The Spin Chronicles (Part 20):Group action on the sphere and the spin idea

 In Part 18 we have met the Klein absolute, which, for the case of the 2D (x,y) plane, happens to be the null cone (ζ₀)²- (ζ₁)² - (ζ₂)² - (ζ₃)² = 0 in the 4D Minkowski space. As it is usual in projective geometry we remove the origin ζ=0 from this null cone - we do not need it. 

Projective geometry is about light rays

Let us recall that we have added two extra dimensions x₀ and x₃. We added x₃ for the stereographic projection, and then we added x₀ to get rid of denominators and, instead of doing the stereographic projection we employed the projective space.

Our null cone  ζ² = 0 is invariant under the action ζ⟼g ζ τ(g) of the Clifford group G of the Clifford algebra Cl(V). The action is linear. We already know that g ζ τ(g) = Λ(g)ζ, where Λ(g) is a Lorentz transformation. We have calculated explicitly these transformations for typical one-parameter subgroups of G and obtain either rotations or Lorentz boosts. Now, since the action is linear, it defines the action on equivalence classes of the equivalence relation "∼" defining the projection (see Part 18). If ζ=λη, λ>0, then Λ(g)ζ = λΛ(g)η. Therefore g acts on the equivalence classes of "∼". Let us take a closer look at the structure of the set of these equivalence classes. What do we get? We rewrite ζ² = 0 as

(ζ₁)² + (ζ₂)² + (ζ₃)² = (ζ₀)².

Since we have removed the origin from the null cone, ζ₀ must be non zero.

Exercise 1. Why?

Thus either ζ₀ > 0, or ζ₀ < 0. We choose λ = 1/|ζ₀| and obtain a unique point in the equivalence class of
ζ with the zero coordinate equal +1 or -1. Let us concentrate on the case +1. We obtain the equation
(ζ₁)² + (ζ₂)² + (ζ₃)² =1 for this unique point. It represents a point on the two-dimensional sphere. It stereographically projects onto a point in (x,y) plane except for the South Pole of the sphere with ζ₃ = -1.

Let us return to our embedding formula from Part 18.

After some thinking, and prompted by Bjab's observation I have changed the embedding there to get rid of unnecessary minuses. It became

ζ₀ = (1+x·x)/2,

ζ₁ = x₁,

ζ₂ = x₂,

ζ₃ = (1-x·x)/2.

Now ζ₀ + ζ₃ = 1. If we have any point ζ' on the cone with ζ'₀ + ζ'₃ > 0, we can always rescale it by a unique positive λ to get ζ₀ + ζ₃ = 1, and then read the coordinates from ζ (the rescaled ζ') the two coordinates on the plane. The line with ζ₀ + ζ₃ = 0 form an equivalence class defining one point on the sphere - its South Pole, the Infinite Point on the plane.

Exercise 2. If ζ₀ + ζ₃ = 0 on the null cone, then, necessarily ζ₁ = ζ₂ = 0. Why?

Now we can return to the action of the Clifford group on Minkowski space, its null cone, the sphere, and the plane. We have a distinguished point on the sphere - its South Pole. Thus we can extract a special subgroup of transformations, namely those that do not move that distinguished point. These transformations will transform the (x,y) plane into itself - they will be easy to interpret. But there will be also other transformations that move the infinity point into some other point. Inverse of such a transformation will move a point on the plane into Infinity Point on the sphere, producing a singularity on the plane. We will analyze all this in the next post.

Where Have All the Spinors Gone?

We’ve got "spinors" boldly declared in the title of this series, yet somewhere along the way, they’ve managed to slip out of sight. This is unacceptable! Spinors should be front and center, the main act, not some backstage crew hiding in the shadows. So, just to keep them from becoming the forgotten middle child of mathematical objects, let’s bring them back into focus.

Recently, I received a seven-page paper from V.V. Varlamov. The topic? Clifford algebras and spinors, inspired by Rozenfeld's work on non-Euclidean geometry. I opened it eagerly, confident that by page seven, I’d emerge enlightened, ready to declare, "At last, I understand spinors!"

I did not.

The math in this paper was so advanced, it left me feeling like a freshman who wandered into a graduate-level seminar by mistake. My head began to spin—not in the cool quantum way, but in the “I need aspirin” way. The formulas weren’t just over my head; they were in orbit. It became clear that to keep my sanity intact, I needed to ditch the "follow the paper" approach and start thinking in my way.

I needed to ditch the "follow the paper" approach and start thinking in my way

So, here’s the mental road trip I embarked on:

We’ve got our trusty geometric Clifford algebra of space. It’s an elegant, eight-dimensional creature, cozy with bi-quaternions and complexified Minkowski space. Lovely. What’s more, this algebra isn’t just lounging around—it acts. And it acts on itself, no less. How?

• From the left: $L(u)v=uv$
• From the right: $R(u)v = vu$
  

Now, here’s the juicy part: when you let a Clifford group element $g$ act on something from the left, $g \cdot x$, it drags you out of the real Minkowski space. (Thanks a lot, $g$.) To reel things back in, you need to hit it from the right with something like $\tau(g)$. This leads to a beautiful commuting relationship:

$$L(u)R(w)=R(w)L(u)\mathrm{.}$$

This is clean, elegant math. But what does it have to do with the quantum physics of spin? Here’s where my mind started making some leaps—and possibly doing a few backflips.

Left, Right, and Quantum Duality

In quantum theory, there’s always this duality: the "observer" versus the "system under observation." For spins, we’ve got the laboratory frame with its neat axes, along with a physicist who’s busy assigning complex spin vectors in Hilbert space. And then we have the spin itself—precessing, pirouetting, doing its quantum dance.

In my mental model, I associate the left action with the quantum system itself and the right action with the observer. Or maybe it’s the other way around? I’ll admit, this part’s still a work in progress. The details need ironing out, like a wrinkly shirt you’re not sure is clean or dirty, but you’re wearing it anyway.

Enter the Hairy Ball Theorem

Somehow, all these thoughts led me to the Hairy Ball Theorem. Yes, that theorem from topology—the one that proves you can’t comb a hairy ball flat without creating a cowlick. If you’re wondering what this has to do with spinors, quantum dualities, or Clifford algebra, congratulations—you’re just as confused as I am.

But don’t worry, this will all (hopefully) become clearer in future posts. For now, I’ll leave you with this cliffhanger: Can spinors help us avoid cowlicks in quantum mechanics?

P.S. 02-12-24 11:30 This morning a friend pointd to me a paper by Jesús Sánchez, available  on Researchgate:

https://www.researchgate.net/publication/371274543_We_live_in_eight_dimensions_and_no_they_are_not_hidden

The title is  "We live in eight dimensions and no, they are not hidden" and the abstract:reads:

"In this paper, we will show how Geometric Algebra expand the three spatial dimensions into entities of 8 degrees of freedom. It is also explained that one of these degrees of freedom (the trivector) can be considered to be the time (so no ad-hoc extra dimension is necessary). The square of the trivector is negative, solving this way the issue of the negative signature of the time (not necessary any ad-hoc metric indicating this, it is a property of time that appear naturally). Also, we will show that we can try to prove this experimentally looking for the electromagnetic trivector, an entity that should exist according to GA.   Also, some comments regarding the similarities with E8 theory are given. Mainly that E8 theory considers 8 dimensions, exactly the same, emerging naturally in this paper. But not only that, also some similarities regarding how gravity can be understood, and others are presented. "

Keywords 

Geometric Algebra, Eight Dimensions, Dirac Equation, Gravitation, E8 theory 

I didn't read it yet, but I will certainly study it carefully, as the content of this paper overlaps with the ideas we are discussing on my blog. Coincidence? Or more than that?

On the last page of this paper I am reading: 

"this case, Cl 3,0  or Cl 0,3  seems sufficient to explain most of the disciplines of physics, if we accept the issue that the time could be an emergent dimension coming from the three spatial dimensions in the form of scalar or trivector and not an ad-hoc added dimension to the reality. "

And that is also what I was bringing our attention to on this blog. Coincidence? Or "something is in the air"?

P.S. 02-12-24 15:03 Here are two youtube videos by Alain Cagnati on the related subject. They are in French, but one can download them with English subtitles. The description reads:

"This is a video on the unification of the 4 interactions... It's the same as the one called “unification” but with a different framing...

In this first of two videos, I show that Clifford's algebra on R^3 (Geometric Algebra) is a very good candidate to be a unifying algebra for physicists. Indeed, this 8-dimensional algebra has all the properties needed to represent reality. Hodge self-duality, Hopf algebra, etc.

In a second video, I'll show what fundamental implications this could have for physics.

Finally, I'll show how the fundamental structure of space (mirror symmetry, etc.) and the structure of the (human) psyche might be related."

P.S. 02-12-24 18:53 I am preparing a pdf document containing in a condensed form all the main concepts and formulas about Cl(V) (renamed to Cl(E) in the document).

P.S. 03-12-24 11:08 Here is the beginning pdf.