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.
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:
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:
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.
Dressing up τ is not too difficult once we realize that Pauli matrices are Hermitian, σi* = σi, 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 σi. 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 = σi, 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:
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.
