The Spin Chronicles (Part 15): The Action starts

Understanding the meaning of the Universe—and, by extension, our purpose in this dizzying 3D playground—is impossible without first grasping the concept of spinors. Yes, spinors. Everything spins. Planets pirouette around stars, galaxies swirl like cosmic pinwheels, and deep inside you, photons and electrons are spinning their little hearts out. Without spin, there would quite literally be "no nothing." But what exactly is this mysterious "spin"? And what, pray tell, is a "spinor"?

To get the most precise answers, we must summon the sharpest tool in humanity’s intellectual shed: mathematics. Forget poetry or philosophy for now (though they’re lovely companions); math is the only game in town when it comes to rigor. The groundwork for this particular game was laid by none other than the French mathematical collective known as Nicolas Bourbaki. This was not an  ordinary group of thinkers—they were the masterminds who struck the match, poured the fuel, and launched the metaphorical rocket of modern mathematical formalism.

The Bourbakists

I, too, am a humble traveler on this rocket of discovery, squinting over the shoulders of these giants to glimpse what lies ahead. 

The Bourbaki

And let me share with you one of the gems they left behind, a definition from Bourbaki’s magnum opus, ÉLÉMENTS DE MATHÉMATIQUE, ALGÈBRE CHAPITRE IX:

DÉFINITION 2: On appelle groupe de Clifford de Q (resp. groupe de Clifford spécial de Q), le groupe multiplicatif des éléments inversibles de C(Q) (resp. C+(Q)) tels que sEs⁻¹ = E.
Dans ce no nous noterons G et G⁺ le groupe de Clifford et le groupe de Clifford spécial de Q.

If you’re already scratching your head, don’t worry—it’s not your fault. Mathematics is as much a language of symbols as it is a puzzle of patience. But let’s break it down: Their E is our V, their Q is the Euclidean quadratic form of V, and their C(Q) corresponds to our Cl(V). Meanwhile, their C⁺(Q) is a subalgebra of Cl(V), comprising those nice even elements that remain unchanged by π.

Here’s the twist: where Bourbaki insists on the condition  sEs⁻¹ = E. we’re a bit more liberal. We drop that requirement, making our Clifford group G bigger and, dare I say, more fun. Why stop at “sensible” constraints when you can explore the wild possibilities of mathematical freedom?

And here’s where it gets spicy: this expansion of the Clifford group isn’t just a technical curiosity—it unlocks surprises. It’s a reminder that while mathematicians love their neat, self-contained universes, Nature herself isn’t so tidy. She operates with her own version of simplicity, one that often leaves mathematicians muttering into their coffee mugs.

In short, if you think Bourbaki nailed it, wait until you see what happens when we take the leash off. Welcome to the uncharted territory where Clifford algebras meet Nature’s playful chaos.

Note: The idea of enlarging the Clifford group is not new (is there anything really-truly new under the Sun?). C.f. for instance William Baylis, "Lecture 4: Applications of Clifford Algebras in Physics, 4.3 Paravectors and relativity", in "Lectures on Clifford (geometric) algebras and applications" [edited by] Rafal Ablamowicz and Garret Sobczyk], Springer 2004, and references therein.

This post is a continuation from The Spin Chronicles Part 14, where we have introduced our "Clifford group G" defined as the set of elements u = (p₀,p)  in Cl(V) for which (p₀)²-p² = 1. Another form of the definition of G is

G = {g ∈ Cl(V): g ν(g) = 1}.

Exercise 1: use the definition of the multiplication in Cl(V) written in a bi-quaternion form to verify that these two definitions are indeed equivalent

In this post we will take a closer look at the action of G on Cl(V) defined by

g: u ⟼ guτ(g).

To this end we will analyze the action of one-parameter subgroups g(t) = exp(t X). In order for g(t) to be in G, we must have X + ν(X) = 0.

Exercise 2. Verify the last statement.

First we take the case of X even. Thus we have two conditions on X:

a) ν(X) = -X,

b) π(X) = X.

A general solution of these two conditions is X = in, where n is a real vector.

Exercise 3. Verify the last statement. Without loss of generality we can choose n to be a unit vector. Remember that n is a vector in Cl(V), therefore we have nn=1. Using this property calculation of exp(itn) is easy. We get:

exp(itn) = cos(t)1 + i sin(t) n.

Exercise 4. Use the power series expansion of exp(itn) to derive the above formula.

We can easily verify that we have

τ(g(t)) = g(-t).

Indeed, τ(exp(itn)) = exp(τ(itn)) = exp(-iτ(tn)) = g(-t), where we used the fact that τ(i) = -i, and that t and n are real, t - scalar, and n - vector.

Taking now u=(x₀,x), where (x₀,x) are complex, and setting it is a matter of simple algebra using the multiplication formula for bi-quaternions (c.f. Exercise 1, Part 11) to get for u(t) = g(t) u g(-t) the formula

x₀(t) = x₀,

x₁(t) = cos(2t)x₂ + sin(2t) x₃,

x₂(t) = -sin(2t)x₂ +cos(2t) x₃

x₃(t) = x₃.

It follows that this action of g(t) on Cl(V) is nothing else but a simple rotation by the angle 2t of the vector part. It rotates the same way both real and imaginary components of the vector. The scalar component x₀ does not change under these rotations.

In the next post we will consider the case of odd X.

The Spin Chronicles (Part 14): The Universe and Clifford group actions

 I can't refrain myself from starting this post with a quote from the Introduction to the paper "Conformal Mappings, Hyperanalyticity and Field Dynamics" by V.V. Kassandrov (1993): Here it goes:

"1. Introduction

It seems impossible to imagine that the primary Void, which contains the divine diversity of forms, determines the dynamics of the World. However, we have to believe and try to understand it, else we are destined to stay on the primitive level of ‘geographical’ phenomenology.

Maybe it is wise to give up, for a while, all the principles and laws established by our great predecessors, even such ‘fundamental’ laws as Lorentz invariance, gauge symmetry or the minimal action principle. We should not flatter ourselves with their apparent beauty – they are neither basic nor primary, since they themselves are to be explained and grounded. It is also valid for most remarkable empirical facts, such as the invariance of light velocity, the quantization of electric charge, the universality of the inverse square law.

We may regard these facts as axioms and then proceed as far as possible in constructing a fruitful physical theory. This is just the way Einstein created his special theory of relativity (STR).

It seems more attractive to find a new interpretation of well-known physical phenomena, where the existence of the correspondent correlation becomes self-evident (as the equality of gravitational and inert masses after a geometric treatment of gravitation in general theory of relativity (GTR)). There still exists another way of thought, the most thankless one but promising. This is the way bequeathed by Pythagoras, Hamilton, Eddington and other Grands, the way chosen by Einstein in the last half of his life. Here we suppose the existence of exceptional relational structures of a purely mathematical nature and intend to discover them and formulate a unique abstract principle on their basis. The characteristic differential equations, algebro-geometrical properties and other relations should then follow from this principle only. Here it is necessary to bring in conformity all the mathematical categories and solutions with the

quantities describing real objects. If and only if the complete correspondence is achieved, we should consider the Universe as a ‘reflection’, a ‘realization’ of this abstract Principle! 

 

The Universe as a ‘reflection’, a ‘realization’ of this abstract Principle! 

One has to understand that such ideology contradicts the widespread treatment of mathematics as a merely subjective formalism. On the contrary, one has to assume that there do exist completely objective laws in its very structure (see [1]), which have to be discovered and identified with physical laws. In this article we try to show that the afore-stated approach is not so fabulous as it seems at the first sight. Naturally, at once it is impossible to establish that very Principle whose structure carries the main part of physical phenomenology.

Nevertheless, even within the frame work of Hamilton’s and his followers’ well-known ideas, it becomes possible to reveal the wonderful correlations between the properties of an exceptional mathematical structure – the algebra of quaternion-like type – and the physics and geometry of space-time.[...]"

Число, время, свет || Владимир Кассандров || Программа Гордона №313 (04.11.2003)

This is exactly the philosophy I am trying to follow in this series. In The Spin Chronicles (Part 13) we have introduced the group G:

G = {g∈Cl(V): ∆(u) = 1},

where ∆(u) = B_ν(u,u) = (p₀)² - p².

In fact we do not even need ∆(u) in this definition - we need only the Clifford conjugation ν itself, since the element ν(u)u automatically belongs to the center of Cl(V), and we have, as can be verified from the form of the product, and the form of  ν(u) (Part 11):

uv = (p₀q₀ + p·q,  p₀q + q₀p ₊ i p⨯q),

ν(u) = (p₀,-p).

Therefore

ν(u)u = uν(u) =  ((p₀)² - p²)1.

Exercise 1. Verify the last formula.

Once we have a group, we analyze its natural actions. By definition we say that a group G acts on a set S if there is a map G ⨯ S ⟶S, denoted (g,s) ⟼ g·s, such that, for all s we have e·s = s, where e is the identity of the group, and for  all g,h in G and all s in S we have g·(h·s) = (gh)·s. In our case for the construction of natural actions we have at our disposal the automorphism π, and two anti-automorphisms ν, and τ. The following four actions are natural:

1). g·u = g u τ(g),

2). g·u = g u ν(g),

3). g·u = π(g) u τ(g),

4). g·u  = π(g) u ν(g).

Exercise 2. Verify that all four cases are indeed group actions.

We will consider them one by one.

Case 1. We recall the form of τ:

τ(u) = (p₀,p)*,

where "*" denotes the complex conjugation. Every element of Cl(V) can be decomposed as

u = (u+τ(u))/2 + (u-τ(u))/2.

We can write it as u = u_R + u_I. Then  τ(u_R) = u_R,  τ(u_I) = - u_I. In other words we split Cl(V) into a direct sum of two eigensubspaces of τ. It can be easily checked that these eigenspaces are invariant under the action 1).

Exercise 3. Verify the last statement.

In order to analyze the group action we will consider its one-parameter subgroup of the form g(t)=exp(t X), where X is in Cl(V), t is real parameter. To have ν(g(t))g(t)=1, since

ν(exp( t X)) = exp(t ν(X)),

we must have ν(X)=-X. Therefore X must be of the form (0,v+iω), where v and ω are real vectors.

Exercise 4. Prove the last statement.

We will continue our analysis in the next post.

The Spin Chronicles (Part 13): Norms, Spinors, and Why Mathematicians Need Better Nature Walks

 Welcome back to The Spin Chronicles! If you’ve been following along (and if you haven’t, shame on you—catch up on Part 12 Geometry, Kant, and the Limits of Physics), you already know we’ve been dancing around the concept of "spinors" for quite a while. A long while.

Well, good news! Today, we’re finally gearing up to meet spinors. But—and there’s always a but in math—we first need to take a small detour into something called the "spinorial norm." Yes, it’s as thrilling as it sounds.

For those of you keeping score at home, we’re still working with the geometric Clifford algebra Cl(V) of 3D Euclidean space V. This notation might seem intimidating, but don’t worry—it’s just math’s way of keeping things exclusive, like a secret handshake for nerds. Now, let’s roll up our sleeves and dive into why norms matter (and why mathematicians, for all their brilliance, could use a little more guidance from Nature).

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The Trouble with Mathematicians

Here’s the thing about mathematicians: they think they’re free spirits, crafting elegant concepts out of thin air, but they’re like poets writing odes to imaginary sunsets. Beautiful? Sure. Practical? Meh. Sometimes, they get so lost in their own abstractions that they forget to look around and ask, “Hey, does any of this actually match up with the real world?” Playing soccer can help in this respect a lot!

Nature, however, is the ultimate editor. She’s brutally practical and has no patience for fluff. Case in point: our physical space is three-dimensional. Why? Who knows, but it’s a fact. And while mathematicians are busy dreaming up n-dimensional geometries, Nature keeps things grounded with good old-fashioned threes. Maybe she’s just stubborn. Or maybe she knows that going beyond three dimensions gives true (i.e. experimental) physicists migraines.

Physicists against mathematicians

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Space, Electromagnetism, and That Pesky Thing Called Time

Speaking of Nature, let’s not forget how much of our sensory experience relies on electromagnetic interactions. That’s right, folks—our ability to see, touch, and generally function as vaguely competent humans is powered by electromagnetism. This makes it pretty important. Yet, time, that slippery little troublemaker, loves to throw a wrench into our neatly ordered perceptions.

Unlike space, time doesn’t play fair. Sometimes it feels like it’s rushing past; other times, it slows to a crawl (usually during committee meetings or while waiting for your code to compile). This quirk makes time fundamentally different from space, and perhaps it’s not just one time but an entire family of "species of time." That’s a philosophical can of worms we’ll save for later. For now, let’s stick to what we can handle: 3D space and its lovely, orderly Clifford algebra.

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Two Norms, One Algebra

Today’s mission is to introduce not one but two norms on Cl(V). Why two? Because math is like that one friend who insists on ordering both the tiramisu and the cheesecake—you’re not really sure why, but they swear it’s important. One of these norms is sometimes called the "spinorial norm," but I’ll let you in on a secret: I’m not married to the standard terminology here. Why? Because mathematicians don’t own the truth—they just rent it from reality.

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So, grab your coffee (or something stronger), and let’s explore these norms. With a little luck—and a lot of algebra—we’ll finally get one step closer to understanding spinors. And who knows? Maybe we’ll learn something about why Nature prefers her math practical, her space three-dimensional, and her time as confusing as a modern art installation.

The first norm is the standard one, usually called the Hilbert-Smith norm, and denoted || ||_HS (see e.g. "Why does submultiplicativity hold for the Hilbert-Schmidt norm"), but we shall denote it here simply by || ||. It is defined by

||u||² = B_τ(u,u) = |p₀|² + |p|².

Here |p|² stands for |p₁|²+|p₂|²+|p₃|² - we remember that p₀ and p are complex! This norm is positive-definite, and it has the nice algebraic submultiplicative property:

||uv|| ≤ ||u|| ||v||.

The second "norm" we denote by ∆(u). It is defined as

∆(u) = B_ν(u,u) = (p₀)² - p²,

where p² = (p₁)²+(p₂)²+(p₃)². 

In fact, we can easily check that in the matrix representation, using the Pauli matrices, we have that

∆(u) = det(u),                     (*)

from which it follows instantly that

∆(uv) = ∆(u)∆(v).              (**)

But, according to our definition,  ∆(u) is complex valued! It is called "spinorial norm", even though calling it a "norm" can be sometimes, misleading.

Exercise 1. Use the property (**), together with ∆(1) =1, to show that u is invertible if and only if ∆(u) ≠ 0.

It follows then instantly that the set G of elements of Cl(V) with ∆(u) = 1 is a group! We can easily identify this group in the matrix representation using (*). G is isomorphic to  SL(2,C) - the group of complex two by two matrices of determinant one - the double cover of the orthochronous proper Lorentz group of special relativity!

Stay tuned!

The Spin Chronicles (Part 12) - Geometry, Kant, and the Limits of Physics

 Welcome back to the odyssey of geometric algebra, where the math gets deep and the philosophy… well, it occasionally dives off the deep end. This post picks up from "The Spin Chronicles (Part 11)" and continues our foray into the natural bilinear forms that emerge within the Clifford algebra of our 3D space. Don't worry if you’re still wondering what that is; just keep in mind that it's an essential way to understand how space itself “behaves.” And no, we won’t be referencing the cosmic dance between the stars and planets—but this will get philosophical enough to warrant a helmet.

Immanuel Kant, who preferred his mornings with a side of metaphysics, once said:

"Space is not something objective and real, nor a substance, nor an accident, nor a relation; instead, it is subjective and ideal, originating from the mind’s nature in accord with a stable law as a scheme, as it were, for coordinating everything sensed externally."

Which is to say, Kant thought of space not as some grand arena laid out by a divine hand, but as a sort of mental wallpaper, perfectly crafted to make sense of the world without needing to be “real” in the way rocks or trees are real. Then he doubles down:

"Space is not an empirical concept derived from outer experiences. In order for sensations to refer to something outside me, and for them to be represented as outside and alongside one another in different places, the concept of space must already exist within us."

Basically, Kant argues that space isn’t something we pick up by wandering around the world and mapping it in our heads; rather, it’s a kind of mental scaffolding that’s there from the get-go, allowing us to experience everything else in relation to it.

Now, whether or not you buy into Kant’s perspective here is a different story. Space, after all, feels pretty solid when you bump into a coffee table in the dark. We intuitively “know” space, and our DNA may even hold clues to why this intuitive knowledge works so well. Our spatial instincts aren’t simply musings; they come from somewhere deeply ingrained, refined by eons of needing to dodge pointy sticks and hungry animals.

But just as we start to think space is on our side, the physicists arrive, white coats flapping, to tell us: “Well, space (and time) aren’t absolute. They’re relative, dependent on the observer’s inertial frame of reference.” Einstein’s Special Relativity was a real buzzkill for those who thought they’d finally “figured out” space. Yet, as bold as these physics claims are, they’re always at the mercy of the next scientific revolution. Mathematics, by contrast, is a steady old friend—unchanging, consistent, and reliably grounded in the Euclidean spaces we’re exploring here.

Well, space (and time) aren’t absolute...

So, let's put the physics opinions aside for now and dive back into mathematics, where truths stay put and constants stay constant. Today, our quest involves understanding the natural bilinear forms within the geometric Clifford algebra of our classic 3D Euclidean space. Yes, we’re talking about the world where the Pythagorean theorem reigns supreme and the ratio of a circle's circumference to its diameter is that oh-so-familiar π ≈ 3.1415926...

So, let’s roll up our sleeves and get back to numbers, forms, and figures that don’t play hide-and-seek depending on who’s looking at them.

We have already discussed the bilinear forms B₀ and B_τ. Next in order is B_ν(u,v) = t_n(ν(u)v),

where 

ν(u) = (p₀,-p).

We are using a real basis E_A (A=0,1,...,7) in Cl(V):

E₀ = 1, E₁ = e₁, E₂ = e₂, E₃ = e₃, E₄ = ie₁, E₅ = ie₂, E₆ = ie₃, E7 = i.

In this basis the matrix of the real and imaginary parts of B_ν are given by:

Re(B_ν)

Im(B_ν)

Both are of neutral signature (++++----).

Finally B_π, with π given by 

π(u) = (p₀,-p)*, where "*" stands for the complex conjugation:

Re( B_π)

Im( B_π)

The real part is symmetric, with eigenvalues (+1,-1,-1,-1,-1,-1,-1,+1). The imaginary part is anti-symmetric.

Since Cl(V) carries a natural complex structure, it is even more instructive to consider our bilinear forms as complex valued. For this it is convenient to use the matrix representation with Pauli matrices. Then the complex basis consists of matrices (I, σ₁, σ₂, σ₃), as discussed in The Spin Chronicles (Part 9): Matrix representation of Cl(V) and The Spin Chronicles (Part 10) - Dressing up the three involutions. In this basis we calculate the quadratic forms B₀(u,u), B_τ(u,u), B_ν(u,u) and B_π(u,u), for u = (p₀,p). Here p₀ is a complex scalar, p is a complex vector. We can easily obtain:

B₀(u,u) = (p₀)² + p²,

B_τ(u,u) = |p₀|² + |p|²,

B_ν(u,u) = (p₀)² - p²,

B_π(u,u) = |p₀|² - |p|².

It is somewhat surprising that  for B_ν and B_π(u,u) we are getting a form resembling the 4D Minkowski metric of special relativity, but that's what mathematics leads to, for its own strange reasons. We will return to this issue in the future posts. Stay tuned...