Arkadiusz Jadczyk - Open system

Friday, October 11, 2024

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.

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:

$x = x^{1} i + x^{2} j + x^{3} k + x^{4}$

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:

$i x = - x^{1} + x^{2} k - x^{3} j + x^{4} i$

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:

$(T (u) f) (x) = f (u^{- 1} x)$

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:

$X 1 (f) = \frac{d}{d t} f (e x p (- t i) x) ∣_{t = 0}$

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!

Sunday, October 6, 2024

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.

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

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.

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)

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.

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?

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!

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:

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.

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.

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.

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.

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?

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.

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!

Wednesday, October 2, 2024

Cracking the Spin Code: A Geometric Dance in Quantum Space

Algebra vs. Geometry: The Battle Begins

Today, we venture into the world of quantum spin, but with a twist. Instead of taking the usual algebraic approach, I’m opting for geometry. Why? Because pictures! Some will love this; others might feel it’s a betrayal to the cold, calculating nature of algebra. Algebra is safe—you follow rules, plug in numbers, and voilà, answers magically appear. Geometry, on the other hand, is risky. It invites you into a world of imagination, where you "see" things that may or may not exist as they seem.

But beware: geometry is seductive. It gives you the illusion of understanding. Algebra is like a loyal friend—predictable, reliable. Geometry? It's the mischievous artist in the room, painting pictures that may stir your imagination but might also lead you down the wrong path. Ideally, we want to befriend both—embrace the devilish precision of algebra and the angelic visuals of geometry.

The devilish precision of algebra and the angelic visuals of geometry.

Under the Hood of Spin: A Sneak Peek

So what exactly is spin? Let’s roll up our sleeves and take a peek inside the quantum gearbox. We’re not going for the full reality show just yet—think of this as the "behind the scenes" tour. Reality? It’s still up for debate. We’ve got theories, we’ve got speculations, but no one’s seen the full episode yet.

Before we dive into the wild, speculative frontier, let’s stick to the script for now. I’m going to explain spin mechanics in a conventional way, but with a sprinkle of fun (because quantum mechanics shouldn’t always feel like quantum suffering).

Spin State vs. State Vector: What’s the Difference?

Previously, we talked about the difference between the “spin state” and the “state vector.” Quick recap: the spin state is something we can actually experiment with—like the direction of spin for a particle. Easy, right? For spin ½, it’s just a direction in space, much like right ascension and declination for a planet position in the sky.

But the state vector is trickier. It holds some extra secret sauce—something called the “internal phase,” which we can’t see. You can represent a spin state by giving it two coordinates, like latitude and longitude on a sphere. But to represent the state vector? You need a third coordinate. That’s where things get juicy (and geometrically delightful).

Welcome to Spin Space: A Gearbox Tour

Imagine you're in a lab with a globe. You’ve got latitude, longitude, all nicely lined up. Now, spin (let’s say of an electron) lives in its own quirky little internal space. Think of it like a hidden compartment with gears that connect to the real world. I’m going to break down these gears and show how they lock into place with our space.

In this internal spin space, we’ll set up some coordinate axes. It’s just like our 3D space, but with a twist. The north-south axis here is like a parallel universe version of our own. Inside this space lives the spinor, which is basically the state vector’s alter ego.

Imagining the Spinor: Two Points, One Mystery

Let’s get creative! How do we imagine a spinor? Picture this: instead of just one point on a sphere, imagine a pair of points—one red, one blue—both chilling on the same parallel in our internal spin space. (Yes, I’m color-coding the quantum world for fun!)

Imagine looking down at these points from above. What do you see? The red and blue points are not static—they can spin around.

But the secret is, this rotation is invisible to us unless the points rotate relative to each other. Their orientation in the internal space is the elusive "phase" that we can’t measure directly. It's like knowing the world is spinning but not feeling dizzy—until the ground shifts unexpectedly.

Latitude, Longitude, and the Spin Connection

So how does this hidden spin machinery connect with the familiar latitudes and longitudes we know? Simple! The common latitude of the red and blue points tells us the spin direction in our space. Meanwhile, the difference in longitude between the two points determines the spin’s longitude in the lab. So the picture below represents the same spin state as the picture above.

What’s Next?

In the next post, we’ll throw in some math—specifically, sines and cosines—to tie these lovely images back to textbook formulas. But for now, bask in the simplicity of geometric intuition. Even if it’s not the full story, it’s a colorful and entertaining glimpse into the quantum world.

Until next time, keep spinning!

In Search of the Third Factor: Reflections on Science, Philosophy, and Purpose

Returning to Old Notes

Every so often, I dive back into my notes from years gone by. It’s a way to ensure I haven’t strayed too far from the path I once set for myself. Although only a handful of people seem interested in these private reflections, a handful is not the same as no one at all. Occasionally, a message from a curious mind lands in my inbox, reminding me that these thoughts resonate beyond my own head.

Yesterday, I felt compelled to revisit an entry from October 1993. Below are a few snippets from those musings, still as relevant to me now as they were then.

October 1993: The World of Information

“The beginning of the world and the Big Bang. Should I believe this? Should you? Where is the flaw in the narrative? What’s missing, misunderstood, or understated?

Here’s my take (borrowing from Popper): beyond the world of matter and the realm of geometry, there exists a world of information. We don’t fully understand this world yet, but we sense it—just the tip of an iceberg. New, bold ideas are needed, new mathematical structures to chart this realm. Today, we are only scratching the surface.

This third world—the world of information—must integrate with the other two. It’s not some distant future; this new paradigm is already here, hanging in the air like a ghost. And this is where I must focus: full steam ahead toward information, toward algorithms, toward the structure tree. This has become the content of my life, my calling.

This has become the content of my life, my calling.

Who knows what I’ll discover along the way? Searching for information, I might stumble upon something even greater. But nothing is more important than this pursuit. Information is life. Information is complexity."

On Tactics and Purpose

“But strategy is not enough. Tactics matter, too. I embraced the audacity of youth. Be bold. Be rowdy. Life demands action, constant action. And with that, comes the realization: I must shed responsibilities that don’t serve this purpose. I am a researcher, not a teacher. A scholar, not an administrator. And certainly not a director. My task is not to manage or organize but to build knowledge, to create something unknown that will one day be known. That is my holy calling. So help me, God.”

April 1994: The Anthropic Principle and Life’s Questions

A year later, in April 1994, I was grappling with the anthropic principle:

"Does a theory make sense when it leaves such a narrow window for the emergence of life? Can something as organized as life arise from chaos? The anthropic principle provides no real answers. It's no explanation to say that the universe must be the way it is because otherwise, we wouldn’t exist to observe it.

Where did the laws of nature come from? Why these laws and not others? The argument that ‘it couldn't be any other way’ falls flat. What is explanation anyway?

And what is chance?"

A Personal Shift: Technology and Self-Reflection

"On a more mundane note, I received a grant. It was just enough to buy a notebook. Will life be easier with a notebook? A little. But is that enough to change everything? Hardly."

Navigating Blindness: The Search for Free Will

The challenge looms: how can the blind lead the blind? How do we reach a generator of free will without the freedom to reach it? Mechanically, I can only detect what is mechanical, and even that requires tremendous effort. Yet here I stand, the best possible laboratory at my disposal: myself. I am both the experimenter and the most sensitive instrument.

My goal has become clear. I must produce a device—a mind—that won’t need external forces to guide it. But is that what I want? I want to take stock of everything I know and push further. I want to gather the fragments of my understanding and draw meaningful conclusions.

Years ago, I didn’t know what I was searching for as a physicist. I was drawn to the idea of additional dimensions, but for reasons that remain unclear, I got sidetracked. I lost more than a year in uncertainty until I encountered the works of Sheldrake, Popper, Eccles, and Jeans. Slowly, the fog lifted, and my mission crystallized: to discover a third factor that exists beyond matter and geometry.

This third factor is information—knowledge. My purpose now is to bridge the gap between modern physics and this new reservoir of understanding. The challenge lies in finding a safe path forward, one that doesn’t fall into the murky waters of philosophy. It’s clear to me now that quantum theory is the key.

Today: The Premonition Endures

Here I am, years later, still chasing that same premonition I had in the '90s. Consciousness, I believe, cannot change the facts themselves, but it can influence their probabilities. How? That’s where the devil hides—in the details. And that’s what I’m still working on.

That’s where the devil hides—in the details.

The journey continues.

This mix of self-reflection, science, and philosophy still defines my work today. The questions I grappled with decades ago remain as crucial and unsolved as ever, but the pursuit is what drives me forward—full steam ahead.

Full steam ahead

P.S. 29-09-24 11:42 This is a partial answer to Igor Bayak asking in his comment what do I mean by "information"? Reading an autobiography of Stefan Osswiecki:

"Stefan Ossowiecki (1877-1944), a businessman by profession, was also a psychic whose clairvoyant abilities made him famous in his native Poland. He became known internationally as a result of numerous successful experiments with Polish, French and British psychical researchers."

Here is an illustration of his psychic abilities taken from the CIA Reading Room document (released in 1977) "Israeli Secret Services":

In his 1944 autobiography "The World of My Soul and Visions of the Future" (written in Polish), Ossowiecki wrote these prophetic words:

"Today's knowledge, based on intellectualism and experimentation, has been profaned as it has become available to both heroes and criminals alike. Not everything should be shared with everyone. Modern man has forgotten the warning of the greatest Sage: “Do not cast your pearls before swine, lest they turn on you and trample them underfoot, and then attack you.” Who knows whether this profanation of nature’s mysteries, this brutal intrusion of modern technology into the subtlest vibrations of matter, this apparent triumph of the intellect, will one day mark the beginning of an unprecedented catastrophe in the history of the modern world?

The ancient Atlanteans unleashed the forces of the elements, and, unable to control them, perished. We today are no different. We are on the brink of collapse, for our civilization has also violently intruded into nature’s secrets without sufficiently cultivating morality in heart and spirit. Today, while flying through the skies and plumbing the ocean’s depths, the people of the 20th century consider themselves victors over nature. But in reality, they are spiritually bankrupt, selfish in their bloody struggle for existence, blinded by their own arrogance, cold and empty-hearted. Their airplanes, submarines, radios, and wireless telegraphs—tools that could be the glory of life if driven by a noble will—may soon become a curse. When hatred and greed unleash yet another war, all of this technology will become instruments of death, not life"

P.S. 01-10-24 11:57 While working on writing a report on a paper on photon's position operator for a physics journal, I decided to check a Referee Report of my own paper on Clifford algebras, that after many changes have been finally happily published. I would like to quote a particular piece from this last report, as it makes me smile. The Referee writes:

"... As a mathematician, I do not at all agree with the Author's point of view. and I am not at all ready to consider the tensor algebra as "the mother" of the other algebras. But as a reviewer, I observe that the Author supports his thesis with interesting arguments (in other words, it is not just propaganda), and I wish his paper to become a useful contribution to the debate."

Friday, September 27, 2024

Decoding the Spin of Electrons: A Beginner’s Guide to Quantum Mechanics

What Exactly is Spin?

Let's talk about spin. When we think of an electron, proton, or other elementary particles, we often imagine them somehow “spinning.” But what exactly is spinning, and how it works – we don't know for sure.

In classical physics, spinning objects have something called angular momentum. The faster they spin, the greater the angular momentum. Similarly, the heavier the object, the greater its angular momentum. Electrons and protons have something like "intrinsic angular momentum", which we call "spin". However, the value of this spin isn't just any number – it takes discrete values, multiples of half the Planck constant.

I consider it as quite possible that if we can one day fully understand what spin is, we might unlock the entire mystery of quantum mechanics.

It is quite possible that if we can one day fully understand what spin is, we might unlock the entire mystery of quantum mechanics.

For now, though, we have to be content with its mathematical description – which, unfortunately, is not quite the same as understanding the essence of the phenomenon.

Understanding Electron Spin

So, let’s dive into the mathematical description of electron spin, and I’ll try to make it a bit more accessible. An electron's spin is equal to half of a Planck constant. This is why we say the electron has a spin of ½.

In experiments, we can align the electron's spin axis in a specific direction, for example, upwards along the z-axis. This alignment defines the spin state, but it doesn’t fully describe the state vector. In quantum mechanics, we make a distinction between states and state vectors.

State: What we observe.
State Vector: Information that includes both what we see and what is invisible, yet still necessary.

A Model for Spin: Visible and Invisible Wheels

Can we visualize this? Maybe. But let’s remember, models can be misleading. What I propose is simply a mental tool – it might help, but it could also lead us astray.

Imagine an electron as a blue cog with a visible mark. The position of this mark in relation to an external coordinate system represents the electron's state – what we can directly control. However, alongside this visible cog, there’s an invisible gray cog, also marked. This cog is hidden from our view but plays a crucial role.

The Internal Phase

To fully describe the state vector of the electron, we not only need to know the position of the visible mark, but also the angle it makes with the mark on the invisible cog. Let’s call this angle the internal phase.

Knowing the state (the position of the visible mark) is important.
But we also need to know the internal phase, the relationship between the visible and invisible marks.

Now, let’s make this more interesting.

A 720-Degree Rotation

Imagine the gray, invisible cog is twice the size of the blue, visible one. If you rotate the blue cog 360 degrees, the gray cog only rotates by 180 degrees. To return both cogs to their original alignment, you would need to rotate the blue cog a full 720 degrees.

Think of it like this:

A More Detailed Model

For those of you following closely, I need to add a bit of complexity. My earlier analogy of two cogs is a bit too simple. Ideally, I should color the gray cog (instead of keeping it plain) to emphasize that the internal phase is relative and subjective. One person might define the "zero" phase when the marks align; another might choose red or green as the reference point. The key idea is that it takes a full 720-degree rotation of the visible cog for both to return to their original states.

Exploring Other Models

This model is just one of many. There are also examples in the literature involving cubes tied together with strings, or twisted strips resembling Möbius bands. However, I prefer my cog analogy – it’s simple and relatable. But again, it's a rough analogy. There’s something happening with the topology of space within the electron itself. It’s as if the electron “screws” itself into space when we rotate it.

Think of it like this: part of the electron exists in our space, while another part is in a sort of “anti-space,” where time flows in the opposite direction, some topologically twisted Einstein-Rosen bridge. When you rotate the electron by 360 degrees, part of it moves into anti-space, and vice versa. To return everything to the initial state, you need to rotate it another 360 degrees. It’s fascinating, but we’re far from fully understanding this yet.

The Spin State on a Sphere

Now that we've explored the analogy, let’s focus on the mathematical representation. We can describe the electron (the “visible” part) as a point on a unit radius sphere in three-dimensional space (x, y, z). This point indicates the spin direction. It can be described by using:

Latitude and longitude, or
Cartesian coordinates (nx, ny, nz) of a vector n of length 1.

Here’s the basic relationship:

nx²+ny²+nz²= 1

The conversion between spherical and Cartesian coordinates follows these formulas:

$n x = \sin (ϕ) \cos (θ)$
$n y = \sin (ϕ) \sin (θ)$
$n z = \cos (ϕ)$

In this model, ϕ (latitude) ranges from 0 to Pi, andθ (longitude) ranges from 0 to 2 Pi. At the poles (where phi = 0 or Pi), theta is undefined, but we often just assign it a value of zero for convenience.

Wrapping It Up

To summarize: the spin state (the visible part) is a point on the sphere that indicates the direction of the electron’s spin axis. Think of this spin axis as an arrow rather than a simple straight line.

In future posts, we’ll dive deeper into the concept of the state vector, its mathematical representation, and its relationship to the spin state. We’ll also explore how to project from four-dimensional space to three-dimensional space, allowing us to visualize the invisible internal phase, which, while not directly observable, plays a vital role in understanding the electron’s behavior.

What's Next?

In the upcoming posts, expect more formulas and visual aids as we continue unraveling the mysteries of quantum mechanics.

Will we discover new insights? Only time will tell. But one thing is for sure – the journey is just beginning.