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).
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.
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.
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.
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 = Bτ(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) = Bν(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!
