The Quirks of Quaternions

The Spark of Curiosity

This post is inspired by a fascinating conversation I had with Igor Bayak and Bjab in the comments of my previous blog. Their input sparked my curiosity to dive deep into quaternions and vector fields on a three-dimensional sphere. 

These concepts aren’t just abstract math—they could be quite handy for my spinor studies. Plus, there’s something aesthetically satisfying about painting these mathematical fields. And let’s be honest, who doesn’t love a little visual beauty in math?

So, here’s a look at the world of quaternions through my curious eyes.

---

Quaternions 101: Meet i, j, and k

Let’s start with our three quirky quaternion friends: i, j, and k. These are the building blocks of quaternions. A general quaternion, x, can be written like this:

$\mathrm{<b>x</b>}=x^1\mathrm{<b>i</b>}+x^2\mathrm{<b>j</b>}+x^3\mathrm{<b>k</b>}+x^4$

Simple enough, right? Now, it’s time to turn these quaternions into something a little more structured—a matrix representation. (Cue dramatic music.)

---

Quaternion Matrix Magic

We’re going to multiply x by i, j, and k from the left and see what matrices pop out. This is where the magic happens. Let’s start with i.

When we multiply i by x, we get:

$\mathrm{<b>i</b>}\mathrm{<b>x</b>}=-x^1+x^2\mathrm{<b>k</b>}-x^3\mathrm{<b>j</b>}+x^4\mathrm{<b>i</b>}$

Now, for the matrix interpretation:

• At i, we have x^4, which gives us the first row of the matrix: {0, 0, 0, 1}.
• At j, we have -x^3, giving the second row: {0, 0, -1, 0}.
• At k, we have x^2, producing the third row: {0, 1, 0, 0}.
• Finally, at unity, we get -x^1, completing the fourth row: {-1, 0, 0, 0}.

Putting it all together, the matrix that represents multiplication by i on the left is:

$L1 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$

Matrix magic! (Applause, please.)

---

Now Multiply by j and k

Using the same process, we get matrices for multiplication by j and k on the left:

For j, we get:

$L2 = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix}$

For k, we have:

$L3 = \begin{pmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix}$

---

What Happens on the Right Side?

Not to be left out (pun intended), we can also multiply quaternions from the right. When we do this, we get the following matrices:

For i on the right:

$R1 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$

For j:

$R2 = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix}$

For k:

$R3 = \begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix}$

So now we’ve got both the left and right multiplication matrices. Our quaternion friends are getting quite versatile.

---

Quaternions, SU(2), and a Three-Dimensional Sphere

Now things get even more interesting. Quaternions with a norm of 1 form a three-dimensional sphere. And not only that, but they form a group that is isomorphic to SU(2)—fancy math speak for “this group behaves like SU(2).”

To make this more concrete, consider the action of this group on the space of all quaternions. Let’s define an action u as:

$u: x \rightarrow ux$

We now have a representation of this group acting on functions, written as:

$(T(\mathrm{<b>u</b>})f)(\mathrm{<b>x</b>})=f({\mathrm{<b>u</b>}}^{-\mathrm{<b>1</b>}}\mathrm{<b>x</b>})$

Are you still with me? Good! Let’s move on.

---

Vector Fields: Expanding the Fun

Consider a one-parameter subgroup, say exp(ti). This subgroup generates a vector field, which we’ll call X1:

$X1(f)=\frac{d}{dt}f(exp(-t\mathrm{<b>i</b>})x){\mid }_{t=\mathrm{0<span\; style="font-size:\; large;"></span>}}$

When we calculate the action of X1 on the coordinate functions x^i, we get the components of X1:

$X^i (x) = - L1^i_j x^j$

In the same way, we can derive the vector fields X2 and X3. 

Here are the results:

The Takeaway

Quaternions may sound intimidating at first, but once you break them down, they’re not only manageable—they’re downright fascinating. Their matrix representations, their connection to SU(2), and the rich vector fields they generate all have deep implications in both math and physics. And who knew matrices could be so much fun?

So, the next time you’re pondering the mysteries of the universe (or trying to impress someone at a party), just casually drop some knowledge about quaternion vector fields. It’s sure to be a hit!