The n-Category Café

One great thing is that Arkani-Hamed keeps the prerequisites to a minimum. Well, okay: if don’t know quantum field theory don’t watch this video. You need to know the Feynman rules, and how Feynman diagrams have poles when the energy-momentum $p$ of a virtual particle in the diagram goes ‘on shell’ — i.e., takes a value that a real particle could have, namely $p^2 = m^2$ for a particle of mass $m$. It also helps to have seen how Feynman diagrams for gauge bosons can be drawn as ‘ribbon graphs’. But he’s remarkably good at keeping things simple.

Beginners may, however, not fully understand his assumptions. He says he’s talking about the $Tr(\Phi^3)$ theory. This is the theory of a spin-0 field $\Phi$ of mass $m$ that takes values in the Lie algebra $\mathfrak{su}(n)$, with a cubic interaction Lagrangian

$$tr(\Phi^3) = \sum_{i,j,k} \Phi_{i j} \Phi_{j k} \Phi_{k i}$$

At this point some raw beginners should be told: this describes a universe unlike our own! When he sets $m = 0$, this theory would apply to nonexistent ‘spin-zero gluons’.

More importantly, he doesn’t say why he’s only considering planar, tree-shaped Feynman diagrams! This restriction kicks in naturally only when we consider the limit $n \to \infty$, so I assume that’s what he’s implicitly doing. There’s a long line of work on the large-$n$ limit of $SU(n)$ Yang–Mills theory, and how in this limit we can do a $1/n$ expansion where the terms of order $1/n^k$ correspond to Feynman diagrams that can first be drawn on a surface of genus $k$.

Once Arkani-Hamed restricts himself to planar, tree-shaped Feynman diagrams, he is studying trees drawn on a disk, with 3 edges meeting at each vertex because of the 3 in the $\mathrm{Tr}(\Phi^3)$ theory. At this point associahedra becomes inevitable — because each such tree is a vertex in some associahedron! Or we can draw them dually as ways of connecting up the vertices of a polygon to chop it into triangles:

Restricting to tree-shaped Feynman diagrams is an extreme simplification. (Luckily I just listened to his second lecture, and he said that in the third lecture he’d consider diagrams with loops.)

Unfortunately, by the time the pop science media get ahold of this work, they drop all inhibitions and say stuff like “physicists have discovered a jewel-shaped geometric object that challenges the notion that space and time are fundamental constituents of nature”. Why does everything need to assume cosmic significance? Why can’t some work just be interesting?

But enough complaining! Arkani-Hamed’s talk has excited me to the point where I’m having a new fantasy about the future of fundamental physics. Attempts to cook up a theory of quantum gravity remain stalled. People turn to studying what already works: quantum field theory. They start finding more and more new structures there. Eventually this changes our picture of what quantum field theory is to the point where it breaks the logjam we’re experiencing today. Not only do people figure out how to make nonperturbative quantum field theory fully rigorous, they figure out things we can’t imagine yet… and eventually this leads to really earth-shaking developments.

Indeed, many exciting things are already going on. We’ve got Costello, Gwilliam and others bringing modern math to bear on quantum field theory using factorization algebras, we’ve got Connes, Marcolli and others studying the connection between quantum field theory and motives, we’ve got work on how perturbative quantum gravity is like a ‘double’ of Yang–Mills theory, we’ve got attempts to find Yang–Mills theory hiding inside the $\mathrm{Tr}(\Phi^3)$ theory, we’ve got Wang and others applying TQFT ideas to the Standard Model and GUTs, and a lot more.

So, I want to keep a bit of an eye on this stuff. What are the most exciting developments you’ve seen lately in quantum field theory on Minkowski spacetime?