The Spin Chronicles (Part 10) - Dressing up the three involutions

 

This post is a continuation of The Spin Chronicles (Part 9): Matrix representation of Cl(V).

In Greek mythology, the three Graces—Aglaea, Euphrosyne, and Thalia—were goddesses of charm, beauty, and creativity. Let’s imagine these sisters lounging on an ancient hillside, as they engage in a playful but philosophical debate on the nature of reality.

Aglaea (the Grace of splendor and brilliance) takes a sip from her goblet of ambrosia and says, “Sisters, I’ll tell you what I believe to be the foundation of all things: matter itself. Look around us! The hills, the olive trees, the sky—everything is made of something. Without matter, there would be nothing to perceive or to enjoy. All reality begins with this.”

Euphrosyne (the Grace of joy and mirth) chuckles, swirling her own drink thoughtfully. “Dear Aglaea, that’s such a, well, grounded take! But think of the elegance behind things, not merely their substance. I argue that geometry is what truly underlies reality. The symmetry of the flowers, the curve of a river, even the shape of the stars in their constellations—geometry organizes matter, gives it purpose and beauty. It’s as if shapes themselves are the mold from which everything springs.”

Thalia, the muse of blooming and abundance, raises an eyebrow, already chuckling in anticipation of her own argument. “Oh, my beautiful, grounded, geometric sisters,” she teases. “How can you talk of matter and form without acknowledging the importance of information? The patterns, the codes, the language that organizes and defines everything we perceive—these are what really hold things together. Geometry and matter might give us the frame, but information is the artist with the brush, deciding what each piece will mean.”

Aglaea sets down her goblet, leaning forward with a glint in her eye. “So you think all things are simply... patterns of knowledge? A set of instructions that give order? But without matter, there’s nothing to know or organize.”

“True,” Euphrosyne nods. “And without geometry, that matter would be a chaotic jumble! It’s through geometry that we get structure, form, coherence.”

Thalia laughs, throwing her arms out as if embracing the universe. “And without information, both of those would be nothing but potential! You need a blueprint, a guide that says, ‘You, little particles, arrange yourselves like so to create something magnificent!’”

The sisters fall quiet, each musing over the others’ words. The sunset casts a golden light across their faces, and in that moment, they feel as if they've touched upon a truth that binds them all.

Finally, Aglaea speaks softly. “You know, there might be something greater than matter, geometry, or information alone. What if they’re all aspects of the same thing?”

“Yes,” Euphrosyne says, her eyes lighting up. “Something that encompasses both the structure of shapes, the substance of matter, and the language of patterns…”

Thalia smiles and whispers, “Geometric algebra.”

The three sisters exchange knowing looks, each filled with a newfound appreciation for the others’ ideas. They raise their goblets to the concept that unifies their philosophies, and in a toast to geometric algebra, they bask in the sweet realization that the foundation of everything is, indeed, a beautiful dance of shapes, substance, and meaning.

Thalia smiles and whispers, “Geometric algebra.”

The Clifford geometric algebra Cl(V) of a 3-dimensional Euclidean real vector space V, "our space", is naturally endowed with three involutions:

• 1. The main automorphism π, changing the signs of the vectors.
• 2. The main anti-automorphism   τ, reversing the order of multiplication in Cl(V).
• 3. Their composition π∘τ = τ∘π, which we will call ν.

As they are born, they are "naked", as the three graces in this Raphael painting:

Raphael Three Graces

It is possible, but not easy to work with them. So, sometimes, it is useful to see them dressed, as it has been done by Botticelli:

Botticelli Three Graces

Today we will dress them in the matrix form. Using the notation of the previous post, we select an orthonormal basis in V and represent the basic vectors e¹,e²,e³ by the Pauli matrices σ₁, σ₂, σ₃, satisfying σ₁σ₂ = iσ₃.

Pauli matrices

Dressing up τ  is not too difficult once we realize that Pauli matrices are Hermitian,  σ_i* = σ_i, and that, for Hermitian matrices X,Y, the Hermitian conjugation reverses the order (XY)* = YX. Thus, in the matrix representation of Cl(V) the main anti-automorphism τ is represented by the Hermitian conjugation. This was part of the Exercise in the previous post, and Bjab solved it right: for every X in Mat(2,C):

τ(X) = X*.

With π we need to be tricky. We need to reverse the signs of vectors, and vectors are represented by σ_i. Pauli matrices anti-commute σ₁σ₂ = -σ₂σ₁, etc, and their squares are I.  Thus σ₂σ₁σ₂ = - σ₁, and σ₂σ₃σ₂ = - σ₃, but, unfortunately, σ₂σ₂σ₂ =  σ₂. On the other hand, this time "fortunately", σ₂ is the only Pauli matrix that is imaginary! Therefore adding the complex conjugation will do the job.

We try: for every X in Mat(2,C) we define

π(X) = complex_conjugate(σ₂Xσ₂) = σ₂ complex_conjugate(X) σ₂.

We check for X = I, for X = σ_i, for X = σ₁σ₂, etc,  for X = σ₁σ₂σ₃ - it works! So, we managed to dress our π! Now dressing ν is easy. It is the composition of π and τ. We need to take

(σ₂ complex_conjugate(X) σ₂)* = σ₂ (complex_conjugate(X))* σ₂

But Hermitian conjugation is a composition of complex conjugation and matrix transpose! Therefore we have the answer:

ν(X) = σ₂ X^T σ₂.

And so we managed to dress up all of our three graces. They can go out and work for us in a useful way! Which we will do in the following posts.

P.S. 08-11-24 10:31 Laura noticed: 

"“So you think all things are simply... patterns of knowledge? A set of instructions that give order? But without matter, there’s nothing to know or organize.”

That's kind of what Ibn al-Arabi says."

P.S. 08-11-244 13:12 Yesterday, I participated in a seminar  «Основания фундаментальной физики» под руководством проф. Ю.С. Владимирова, at Moscow State University (thanks to A.S. for informing me about it). The seminar was online. The title was "Понятие «Охватывающее настоящее» у К. Ф. фон Вайцзеккера". Автор: Севальников Андрей Юрьевич, физик-теоретик и доктор философских наук, профессор, заведующим сектором в Институте философии Российской Академии Наук. Different philosophical concepts of "time" were the main subjects of this talk. But Weizsacker's "Ur" is nothing else but an irreducible module for CL(V). We will discuss it later on.

Понятие «Охватывающее настоящее» у К. Ф. фон Вайцзеккера

For me it was a real pleasure to see Gennady Shipov as a participant of the discussion. When the discussion touched the subject of Quantum Mechanics, Gennady, with a smile on his face,  made a very good remark to the participating mainly philosophers of science. He remarked that there is no single proof that QM (extensively quoted in the talk) is a complete theory. Other participants could not really argue his point!