Amplituhedron

I’ve been depressed about physics for some time. I hadn’t noticed much progress in either theory or experiment for a while, but a paper from last year by Nima Arkani-Hamed and co-authors has called attention to a remarkable new idea called (somewhat unfortunately, I think) an amplituhedron. I learned about it from this nice article in Quanta. The opening lines of Arkani-Hamed et. al’s paper:

The traditional formulation of quantum field theory—encoded in its very name— is built on the two pillars of locality and unitarity [1]. The standard apparatus of Lagrangians and path integrals allows us to make these two fundamental principles manifest. This approach, however, requires the introduction of a large amount of unphysical redundancy in our description of physics. Even for the simplest case of scalar field theories, there is the freedom to perform field-redefinitions. Starting with massless particles of spin-one or higher, we are forced to introduce even larger, gauge redundancies, [1].

Over the past few decades, there has been a growing realization that these re- dundancies hide amazing physical and mathematical structures lurking within the heart of quantum field theory.

The amplituhedron is a geometric object, the positive Grassmannian, which apparently encapsulates the physics minus the redundancies, with huge gains in computational and (maybe) conceptual efficiency.

The Grassmannian is a sort of generalization of projective space, and, if I understand correctly, positivity is linked to convexity of some underlying polyhedron and enforcement of nice conditions on the calculational results. If this works out, it could be the most important idea since superstrings, and maybe since Feynman diagrams.

It’s interesting that so many of the core ideas in the past century (or so) of physics have proven to be geometric. The amplituhedron has been described as a “jewel in infinite dimensional space” so it clearly fits the category. The sacrifices permitting elimination of the redundancies are locality and unitarity. Locality was already in a certain amount of danger, but I don’t yet understand anything about the sacrifice of unitarity.

On a less giddy note, wikipedia notes:

Since the N=4 supersymmetric Yang-Mills theory is a toy theory that does not describe the real world, the relevance of this theory to the real world is currently unknown, but it provides promising directions for research into theories about the real world.

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UPDATE: Peter Woit casts a somewhat jaundiced eye on the whole thing here. And Lumo has a take here.