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....

P.S. 22-11-24 18:50 Notice that the real and imaginary part of x transform independently.

Disagree? Good! Nature thrives on competition, and so should ideas. Just don’t expect her to hand you the answers. She prefers to keep her secrets—and her sense of humor—intact.