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.
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 Bτ. Next in order is Bν(u,v) = tn(ν(u)v),
where
We are using a real basis EA (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 Bν are given by:
Both are of neutral signature (++++----).
Finally Bπ, with π given by
π(u) = (p0,-p)*, where "*" stands for the complex conjugation:
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), Bτ(u,u), Bν(u,u) and Bπ(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,
Bτ(u,u) = |p0|2 + |p|2,
Bν(u,u) = (p0)2 - p2,
Bπ(u,u) = |p0|2 - |p|2.
It is somewhat surprising that for Bν and Bπ(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...
