The Spin Chronicles (Part 21): Magritte and Flammarion

 

We’re on a journey. A journey to where, you ask? Honestly, I’m not entirely sure myself. Is the ultimate goal to solve all the mysteries of the universe? (No pressure, right?) Or maybe just crack open one of them, like an intellectual piñata? Or perhaps it's something subtler, something closer to René Magritte's enigmatic wisdom: “Each thing we see hides something else we want to see.”

But Magritte, ever the paradoxical philosopher, also declared: “I think we are responsible for the universe, but that doesn’t mean we decide anything.” Really, René? Nothing? Well, I’ll take my victories where I can, starting with today’s decision to write this blog. I have no clue what cosmic ripples this choice will send out (but hey, I hear butterflies start hurricanes), yet I’ll cling to the notion that I’m somehow fulfilling my obligations to the universe by clicking away at my keyboard.

I have no clue what cosmic ripples this choice will send out 

So, where were we? Ah, yes—the journey. It turns out, we’re not exactly on the straight path. We’ve taken a detour. A scenic route, if you will. After Part 17, the plan was to dive straight into options (3) and (4) for two-sided actions of the Clifford group on Cl(V):

1. $g \cdot u = g u \tau(g)$
2. $g \cdot u = g u \nu(g)$
3. $g \cdot u = \pi(g) u \tau(g)$
4. $g\cdot u=\pi (g)u\nu (g)$

Instead, I decided to veer off-road. Why? Because sometimes you’ve got to set aside the GPS of mathematical rigor and take a little detour to explore something... different. Something fun. So, let’s play with the action of the Clifford group on the null cone in Minkowski space and the induced action on the $(x, y)$-plane.

At first blush, this might sound like one of those “because I can” exercises in pure abstraction. But hold on—this isn’t some random doodle in the margins of physics. It has a gorgeous physical interpretation! The null cone in Minkowski space represents light emanating from a single spacetime point. Picture a laboratory at time $t=0$, where light bursts forth in all directions. By $t = 1$, that light forms a sphere around the lab. Think of it as the universe’s way of saying, “Let there be geometric algebra!”

On this sphere, we can project the stars of the universe—the celestial sphere—much like Flammarion’s famous engraving, which tantalizingly invites us to peek beyond the cosmic veil:

Now, let’s spice things up a bit. What happens if we apply a Lorentz transformation to our lab? A rotation is straightforward—the sky spins, and the stars gracefully whirl like a cosmic waltz. But a Lorentz boost? That’s a whole different party. Enter relativistic aberration: the stars’ positions shift dramatically, bending and dancing under the influence of spacetime’s warping lens.

We’re going to calculate this starry choreography using the Clifford group. (And if you’re getting déjà vu, it’s because I hinted at this back in my September 4, 2024 post, “Monkeys, UFOs, and Quantum Fractals: A Mind-Bending Journey through Relativity and Imagination.”)

Here’s the twist: we’ll go beyond. We’ll project the celestial sphere onto the $(x, y)$-plane of our lab. But don’t expect a straightforward mapping. The projection originates from the South Pole of the celestial sphere—a point conspicuously absent in Flammarion’s image. (That’s because it’s “beneath” the feet of our starry-eyed astronomer. Practical, but disappointing.)

This is the plan. And in the next post, we’ll dive into the algebra to unravel what happens to the projected images of stars. Will they remain stars? Will they smear into streaks of cosmic glory? Stay tuned. Who knows—this little exploration might even shake up your view of the heavens. Or at least provoke a disagreement or two.

P.S. 04-12-24 Updated the pdf file.

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?

The Spin Chronicles (Part 15): The Action starts

Understanding the meaning of the Universe—and, by extension, our purpose in this dizzying 3D playground—is impossible without first grasping the concept of spinors. Yes, spinors. Everything spins. Planets pirouette around stars, galaxies swirl like cosmic pinwheels, and deep inside you, photons and electrons are spinning their little hearts out. Without spin, there would quite literally be "no nothing." But what exactly is this mysterious "spin"? And what, pray tell, is a "spinor"?

To get the most precise answers, we must summon the sharpest tool in humanity’s intellectual shed: mathematics. Forget poetry or philosophy for now (though they’re lovely companions); math is the only game in town when it comes to rigor. The groundwork for this particular game was laid by none other than the French mathematical collective known as Nicolas Bourbaki. This was not an  ordinary group of thinkers—they were the masterminds who struck the match, poured the fuel, and launched the metaphorical rocket of modern mathematical formalism.

The Bourbakists

I, too, am a humble traveler on this rocket of discovery, squinting over the shoulders of these giants to glimpse what lies ahead. 

The Bourbaki

And let me share with you one of the gems they left behind, a definition from Bourbaki’s magnum opus, ÉLÉMENTS DE MATHÉMATIQUE, ALGÈBRE CHAPITRE IX:

DÉFINITION 2: On appelle groupe de Clifford de Q (resp. groupe de Clifford spécial de Q), le groupe multiplicatif des éléments inversibles de C(Q) (resp. C+(Q)) tels que sEs⁻¹ = E.
Dans ce no nous noterons G et G⁺ le groupe de Clifford et le groupe de Clifford spécial de Q.

If you’re already scratching your head, don’t worry—it’s not your fault. Mathematics is as much a language of symbols as it is a puzzle of patience. But let’s break it down: Their E is our V, their Q is the Euclidean quadratic form of V, and their C(Q) corresponds to our Cl(V). Meanwhile, their C⁺(Q) is a subalgebra of Cl(V), comprising those nice even elements that remain unchanged by π.

Here’s the twist: where Bourbaki insists on the condition  sEs⁻¹ = E. we’re a bit more liberal. We drop that requirement, making our Clifford group G bigger and, dare I say, more fun. Why stop at “sensible” constraints when you can explore the wild possibilities of mathematical freedom?

And here’s where it gets spicy: this expansion of the Clifford group isn’t just a technical curiosity—it unlocks surprises. It’s a reminder that while mathematicians love their neat, self-contained universes, Nature herself isn’t so tidy. She operates with her own version of simplicity, one that often leaves mathematicians muttering into their coffee mugs.

In short, if you think Bourbaki nailed it, wait until you see what happens when we take the leash off. Welcome to the uncharted territory where Clifford algebras meet Nature’s playful chaos.

Note: The idea of enlarging the Clifford group is not new (is there anything really-truly new under the Sun?). C.f. for instance William Baylis, "Lecture 4: Applications of Clifford Algebras in Physics, 4.3 Paravectors and relativity", in "Lectures on Clifford (geometric) algebras and applications" [edited by] Rafal Ablamowicz and Garret Sobczyk], Springer 2004, and references therein.

This post is a continuation from The Spin Chronicles Part 14, where we have introduced our "Clifford group G" defined as the set of elements u = (p₀,p)  in Cl(V) for which (p₀)²-p² = 1. Another form of the definition of G is

G = {g ∈ Cl(V): g ν(g) = 1}.

Exercise 1: use the definition of the multiplication in Cl(V) written in a bi-quaternion form to verify that these two definitions are indeed equivalent

In this post we will take a closer look at the action of G on Cl(V) defined by

g: u ⟼ guτ(g).

To this end we will analyze the action of one-parameter subgroups g(t) = exp(t X). In order for g(t) to be in G, we must have X + ν(X) = 0.

Exercise 2. Verify the last statement.

First we take the case of X even. Thus we have two conditions on X:

a) ν(X) = -X,

b) π(X) = X.

A general solution of these two conditions is X = in, where n is a real vector.

Exercise 3. Verify the last statement. Without loss of generality we can choose n to be a unit vector. Remember that n is a vector in Cl(V), therefore we have nn=1. Using this property calculation of exp(itn) is easy. We get:

exp(itn) = cos(t)1 + i sin(t) n.

Exercise 4. Use the power series expansion of exp(itn) to derive the above formula.

We can easily verify that we have

τ(g(t)) = g(-t).

Indeed, τ(exp(itn)) = exp(τ(itn)) = exp(-iτ(tn)) = g(-t), where we used the fact that τ(i) = -i, and that t and n are real, t - scalar, and n - vector.

Taking now u=(x₀,x), where (x₀,x) are complex, and setting it is a matter of simple algebra using the multiplication formula for bi-quaternions (c.f. Exercise 1, Part 11) to get for u(t) = g(t) u g(-t), and for n=(0,0,1),  the formula

x₀(t) = x₀,

x₁(t) = cos(2t)x₁ + sin(2t) x₂,

x₂(t) = -sin(2t)x₁ +cos(2t) x₂

x₃(t) = x₃.

It follows that this action of g(t) on Cl(V) is nothing else but a simple rotation by the angle 2t of the vector part. It rotates the same way both real and imaginary components of the vector. The scalar component x₀ does not change under these rotations.

In the next post we will consider the case of odd X.

The Spin Chronicles (Part 14): The Universe and Clifford group actions

 I can't refrain myself from starting this post with a quote from the Introduction to the paper "Conformal Mappings, Hyperanalyticity and Field Dynamics" by V.V. Kassandrov (1993): Here it goes:

"1. Introduction

It seems impossible to imagine that the primary Void, which contains the divine diversity of forms, determines the dynamics of the World. However, we have to believe and try to understand it, else we are destined to stay on the primitive level of ‘geographical’ phenomenology.

Maybe it is wise to give up, for a while, all the principles and laws established by our great predecessors, even such ‘fundamental’ laws as Lorentz invariance, gauge symmetry or the minimal action principle. We should not flatter ourselves with their apparent beauty – they are neither basic nor primary, since they themselves are to be explained and grounded. It is also valid for most remarkable empirical facts, such as the invariance of light velocity, the quantization of electric charge, the universality of the inverse square law.

We may regard these facts as axioms and then proceed as far as possible in constructing a fruitful physical theory. This is just the way Einstein created his special theory of relativity (STR).

It seems more attractive to find a new interpretation of well-known physical phenomena, where the existence of the correspondent correlation becomes self-evident (as the equality of gravitational and inert masses after a geometric treatment of gravitation in general theory of relativity (GTR)). There still exists another way of thought, the most thankless one but promising. This is the way bequeathed by Pythagoras, Hamilton, Eddington and other Grands, the way chosen by Einstein in the last half of his life. Here we suppose the existence of exceptional relational structures of a purely mathematical nature and intend to discover them and formulate a unique abstract principle on their basis. The characteristic differential equations, algebro-geometrical properties and other relations should then follow from this principle only. Here it is necessary to bring in conformity all the mathematical categories and solutions with the

quantities describing real objects. If and only if the complete correspondence is achieved, we should consider the Universe as a ‘reflection’, a ‘realization’ of this abstract Principle! 

 

The Universe as a ‘reflection’, a ‘realization’ of this abstract Principle! 

One has to understand that such ideology contradicts the widespread treatment of mathematics as a merely subjective formalism. On the contrary, one has to assume that there do exist completely objective laws in its very structure (see [1]), which have to be discovered and identified with physical laws. In this article we try to show that the afore-stated approach is not so fabulous as it seems at the first sight. Naturally, at once it is impossible to establish that very Principle whose structure carries the main part of physical phenomenology.

Nevertheless, even within the frame work of Hamilton’s and his followers’ well-known ideas, it becomes possible to reveal the wonderful correlations between the properties of an exceptional mathematical structure – the algebra of quaternion-like type – and the physics and geometry of space-time.[...]"

Число, время, свет || Владимир Кассандров || Программа Гордона №313 (04.11.2003)

This is exactly the philosophy I am trying to follow in this series. In The Spin Chronicles (Part 13) we have introduced the group G:

G = {g∈Cl(V): ∆(u) = 1},

where ∆(u) = B_ν(u,u) = (p₀)² - p².

In fact we do not even need ∆(u) in this definition - we need only the Clifford conjugation ν itself, since the element ν(u)u automatically belongs to the center of Cl(V), and we have, as can be verified from the form of the product, and the form of  ν(u) (Part 11):

uv = (p₀q₀ + p·q,  p₀q + q₀p ₊ i p⨯q),

ν(u) = (p₀,-p).

Therefore

ν(u)u = uν(u) =  ((p₀)² - p²)1.

Exercise 1. Verify the last formula.

Once we have a group, we analyze its natural actions. By definition we say that a group G acts on a set S if there is a map G ⨯ S ⟶S, denoted (g,s) ⟼ g·s, such that, for all s we have e·s = s, where e is the identity of the group, and for  all g,h in G and all s in S we have g·(h·s) = (gh)·s. In our case for the construction of natural actions we have at our disposal the automorphism π, and two anti-automorphisms ν, and τ. The following four actions are natural:

1). g·u = g u τ(g),

2). g·u = g u ν(g),

3). g·u = π(g) u τ(g),

4). g·u  = π(g) u ν(g).

Exercise 2. Verify that all four cases are indeed group actions.

We will consider them one by one.

Case 1. We recall the form of τ:

τ(u) = (p₀,p)*,

where "*" denotes the complex conjugation. Every element of Cl(V) can be decomposed as

u = (u+τ(u))/2 + (u-τ(u))/2.

We can write it as u = u_R + u_I. Then  τ(u_R) = u_R,  τ(u_I) = - u_I. In other words we split Cl(V) into a direct sum of two eigensubspaces of τ. It can be easily checked that these eigenspaces are invariant under the action 1).

Exercise 3. Verify the last statement.

In order to analyze the group action we will consider its one-parameter subgroup of the form g(t)=exp(t X), where X is in Cl(V), t is real parameter. To have ν(g(t))g(t)=1, since

ν(exp( t X)) = exp(t ν(X)),

we must have ν(X)=-X. Therefore X must be of the form (0,v+iω), where v and ω are real vectors.

Exercise 4. Prove the last statement.

We will continue our analysis in the next post.