Spin Chronicles Part 37: What we do with states

Hamilton discovered the algebra of quaternions on October 16, 1843. He wrote about this:

"They started into life, or light, full grown, on the 16th of October, 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge. "

They were strange creatures at this time. Euler was very close to their discovery already 100 years before. Gauss discovered them 20 years before. But these two great mathematicians did not realize that they, quaternions, will be important. Hamilton discovered them all by himself and realized their importance. Of course not without obstacles. Lord Kelvin, the famous Scottish physicist, commented on Hamilton's discovery with a warning: "Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way."

Algebra is an unmixed evil indeed. What to do with an algebra? Multiply its elements one by another? Divide one by another? In 1858 Cayley discovered matrices and realized the abstract algebra of quaternions as a particular subalgebra of Mat(2,C). Quaternions became ready to do some real work. Matrices act on vectors creating new vectors. Cayley has found a representation of Hamilton's algebra.

Representations of abstract algebras as "algebras of operators" is today big branch of mathematics, with applications to physics, in particular to quantum physics. To the set of all (equivalence classes of) irreducible representations of a C*-algebra A has been given a name: the spectrum of A. So algebras have their spectra. Studying spectra of atoms gave birth to quantum theory. But algebras, as it seems, have their own ways of producing some kind of "light".

Here we are studying just one monadic-type of algebra - the geometric algebra of space. We should not expect more than one light ray coming from it. Perhaps two. Our algebra is isomorphic to the algebra of biquaternions (complex quaternions), thus double the size of Cayley algebra. We have already found its irreducible representations by looking for non-trivial left ideals. The left regular representation of our A can be decomposed into a direct sum of two equivalent representations (see Part 31)

A = F⊕F^⟂,

where F is the left ideal Ap, p=(1+n)/2, F^⟂=A(1-p).

As announced in my previous post we will take the same task, but using a different perspective - will use the Gelfand-Neumark construction (today known under the name GNS-construction). The GNS machine has been developed for the much more interesting infinite-dimensional case. Here  we will use it for the simplest possible case of a 4-dimensional complex algebra. In finite dimension we can disregard all talk about continuity, because all finite dimensional linear maps are automatically continuous. We can disregard all talk about one subspace being dense in another, because every finite-dimensional linear subspace is automatically closed. What remains of the Gelfand-Neumark construction is a pure algebra. However GNS construction is more "physics oriented", where by "physics" I mainly mean the lessons we have learned from quantum mechanics. And so, we will meet the important concept of "positivity". Here by positivity I shall always mean "non-negativity". Positivity is related to the fact that probabilities are usually being considered as positive, taking values in the interval [0,1]. For real numbers being positive can be defined as being a square of some other real number. For complex numbers being positive is the same as being of the form a*a, where a is another complex number, and a* denotes the complex conjugate of a. Quantum theory suggests us that this last definition also seems to work well with noncommutative *-algebras. So we define an element of our *-algebra A to be positive if it can be written as a*a, where now a* is the antilinear anti-automorphism in A. Now if b is positive, it is automatically Hermitian : b*=b (Why?). Hermitian elements in quantum theory are usually called "observables", positive observables are those that have positive eigenvalues - their spectrum is on the positive real axis. This is not evident from the definition, but it can be proved without great difficulties.

Note: It is rather intuitive, but it takes some real effort to prove it from the above definition that the sum of two positive elements is positive. But it is so.

Of course here we meet a big interpretational problem: what is the meaning of the algebra product for two noncommuting algebra elements a,b? Or even what is meaning of a+b when a and b do not commute? It is not a surprise that Feynman declared that nobody understands quantum theory. Of course many physicists and mathematicians work hard to find a way around these problems (quantum logic, Jordan algebras, noncommutative probability, "effects", etc.), but none of these many proposals has been generally accepted as "the solution". The problem still exists, and waits for a satisfactory answer. "Shut up and calculate "is not a fully satisfactory answer.

The next important ingredient of the construction is the concept of state. A state is a normalized positive linear functional on the algebra A. Linear functional means a linear map from the algebra A to complex numbers C. For every (finite-dimensional) vector space E we have its dual E' - the space of all linear functionals on E.  If v_i are the components of vector v, and fⁱ is any sequence of complex numbers, then f(v) = fⁱv_i defines an element f of E', and any element of E' is of this form (Why?). Here the fact that A is not just a vector space, but an algebra, plays no role. But we want f to be positive, which is defined as: f(v) is positive for all positive v. That is

f(a*a) ≥ 0 for all a in A.

Here the algebra structure and its star operation  play their role.

Finally state must be normalized. Here we use the fact that our algebra has a unit, which we denote simply by 1. Normalization means that we require f(1) = 1. On the left hand side 1 is the unit in the algebra, on the right-hand-side it is the number 1.

We usually interpret f(a) in a probabilistic way as an "expectation value" of a in the state f. So our requirement on f are: expectation value of a positive observable should be positive, and expectation value of an observable taking only value 1 is 1. Intuitive, but, perhaps, misleadingly simple.

So states provide numbers to algebra elements: complex numbers to general elements, real numbers to self-adjoint (a=a*) elements, positive numbers to positive elements. Numbers we understand better than abstract algebra elements. We call these numbers "expectation values" and instantly feel much better. What can we do with states? The same we do in the kitchen with the ingredients: we mix them. If f₁ and f₂ are states, and t is a real number in the interval [0,1], we can forma new state tf₁+(1-t)f₂. We can proceed with mixing adding to the mixture more and more states. By mixing states we lose information - this is known from classical probability, where we mix probability measures. Going in the reverse direction we can try to "un-mix" states. If our state can be decomposed into f₁ and f₂, we try to decompose f₁ and f₂ further, and continue until we finally arrive at states that are not mixtures of other states. These are called "pure states". They contain maximal information about the system, maximal within a given statistical model. This is common to both classical and quantum physics. The main difference between classical and quantum, in this respect, is the fact that in classical statistical mechanics the decomposition of a mixed  into pure states is unique (we say that in classical physics the statistical figure - the convex set of states - is a "simplex"), while in quantum mechanics there is no such uniqueness. This is perhaps one of the main puzzles of quantum theory. Where is this non-uniqueness coming from? And what does it mean? We will meet  this non-uniqueness on an example later on.

And then we can use states (mixed or pure) to construct representations of the algebra as algebras of operators acting on Hilbert spaces. Why do we need this? Can't we simply work with "expectation values" and be happy forever? Here comes another quantum mystery. Louis de Broglie associated waves with particles. Waves are famous for the phenomenon of "interference". Waves can "superpose". This is not the same as statistical mixing. Then came Heisenberg with his matrix quantum mechanics saying bye-bye to the wave picture, but the superposition principle was preserved: we can make superpositions of vectors in the space on which our matrices act. We can treat the superposition within the Hilbert space formalism, but they do not fit the abstract algebra framework. So, by looking for a representation of the algebra, we move from "states" to state vectors. What these state vectors represent beyond reproducing expectation values given by states - that is again a mystery.

Gelfand-Neumark construction takes a state and uses it to construct a Hilbert space and a representation of algebra as an algebra of operators in this space. It realizes this particular state used for the construction as one particular vector in a Hilbert space, and it creates a linear "envelope" of this state by acting with operators representing the algebra elements on this one distinguished vector. This is a general picture. It will be better understood when we will do it on several examples using our Clifford algebra as a toy.   

The Spin Chronicles (Part 20):Group action on the sphere and the spin idea

 In Part 18 we have met the Klein absolute, which, for the case of the 2D (x,y) plane, happens to be the null cone (ζ₀)²- (ζ₁)² - (ζ₂)² - (ζ₃)² = 0 in the 4D Minkowski space. As it is usual in projective geometry we remove the origin ζ=0 from this null cone - we do not need it. 

Projective geometry is about light rays

Let us recall that we have added two extra dimensions x₀ and x₃. We added x₃ for the stereographic projection, and then we added x₀ to get rid of denominators and, instead of doing the stereographic projection we employed the projective space.

Our null cone  ζ² = 0 is invariant under the action ζ⟼g ζ τ(g) of the Clifford group G of the Clifford algebra Cl(V). The action is linear. We already know that g ζ τ(g) = Λ(g)ζ, where Λ(g) is a Lorentz transformation. We have calculated explicitly these transformations for typical one-parameter subgroups of G and obtain either rotations or Lorentz boosts. Now, since the action is linear, it defines the action on equivalence classes of the equivalence relation "∼" defining the projection (see Part 18). If ζ=λη, λ>0, then Λ(g)ζ = λΛ(g)η. Therefore g acts on the equivalence classes of "∼". Let us take a closer look at the structure of the set of these equivalence classes. What do we get? We rewrite ζ² = 0 as

(ζ₁)² + (ζ₂)² + (ζ₃)² = (ζ₀)².

Since we have removed the origin from the null cone, ζ₀ must be non zero.

Exercise 1. Why?

Thus either ζ₀ > 0, or ζ₀ < 0. We choose λ = 1/|ζ₀| and obtain a unique point in the equivalence class of
ζ with the zero coordinate equal +1 or -1. Let us concentrate on the case +1. We obtain the equation
(ζ₁)² + (ζ₂)² + (ζ₃)² =1 for this unique point. It represents a point on the two-dimensional sphere. It stereographically projects onto a point in (x,y) plane except for the South Pole of the sphere with ζ₃ = -1.

Let us return to our embedding formula from Part 18.

After some thinking, and prompted by Bjab's observation I have changed the embedding there to get rid of unnecessary minuses. It became

ζ₀ = (1+x·x)/2,

ζ₁ = x₁,

ζ₂ = x₂,

ζ₃ = (1-x·x)/2.

Now ζ₀ + ζ₃ = 1. If we have any point ζ' on the cone with ζ'₀ + ζ'₃ > 0, we can always rescale it by a unique positive λ to get ζ₀ + ζ₃ = 1, and then read the coordinates from ζ (the rescaled ζ') the two coordinates on the plane. The line with ζ₀ + ζ₃ = 0 form an equivalence class defining one point on the sphere - its South Pole, the Infinite Point on the plane.

Exercise 2. If ζ₀ + ζ₃ = 0 on the null cone, then, necessarily ζ₁ = ζ₂ = 0. Why?

Now we can return to the action of the Clifford group on Minkowski space, its null cone, the sphere, and the plane. We have a distinguished point on the sphere - its South Pole. Thus we can extract a special subgroup of transformations, namely those that do not move that distinguished point. These transformations will transform the (x,y) plane into itself - they will be easy to interpret. But there will be also other transformations that move the infinity point into some other point. Inverse of such a transformation will move a point on the plane into Infinity Point on the sphere, producing a singularity on the plane. We will analyze all this in the next post.

Where Have All the Spinors Gone?

We’ve got "spinors" boldly declared in the title of this series, yet somewhere along the way, they’ve managed to slip out of sight. This is unacceptable! Spinors should be front and center, the main act, not some backstage crew hiding in the shadows. So, just to keep them from becoming the forgotten middle child of mathematical objects, let’s bring them back into focus.

Recently, I received a seven-page paper from V.V. Varlamov. The topic? Clifford algebras and spinors, inspired by Rozenfeld's work on non-Euclidean geometry. I opened it eagerly, confident that by page seven, I’d emerge enlightened, ready to declare, "At last, I understand spinors!"

I did not.

The math in this paper was so advanced, it left me feeling like a freshman who wandered into a graduate-level seminar by mistake. My head began to spin—not in the cool quantum way, but in the “I need aspirin” way. The formulas weren’t just over my head; they were in orbit. It became clear that to keep my sanity intact, I needed to ditch the "follow the paper" approach and start thinking in my way.

I needed to ditch the "follow the paper" approach and start thinking in my way

So, here’s the mental road trip I embarked on:

We’ve got our trusty geometric Clifford algebra of space. It’s an elegant, eight-dimensional creature, cozy with bi-quaternions and complexified Minkowski space. Lovely. What’s more, this algebra isn’t just lounging around—it acts. And it acts on itself, no less. How?

• From the left: $L(u)v=uv$
• From the right: $R(u)v = vu$
  

Now, here’s the juicy part: when you let a Clifford group element $g$ act on something from the left, $g \cdot x$, it drags you out of the real Minkowski space. (Thanks a lot, $g$.) To reel things back in, you need to hit it from the right with something like $\tau(g)$. This leads to a beautiful commuting relationship:

$$L(u)R(w)=R(w)L(u)\mathrm{.}$$

This is clean, elegant math. But what does it have to do with the quantum physics of spin? Here’s where my mind started making some leaps—and possibly doing a few backflips.

Left, Right, and Quantum Duality

In quantum theory, there’s always this duality: the "observer" versus the "system under observation." For spins, we’ve got the laboratory frame with its neat axes, along with a physicist who’s busy assigning complex spin vectors in Hilbert space. And then we have the spin itself—precessing, pirouetting, doing its quantum dance.

In my mental model, I associate the left action with the quantum system itself and the right action with the observer. Or maybe it’s the other way around? I’ll admit, this part’s still a work in progress. The details need ironing out, like a wrinkly shirt you’re not sure is clean or dirty, but you’re wearing it anyway.

Enter the Hairy Ball Theorem

Somehow, all these thoughts led me to the Hairy Ball Theorem. Yes, that theorem from topology—the one that proves you can’t comb a hairy ball flat without creating a cowlick. If you’re wondering what this has to do with spinors, quantum dualities, or Clifford algebra, congratulations—you’re just as confused as I am.

But don’t worry, this will all (hopefully) become clearer in future posts. For now, I’ll leave you with this cliffhanger: Can spinors help us avoid cowlicks in quantum mechanics?

The Spin Chronicles (Part 16): The Action surprises

 This is a continuation of Part 15, where we have discussed the action of the Clifford group G of Cl(V) on Cl(V), defined by

g: x⟼ g x τ(g).

So far we have considered one-parameter subgroups of G of the form g(t) = exp(tX), with X = -ν(X) and X even. These were X of the form X = in, where n is a real vector in V. Today we will consider the case of X odd. Thus we take X=n, where n is a unit vector in V. Since ν(n) = -n, and π(n) = -n, it follows that τ(n)=n, and therefore τ(exp(tn))=exp(tn). For x = x₀+x, x₀ - complex scalar, and x - complex vector, we need to calculate

x' = (x'₀,x') = exp(tn) x exp(tn).

Again it is a question of a straightforward algebra, even somewhat easier than in the previous case, so I will give just the result. For this it is convenient to introduce the following notation. For any x in V we define x_∥ as the component of x parallel to n, and x_⟘ as the complementary component perpendicular to n:

x_∥ = (x·n)n,
x_⟘ = x - (x·n)n.

With this notation, for x'=exp(tn) x exp(tn), we obtain:

x'₀ = cosh(2t)x₀ + sinh(2t) (x·n),

x'_∥ = cosh(2t)x_∥ + sinh(2t)x₀n,

x'_⟘ = x_⟘.

These are the well-known formulas for the Lorentz boost from Special Relativity. But here they arise almost effortlessly, as if conjured by some hidden symmetry. Coincidence? Or perhaps this is another case of Nature’s fondness for recycling its best ideas—its cosmic version of "As above, so below" and "As below, so above."

As above, so below

Nature seems to have a habit of finding a useful tool or building block and reusing it ad nauseam, like that one song you can’t get out of your head. Consider the "eye"—an evolutionary smash hit that popped up independently across the animal kingdom. Or the cell, biology's equivalent of the Lego brick. Could Clifford algebra, $Cl(V)$, be another one of Nature's favorite gadgets? It shows up everywhere: in the microcosm (spin of electrons and protons) and the macrocosm (Minkowski spacetime in Special Relativity and the tangent spaces of General Relativity). It's the ultimate multitasker—kind of like duct tape, but for the fabric of reality.

If that’s the case, maybe it’s not so far-fetched to expect quantum phenomena at a macrocosmic scale. The catch? We might just be staring them in the face without realizing it, like someone trying to find their glasses while wearing them. To uncover such phenomena, we need more than just better technology or theories—we need a fresh perspective, a new way of asking questions. After all, the universe has a funny way of hiding its best tricks in plain sight, daring us to figure out the rules of its game. There may be new phenomena on the cosmic scale as well. Think of a "condensate (infinite continuous tensor product) of Clifford algebras" etc. "More" is sometimes essentially different....

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

 

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

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

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

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

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

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

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

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

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

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

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

Thalia smiles and whispers, “Geometric algebra.”

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

Thalia smiles and whispers, “Geometric algebra.”

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

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

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

Raphael Three Graces

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

Botticelli Three Graces

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

Pauli matrices

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

τ(X) = X*.

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

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

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

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

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

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

ν(X) = σ₂ X^T σ₂.

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