concept of the spin structure - representing the universe as a set of interlocked rotations - described below.

Spin structure

Where Do Spinors Live?

Imagine a strange and mysterious land—let’s call it V—a two-dimensional complex vector space. It’s not just any space, though. It has "structure." Picture it equipped with a hermitian non-degenerate scalar product that we’ll denote by $(u,v)$, and a bilinear antisymmetric non-degenerate form we’ll call $\epsilon$. Sounds fancy, right? These two should be related in a specific way, and here’s how:

There exists a special kind of basis for this space, which we’ll call “special orthonormal.” It satisfies:

1. $(e_1,e_1) = (e_2,e_2) = 1$,
2. $(e_1, e_2) = 0$,
3. $ϵ(e_1,e_2)=1.$

“Special” indeed. The first natural question is: Can this magical basis even exist?

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The Magic of C²: Relax, It’s All Possible

Yes, it exists! Here’s why. Let’s choose any basis in V. Using this, we can identify V with $C^2$ (our familiar two-dimensional complex space). In $C^2$, the scalar product $(u,v)$ is defined as:

$$(u,v)=u^{\mathrm{1\ast }}{v}^1+u^{\mathrm{2\ast }}{v}^2$$

where the star (*) represents complex conjugation. As for $\epsilon$, we define it as:

$$ϵ(u,v)=u^1{v}^2-u^2{v}^1$$

Now, using a standard basis $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, it’s easy to check that our special orthonormal basis conditions hold. So, crisis averted! We don’t need to worry about whether this basis exists—it does. Phew.

Oh, and here’s a fun fact: if $(e_1, e_2)$ is a special orthonormal basis, then $(-e_1, -e_2)$ is also a special orthonormal basis. Isn’t symmetry beautiful?

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Back to the Mysterious World of V

So, we’ve got our scalar product, our fancy $\epsilon$, and we’re cruising in V—home to the enigmatic spinors. But are there other “special orthonormal” bases? And if so, how many?

Let’s say $e'_1, e'_2$ is another basis. There exists some matrix $A$ such that:

$$e_i^{\mathrm{\prime }}=e_j{A}_i^j(i,j=1,2)$$

(Yes, that’s a summation over “j”.)

Now, here’s the neat part: $e'_1, e'_2$ will also be special orthonormal if and only if the matrix $A$ is unitary with determinant 1. Matrices like this form a group called SU(2).

By the way, SU(2) is isomorphic to the group of unit quaternions. I won’t dive into that now, but it’s a pretty cool connection between algebra and geometry.

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The Spinor Connection

Let’s denote by O(V) the space of all these special orthonormal bases. The group SU(2) acts on O(V) from the right, both transitively and freely. The elements of O(V) are what we call spinor bases, and the elements of V are simply spinors. If $u$ is a spinor, and you change the basis using an SU(2) matrix $A$, the components of the spinor change according to this formula:

$$u^{\mathrm{\prime }i}=A_j^i{u}^j$$

Got it? Good! Now, let’s connect this with something real, like the lab—where actual experiments happen (sometimes with coffee stains on the data sheets).

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What About Our Lab?

To bridge this to the physical world, we need some more algebra (don’t worry, just a little more). Quaternions are often used for handling 3D rotations, but since we’re sticking with matrices, here’s what we do in quantum mechanics:

For each point $x = (x^1, x^2, x^3)$ in $R^3$, associate the following 2x2 Hermitian matrix, $\sigma(x)$:

$$\sigma (\mathrm{<b>x</b>})=\left(\begin{array}{cc}{x}^3& x^1-i{x}^2\\ x^1+i{x}^2& -x^3\end{array}\right)$$

Now, for any SU(2) matrix $A$, it turns out:

$$A\sigma (\mathrm{<b>x</b>})A^{\ast }=\sigma (R(A)\mathrm{<b>x</b>})$$

where $R(A)$ is a 3x3 real rotation matrix, an element of the special orthogonal group SO(3). In fact, the map from $A\to R(A)is\; a\; homomorphism.\; And\; every\; element\; of\; SO(3)\; can\; be\; generated\; this\; way.\; The\; catch?\; If$$A$ and $B$ are two SU(2) elements, then $R(A) = R(B)$ if and only if $B = \pm A$.

Hence, SU(2) is a double cover of SO(3). It’s like the universe giving you two chances to rotate.

Note: With quaternions it would look more "natural". To each x we would associate a pure imaginary quaternion 

q(x) = xi+yj+zk, 

and for each unit quaternion u, we would have 

uq(x)u* = q(R(u)x), 

where R(u) is in SO(3). 

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The Spin Structure: So What About That Spin Axis?

Let’s bring this back to the real world—our lab, where we have Cartesian coordinates. Any two right-handed orthogonal systems (say, different setups in the same room) are related by a unique rotation matrix from SO(3). The set of all such frames forms a space O(L), which SO(3) acts on.

Now, here’s the kicker: nature somehow maps each element of O(V) (the spinor world) to an element of O(L) (our lab’s coordinate system) in a way that respects the rotations. We call this map the spin structure.

In other words, when you rotate an element of O(V) by an SU(2) matrix A, the corresponding lab element rotates, too, By and SO(3) matrix R(A). It’s like a cosmic interlock between spinors and real-world rotations.

Note (for experts): This is a baby version of a spin structure, good for a 3D flat space or, more generally, for a space with a distant parallelism. But this baby version is all that we need for now.

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What’s Next?

All this brings us closer to answering the question: what do I mean by a spin axis? Stay tuned, because we’ll tackle that mystery in the next post! We will also take under the loop the spin structure - the most important point here.

P.S. I am experimenting with the mathematical notation on this blog. Sometimes I mess up, then correct what can be corrected.

P.S. I have uploaded the pdf version of this post to academia.edu.

P.S. 18-10-24 9:33 Igor Bayak (1.1.6)