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.

The Spin Chronicles: Painting Quantum Tori (Part 2)

 The Mysterious Phase of Quantum Mechanics: A Spin on Reality

In quantum mechanics, the phase of a wave function is treated like that distant cousin you only hear about but never meet—it's assumed to be unobservable. The star of the show is the amplitude, or more precisely, the square of its modulus. That’s the part we can see and measure. Textbooks say: "Only this, and nothing else, is observable." And sure, that’s one way of looking at it.

But let's face it—opinions in quantum mechanics are as varied as the stars in the sky. Physicists are like travelers in a big, wide world, each with their own unique map of how quantum mechanics works. In fact, no two physicists share exactly the same view—unless, of course, one of them is just crunching numbers from a cookbook of equations. And even then, not all cookbooks are created equal!

The Wave Function and the Art of Spin

We often talk about the "wave function" when we’re trying to pinpoint the location of an object. You know, that quantum object that seems to be everywhere and nowhere at the same time? But today, let's zoom in on something a little more grounded: spin.

Now, spin is a different beast. The object in question stays put, but its spin axis can change direction. This is where we enter the world of the state vector. For spin-½ particles, this vector is made up of four real numbers: $X$, $Y$, $Z$, and $W$. Their squares add up to one, which is pretty neat. If you’re a fan of complex numbers (and let’s be real, who isn’t?), it’s just two complex numbers whose modulus squares also sum to one. The conversion between the two is surprisingly simple:

$$z_1=X+iY\text{and}{z}_2=Z+iW$$

This lets us break the real part and imaginary part down like so:

$$X=\text{Re}(z_1),Y=\text{Im}(z_1),Z=\text{Re}(z_2),W=\text{Im}(z_2)$$

Easy, right? Well, it gets more interesting.

From Numbers to Angles: Theta, Phi, and Psi

Instead of juggling numbers like $X$, $Y$, $Z$, and $W$, it’s often more intuitive to use angles—

θ, ϕ, and ψ. 

These angles help specify the position of our vector on a three-dimensional sphere that lives in four-dimensional space. So, the coordinates transform into:

$$X = \sin\left(\frac{\phi}{2}\right)\cos(\psi), \quad Y = \sin\left(\frac{\phi}{2}\right)\sin(\psi)$$
$$Z = \cos\left(\frac{\phi}{2}\right)\cos(\psi + \theta), \quad W = \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)$$

The Projection into 3D Space

At this point, we need to bring things back into the three-dimensional world we're used to. To do that, we use something called stereographic projection, which I’ve covered in detail a couple of blog posts ago. This allows us to map the points from four-dimensional space into our three-dimensional world. The result gives us familiar coordinates:

$$x(\theta, \phi, \psi) = \frac{\sin\left(\frac{\phi}{2}\right)\cos(\psi)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$ $$y(\theta, \phi, \psi) = \frac{\sin\left(\frac{\phi}{2}\right)\sin(\psi)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$ $$z(\theta, \phi, \psi) = \frac{\cos\left(\frac{\phi}{2}\right)\cos(\psi + \theta)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$

What's the Deal with Theta, Phi, and Psi?

So what’s the physical meaning behind these angles? In our lab, the $z$-axis points upwards (think of it as "North" on a globe), and the ϕ angle measures the tilt of the spin axis—essentially, it's latitude. θ is the longitude, with the x-axis as 0 degrees and the y-axis at 90 degrees (or π/2 if you’re feeling mathematically fancy).

But the ψ angle? That’s where things get interesting. Psi is like the "invisible phase," a ghostly presence that Richard Feynman himself couldn't ignore when he wrote about spin. And today, we’re going to shine a light on this elusive angle.

Painting the Invisible: Great Circles and Wheels

Our mission here is to bring these invisible aspects of the state vector to life. As we vary psi from 0 to $2\pi$, something magical happens—a closed curve appears. This curve, on a three-dimensional sphere in four-dimensional space, is known as a "great circle."

You’ve seen great circles before—they’re the big loops you get when slicing through a globe. Meridians and the equator are great circles. The same idea applies here, except now we’re in four dimensions. These invisible great circles can be projected back into our familiar 3D space as... wait for it... circles! That’s why this blog post is titled Invisible Wheels.

A Torus of Invisible Wheels

Now, imagine a large torus made of these invisible circles—these are called Villarceau circles, similar to what you’d find in the architecture of Strasbourg Cathedral. But here’s the catch: each of these circles is perceived as just a single point—a spin pointing in a particular direction ($theta, phi$). The psi phase, which determines where we are on that circle, remains hidden from us.

Want to see what these circles look like? I whipped up some visualizations in Mathematica, and here’s what I got: 

X[theta_, phi_, psi_] = Sin[phi/2]*Cos[psi];

Y[theta_, phi_, psi_] = Sin[phi/2]*Sin[psi];

Z[theta_, phi_, psi_] = Cos[phi/2]*Cos[psi + theta];

W[theta_, phi_, psi_] = Cos[phi/2]*Sin[psi + theta];

ParametricPlot3D[Table[

  {X[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p]), 

   Y[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p]), 

   Z[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p])}, {i, 0, 35}], {p, 0,

   2 Pi}, PlotRange -> All, PlotStyle -> White, Background -> Black]

Still not satisfied? I’m currently experimenting with MathMod to generate even better images, but for now, this will have to do.

Wrapping Up

In the next post, we’ll explore how these tori and invisible wheels interact with each other. We’ll dive even deeper into their physical interpretation, so stay tuned!

For those of you curious about 4D visualizations, check out Dimensions Math. Search for "Hopf fibration"—it’s the key to everything we’ve discussed, even though I haven’t officially mentioned it... yet.

Until next time, keep spinning those wheels—visible or not!

The Spin Chronicles: Painting Quantum Tori (Part 1)

 Introduction: Spin, Tori, and Confusion Ahead!

Welcome, dear Reader, to another journey through the wild and wonderful world of quantum mechanics! Today, we dive headfirst into the realm of spin state vectors, grappling with a particularly beautiful concept: the spin tori. I promise we'll have some fun along the way, though I must issue a warning—things may get... twisty. 

Things may get... twisty

Especially when I casually decide to switch conventions that everyone else uses. Just a heads up!

A Quick Note on My Notation Shenanigans

Before we proceed, a quick confession. When I mention spherical coordinates, by φ, I mean latitude (measured from the North Pole), and by θ, I mean longitude. Now, the rest of the world? Well, they decided to do it the other way around! So if you're comparing my formulas to the ones in textbooks or online, remember that my φ is their θ and vice versa. Confused yet? Good. That's part of the charm.

Real Numbers vs. Complex—The Programmer's Dilemma

To keep things programmer-friendly, I’m sticking with real numbers, sines, and cosines. However, if you've cracked open a textbook or searched the internet, you've probably encountered those pesky complex numbers, accompanied by exp(iφ). While it's easy to translate between the two systems, the process can feel like translating Shakespeare into a meme—you're bound to lose some of the elegance along the way. So, I'll stick to the real stuff here and leave the complex translations as homework for the brave souls among you.

Spin Vectors: Real Numbers Edition

Now, let's get down to business. For us, the state vector is a column of four real numbers: X, Y, Z, and W, where the sum of their squares equals one. Textbooks, however, use two complex numbers a and b:

• a = X + iY
• b = Z + iW

It’s all the same thing, really—just more exciting when you throw in some complex numbers.

The Iconic State Vector |1)

Enter the state vector |1). Textbooks love this notation, but I’ll stick with using ) instead of the sharp "ket" symbol to avoid HTML shenanigans. It’s just a column with a 1 at the top and 0 at the bottom. For us, it looks like X=1, Y=Z=W=0. In quantum mechanics, this corresponds to a spin pointing along the z-axis. Pretty straightforward, right? Just don’t ask me to visualize it before my morning coffee.

Where the Math Gets Funky: Probabilities and Scalar Products

Now for the juicy part: if |u) is any state vector, the square of the scalar product |(1|u)|² gives us the probability that, when we measure the z-axis spin, we’ll get the result h/2 (the Planck constant divided by two). If u is our vector (X, Y, Z, W), and we write it as (X + iY, Z + iW), then the dot product of (1, 0) with this vector is X + iY. The square of the magnitude of this number is just X² + Y².

Angles, Angles, and More Angles

Let’s bring in some angles! Remember, I’m using φ, θ, and ψ. Here’s how they fit in:

• X = cos(φ/2)cos(ψ)
• Y = cos(φ/2)sin(ψ)
• Z = sin(φ/2)cos(ψ + θ)
• W = sin(φ/2)sin(ψ + θ)

Thus, X² + Y² = cos²(φ/2). And voilà! The probability that the spin is pointing up along the z-axis (with value h/2) is cos²(φ/2). And there you have it, folks—our φ angle now has physical meaning! We can even say, in true quantum mechanic fashion, that the "probability of transition" from state |u) to state |1) is cos²(φ/2). 

A 3D Quest: Visualizing the State Vectors

Now let’s level up. Suppose we want to visualize, in 3D, the set of state vectors where the probability of transition to the |1) state is 1/2. Easy enough: just take φ = π/2, since cos(π/4) = 1/√2, and cos²(π/4) = 1/2.

Let’s also recall our stereographic projection formulas:

• x = X/(1 - W)
• y = Y/(1 - W)
• z = Z/(1 - W)

Plug these into the expressions for X, Y, Z, and W, using a fixed φ, and we get some fancy formulas that lead to a very special surface. And guess what? That surface happens to be a torus! (Actually, it’s three nested tori, because one is never enough.)

Let’s Draw Some Tori (With MathMod)

Now, if you’re like me, staring at all these formulas will eventually lead to some existential questions, like “Why am I doing this?” But don't worry—I’ve got your back. To visualize these tori, I suggest you download and install MathMod. It’s free, it’s multiplatform, and, well, it works once you wrestle it into submission. I even wrote a script for you! Just save it as a .js file,copy and paste into Script Edit window,  run it, and let MathMod do the heavy lifting. With the mouse you can move the tori around.

Here is my script (you can also download it from here):

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.

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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.)

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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.)

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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:

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

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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.

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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.

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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!

Navigating the Quantum Maze: Spin State Vectors and the Magic of Projection

Welcome to the Spin Vector Universe

Today, we embark on a cosmic adventure, learning how to navigate through a space where the points are none other than spin ½ state vectors. These vectors, much like secret agents, carry both the “observable information” that we can detect and a hefty dose of hidden data that remains mysterious—at least for now. Who knows what tomorrow’s physicists will uncover about these enigmatic vectors? Perhaps one day, even consciousness itself (and even  "psi" phenomena) will be written into the math.

These vectors, much like secret agents, carry both the “observable information”...

That’s right, physicists may finally crack the "consciousness code," but until then, let's stick to the basics—like juggling four numbers in a multidimensional space.

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The Joy of Four Numbers

We initially presented the state vector as a pair of points on a two-dimensional sphere. But why settle for simplicity when we can dive into complexity? We now represent it by four real numbers: X, Y, Z, and W. And if you’re keeping score, the squares of these four beauties add up to 1:

X² + Y² + Z² + W² = 1

But wait, there's more! We can rewrite these numbers using angular coordinates φ, θ, and ψ. Let’s get fancy:

• X = sin(φ/2) cos(ψ)
• Y = sin(φ/2) sin(ψ)
• Z = cos(φ/2) cos(ψ+θ)
• W = cos(φ/2) sin(ψ+θ)

The ranges of variation of these angles are:

•  φ from 0 to π. (latitude)
• θ, ψ from 0 to 2 π.

 Cue the applause from the programmers in the room! Of these three angles, only θ and φ are observable in our space; ψ remains the mysterious guest who refuses to show their face.

Note: In quantum mechanics we usually present the spin vector as a column of two complex numbers, the "Pauli spinor":

X + iY

Z + iW

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A Tour of Spherical Spaces

To keep things simple (or at least as simple as quantum mechanics gets), here’s a quick refresher:

• X² + Y² = 1: Points on the plane (X, Y) lie on a one-dimensional circle.
• X² + Y² + Z² = 1: Points in three-dimensional space lie on a two-dimensional sphere.
• X² + Y² + Z² + W² = 1: Points with coordinates (X, Y, Z, W) lie on a… wait for it… three-dimensional sphere embedded in four-dimensional space.

Wrap your head around that one! But don’t worry, mathematicians have our backs. Using stereographic projection, they can help us visualize these four-dimensional wonders in our humble three-dimensional world.

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Stereographic Projections: Cartography for the Quantum Age

Remember those flat maps of the Earth that distort everything around the poles? That’s stereographic projection in action. We can apply the same trick to our three-dimensional sphere, squishing it down into a more manageable three-dimensional space. Here’s the cartographer’s recipe:

• If X, Y, Z are the coordinates on a 2-sphere, the projection onto a plane gives:
  • x = X / (1 - Z)
  • y = Y / (1 - Z)

To up the ante for our four-dimensional friends:

• For X, Y, Z, W, the projection into 3D space becomes:
  • x = X / (1 - W)
  • y = Y / (1 - W)
  • z = Z / (1 - W)

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What Do We Gain? What Do We Lose?

Through stereographic projection, we lose just one point: the North Pole, or in this case, the point where W = 1. When W = 1, X, Y, and Z all vanish into thin air, leaving us with a single state vector that escapes to infinity. But don’t worry, we’ll survive this minor loss.

On the plus side, we get to visualize the structure of spin state vectors in all their three-dimensional glory, and the angles between intersecting lines remain preserved. Cartographers everywhere are cheering in solidarity.

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The Aesthetics of Spin Space

Is it worth painting a picture of the spin state vector space? Absolutely! 

Both for its educational value and sheer aesthetic beauty. We can craft these images with relative ease (shout-out to the programmers) and even delve into the lovely world of Villarceau circles, which have graced the stairs of Strasbourg Cathedral for centuries. Talk about a deep cut!

In our next post, we’ll dive into how to create such images and translate them into physicist-friendly terms. For now, imagine a torus—a mathematical donut—and the intricate circles that define its geometry. Doesn’t quantum physics just make you hungry?

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PS: For Your Viewing Pleasure

If you’re curious about more stunning images and videos (probably better than the ones I can whip up), check out the “Dimensions” website: Dimensions. Just a heads-up, though—they skip the spin physics. You’ll have to come back here for that!

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And there you have it—our journey through the quirky, brain-bending world of spin state vectors. Let’s call it the appetizer for the main course, which will be served in the coming posts. Until then, keep your angles sharp and your vectors spinning! 

Untangling the Mysteries of Spinors: A Wild Ride Through 4D Geometry (Who Said Math Wasn't Fun?)

Minimizing the Mystery (Kind of)

In my spinor notes, I started with the humble attempt to strip away some of the mystique. You see, a spinor is just a pair of points in "internal space." Nothing too spooky, right? Well, actually, it kind of is spooky—but more on that later. 

Well, actually, it kind of is spooky—but more on that later. 

First, let's take a closer look at this pair of points. I even drew them in a previous note (pretty handy, right?).

Now, let's get serious—or as serious as one can get when doodling 4D geometry. Today, we'll draw these points more precisely, name a few angles, and write out some formulas that will make your head spin (pun intended). To this end let us define vectors a and b ending at the respective points as follows: 

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View from Above: The Pair of Points

Looking from above, our pair of points looks like this:

Fig. 1

And looking from the side:

Fig.2

Feel free to click and enlarge the image, though honestly, it's not that exciting—unless you're an angle enthusiast (no judgment).

In Figure 1, I marked the angle ψ between the red point (the one hanging out closer to the South Pole) and the X-axis of our inner space. The angle θ is the one between the red and blue points, which is kind of a big deal because it's what shows up in the outer space as the longitude of the spin state. Fancy, right?

In Figure 2, I’ve marked φ, which is the latitude shared by both points. We’re measuring it from the North Pole, because unlike geography, where latitude is measured from the equator, in math we like things to be... a bit more north-centric. Makes things more convenient—if you're into convenience, that is.

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The Dance of Phi and Psi: Trigonometry Comes to Play

So here’s a cool tidbit: trigonometry tells us (thank you, middle angle theorem) that point "a" makes an angle of φ/2 with the vertical axis. I decided to make life a bit easier by setting the radius of my sphere to ½. That means the diameter is 1, and thanks to our old buddy Pythagoras, we get the equation a² + b² = 1.

This angle φ also makes its way into the outer space, representing the latitude of the spin state. So what we're really dealing with here is geometry pretending to be physics.

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Circles, Vectors, and... Math?

At the top and bottom of the sphere, I've drawn two circles and two vectors (you know, to keep things interesting). The vector at the bottom has a phase ψ and a length a. Meanwhile, the vector at the top is just trying to be fancy with its phase ψ+ θ and length b.

When you superimpose these two planes—like the mathematical equivalent of stacking pancakes—you get this:

Fig. 3

So, we have two circles, one with radius a and one with radius b. The important part? a² + b² = 1. That’s it! You’ve got yourself a spin state vector—a spinor, if you will.

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The Coordinates: Let’s Break It Down

Let’s describe this using coordinates. For the vector a, let's use X and Y for its plane coordinates. For b, we'll use Z and W. Simple enough, right?

So:

• a = (X, Y)
• b = (Z, W)

But here's the catch: remember, these vectors are hanging out on two planes—one at the top of the sphere and one at the bottom. They’re superimposed, so you’re seeing them both at once.

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Time for Some Equations

From Figure 3, we can deduce:

• X = a cos(ψ)
• Y = a sin(ψ)
• Z = b cos(ψ + θ)
• W = b sin(ψ + θ)

And from Figure 2, we know:

• a = cos(φ/2)
• b = sin(φ/2)

Bringing it all together, we get:

• X = cos(φ/2) cos(ψ)
• Y = cos(φ/2) sin(ψ)
• Z = sin(φ/2) cos(ψ + θ)
• W = sin(φ/2) sin(ψ + θ)

And voila! X² + Y² + Z² + W² = 1. That’s your whole spinor, folks—four numbers whose squares add up to one. The first pair, (X, Y), is vector a, and the second pair, (Z, W), is vector b. Easy, right?

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The Fourth Dimension: Not As Scary As It Sounds

At this point, the mathematically inclined among you might recognize that X² + Y² + Z² + W² = 1 is the equation of a three-dimensional sphere in four-dimensional space! Don’t freak out about the extra dimension. If it makes you feel better, just think of it as two plane vectors, a and b, whose lengths always add up to 1. Phew, problem solved!

If you’re still feeling brave, you can even project this 4D sphere onto 3D space using something called a stereographic projection, which basically compresses it into something we can visualize. We’ll dive into that graphic wizardry in the next note.

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The Algebraic Approach: Complex Numbers to the Rescue

But wait, there’s more! We can also turn our vectors into two complex numbers. Here’s how:

• a₊ = X + iY
• a_- = Z + iW

And just like that, we jump straight into the algebraic description used in quantum mechanics. Don’t worry, we’ll cover this in more depth in future notes, where quantum math takes over and everything gets extra... well, quantum-y.

Stay tuned!