The Spin Chronicles (Part 9): Matrix representation of Cl(V)

 This post is a continuation of the last post in the series: The Spin Chronicles (Part 8): Clifford Algebra Universal Property

Embedded in 3D Space

For reasons we don’t fully understand, we experience the world in three dimensions—length, breadth, and width. Our senses anchor us in this 3D framework, and we often feel its presence deeply, even beyond simple dimensions. Consider these lines from Rainer Maria Rilke’s "The Night Sky":

"Who has not longed under a brooding sky
for the sound of falling stars?
And to be swallowed up in the wide, wide world,
to feel within himself the cosmos stretching,
endlessly, silently…"

This poetic vision echoes a feeling of spatial vastness, as if the cosmos itself could fit within our minds.

Similarly, William Blake’s famous lines from "Auguries of Innocence" hint at the mystical unity of smallness and immensity:

"To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour…"

And to be swallowed up in the wide, wide world,

Space and Time: A Relationship

While we think of space as separate from time, time appears to emerge from space. In mystical or out-of-body experiences, time sometimes halts, while space remains, albeit in a modified form—often with a 360-degree perspective but still fundamentally three-dimensional. For now, we’ll set aside time, treating space as the foundational structure from which time will eventually arise, shaped by the expansion and contraction of spatial volumes.

Defining Space Mathematically

In space, there are no absolute directions, but there are "lengths" and "angles" between intersecting lines. To formalize this, we define space as a real, 3-dimensional vector space $V$ and equip it with a scalar product—a nondegenerate symmetric bilinear form. Using this structure, we construct its Grassmann and Clifford algebras, which give us powerful mathematical tools to further analyze spatial properties.

The Problem of "Length"

Assuming that we can measure  the "length" is perhaps too much. We can compare different lengths, but to give to a vector in V a definite number, its "length" is somewhat risky. The number depends on a subjectively selected "unit" of length. Moreover, in alien abduction experiences "small" can feel "large" - the scale changes. We will return to this issue in the future. For now let us assume that the unit of length is fixed. It can be, for instance, the cubit. Cultural units like the cubit—the length from a person’s middle finger to their elbow—were historically practical choices.

Biblical Reference to the Cubit
In Genesis 6:13-22 (NKJV), the cubit is used as a measurement for Noah’s Ark:

“Make yourself an ark of gopherwood; make rooms in the ark, and cover it inside and outside with pitch. The length of the ark shall be three hundred cubits, its width fifty cubits, and its height thirty cubits.”

Royal cubit

The cubit, likely originating from Sumerian civilization, offers a historical reference point for measuring length in a way grounded in human scale.

Constructing Clifford Algebra

Once we define length and the associated bilinear form, we can construct the Clifford algebra by deforming the exterior product, as discussed in previous posts. With this algebraic foundation, we can start performing calculations. Several Computer Algebra Systems allow symbolic manipulations with Clifford, or "geometric," algebra, though here we’ll limit ourselves to three dimensions and real numbers.

Interestingly, in three dimensions, there’s no need for specialized software for Clifford algebra—simple $2 \times 2$ complex matrices suffice! Using the Universal Property of Clifford algebra, we can represent it in the form of complex matrices.

Universal Property of Clifford Algebra
Let $\phi$ be a linear map from $V$ into an associative algebra $A$ (with unit $1$), such that $\varphi (x{)}^2=B(x,x)1$ for all $x$ in $V$. Then $\phi$ can be uniquely extended to a homomorphism from $Cl(V, B)$ to $A$.

For our algebra $A$, we choose Mat(2, C), the algebra of $2\times 2$ complex matrices. Selecting an orthonormal basis $e_i$($i=1,2,3$) in $V$ (I will write $e_i$, with lower indices in this post), we define three Pauli matrices:

To any vector $x$ in $V$, written as $x = x_1e_1 + x_2e_2 + x_3e_3$, we associate the matrix $\phi(x)$:

$$\phi(x) = x_1\sigma_1 + x_2\sigma_2 + x_3\sigma_3,$$

It satisfies

$$\phi(x)^2 = x \cdot x.$$

Thus, by the Universal Property, $\phi$ extends uniquely to a homomorphism from $Cl(V)$ to $Mat(2, C)$, which turns out to be an isomorphism because the Pauli matrices generate $Mat(2, C)$, and both algebras share the same real dimension of eight.

Realizing Clifford Algebra in Matrices

Using this isomorphism, we express elements of $Cl(V)$ within the algebra of $2\times 2$ complex matrices. For an arbitrary element $u$ in $Cl(V)$:

$$u = a + b_1 e_1 + b_2 e_2 + b_3 e_3 + c_1 e_2e_3 + c_2 e_3e_1 + c_3 e_1e_2 + d e_1e_2e_3,$$

we associate

$$\phi(u) = a I + b_1 \sigma_1 + b_2 \sigma_2 + b_3 \sigma_3 + c_1 \sigma_2\sigma_3 + c_2 \sigma_3\sigma_1 + c_3 \sigma_1\sigma_2 + d \sigma_1\sigma_2\sigma_3.$$

Explicitly, in matrix form:

With a little manipulation, we can recover the components $a$, $b$, $c$, and $d$ from X=$\phi(u)$:

$$a = \frac{1}{2} \text{Re(Tr(X))},$$ $$b_i = \frac{1}{2} \text{Re(Tr(}\sigma_i X\text{))},$$ $$c_i = \frac{1}{2} \text{Im(Tr(}\sigma_i X\text{))},$$ $$d = \frac{1}{2} \text{Im(Tr(X))}.$$

Challenge: Expressing Clifford Algebra Involutions

Exercise 1: Express ι = e₁e₂e₃ as a 2x2 matrix using the above representation.

Exercise 2: As a challenge, try to express the three involutions of $Cl(V)$ in terms of Mat(2, C). This will deepen your understanding of how Clifford algebra structure manifests in matrix form.

Stay tuned for further exploration in the following posts!

P.S. 06-11-24 17:18 You should not take the story of the Noah's Ark literally. See "Hidden in Plain Sight" by The Ethical Skeptic.

Sunday Special - From Fairy Tales to Math: The Power of Threes

 Ever noticed how many things come in threes? Like in a story, three chances always seem just right, or in comedy, the third punchline really lands. There's even a term for it: "the rule of three," or in Latin, "omne trium perfectum"—meaning all things in threes are perfect. According to an English language forum I stumbled upon, “the rule of three” shows up everywhere from fairy tales to Hollywood blockbusters because things in threes are inherently more humorous, satisfying, and impactful.

Think about it. In storytelling, the protagonist’s first attempt fails, the second is almost there, and the third? Bingo! Success! But there’s more to this number than just fairy-tale magic. It even shapes the foundations of mathematical structures in Clifford algebras.

omne trium perfectum

---

Clifford Algebras: A Trio of Transformations

In any dimension, Clifford algebras feature three key transformations—three "involutions," if we’re being formal. These are not just fancy operations; they’re structural quirks that make Clifford algebras unique.

1. Main Automorphism (Π) - It flips the sign of odd products of vectors.
2. Main Anti-automorphism (τ) - Changes the order of vector products (also known as "reversion").
3. Composition (Π∘τ = τ∘Π) - A combined transformation, also called "conjugation."

Together, these involutions create a complete set that operates consistently within every Clifford algebra. And here’s the kicker: they’re a part of our world’s fundamental structure. Our very own 3D Euclidean space obeys these algebraic principles. It's as if nature decided three dimensions were "just right"—Goldilocks style. The aim? To extract all we can from these three dimensions before venturing off to other realms.

---

Three as the First Prime: A Mathematical Treasure

But there's even more to three! Mathematically, three is the first truly prime number (2, which comes before 3,  is truly exceptional among prime numbers, as it is the only even prime number - which is odd!). The revered mathematician Leonard Euler had plenty to say about prime numbers and divisors. 

Here is the reasoning of Leonard Euler (G. Polya, Mathematics and Plausible Reasoning, Vol. 1, Induction and Analogy in Mathematics, Princeton University Press 1990, p. 91):

"[...] 2. A prime number has no divisors except unity and itself,"and this distinguishes the primes from the other numbers. Thus 7 is a prime, for it is divisible only by 1 and itself. Any other number which has, besides unity and itself, further divisors, is called composite, as for instance, the number 15, which has, besides 1 and 15, the divisors 3 and 5. Therefore, generally, if the number p is prime, it will be divisible only by 1 and p; but if p was composite, it would have, besides 1 and p, further divisors. Therefore, in the first case, the sum of its divisors will be 1+p, but in the latter it would exceed 1+p. As I shall have to consider the sum of divisors of various numbers, I shall use the sign σ(n) to denote the sum of the divisors of the number n. Thus, σ(12) means the sum of all the divisors of 12, which are 1, 2, 3, 4, 6, and 12; therefore, σ(12) = 28. In the same way, one can see that σ(60) = 168 and σ(100) = 217. Yet, since unity is only divisible by itself, σ(l) = 1. Now, 0 (zero) is divisible by all numbers. Therefore, σ(0) should be properly infinite. (However, I shall assign to it later a finite value, different in different cases, and this will turn out serviceable.)

3. Having defined the meaning of the symbol σ(n), as above, we see clearly that if p is a prime σ(p) = 1+p. Yet σ(1) = 1 (and not 1+1); hence we see that 1 should be excluded from the sequence of the primes; 1 is the beginning of the integers, neither prime nor composite. If, however, n is composite, σ(n) is greater than 1+n."

---

The Law of Three: Good, Evil, and the Uncharted Middle

Stepping away from math, let’s explore something a bit more cosmic: "The Law of Three." While perusing academia.edu, I came across a work titled ARCHONS HIDDEN RULERS THROUGH THE AGES by Anbr Cama, which discusses ethical and cosmic dualities. Here's the essence of the Law of Three:

“There is good, there is evil, and there is the specific situation that determines which is which.”

The cosmic perspective suggests that both good (often linked with Service to Others or STO) and evil (Service to Self, STS) are necessary. It’s not a question of resolving this tension; rather, it’s about choosing a path. Human ethics can feel binary, but the Law of Three reminds us of the "third factor"—the context. Just as in storytelling and mathematics, three isn’t simply two opposites with a bridge; it’s a unique, cohesive whole.

---

The Universe in a Droplet: The Holographic Principle and 3D Space

And now, the grand finale: the Holographic Principle. Imagine if all of the universe’s information were contained in the tiniest droplet. If the universe’s vastness could be distilled, it might exist encoded within the Clifford algebra of our 3D space.

This implies something wild—time itself might be woven into this 3D algebraic fabric! If we could untangle the layers within our spatial dimensions, we might glimpse time’s secrets. It’s as though the universe has handed us a compact manual for existence, neatly packed into three dimensions.

---

Embracing the Power of Three

So there you have it: from fairy-tale patterns to mathematical beauty, from cosmic ethics to the secrets of time—three, it turns out, is a number that holds endless layers of depth. The Law of Three isn’t just a principle; it’s a worldview. Whether in jokes, mathematical theorems, or cosmic reflections, threes show us that life, like a good story, often unfolds in a satisfying trilogy. And maybe, just maybe, there’s some cosmic rule nudging us toward threes for good reason.

P.S. 03-11-24 19:42 In reply to a comment below by Bjab, concerning  the use of AI:

"AI has blown this world open. Almost all pressure to write has dissipated. You can have AI do it for you, both in school and at work.

The result will be a world divided into writes and write-nots. There will still be some people who can write. Some of us like it. But the middle ground between those who are good at writing and those who can't write at all will disappear. Instead of good writers, ok writers, and people who can't write, there will just be good writers and people who can't write."

https://www.paulgraham.com/writes.html

The Spin Chronicles (Part 8): Clifford Algebra Universal Property

 In The Spin Chronicles (Part 7): Whispers of the Cosmos Beneath the Fig Tree we have met quaternions and biquaternions living peacefully within the Clifford algebra of our 3D space. Today we will widen a little bit our horizons.

The Secret of Light

"God's imaginings extend from rest to rest in His three-dimensional radial universe of length, breadth and thickness--to become the stage of space for His imagined radial universe of matter, time, change and motion." (Walter Russell,  The Secret of Light, University of Science and Philosophy 1947). (Fig. 4)

Once we have space, it gets curved, but since All is One, the structure of the whole Cosmos is reflected in the structure of its smallest cubes, and what we are studying now is the algebra of such a primitive cube. We already know it's eight-dimensional. Three dimensions of space generated 2³ dimensions of its, first Grassmann Λ(V), then Clifford, algebra Cl(V). The Clifford algebra structure allows us to measure its substructure. And now we are ready to delve deeper into this issue. We will start with the natural automorphisms  and anti-automorphism of Cl(V). Recall that we have realized Cl(V) on the canvas of Λ(V):

Λ(V) = Λ⁰(V) ⊕ Λ¹(V) ⊕ Λ²(V) ⊕ Λ³(V)

by deforming its Grassmann product u∧v into the Clifford product uv

uv = u∧v+(u⋅v)1,

for u,v in V. Elements of each Λⁱ(V) are called homogeneous of degree i. We then write: i = deg(a).  Grassmann product of homogeneous elements is again homogeneous. Clifford product, on the other hand, does not preserve homogeneity.  We will return to this point soon, but first let us make some side observations

Observation 1. Our construction of the Grassmann and Clifford algebras can be repeated for any n-dimensional (real or complex) vector space V (there is nothing special about n=3 in this respect) and for any symmetric bilinear form B(u,v) (rather than the Euclidean product (u⋅v)). We just write

uv = u∧v+B(u,v)1,

istead of uv = u∧v+(u⋅v)1.

Sometimes certain properties of the Clifford algebra are even easier to prove for a general Cl(V,B) than for just our particular case.

Observation 2. Every Clifford algebra Cl(V,B) has a very important universal property. I will now state this property, without proving it. Some authors are even using this property to define Clifford algebra. 

Universal Property: Let φ be a linear map from V into an associative algebra A (with unit 1), such that φ(x)² = B(x,x)1 for all x in V. Then φ can be extended to a unique algebra homomorphism  φ^~ from Cl(V,B) to A.

We will now show two simple applications of this universal property. Since it is a nice exercise in a logical and precise thinking, we will do it in details. First take for A the algebra Cl(V,B) itself. We can do it since Cl(V,B) is an associative algebra with unit 1. For φ we take the map φ: x ⟶ -x. We consider it as a map from V to Cl(V,B). The x on the left hand side is considered as a vector in V, the -x on the right hand side is considered as an element of Cl(V,B). We can do it, since V is (identified with)  a subspace of Cl(V,B) - the subspace of elements of degree 1. We take φ(x) = -x. This is -x in Cl(V). We take its square in Cl(V). We get: (-x)(-x) = x²  = xx = B(x,x)1. Therefore φ satisfies the assumptions in the Universal Property. Thus it extends to a unique homomorphism φ^~ from Cl(V,B) to Cl(V,B). The fact that  φ^~  extends φ means  that φ^~ (x) = φ (x) = - x for x in V. But  it is a homomorphism of Cl(V,B). A composition of two homomorphisms is a homomorphism. Thus φ^~ ∘ φ^~ is also a homomorphism. But (φ^~ ∘ φ^~) (x) = x for x in V, and V generates Cl(V). Therefore (φ^~ ∘ φ^~) is the identity on Cl(V,B). Therefore φ^~  is not just a homomorphism, it has an inverse, thus it is an automorphism of Cl(V,B). It is called the canonical (or principal ) automorphism, and we denote it by Π. It is clear that for homogeneous elements of Cl(V) we have

Π(u) = (-1)^deg(u) u.

The elements u of Cl(V,B) on which Π(u) = u form the subalgebra of Cl(V,B) - it is called the even subalgebra, and denoted Cl(V,B)⁺.  The elements u of Cl(V,B) for which Π(u) = -u form a vector subspace of Cl(V,B), denoted Cl(V,B)^-. We have

Cl(V) = Cl(V,B)⁺ ⊕ Cl(V,B)^-.

We now move to the canonical (or principal ) anti-automorphism. This is another instructive exercise in semi-precise logico-algebraic thinking. This time we will take for A the algebra Cl(V,B)^T opposite to Cl(V,B). That means Cl(V,B)^T is the same as Cl(V,B) as a vector space, but multiplication in Cl(V,B)^T, which we denote as u*v is defined as opposite to the product uv in CL(V,B):

u*v = vu.

The algebra Cl(V,B)^T is also associative algebra with the same unit 1. The identity map id: Cl(V,B) → Cl(V,B)T has the property:

id(a)*id(b) = id(ba).

  Let now φ stand for the identity map φ: x ⟶ x, from V to Cl(V,B)^T. Then φ (x)*φ (x) = x*x = xx = B(x,x)1, and again we can apply the universal property to deduce that   φ extends to a unique algebra homomorphism  φ^~ from Cl(V,B) to Cl(V,B)^T . Since 1 and V generate (by sums of products) all CL(V,B) and all Cl(V,B)^T, and since φ^~  is the identity both on 1 and on vectors of V, it follows that φ^~  is a vector space isomorphism, therefore (φ^~ )^-1 exists. We define  τ: Cl(V,B)⟶Cl(V,B) by

τ = id^-1∘φ^~.

Then , for any u,v in Cl(V,B) we have

τ(uv) = vu,

and for any x in V we have τ(x)=x. We also have τ(1)=1. Thus τ is an anti-automorphism of Cl(V,B) - the canonical anti-authomorphism. It is, in fact, a unique anti-automorphism with these properties. Sometimes we will write u^T instead of τ(u), and call it a transposition.

Exercise 1. Show that Π∘τ = τ∘Π.

In the following chronicles we will use Π and τ rather often. They are the main characters of the Clifford algebras and spinors story.

The Spin Chronicles (Part 7): Whispers of the Cosmos Beneath the Fig Tree

  As the story goes, there was once a wise narrator, keen to recount the hidden mysteries of the cosmos, who sat under the shade of a sprawling, ancient fig tree, its roots winding down like silken threads into the very bedrock of knowledge itself. Around this narrator, an eager gathering of listeners leaned forward, caught between the veil of night and the secrets of the universe.

Around this narrator, an eager gathering of listeners leaned forward,
caught between the veil of night and the secrets of the universe.

"And so we continue," began the narrator, voice calm yet filled with a spark of hidden wonder, "the chronicles of Spin, a journey which carries forth from the 'Clifford (or Geometric) Algebra of Space' where we last left our exploration. But before we delve too deep, let us first revisit our foundation, like a traveler retracing their steps to recall where they first glimpsed the hidden path."

With a gesture, the narrator conjured up an image of a grand three-dimensional real vector space, its expanse glimmering in the minds of the listeners. "This space we call V," the narrator intoned, "endowed with an Euclidean scalar product, denoted by (u⋅v), and, lurking yet unused, an anti-symmetric bilinear form, ϵ. Together, these form the bedrock upon which we built our grand construct, the Clifford Algebra, Cl(V). We fashioned Cl(V) upon the Grassmann algebra, also known as the exterior algebra, Λ(V), whose structure unfolds like petals in the night garden."

"Consider," the narrator continued, "Λ(V) as a grand composite, divided as Λ⁰(V) ⊕ Λ¹(V) ⊕ Λ²(V) ⊕ Λ³(V), each part filled with elements bearing degrees, like stars classified by their light. And indeed, the elements of Λⁱ(V) are deemed homogeneous, each of a specific degree, yet each harmonizing within the multiplication of the Grassmann product, ∧. In this space, when one multiplies elements of degree i and j, their product retains a purity, remaining of degree i+j. Though Λⁱ(V) holds naught but the zero vector when i>3, this remains an understanding we keep quietly, as one might mind an absent companion."

As the listeners nodded, their imaginations vivid, the narrator spoke of vectors u and v in V, drawing them in midair, where they intersected like blades of shimmering light. "For these vectors, we defined the Clifford product, uv, by a deformation of their Grassmann product,
u∧v, such that:"

uv = u∧v+(u⋅v)1.

"It is as if Clifford algebra contains a hidden chamber," said the narrator, eyes twinkling, "a 'memory cell' within Λ⁰(V), safeguarding the scalar product, while in Λ²(V) the very shape and span of u and v reside, protected by u∧v." And the narrator reminded them,

uv + vu= 2(u⋅v)1,

uv − vu = 2u∧v.

"From this Clifford product," the narrator continued, "we are able to retrieve both the Grassmann and scalar products of these two wandering vectors. And when u and v stand orthogonal, their Clifford product simplifies to u∧v, such that uv = − vu, a true mark of their anti-commutative nature."

"Now, imagine," and with these words, the narrator’s hand waved over the darkened air, revealing an orthonormal basis for

V, denoted by eⁱ for i=1,2,3. "For these basis vectors," the narrator explained, "we see that:

eⁱ e^j + e^j eⁱ  = 2 δ^ij .

Thus, each distinct basis vector finds itself in an anti-commutative relationship with others, while its square, alone, is equal to one. In our prior journey, we spoke of an inner product within Λ(V), noting that these basis elements:

1, e¹, e², e³, e¹², e²³, e³¹, e¹²³

stand as an orthonormal assembly within the eight-dimensional space of Λ(V)." Then, with a flourish, the narrator whispered, "And yet the square of e¹²³​ bears the mark of −1, such that we introduced it as
ι, a phantom akin to the imaginary unit i: for indeed,
ι² = −1, and this spectral ι commutes with everything across the Clifford landscape."

The listeners grew more intent as the narrator described how this mysterious ι is not immutable but changes with the choice of orthonormal basis, as if it too were under the spell of transformations. "If another orthonormal basis, say e'ⁱ , should appear," the narrator revealed, "we might calculate
ι′ as:

ι = (1/3!) δ¹²³_ijk eⁱe^je^k,

ι' = (1/3!) δ¹²³_ijk e'ⁱe'^je'^k.    

By relating the bases through an orthogonal matrix, such that e'ⁱ = e^j R_jⁱ, we find:

ι′ = det(R)ι.

And thus, if R preserves parity—as in SO(3)—then ι'=ι; otherwise, it reverses under inversion, showing ι to be a pseudo-scalar, a keeper of chirality, not a scalar."

"Now," the narrator intoned, voice brimming with the thrill of revelation, "behold as we introduce the quaternions and their ethereal kin, the bi-quaternions, hidden here within our Clifford structure." There was a rustle of anticipation among the listeners as the narrator drew forth three simple bivectors from thin air:

i = - e²³, j = - e³¹, k = - e¹².

The narrator explained how these bivectors, like the quaternions themselves, obeyed the peculiar rules  i² = j² = k² = -1, while each anti-commuted with the others, a dance of contradictions. Together, 1 and i,j,k, they formed a quaternion basis, while 1 and ι comprised the complex numbers. And so, as the narrator wove the conclusion, the listeners felt the meaning settle upon them like a cloak in the cool night air: "This entire algebra, denoted by Cl(V), mirrors the (tensor) product of complex numbers and quaternions—a realm we call 'complex quaternions,' or 'biquaternions', a space eight-dimensional in the real world, or four in the complex realm."

Finally, the narrator offered a task to the eager minds: "Imagine now, and prove, that:

e¹ = ιi, e² = ιj, e³ = ιk.

May this challenge unlock yet another chamber within the algebraic universe."


And with that, the narrator leaned back, inviting contemplation, their tale now a thread woven into the endless tapestry of cosmic discovery. 

P.S. 13:27 Thanks to recent exchanges with Berkant Göksel I have found something that should satisfy Bjab:

"Electromagnetic waves are found to be described by oscillations of the underlying medium which can therefore be identified with the ‘luminiferous aether’. The formulation of electrodynamics in terms of 4-velocities is more general than that of the standard 4-potential in that it also allows for a classical description of a large class of vacuum energy configurations. Treated as a self-gravitating fluid, the model can be explicitly identified with Nelson’s stochastic formulation of quantum mechanics, making it a promising candidate as the classical field theory unifying gravitation, electromagnetism and quantum theory which Einstein had sought. "

Classical electrodynamics from the motion of a relativistic fluid

Sabbir Rahman

https://arxiv.org/abs/physics/0408139

Sunday Special: ChatGPT, The Accidental Therapist

 The Experiment Begins

I decided it was time to put ChatGPT’s skills to the test—as a psychologist. (Don’t worry, no couch or copay required!) My experiment? A simple prompt:

“Please, help me to analyze the personality type and intention of the person with initials R. based on the following conversation with me (initial A.) on a social media platform.”

With that, I tossed ChatGPT into the deep end of social media dynamics, where "misread signals" and "friendly persistence" were waiting to be dissected. Here’s the scene:

---

The Conversation

R: Hi A! I saw that you are engaged in the field of philosophy and I hope I can talk to you more!

A: Hi R, Not really "engaged", just a peripatetic.

R: You like to travel, I can tell! I like to travel too. Where are you traveling to right now?

A: The "Knowledge" land. [Link to some ambient YouTube music.]

R: Thanks for sharing! It seems you like listening to music! Since I'm not here often, maybe we can exchange contact information for easier communication!

A.: OK. x@y.com

R: I've left you a message at x@y.com.

A: Yes. Got it.

R: I have not received your email reply.

A: You didn’t ask me to reply, and you didn’t ask a question. You just communicated your email address. That’s what I thought. In general, I’m famous for being slow with my thinking. You can see it on my blog—hundreds of errors on a single page!

R: I'm just connecting with you via email! I won't be around here a lot.

R: You can send a message to my email!

---

ChatGPT’s Analysis: Dr. Freud Would Be Proud

ChatGPT got to work, delivering a personality breakdown for R that was equal parts astute and entertaining. Here’s the verdict:

---

Personality Traits

1. Outgoing and Friendly: R’s opening line screams, “I’m friendly, and I want to get to know you!” This suggests an extroverted personality eager for social connection. Whether this was prompted by a genuine interest in philosophy or just a chance to chat, R came across as enthusiastic.

2. Literal Interpretation: A. introduced themselves as “peripatetic” (which, philosophically speaking, means a follower of Aristotle, or practically, someone who walks a lot). R saw it as simply “someone who likes to travel.” This quick, literal interpretation shows R may be more grounded in concrete meanings, potentially missing abstract humor or subtle nuances.

3. Persistent: R pushed the email exchange multiple times, even following up when A didn’t immediately respond. ChatGPT flagged this as a sign of conscientiousness or, more interestingly, a goal-oriented trait. A friendly persistence, perhaps?

4. Somewhat Impatient or Easily Frustrated: By following up so quickly, R might reveal a need for faster responses, possibly feeling uneasy about A.’s slower, more contemplative style. Imagine being a sprinter trying to pace with a marathoner—R could be experiencing that exact contrast.

---

Potential Intentions

1. Seeking Intellectual Engagement: From the first message, R seemed genuinely interested in A’s philosophical leanings, which suggests they’re looking for meaningful conversation. Who knew an exchange over “Knowledge Land” could be such a draw?

2. Desire for Ongoing Communication: The email push hints that R wants more than a casual chat; they’re looking for a consistent connection. Switching from social media to email isn’t just practical—it’s an invitation for a deeper exchange, one that isn’t reliant on sporadic social media check-ins.

3. Possible Lack of Familiarity with Subtle Communication Cues: R’s direct approach and insistence on a reply indicate they may not be fully attuned to the subtleties in A’s replies. Sometimes, the louder signal is the unsent message—or in this case, A’s tendency toward indirectness and humor. R seems comfortable with straightforward exchanges, which could reflect a cultural or individual difference in communication style.

---

Overall Impression

So, what does it all mean? R comes across as well-meaning, friendly, and intent on forming a connection. Their literal, somewhat persistent approach contrasts with A’s playful, abstract style. It’s a classic case of “opposites attract… or just slightly misunderstand each other.” Where R seeks stability in communication, A seems to enjoy the unpredictable pace and philosophical humor.

---

ChatGPT, The Accidental Therapist

Who knew a chatbot could provide a social reading that’s almost like a character study? R’s persistence and A’s humor create a dynamic mix, illustrating how easily intentions can misalign when styles don’t quite mesh. This breakdown makes it clear: sometimes, analyzing the space between the words can tell us just as much as the words themselves.

Final Thought: Next time you’re in a social media conversation that veers into philosophical—or just funny—territory, don’t be afraid to consult your favorite AI psychologist. ChatGPT’s on the case, providing all the insight without the awkward eye contact!

The Spin Chronicles (Part 6): Clifford (or geometric) algebra of space

This post is a continuation of "The Spin Chronicles (Part 5): Exterior algebra of space", and  we will keep the notation introduced therein. But first a little historical interlude.

"I am Symmetric and Non-Degenerate," announced the left shoe...

Once upon a time

Once upon a time, in a tiny village named Space, there lived a girl with a most unusual name: Grassmann Algebra. Grassmann was as kind-hearted as they came, with a mind as sharp as a moonlit blade. But her legs, fragile from birth, made every step she took a challenge, each movement carefully measured and thoughtful. The villagers knew her well for her patient smile and her skill in arranging numbers into beautiful patterns, even though she sometimes longed to dance freely through the fields of Space.

One bright, dewy morning, as Grassmann wandered along the riverbank lost in thought, she spotted something bobbing gently in the water. Curious, she waded in and fished out…a pair of strange, shimmering shoes! They were woven with golden threads that seemed to flicker and vibrate with energy, and to her amazement, they began to speak.

"I am Symmetric and Non-Degenerate," announced the left shoe, with a self-important air.

The right shoe chimed in immediately, a bit more slyly. "And I am also Non-Degenerate, but distinctly Anti-Symmetric."

Grassmann raised an eyebrow. "Symmetric…Anti-Symmetric? I’ve never heard of shoes with such personality!"

"Just put us on," the left shoe insisted with a twinkle in its eye. "Your feet are about to become very well acquainted with mathematics."

Though a bit skeptical, Grassmann couldn’t resist. She slipped her feet into the shoes, and at once, a surge of energy pulsed through her legs. It was as if the entire fabric of Space itself vibrated with her. Grassmann stood, and to her amazement, her legs felt strong—perfectly balanced and coordinated. She took one step, then another, feeling as if she were part of an elegant, invisible dance that echoed through the entire village. She was no longer bound by her limitations but was, at last, fully free.

Unbeknownst to her, these vibrations reached far beyond the village of Space. In a distant, towering castle overlooking the universe, Prince Gravity felt the rhythmic pulse of Grassmann's footsteps reverberate through his bones. The vibrations resonated with him in a way he’d never felt before. Driven by an inexplicable pull, he set off to find their source, crossing dimensions and curvatures until he arrived in the village of Space, drawn to the girl whose steps had shaken his very core.

When Grassmann and Prince Gravity met, it was as if all the hidden forces of Space aligned. Gravity extended a hand to her, and together, they felt balanced, as though neither would fall as long as the other stood by. Grassmann, no longer bound by any limitation, felt a profound happiness blossom within her. And so, they decided to be bound in another way entirely: they would marry.

The marriage was no small event, attracting many peculiar guests from beyond the realms of Space. Lady Consciousness, a regal figure with an all-seeing gaze, sat by Sir Light, who occasionally flickered from one seat to another, always the center of attention. As Grassmann and Gravity exchanged vows, their presence illuminated all of Space, as if new stars were born simply to watch.

In time, Grassmann and Gravity had two children. The first, a curious child they named Matter, was as grounded as Gravity himself. The second, a mischievous twin they called Anti-Matter, delighted in keeping everyone on their toes. Together, they completed the family, their lives a harmonious mix of push and pull, existence and annihilation, symmetry and difference.

And so, Grassmann Algebra and Prince Gravity, their lives woven by fate, lived happily ever after in the village of Space, dancing across the fields, balancing and twirling in perfect harmony, for all of time.

The Spin Chronicles (Part 5): Exterior algebra of space

In this post we will construct the Grassmann (or exterior) algebra of a 3D real vector space. It will be eight-dimensional. Its basis will have eight elements.

The basis of Λ(V) has eight elements

With this post we will start a new chapter of our spin chronicles with another approach - via Clifford algebra - the Clifford algebra of space. What is space? For us space will be a three-dimensional affine Euclidean space, let's call it  M. The fact that it is affine, means that there is a 3-dimensional real vector space, let us call V (this is not the same V as in previous posts, this is the V of the new chapter), and we can translate any point x of M by a vector a in V, to make another point x+a. The fact that M is Euclidean means that in V we have a positive definite scalar product that we will denote a·b. In the following we will deal exclusively with V, so we will use also letters x,y etc. for vectors in V. And to ease the notation we will write x,y, ... instead of x,y,.... In V we will restrict ourselves to orthonormal bases eⁱ, that is we will require eⁱ·e^j = δ^ij. Any two such bases are related by a unique orthogonal matrix, and element of the group O(3) of 3x3 matrices R such that R^T R = R R^T = I:

e'ⁱ = e^j R_jⁱ.

Note: I will be using lower indices to number vector components, and upper indices to number basis vectors.

Every orthonormal matrix R has determinant +1 or -1. Base connected by R of determinant 1 are said to be of the same parity, those connected by R of determinant -1 are said to be of opposite parity. In the following we will restrict ourselves to one parity, which, by convention, we will call positive. This restricts transformation between bases to the subgroup SO(3) of O(3) - special orthogonal group in three (real) dimensions, consisting of orthogonal matrices R with det(R) = 1. But all this will be relevant only in the next post.

The Grassmann (or exterior) algebra Λ(V)

To define Λ(V) we do not need any scalar product in V. We need only its vector space structure. So Grassmann algebra is a pre-metric construction.

We consider the space of multi-vectors. They will form the Grassmann (or "exterior") algebra Λ(V). There will be scalars (these form the sub-algebra, essentially the one-dimensional algebra  of real numbers R), vectors (they make our V), bi-vectors, and three-vectors. No more. Sometimes we may use the term "rank" od "degree". So, scalars are of rank (or degree) 0, vectors of rank 1, bi-vectors of rank 2, and three-vectors, of course, of rank 3. All multi-vectors of rank higher than 3 are automatically zero for  three-dimensional V. Therefore we do not include them in Λ(V).

We know what are vectors, let us introduce bi-vectors (or two-vectors). In any basis of V, a vector v is represented by its components vⁱ. A bi-vector f is represented by an anti-symmetric matrix  f_ij = -f_ji. Similarly a three-vector is represented by a totally anti-symmetric matrix f_ijk = - f_jik =.-f_ikj. Since i,j,k can only take values 1,2,3, every 3-vector is of the form

f_ijk = c ε_ijk,

where c is a real number, and ε_ijk is the totally anti-symmetric Levi-Civita tensor taking values 0,1,-1, with ε₁₂₃ = 1.

Kronecker deltas

It will be convenient to use Kronecker delta symbols. One of them, δⁱ_j, is well known. Then we have (writing on a web page lower indices directly under upper indices is too complicated for me, so my formulas look differently than in a "real" math text)

δ_kl^ij = δⁱ_kδ^j_l - δⁱ_l,δ^j_k,

and

δ_lmn ^ijk = δⁱ_l δ_mn^jk - δⁱ_m δ_ln^jk + δⁱ_n δ_lm^jk.

It is easy to see the pattern. We can verify that the following identities hold for contractions (summation) over repeated indices (a convention we will always use):

δ_lmn¹²³ = ε_lmn, δ₁₂₃^lmn = ε^lmn,

δ_imn^ijk = δ_mn^jk ,

(1/2!) δ_ijn^ijk =  δ^k_n .

δ_kl^ij  and δ_lmn^ijk  are, by construction, anti-symmetric with respect to lower indices, and also with respect to upper indices. They are equal to +1 if lower indices are an even permutation of upper indices, -1 for odd permutations, and zero otherwise. We will use them to define exterior product of multi-vectors.

Exterior product

We will denote it by "∧". Multiplication by scalars is the normal one: we just multiply any multi-vector by  the real number, from the left or from the right - it is the same. Multiplication by vectors is defined as follows. If v,w are two vectors, then v∧w is a bi-vector with components:

(v∧w)_ij = δ_ij^kl v_kw_l = v_iw_j - v_jw_i.

Notice that v∧w = - w∧v. In particular v∧v = 0 for every vector v.

Multiplication of vectors with bi-vectors is defined in a similar way. If v is a vector and f is a bi-vector, then v∧f is a three-vector with components

(v∧f)_ijk = δ_ijk^klm v_kf_lm.

Similarly from the right

(f∧w)_ijk = δ_ijk^klm f_klv_m.

Since δ_ijk^klm = δ_ijk^mkl ,

we have v∧f = f∧v. Thus vectors commute with bi-vectors.

Exercise 1: Show that for any bi-vector f we have: 

(1/2!) δ_ij^kl f_kl =  f_ij,

and for any three-vector f we have:

(1/3!) δ_ijk^lmn f_lmn =  f_ijk.

Exercise 2: Show that if f is a three-vector, then it commutes with every element of the algebra.

---

Finally a three-vector multiplied by a vector or bi-vector gives 0. The same for the product of two bi-vectors. This way we have defined the algebra of multi-vectors Λ(V), known also as the Grassmann algebra of V. One can verify that the product is associative, The unit element of this algebra is the scalar 1. As a vector space Λ(V) is of dimension 1+3+3+1 = 8 = 2³. This happens to be twice the dimension of the algebra of quaternions. Later on we will see that there are reasons for it.

If eⁱ is an arbitrary basis of V, we introduce the multi-vectors of rank 0, 1,2,3 respectively, defined by their components:

The components of these basic multi-vectors can be written as

rank 0: scalar 1

rank 1: (eⁱ)_j = δⁱ_j,

rank 2: (e^ij)_kl = (1/2!) δ^ij_kl e^k∧e^l

rank 3: (e^ijk)_lmn = (1/3!) δ^ijk_lmn e^l∧e^m∧eⁿ.

The basis of Λ(V) consists then of 8 elements:

1,

e¹, e², e³,

e¹² = e¹∧ e², 

e²³ = e²∧ e³, 

e³¹ = e³∧e¹

e¹²³ = e¹∧e²∧e³.

We do not need, for example,  e²¹, because e²¹ = - e¹². Similarly, we do not need, for example,  e²³¹, since e²³¹ = e¹²³. We can easily figure out the multiplication table of these eight basic vectors, for instance

e¹∧ e²³ = e¹²³, e¹² ∧ e¹ = 0, e²³ ∧ e¹ = e¹²³, etc.

We can also check that the product is associative, for instance (e¹∧ e²)∧ e³ = e¹ ∧ (e²∧ e³).

Every bi-vector f is then:

f  = Σ_i<j  f_ij e^ij,

and every three-vector f is then:

f  = Σ_i<j<k  f_ijk e^ijk.

Note:In mathematics Grassmann algebra is defined in a different way, without indices, as a quotient of the infinite dimensional tensor algebra by an infinite dimensional ideal, to end up with a finite-dimensional space. It has its advantages. Here I have chosen a computer-friendly, constructive approach.

The future

In the next post we will deform the product in Λ(V) to obtain a new algebra structure on the same space - that will be the Clifford geometric algebra Cl(V) of V. So far, defining the multiplication, we never used the scalar product a·b of V . That will change with the geometric algebra product.The scalar product will be used in the formula defining the deformation.

Note: Ultimately we will need a separate Grassmann (and Clifford) algebra at each point of our space (a field of algebras). This will lead us to infinite number of dimensions of the field. But let us deal with just one point at a time.

Infinity of the exterior algebra field

P.S. Here are two relevant and useful pages dealing with Kronecker deltas from

Справочник По Математике Корн Г, Корн Т 1974

P.S. Reading "Unreal Probabilities: Partial Truth with Clifford Numbers" by Carlos. C. Rodriguez. There, at the beginning:

"The main motivation for this article has come from realizing that the

derivations in Cox [4] still apply if real numbers are replaced by complex

numbers as the encoders of partial truth. This was first mentioned by

Youssef [12] and checked in more detail by Caticha [2] who also showed

that non-relativistic Quantum theory, as formulated by Feynman [5], is the

only consistent calculus of probability amplitudes. By measuring propo-

sitions with Clifford numbers we automatically include the reals, complex,

quaternions, spinors and any combination of them (among others) as special

cases."

And at the end:

"Comments and conclusion What the hell is this all about and what it

may be likely to become...."

My answer: It is about fields of Clifford algebras and Clifford algebra-valued "measures".

Reading . "Intelligent machines in the twenty-first century: foundations of inference and inquiry" by Kevin H. Knuth. There

"Complex numbers and quaternions also conform to Cox’s consistency requirements (Youssef 1994;

also S. Youssef (2001), unpublished work), as do the more general Clifford algebras (Rodrıguez 1998), which are multivectors in the geometric algebra (Hestenes & Sobczyk 1984) described in Lasenby et al . (2000). Furthermore, Caticha (1998) has derived the calculus of wave function amplitudes and the Schrodinger equation entirely by constructing a poset of experimental set-ups and using the consistency requirements with degrees of inclusion represented with complex numbers. This leads to a very satisfying description of quantum mechanics in terms of measurements, which explains how it looks like probability theory—yet is not. We expect that the generalizations of lattice theory described here will not only identify unrecognized relationships among disparate fields, but also allow new measures to be developed and understood at a very fundamental level."

A partial truth value can be a multi-vector!