Spin Chronicles Part 25: QM ahead

 A few days ago, my neighbor—a passionate enthusiast of Clifford and other algebras—asked me: why do I use Cl(V) instead of Cl(3) like most others? This is a fair question, and my answer lies in a subtle but significant distinction. I am not entirely sure what people mean when they refer to Cl(3). Sometimes, they mean the Clifford algebra of R³, but our Cl(V) is not the same as that.

I equip V, our model of space, with a preferred orientation

So, what exactly distinguishes V from R³? In R³ there exists a canonical, distinguished basis consisting of the vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1). This basis is intrinsic to R³  and is the same for everyone. However, V, as we consider it, has no such distinguished basis. If we were to choose a basis for V, then V could indeed be identified with R³, but this identification depends entirely on the choice of basis. This dependency makes a world of difference, especially if we think of V as a model for physical space.

When we observe our physical space, do we see a distinguished basis? Not at all. One observer might adopt one basis, while another might choose a completely different one. In this sense, the basis is observer-dependent, and this observer dependence plays a crucial role in quantum theory. While in relativity—both special and general—we often strive to separate observer-dependent phenomena from observer-independent (or coordinate-independent, as in general relativity) quantities, quantum theory forces us to grapple directly with this observer dependence.

Therefore, I could use Cl(3) in my notation, and perhaps I should, provided we understand exactly what I mean by Cl(3). However, this might not align precisely with what others mean by the same term. Although the difference is often non-essential, it is critical to remember the freedom of choosing a basis. This subtle detail will soon become very important.

Introducing Cl(V)

Let’s delve into this intriguing construct, Cl(V). Since today is Sunday, we can afford to speak more about generalities and less about technicalities. Of course, technicalities matter—they give us the "real taste" of the tool we’re dealing with. After all, Cl(V) is a mathematical tool. It’s not a tangible object that burns your fingers when you touch it, yet it possesses a certain depth and elegance.

So far, our focus has been on uncovering the hidden structure within Cl(V). Through it, we’ve encountered rotations, translations, Lorentz boosts, space and time, and even electromagnetic field vectors E and B. All of these concepts lie embedded within Cl(V), waiting to be revealed by a proper lens. However, the ultimate goal of this series has been to understand spinors, and so far, they have remained elusive. Spinors, as you might know, originate from quantum theory—a topic we’ve yet to discuss in depth.

Interestingly, spinors are also valuable in classical relativity. They provide insights into certain reformulations of general relativity. Whether this utility is merely coincidental or indicative of something deeper remains an open question.

A Quantum Perspective on Cl(V)

Now, let’s start examining Cl(V) from a quantum perspective. The first thing to note is that Cl(V) is an algebra. This should immediately set off a mental alarm: quantum theory is fundamentally algebraic—at least in its operator formulation. (This stands in contrast to the wave mechanics picture championed by Schrödinger and de Broglie.) So, for now, let’s zero in on this algebraic aspect of Cl(V).

Observer Independence and Orientation

Before moving further, there’s another small but crucial detail worth highlighting. Earlier, I emphasized that our space has no preferred basis or direction. However, there is substantial evidence suggesting that it does have a preferred orientation—or “handedness,” if you prefer that term. This preference manifests in biological systems, in parity violation at the micro-level (and possibly at the scale of spiral galaxies, though proving this is more challenging). It also appears in the asymmetry between matter and antimatter in the observable universe.

This preference manifests in biological systems

For this reason, I equip V, our model of space, with a preferred orientation. This choice has significant algebraic consequences: Cl(V) acquires a natural complex structure when we identify e¹e²e³ with the imaginary unit "i". Importantly, this identification is basis-independent as long as we restrict ourselves to bases with the same orientation. Thus, while Cl(V) is fundamentally a real algebra, it gains a natural complex structure. Recognizing this, we also see that Cl(V) becomes isomorphic to the algebra of complex quaternions. A fascinating realization, indeed!

The Road Ahead

This interplay between algebra and complex structure unmistakably hints at quantum mechanics. In the posts to come, we will dig deeper into this connection, exploring how Cl(V) serves as a bridge between the abstract mathematical framework of Clifford algebras and the physical principles of quantum theory.