Noncommutative Geometry and physics


At least they were able to get to the point of making actual predictions, it's not clear why this attempt of a theory isn't talked about more.



Siarles wrote:

I can catch bits and pieces, but a lot of this is over my head. Can anyone explain it in layman's terms? Or at least explain what the non-layman's terms mean? I tried googling "KO-dimension", but it just sent me back to the same website (which seems sketchy, but I don't want to jump to conclusions about something I don't understand).

Minovskyy wrote:

KO-dimension is the dimensionality of the the KO-theory. A K-theory is sort of like a group structure for vector bundles. Vector bundles are used in physics for e.g. the formal mathematical description of gauge theories.
Mathematically noncommutative geometry quite well founded (and ncatlab is a reputable resource), but I can't say I understand exactly how the physics model works.


Quantum holonomy theory
It's unclear if this theory will indeed give rise to physics as we observe it in our universe but it might still be the best candidate for a theory of everything.

Here's a good medium-level explanation from another comment:

Physicist here:

Thank you for giving Grimstrup & Aastrups work their due attention! Wonderful presentation too, clear exposition and a great channel.

I’d like to give a bit of background on the idea behind QHT also, for those who are curious and want to dive deeper. While their papers definitely open up more questions than they answer, I personally believe that this is some of the best and most interesting research to come out of theoretical physics for the past 20 years:

1. The name “Quantum Holonomy Theory” comes from the fact that Aastrup & Grimstrup (henceforth A&G) use holonomies as their starting point. A “holonomy” is a mathematical concept which describes how to move things around in space. That’s basically it. It appears in many other areas of mathematical physics and the mathematics behind it is well-understood.

2. A&G use holonomies to construct something they call the “Holonomy Diffeomorphism Algebra”. Despite its ominous-sounding name, it’s actually a remarkably simple idea. Put as simply as possible, this is just a set of mathematical operators which can ACT on stuff in space. Whatever such an operator acts on, it MOVES along a certain trajectory, according to the holonomy. Two such operators can be multiplied together. That is, first you act with one operator – moving stuff along one trajectory – and then you act with another operator, moving stuff along a second trajectory. The fact that any two operators can be multiplied together, means that the set of operators form what’s called an “algebra” (very roughly speaking, this is just a set with a notion of multiplication).

That’s it.

From this amazingly simple, almost canonical starting point (an empty space, with a bunch of operators that can move stuff around in this space), A&G are able to obtain a great many results.

3. Among the interesting properties that this algebra has, for instance, is the fact that it encodes what is known as the "canonical commutation relations" from quantum field theory. Roughly speaking, it means that the starting point of the theory is quantum to begin with.

Historically speaking, physicists first built a (classical) theory, which they then try to quantize using all kinds of mathematical machinery. But A&G’s theory is inherently quantum; no “post”-quantization procedure seems needed. From this algebra, one can also obtain what’s called the "holonomy-flux" relations, which is a well-known result from loop quantum gravity. So the theory, using very little and completely standard mathematical stuff (known from ordinary gauge theory), reproduces some fundamental aspects of both quantum Yang-Mills theory and a leading candidate for a theory of quantum gravity!

4. Another point I should add: The Holonomy Diffeomorphism Algebra is noncommutative. Meaning, the order in which you multiply two operators matters. Acting with operator A, followed by B, does NOT give the same result as acting with operator B, followed by A.

So the Holonomy Diffeomorphism Algebra is a noncommutative algebra. Why is that important? Well, it puts A&G into an area known as “noncommutative geometry”. This is a modern and very rich, but also difficult area of mathematics, which was pioneered by the French mathematician Alain Connes. What Connes showed (back in the 1980’s) is essentially the following: Any algebra (a set with a notion of multiplication) will – under certain conditions – describe a space with a geometry. In fact, any notion of a geometric space that physicists use in their models (compact Riemannian spin manifolds, for all you cool kids out there) can be SWAPPED in favor of an algebra, alongside some additional ingredients. These extra ingredients, together with the algebra, form something called a SPECTRAL TRIPLE.

So a spectral triple is essentially just another description for a piece of geometry. Any spectral triple describes a geometric space and any geometric space has an associated spectral triple. OK, so why do we care? Well, it turns out that the spectral triple formalism allows us to greatly expand our notion of geometry. You see, the geometric spaces that physicists use correspond to COMMUTATIVE spectral triples – that is, spectral triples built from commutative algebras. But we can now built NONCOMMUTATIVE spaces via noncommutative algebras, thereby gaining a much broader conceptualization of space. This is not just a funky mathematical idea. In fact, what Connes has shown is that the entire Standard Model of Particle Physics can be fully realized as a theory of gravity on a noncommutative space(!!!).

I say that again: Using spectral triples, one can write up the entire Standard Model combined with General Relativity, as a theory of gravity on a noncommutative space. “That sounds like a Theory of Everything”, you say. Well, no. You see, the theory is not quantum. So while it is possible to unify Einsteins theory with the Standard Model in a very natural way, using noncommutative geometry as the setting, the theory is essentially classical. There’s a lot of subtleties here which I’ve skipped (though they are cool) and I've simplified this dramatically, but this is essentially the idea.

5. Back to A&G: Using the Holonomy Diffeomorphism Algebra, they’ve been able to construct a (certain kind of) spectral triple; that is, a certain kind of noncommutative space. What’s interesting about their spectral triple is the following: A spectral triple, as mentioned, describes a space. It consists of an algebra, alongside some additional ingredients. One of these additional ingredients is something called a Dirac operator. This operator is essentially what carries the information about the geometry (its curvature, for instance). Now, A&G can pick something called the Bott-Dirac operator, which is a “natural” choice to go along with their Holonomy Diffeomorphism algebra. The Bott-Dirac operator is interesting, not only for the reasons mentioned in this video, but because it’s a mathematical description of an infinite-dimensional harmonic oscillator from quantum field theory. So the noncommutative space that A&G have come up with, is very different than the ones studied by Connes and his peers; it’s built using holonomies and the infinite-dimensional configuration space of gauge fields, and the associated Bott-Dirac operator, which comes out of quantum field theory.
This all smells a lot like noncommutative geometry, but thrown into a quantum setting!

6. Finally, in their newer papers (which is mostly what this video talks about), A&G look at the configuration space of gauge fields itself.
Using the Bott-Dirac operator, they can provide this space with a geometry: Now (they claim), the fermionic sector of quantum field theory emerges from the Bott Dirac operator. What does that mean? Well, fermions are the kinds of particles that describe matter in the universe. So essentially, matter comes from the geometry of the configuration space of gauge fields!"

It’s a shame that their ideas are not more well-known or supported by the academic community. Grimstrup’s book, “Shell Beach”, is highly recommended. It’s a wonderful exposition on physics, the academic world and the personal journey that goes along with being preoccupied with “big questions”!