Notes on the Revolution, 5
I’ve just now had a look at Wilczek’s new book* and was fascinated to discover that he’d also echoed my ideas on color space v. the higher dimensions of M-theory. He seems to be still working through these matters, though; at a glance, he seems to prefer the color cube,whereas the color sphere makes a lot more sense.
Aside from the simple visual analogy, there is the suggestive example of the unit sphere in quantum theory. Heisenberg’s formulation of quantum mechanics assigns all physical states to vectors, which live on such a sphere in Hilbert space. Changing a state entails rotating its vector. [PDF]
Curiously, we use matrices to operate on vectors, which suggests a link to M(atrix)-theory — a hot topic in physics today.
Although this isn’t the place to get into the details, there are compelling reasons for choosing the sphere, bearing on the importance of phase in quantum theory, which informs the vast subject of gauge theory.
I’ve explained all this here, in a paper written at about the level of a Scientific American article.
Here’s a handy table from Ryder’s easy-going book on quantum field theory (QFT). You don’t need to know much about relativity or gauge theory for the parallels to jump out at you. John Baez, who has a knack for explaining abstruse topics, says this:
In the language of physics, theories where forces are explained in terms of curvature are called `gauge theories’. Mathematically, the key concept in a gauge theory is that of a `connection’ on a `bundle’. The idea here is to start with a manifold M describing spacetime. For each point x of spacetime, a bundle gives a set E of allowed internal states for a particle at this point.
In simple models of gauge theory, as well as in M-theory, those internal states are often represented as spheres “sitting over” (fibering over) every point in spacetime. Now, this is quite suggestive, given that color space appears to fiber over our visual world, much like the pixels on a TV screen. Wittgenstein both clarifies and broadens this observation in his Tractatus.
A speck in the visual field, though it need not be red, must have some color; it is, so to speak, surrounded by color-space.
*Wilczek doesn’t give me credit, so far as I can tell, so I thought I’d address this deficit. I doubt whether he’s trying to steal my thunder, as that notion doesn’t seem to square with his apparent good character. But then, academics are, of course, notorious for that kind of thing. Then again, two of the most compelling reasons in favor of my theory have to do with naturalness and simplicity, and so it’s quite possible that he arrived at this POV independently.
Originally published at www.linkedin.com.