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.

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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).

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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).

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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.

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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.

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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.

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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!

P.S. 02-10-24 17:32 Reading the autobiography of Stefan Ossowiecki (in Polish). The following experiments really strikes me! What kind of new physics will explain it? Here is the English translation:

Experiment 56.

"The Psycho-Physical Society in Warsaw wanted to conduct a psychometric experiment under conditions that would completely eliminate the possibility of telepathy. The responsibility of recording the protocol was entrusted to Prof. Szmurło.

To carry out this type of experiment, the Committee of the Society decided to ask only one person to prepare the material for the experiment. However, even that person should not know what material would be given to Engineer Ossowiecki.

For this reason, we turned to someone who was not even a member of the Psycho-Physical Society, namely Mr. Marian Wawrzeniecki, a well-known painter who was also involved in archaeology and anthropology, and who did not know Mr. Ossowiecki. Mr. Wawrzeniecki was to place several objects in various small boxes of the same shape, with the same design on the surface of the box. Each box was to contain a slip of paper with a number. Sealed envelopes recorded the numbers corresponding to the objects, with a description of the objects. Mr. W. received four identical-sized boxes and a packet of cotton wool, provided by us.

On February 1, 1923, Mr. M. Wawrzeniecki sent back these four boxes, tied crosswise with a string sealed with his signet ring, and four envelopes also sealed and numbered: 1, 2, 3, 4. All these items were locked away.

We contacted Engineer Ossowiecki on February 28, asking him to set an evening for the experiment. On March 10, we received a response in which Engineer Ossowiecki informed us that he would be waiting at his apartment on March 12 at 8 p.m. Wawrzeniecki was not informed.

On the appointed day, we went to Engineer Ossowiecki, accompanied by our vice-president, Chief of the Sanitary Department of the Ministry of Internal Affairs, General Dr. J. Trzemeski. Apart from Mr. Ossowiecki, the following were present: the Latvian envoy, Mr. M. Nuksza with his wife, and Mr. Szmurło.

The engineer took one of the small boxes, randomly chosen, wrapped in several pieces of paper, tied, sealed, and sat on the sofa next to me. Everything Mr. Ossowiecki said was continually recorded by me.

‘This is a black box. Inside are about twelve photographic plates, one of them is broken. All of them were taken out by a slender lady who likes music. The box was taken from the vicinity of Chmielna Street in Warsaw, but it was made abroad, in Germany. On the cover of the box, there is a label in the Egyptian style. I see a factory, a workplace, many girls are working there. There are no plates in the box anymore. It contains a gray object made of fired clay, it is a fragment of something. It is ancient, hundreds of years old. It is a fragment of a prehistoric urn, broken. I can also see the urn, I will draw its shape.’ Mr. Ossowiecki described its shape. ‘They found it deep in the ground. I also see something else in the box. I don't understand what this box has to do with a pharmacy. I see where the cotton wool was bought, in a pharmacy on Marszałkowska Street, that lady bought it.’

The experiment lasted 50 minutes, and before the box was opened, Mr. Ossowiecki outlined the contours of this object with a pen, even drawing what was on it. We unwrapped the paper, seal, string, and to our amazement—there was the black box from which I had removed several photographic plates months ago from the Szalay company located on Chmielna Street in Warsaw. This box was given to me by a slender woman, a clerk in the warehouse. As I later found out, she plays the piano. The box contained 13 diapositive plates, measuring 8 ½ x 8 ½. The plates came from the Ernemann factory in Dresden, Germany. On the label, there was a female head—Isis—in the Egyptian style. In the box, we found a fragment of an urn with a drawing identical to the one made by Mr. Ossowiecki. The dimensions and shape were the same. The box was labeled number 2, so we took the envelope marked with the same number and read its contents:

‘A fragment of a prehistoric urn, found by Mr. Wawrzeniecki in 1904 in the vicinity of Warsaw.’

The cotton wool, in which the urn fragment was wrapped, was bought by a lady in a pharmacy on Marszałkowska Street.

We consider this experiment one of the most interesting and scientifically significant because it is purely psychometric and entirely negates the factor of telepathy. Mr. Ossowiecki spoke everything from the beginning, including when the box was made, where the plates were bought, who brought the cotton wool, and finally, described and drew the excavation fragment in the finest detail, identifying it as part of an urn. Neither I nor General Trzemeski could have imagined that Mr. Ossowiecki would conduct the experiment with such meticulousness. We assumed he would describe rather the object that was inside the sealed box. It was a great surprise for us. In this case, there can be no talk of suggestion. Mr. Ossowiecki's abilities are indeed universal.’

(Signed) P. Szmurło President of the Psycho-Physical Society (Signed) General Trzemeski Vice President of the P.-F. Society"

P.S. 03-10-24 14:06 James Webb telescope watches ancient supernova replay 3 times — and confirms something is seriously wrong in our understanding of the universe.