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Stream: theory: category theory

Topic: Amplituhedron, associahedron, and related approaches

John Baez (Oct 15 2024 at 20:13):

I found this lecture to be quite informative and delightfully fun to watch:

• Nima Arkani-Hamed, Advanced class in amplitudes (part 1 of 5).

.

He has a nice way of presenting things as a story, here one where imagined particle physicists discover the Stasheff polytope or [[associahedron]] as a way of organizing S-matrix amplitudes, simply by keep track of the poles in the S-matrix as a function of the momenta of the incoming and outgoing particles!

He's also great at keeping prerequisites to a minimum. I guess to enjoy this lecture one needs to know a bit of quantum field theory: the Feynman rules, and how Feynman diagrams tend to have poles when the energy-momentum of a virtual particle in the diagram is "on shell" - i.e. takes a value that a real particle can have (namely with $p^2 = m^2$ for a particle of mass $m$). Also it helps to have seen how Feynman diagrams for gluons look like ribbon graphs... though that shows up only at first, and if you grit your teeth it will go away later!

John Baez (Oct 15 2024 at 20:27):

What I don't love is how he covers up some of the implicit assumptions that limit the scope of his work. You can see them if you know what to look for. He is talking about a spin-0 field theory where a field $\Phi$ of mass $m$ takes values in the Lie algebra $\mathfrak{u}(n)$ or $\mathfrak{su}(n)$ and the interaction is

$\mathrm{Tr}(\Phi^3) = \sum_{i, j, k = 1}^n \Phi_{ij} \Phi_{jk} \Phi_{ki}$

This is not a theory that applies to our universe. When he sets $m = 0$ this theory would apply to nonexistent 'spin-zero gluons'.

More importantly, in this talk, he only considers planar, tree-shaped Feynman diagrams. This is what gets him to start drawing trivalent trees, and then dually triangulations of a polygon. At that point the associahedron is inevitable.

But I believe this restriction to planar, tree-shaped Feynman diagrams naturally only when we consider the limit $n \to \infty$. (The restriction to tree-shaped diagrams also happens when we consider the classical limit $\hbar \to 0$, but I don't think that limits us to planar diagrams, despite some vague remark he makes to that effect. [Edit: hmm, maybe it does - I'd be happy to discuss this.]

So, while the math is beautiful and this is the best talk I've heard in a long time, I feel there's a bit of salesmanship involved here. This becomes much more pronounced in the pop science media, where we see bullshit headlines like "physicists have discovered a jewel-shaped geometric object that challenges the notion that space and time are fundamental constituents of nature".

Federica Pasqualone (Oct 15 2024 at 21:24):

Thank you for this! It's on my list! :sunglasses:

Federica Pasqualone (Oct 16 2024 at 00:35):

P.s. The speaker is very Federica btw hahah [Lemme get to the last lecture to have the total pic of this enterprise.]

Madeleine Birchfield (Oct 16 2024 at 03:25):

A different group of researchers have applied the newer techniques to a theory called "colored Yukawa theory": https://link.springer.com/article/10.1007/JHEP09(2024)160

They seem to be calling these techniques the "curve integral formalism".

One of the authors gave a talk about their work here: https://www.youtube.com/watch?v=_SeFPHPqTbY

Madeleine Birchfield (Oct 16 2024 at 03:41):

There's also this presentation by Nima Arkani-Hamed titled "Surface Kinematics and THE all-loop integrand for gluon amplitudes"

• https://www.youtube.com/watch?v=kqdICU54FFI
  

John Baez (Oct 16 2024 at 05:25):

Madeleine Birchfield said:

A different group of researchers have applied the newer techniques to a theory called "colored Yukawa theory": https://link.springer.com/article/10.1007/JHEP09(2024)160

Nice! The original Yukawa theory was invented in 1930s by Yukawa as a theory of how protons and neutrons interact. In the simplest version you have a spin-1/2 particle $\Psi$ (say either a proton or neutron) and a spin-0 particle $\Phi$ and an interaction Lagrangian $\overline{\Psi} \Psi \Phi$. Using a simple version of this theory he predicted that the force between protons and neutrons was mediated by a scalar field of a mass around 1/7 times that of the proton. This particle was indeed found: it's the pion.

In fact the pion comes in 3 forms, and Heisenberg used this and other facts to argue that $\Phi$ should be an $\mathfrak{su}(2)$-valued field, while $\Psi$ is $\mathbb{C}^2$-valued to accomodate the proton and neutron as two states of a single particle, the 'nucleon'. The $\mathrm{SU}(2)$ symmetry group of this theory was called 'isospin' $\mathrm{SU}(2)$, where 'iso-' comes from the word 'isotope'.

Later Gell-Mann introduced a very successful $\mathrm{SU}(3)$ version of Heisenberg's theory.

The colored Yukawa theory in this new paper is very similar to Heisenberg's! However, it includes a cubic self-interaction for the $\Phi$ field: the Lagrangian includes the $\mathrm{Tr}(\Phi^3)$ term we've been talking about earlier.

So, I'd say this new work is an expansion of Arkani-Hamed et al's original $\mathrm{Tr}(\Phi^3)$ theory to include a fermionic $\Psi$ field as well as the bosonic $\Phi$ field. I haven't read what they actually do with this theory - but it's getting closer to a physically realistic theory, since it involves spin-0 bosons and spin-1 fermions interacting in a way similar to the early theories I mentioned. (In modern theories, called gauge theories, the bosons are instead spin-1 particles, which really come from connections: this brings in more powerful math.)

John Baez (Oct 16 2024 at 05:30):

Their $\Phi$ field is an $\mathfrak{su}(n)$-valued function, while the $\Phi$ field is a $\mathbb{C}^n$-valued spinor field. I fear the nonrealistic aspect will be that they - perhaps implicitly! - take the $n \to \infty$ limit to make all but planar Feynman diagram negligible. That seems to be a crucial trick in this game.

But I haven't read the paper, just looked at the Lagrangian in equation (3.1).

John Baez (Oct 16 2024 at 05:38):

Hmm, they do mention some non-planar Feynman diagrams, and they relate these to triangulations to surfaces of higher genus (while planar trivalent graphs are Poincare dual to triangulations of a disk).

This suggests that people could go ahead and look at an expansion by genus, where each surface summarizes infinitely many Feynman diagrams, namely those Poincare dual to its triangulations. This general idea is already familiar in 2d quantum field theories, where however the math is simpler, yet very beautiful. For example in 1986 Penner wrote a paper The moduli space of a punctured surface and perturbative series which showed how to do some integrals over the moduli stack of conformal structures on a surface as a sum over triangulations of that surface, and used this to give a formula for the Euler characteristic of that moduli stack in terms of Bernoulli numbers!

James Deikun (Oct 16 2024 at 06:04):

I'm far from an expert in this stuff, but by any chance, does this sum-over-genuses approach not make calculations for theories of particles look a lot more like calculations for theories of strings?

John Baez (Oct 16 2024 at 06:22):

Yes!!! And Arkani-Hamed explains this very beautifully in his nice lecture. In his imaginary scenario where experimental physicists start noticing patterns in the poles in the S-matrix, he imagines two theorists "Feynman" and "Deligne" who give two different explanations of these patterns. "Feynman" deduces that they're explained by particles, while "Deligne" deduces that they're explained by strings. Go to the time 59:30 in his talk, if you're interested.

John Baez (Oct 16 2024 at 06:28):

As he briefly hints, this is connected to the origin of string theory in the first place: people were trying to dream up formulas for an S-matrix obeying some nice properties, and Veneziano noticed that a certain function had those properties. It turned out to be a known special function, and eventually it was realized that it equaled an integral over the moduli space of conformal structures on a 4-punctured sphere. Or something like that.