I was going to say that's extremely interesting, until I realized to just get what this is talking about I have to go at least 3 levels deep into links describing the terminology. I dropped mathematics somewhere around modular forms and number theory, where everything became so abstract it was hard to keep any interest without an external driver. Modern physics is even crazier in that regard, but despite it being more interesting per se, those guys lost me some time after Hamiltonian mechanics (which was in itself fun).
That's an amusing to read, though:
In the approach, the
on-shell scattering process "tree" is described by a positive
Grassmannian, a structure in
algebraic geometry analogous to a
convex polytope, that generalizes the idea of a
simplex in
projective space.
[3] A
polytope is the
n-dimensional analogue of 3-dimensional
polyhedrons, and the values being calculated in this case are
scattering amplitudes, and so the object is called an
amplituhedron.
[1]
Why does Star Trek have to come up with nonsensical technobabble, when they can simply quote Wikipedia articles on math?
“A quantum affine algebra is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang-Baxter equation occurs with a spectral parameter.”
“Ugh, I hate that class, this is so last century. When are we going to study warp theory?”
...
On a more serious note, I'm ambivalent about the expectation that a theory of everything will actually ever emerge. Not because I think our present understanding of physics is complete, or accurate, or the most simple model of the universe, or we need to stop looking, or these aren't amazing people who I envy, but because I kind of question the rationale for this expectation. We assume that the current fundamental laws of physics are too complex, patchy, and a convoluted mess with holes in it. So it can't be fundamental, it has to be caused by something else. Which feels like another way of saying we don't like it, so we
will be able to further explain it.
Looking for further explanation in itself is always a good thing. (You can see me green with envy other people are doing it, and I can barely follow the conversation.) We would have never gotten this far if we thought atoms are really atomic, hairy chemistry is the root of it all, and let's be done with it. But we can't expect to keep finding more explanations. At some point we will hit rock bottom and get to the actual foundation. There is zero reason to expect it will be simple, let alone elegant when we reach it. The current complex, patchy convoluted mess with holes in it that I can't wrap my head around is exactly what I expect the foundation is going to be like. So it might be that we already hit it, we simply haven't figured out all the quirks.
Ultimately, our expectation should be that the fundamental laws of physics are the simplest ones that support intelligent life (the day survivor bias cut itself on the Occam's razor). That can be philosophically explained away by invoking infinite universes, and imagining the cosmic equivalent of natural selection. Under such premise, and given how messy the results of biological evolution are, its cosmic evil twin wouldn't yield anything more pleasing. Not to mention that we have literally evolved to enjoy our own messiness (we like sex, for heaven's sake), but we have not evolved to like the universe's. So it might just be as nasty as it was intended to be.
Injecting philosophical drivel like multiple universes in itself sounds questionable, but whenever you reach the actual fundamental laws of the universe – which are by their own definition unexplainable (no other other thing causes them, duh), you have to descend into philosophy. I kind of hate the phrase “fine tuning”, as it simply says “it's too complex still”. Well, what's the threshold? It's not like life is something trivial that can just pop anywhere – none of my Game of Life patterns have invented the wheel yet. Sure, the universe makes up for some of that by being gargantuan in size, so if we're the only life anywhere in it, maybe we can expect some simplicity to be found in the rules. It still won't work without
some fine tuning, and some might still be
a lot. We can't even get planets without fine tuning.
Sure, you can have a theory of everything that in itself implies multiple universes in some fashion, so it's no longer philosophical (and sound more sciency). But then where did the multiverse fine tuning that implies this specific set of universes come from? Then you'd have to replace the theory of everything with a theory of anything, and you're back to square one.
So it could be that GR and QFT can't really be unified in a single theory, aren't really incompatible e.g. because the insides of black holes don't really exist in our observable universe – you could have two identical universes with a different set of laws governing the insides of black holes, and you wouldn't be able to tell which you're in because of the No Hair Theorem, leaving the conflict zone of GR and QFT untestable (you can even wonder if you're simultaneously in both universes). It's a preposterous suggestion, but only because it's extremely ugly. It could still match observation.
