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:

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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:

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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.

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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!

Sunday Special - Consciousness, Quanta, and Meaning

 There’s no shortage of theories trying to explain the mysteries of quantum mechanics, and consciousness often gets dragged into the mix. Some even claim it has a special role in state vector reduction—those quirky “quantum jumps.” Roger Penrose is one such advocate, but he’s not alone. When you ask these consciousness enthusiasts if physics behaves differently on the far side of the Moon (where no one’s watching) or if atoms over there just stop jumping, the answers get...let’s say, "evasive."

Who's Consciousness Is It Anyway?

Now, I'm not here to knock anyone’s ideas. Everyone is entitled to their opinions. But, honestly, this view feels a bit like trying to peer through a foggy window. I mean, whose consciousness are we talking about here? When John Wheeler, another big name in the "consciousness club," was pushed on this point, he eventually said it was “consciousness in general.” Sounds poetic, right? The problem is, no one knows what “consciousness in general” actually means.

And let’s be real—science knows more about quarks and gluons than it does about human consciousness. So, no, I don’t think consciousness is irrelevant, but I do think it’s being overhyped in quantum mechanics.

Is the Universe Watching Itself?

Measurements and observations in the universe happen, whether we’re looking or not. The cosmos doesn’t wait for us to wake up in the morning to start its business. Atoms are constantly observing other atoms, cats are keeping tabs on mice, and things happen regardless of our awareness.

This brings me to my simple, no-nonsense perspective: consciousness doesn’t create reality, it just interprets it. Consciousness, in my view, is there to give meaning to what we observe. The more complex the consciousness, the deeper and richer the meaning.

The meaning of it all?

The Meaning of Consciousness

Here’s the kicker: the higher your level of consciousness, the more meaning you can extract from an experience. A cat sees the world differently than bacteria, and we humans have the potential to see meaning on levels a cat can’t even dream of. This ability to find meaning ties into free will—we see meaning in something, and it allows us to make conscious decisions, changing ourselves and reality. It’s like designing your own narrative rather than just playing out some mechanical script.

So, no, I don’t believe in consciousness collapsing wave functions. But that doesn’t mean it’s useless—it’s just not responsible for playing referee in quantum mechanics.

Gravity vs. Consciousness: The Final Showdown

Now, if consciousness isn’t out there reducing wave packets, then what is? Gravity, maybe? Some theories suggest gravity might be responsible for collapsing quantum states. But here’s the twist—what if wave function collapse actually explains gravity? It’s one of those chicken-and-egg scenarios, except the chicken is quantum mechanics and the egg is the entire fabric of spacetime.

I don’t claim to know the definitive answer to what reduces wave packets, but I do know how to describe the process. And that’s what matters. A simple, quantitative description that satisfies both quantum theory and the observations of a humble peasant (equipped with the finest metaphorical pitchfork, of course).

The Spin Chronicles (Part 4): spin up and down

This post is a continuation of "The Spin Chronicles (Part 3): Spin frames". It is purely technical. This previous post ended with:

"In other words, when you rotate an element of O(V) by an SU(2) matrix A, the corresponding lab element rotates, too, by and SO(3) matrix R(A). It’s like a cosmic interlock between spinors and real-world rotations."

Today we will see how this can be realized, and, in fact, is realized, in quantum mechanics. I should say that I am not happy with this way of dealing with spin, but, at the moment, this is the best I can do. What I present here is just one way of dealing with spinors. There are other formulations possible, but, in fact, they are all equivalent and equally unsatisfactory.

I should say that I am not happy with this way of dealing with spin

We will treat V as a Hilbert space, with its scalar product (u|v).

Note: Here, more in the spirit of quantum physics we will use the notation (u|v) instead of (u,v) for the scalar product.

Once we have a Hilbert space, we have linear operators acting on this space. Let us denote the space of linear operators  by L(V). Then L(V) becomes itself a Hilbert space if we define the scalar product by

(A|B) = Tr(A*B),

where (*) denotes the Hermitian conjugation, and "Tr" stands for "trace". In an orthonormal basis operators are represented by matrices, and (*) becomes the ordinary Hermitian conjugation of matrices (complex conjugate and transpose), while trace becomes the sum of diagonal elements of the matrix. Notice that (A|B)* = (B|A), where now (*) is applied to a complex number, and denotes the ordinary complex conjugation, usually denoted by a "bar" over that number. But "bar" is easy in Latex, and not really supported on the web page. So I will stick with the (*) notation.

In L(V) we have, in particular, Hermitian operators. The set of all Hermitian operators is a real vector space. We denote it by H(V). For Hermitian A,B the scalar product (A|B) is real. So H(V) is a real vector space, with a real scalar product. In order to see what kind of scalar product it is, we need a basis in H(A), So, let us construct one. First of all, given any two unit vectors u,v in V, where where "unit" means (u|u) = (v|v) = 1,we denote by |u)(v| the operator in L(V) defined as

|u)(v| w = (v|w) u.

Then (|u)(v|)*= |v)(u|

Then |u)(u| is the orthogonal projection operator on u. It acts, by definition, as

|u)(u| v =(u|v)u

It is Hermitian (thus in H(V)), and idempotent (that means ( |u)(u| )² = |u)(u|. Its eigenvalues are 1 and 0. Its eigenvectors are u (for eigenvalue 1), and 0 for any non-zero vector orthogonal to u). Sometimes it is easier yo write P_u instead of |u)(u|. So we have P_u=P_u*, and P_u2 = P_u, (P_u|P_u) = 1. We will construct a basis in H(V) from a basis in V. But first let us talk some "physics". Let us think of "spin in a given direction" as a binary quantity. It can be "up" or "down". So, we want to associate with its "measurement" a Hermitian operator with eigenvalues +1 and -1, +1 for "up" and -1 for "down". But which one? The answer, as far as I know, is this: "anyone", but it should be done in a "consistent" way. We can do it as follows: first choose an orthonormal basis e₁,e₂ in V. Then we have the following (easy to prove) Theorem:

Theorem: Every Hermitian operator with eigenvalues ±1 can be uniquely represented as σ(n) = n₁σ₁+n₂σ₂+n₃σ₃, where
σ₁,σ₂,σ₃ are the Pauli matrices

Pauli matrices

and n is a unit vector in R³. Conversely, if n is a unit vector in R³, then σ(n) represents a Hermitian operator with eigenvalues ±1 in the basis e₁,e₂. Moreover, if A is any matrix from SU(2), then

A σ(n) A* = σ( R(A)n ).

Now we associate spin "up" in "z" direction in the lab with n =(0,0,1). The formula above allows us to consistently associate Hermitian operator, namely σ(n), with "spin up" in any direction.

Why is that not quite satisfactory? The problem is that we have chosen the particular Pauli matrices above. Why these and not other? If U is any unitary matrix, then the matrices U σ_i U* would do as well (with a modified R(A)) . This seems to tell us that these "spin directions" are in a space that looks and behaves as "our space", but it is not "our space". It is a mystery that needs to be understood. 

It is a mystery that needs to be understood.  

We will continue these considerations in the next post.

P.S. 19-10-24 This is my reply to the comment Bjab October 19, 2024 at 10:11 AM. O(V) was defined in the previous post as the set of special orthonormal frames. Taking C² for V, as in the example in that post, for special orthonormal frame we require

1) (e_i|e_j) = δ_ij,

2) ε(e_i,e_j) = ε_ij.

where

(u|v) = u₁* v₁ + u₂* v₂, =u*v,
ε(u,v) = u₁ v₂ - u₂ v₁.= u^T ε v,

where ε on the right hand side denotes the matrix

0	1
-1	0

One special orthonormal basis is e₁ = (1,0)^T, e₂ = (0,1)^T. If e'₁,e'₂ is another special orthonormal frame, they are related by some invertible complex matrix A: e'_i = ej A^j_i. Then from 1) it follows that the matrix must be unitary, while from 2) it follows that

ε = A^T ε A,

which implies that det(A) = 1. Therefore A is SU(2).  Conversely, any such A transforms special orthonormal frame into a special orthonormal frame. Thus the space of all special orthonormal frames has the same dimension as SU(2), that is 3.

The Spin Chronicles (Part 3): Spin frames

What’s a Spin Axis, Anyway?

This post was inspired by a question from Bjab, who asked: "What do you mean by ‘spin axis’?" Well, buckle up because I’m about to explain that! I’m not entirely thrilled with the answer I have right now, but hey—it works (kind of). Hopefully, in the near future, I'll come up with something that satisfies both Bjab and my restless curiosity. For now we have to be happy with the standard concept of the spin structure - representing the universe as a set of interlocked rotations - described below.

Spin structure

Where Do Spinors Live?

Imagine a strange and mysterious land—let’s call it V—a two-dimensional complex vector space. It’s not just any space, though. It has "structure." Picture it equipped with a hermitian non-degenerate scalar product that we’ll denote by $(u,v)$, and a bilinear antisymmetric non-degenerate form we’ll call $\epsilon$. Sounds fancy, right? These two should be related in a specific way, and here’s how:

There exists a special kind of basis for this space, which we’ll call “special orthonormal.” It satisfies:

1. $(e_1,e_1) = (e_2,e_2) = 1$,
2. $(e_1, e_2) = 0$,
3. $ϵ(e_1,e_2)=1.$

“Special” indeed. The first natural question is: Can this magical basis even exist?

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The Magic of C²: Relax, It’s All Possible

Yes, it exists! Here’s why. Let’s choose any basis in V. Using this, we can identify V with $C^2$ (our familiar two-dimensional complex space). In $C^2$, the scalar product $(u,v)$ is defined as:

$$(u,v)=u^{\mathrm{1\ast }}{v}^1+u^{\mathrm{2\ast }}{v}^2$$

where the star (*) represents complex conjugation. As for $\epsilon$, we define it as:

$$ϵ(u,v)=u^1{v}^2-u^2{v}^1$$

Now, using a standard basis $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, it’s easy to check that our special orthonormal basis conditions hold. So, crisis averted! We don’t need to worry about whether this basis exists—it does. Phew.

Oh, and here’s a fun fact: if $(e_1, e_2)$ is a special orthonormal basis, then $(-e_1, -e_2)$ is also a special orthonormal basis. Isn’t symmetry beautiful?

---

Back to the Mysterious World of V

So, we’ve got our scalar product, our fancy $\epsilon$, and we’re cruising in V—home to the enigmatic spinors. But are there other “special orthonormal” bases? And if so, how many?

Let’s say $e'_1, e'_2$ is another basis. There exists some matrix $A$ such that:

$$e_i^{\mathrm{\prime }}=e_j{A}_i^j(i,j=1,2)$$

(Yes, that’s a summation over “j”.)

Now, here’s the neat part: $e'_1, e'_2$ will also be special orthonormal if and only if the matrix $A$ is unitary with determinant 1. Matrices like this form a group called SU(2).

By the way, SU(2) is isomorphic to the group of unit quaternions. I won’t dive into that now, but it’s a pretty cool connection between algebra and geometry.

---

The Spinor Connection

Let’s denote by O(V) the space of all these special orthonormal bases. The group SU(2) acts on O(V) from the right, both transitively and freely. The elements of O(V) are what we call spinor bases, and the elements of V are simply spinors. If $u$ is a spinor, and you change the basis using an SU(2) matrix $A$, the components of the spinor change according to this formula:

$$u^{\mathrm{\prime }i}=A_j^i{u}^j$$

Got it? Good! Now, let’s connect this with something real, like the lab—where actual experiments happen (sometimes with coffee stains on the data sheets).

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What About Our Lab?

To bridge this to the physical world, we need some more algebra (don’t worry, just a little more). Quaternions are often used for handling 3D rotations, but since we’re sticking with matrices, here’s what we do in quantum mechanics:

For each point $x = (x^1, x^2, x^3)$ in $R^3$, associate the following 2x2 Hermitian matrix, $\sigma(x)$:

$$\sigma (\mathrm{<b>x</b>})=\left(\begin{array}{cc}{x}^3& x^1-i{x}^2\\ x^1+i{x}^2& -x^3\end{array}\right)$$

Now, for any SU(2) matrix $A$, it turns out:

$$A\sigma (\mathrm{<b>x</b>})A^{\ast }=\sigma (R(A)\mathrm{<b>x</b>})$$

where $R(A)$ is a 3x3 real rotation matrix, an element of the special orthogonal group SO(3). In fact, the map from $A\to R(A)is\; a\; homomorphism.\; And\; every\; element\; of\; SO(3)\; can\; be\; generated\; this\; way.\; The\; catch?\; If$$A$ and $B$ are two SU(2) elements, then $R(A) = R(B)$ if and only if $B = \pm A$.

Hence, SU(2) is a double cover of SO(3). It’s like the universe giving you two chances to rotate.

Note: With quaternions it would look more "natural". To each x we would associate a pure imaginary quaternion 

q(x) = xi+yj+zk, 

and for each unit quaternion u, we would have 

uq(x)u* = q(R(u)x), 

where R(u) is in SO(3). 

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The Spin Structure: So What About That Spin Axis?

Let’s bring this back to the real world—our lab, where we have Cartesian coordinates. Any two right-handed orthogonal systems (say, different setups in the same room) are related by a unique rotation matrix from SO(3). The set of all such frames forms a space O(L), which SO(3) acts on.

Now, here’s the kicker: nature somehow maps each element of O(V) (the spinor world) to an element of O(L) (our lab’s coordinate system) in a way that respects the rotations. We call this map the spin structure.

In other words, when you rotate an element of O(V) by an SU(2) matrix A, the corresponding lab element rotates, too, By and SO(3) matrix R(A). It’s like a cosmic interlock between spinors and real-world rotations.

Note (for experts): This is a baby version of a spin structure, good for a 3D flat space or, more generally, for a space with a distant parallelism. But this baby version is all that we need for now.

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What’s Next?

All this brings us closer to answering the question: what do I mean by a spin axis? Stay tuned, because we’ll tackle that mystery in the next post! We will also take under the loop the spin structure - the most important point here.

P.S. I am experimenting with the mathematical notation on this blog. Sometimes I mess up, then correct what can be corrected.

P.S. I have uploaded the pdf version of this post to academia.edu.

P.S. 18-10-24 9:33 Igor Bayak (1.1.6)

The Spin Chronicles: Painting Quantum Tori (Part 2)

 The Mysterious Phase of Quantum Mechanics: A Spin on Reality

In quantum mechanics, the phase of a wave function is treated like that distant cousin you only hear about but never meet—it's assumed to be unobservable. The star of the show is the amplitude, or more precisely, the square of its modulus. That’s the part we can see and measure. Textbooks say: "Only this, and nothing else, is observable." And sure, that’s one way of looking at it.

But let's face it—opinions in quantum mechanics are as varied as the stars in the sky. Physicists are like travelers in a big, wide world, each with their own unique map of how quantum mechanics works. In fact, no two physicists share exactly the same view—unless, of course, one of them is just crunching numbers from a cookbook of equations. And even then, not all cookbooks are created equal!

The Wave Function and the Art of Spin

We often talk about the "wave function" when we’re trying to pinpoint the location of an object. You know, that quantum object that seems to be everywhere and nowhere at the same time? But today, let's zoom in on something a little more grounded: spin.

Now, spin is a different beast. The object in question stays put, but its spin axis can change direction. This is where we enter the world of the state vector. For spin-½ particles, this vector is made up of four real numbers: $X$, $Y$, $Z$, and $W$. Their squares add up to one, which is pretty neat. If you’re a fan of complex numbers (and let’s be real, who isn’t?), it’s just two complex numbers whose modulus squares also sum to one. The conversion between the two is surprisingly simple:

$$z_1=X+iY\text{and}{z}_2=Z+iW$$

This lets us break the real part and imaginary part down like so:

$$X=\text{Re}(z_1),Y=\text{Im}(z_1),Z=\text{Re}(z_2),W=\text{Im}(z_2)$$

Easy, right? Well, it gets more interesting.

From Numbers to Angles: Theta, Phi, and Psi

Instead of juggling numbers like $X$, $Y$, $Z$, and $W$, it’s often more intuitive to use angles—

θ, ϕ, and ψ. 

These angles help specify the position of our vector on a three-dimensional sphere that lives in four-dimensional space. So, the coordinates transform into:

$$X = \sin\left(\frac{\phi}{2}\right)\cos(\psi), \quad Y = \sin\left(\frac{\phi}{2}\right)\sin(\psi)$$
$$Z = \cos\left(\frac{\phi}{2}\right)\cos(\psi + \theta), \quad W = \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)$$

The Projection into 3D Space

At this point, we need to bring things back into the three-dimensional world we're used to. To do that, we use something called stereographic projection, which I’ve covered in detail a couple of blog posts ago. This allows us to map the points from four-dimensional space into our three-dimensional world. The result gives us familiar coordinates:

$$x(\theta, \phi, \psi) = \frac{\sin\left(\frac{\phi}{2}\right)\cos(\psi)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$ $$y(\theta, \phi, \psi) = \frac{\sin\left(\frac{\phi}{2}\right)\sin(\psi)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$ $$z(\theta, \phi, \psi) = \frac{\cos\left(\frac{\phi}{2}\right)\cos(\psi + \theta)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$

What's the Deal with Theta, Phi, and Psi?

So what’s the physical meaning behind these angles? In our lab, the $z$-axis points upwards (think of it as "North" on a globe), and the ϕ angle measures the tilt of the spin axis—essentially, it's latitude. θ is the longitude, with the x-axis as 0 degrees and the y-axis at 90 degrees (or π/2 if you’re feeling mathematically fancy).

But the ψ angle? That’s where things get interesting. Psi is like the "invisible phase," a ghostly presence that Richard Feynman himself couldn't ignore when he wrote about spin. And today, we’re going to shine a light on this elusive angle.

Painting the Invisible: Great Circles and Wheels

Our mission here is to bring these invisible aspects of the state vector to life. As we vary psi from 0 to $2\pi$, something magical happens—a closed curve appears. This curve, on a three-dimensional sphere in four-dimensional space, is known as a "great circle."

You’ve seen great circles before—they’re the big loops you get when slicing through a globe. Meridians and the equator are great circles. The same idea applies here, except now we’re in four dimensions. These invisible great circles can be projected back into our familiar 3D space as... wait for it... circles! That’s why this blog post is titled Invisible Wheels.

A Torus of Invisible Wheels

Now, imagine a large torus made of these invisible circles—these are called Villarceau circles, similar to what you’d find in the architecture of Strasbourg Cathedral. But here’s the catch: each of these circles is perceived as just a single point—a spin pointing in a particular direction ($theta, phi$). The psi phase, which determines where we are on that circle, remains hidden from us.

Want to see what these circles look like? I whipped up some visualizations in Mathematica, and here’s what I got: 

X[theta_, phi_, psi_] = Sin[phi/2]*Cos[psi];

Y[theta_, phi_, psi_] = Sin[phi/2]*Sin[psi];

Z[theta_, phi_, psi_] = Cos[phi/2]*Cos[psi + theta];

W[theta_, phi_, psi_] = Cos[phi/2]*Sin[psi + theta];

ParametricPlot3D[Table[

  {X[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p]), 

   Y[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p]), 

   Z[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p])}, {i, 0, 35}], {p, 0,

   2 Pi}, PlotRange -> All, PlotStyle -> White, Background -> Black]

Still not satisfied? I’m currently experimenting with MathMod to generate even better images, but for now, this will have to do.

Wrapping Up

In the next post, we’ll explore how these tori and invisible wheels interact with each other. We’ll dive even deeper into their physical interpretation, so stay tuned!

For those of you curious about 4D visualizations, check out Dimensions Math. Search for "Hopf fibration"—it’s the key to everything we’ve discussed, even though I haven’t officially mentioned it... yet.

Until next time, keep spinning those wheels—visible or not!