All possible QFTs from geometry?

Physicist Nima Arkani-Hamed has taken an approach to understand fundamental physics based on geometry (specifically, positive geometry). This started with his work with Jaroslav Trnka in the amplituhedron [1] and later it was generalised to the associahedron [2],the EFT-hedron [3]...

I was reading an interesting presentation made by one of Arkani-Hamed's collaborators called "Inside the walls of positive geometry: the space of consistent QFTs" [4] based on recent work done by Arkani-Hamed and other co-authors.

There, they suggest that positive geometry is the underlying property of general QFTs.

If that were true, then, would all possible theories proposed so far (including all possible QFTs and all possible theories of everything proposed so far) emerge from positive geometric constructions?

[1]: https://arxiv.org/abs/1312.2007
[2]: https://arxiv.org/abs/1912.11764
[3]: https://arxiv.org/abs/2012.15849
[4]: https://member.ipmu.jp/yuji.tachikawa/stringsmirrors/2018/107.pdf

Suekdccia said:

There, they suggest that positive geometry is the underlying property of general QFTs.

If that were true, then, would all possible theories proposed so far (including all possible QFTs and all possible theories of everything proposed so far) emerge from positive geometric constructions?

It's not going to be "all possible theories of everything" because you always have e.g. cellular automata theories, or other discrete computational models, that just aren't QFTs. A framework capable of encompassing all possible theories would need to be as general as mathematics itself (like set theory or category theory or "the set of all possible programs").

However, it's certainly possible that e.g. the coefficients of all well-defined QFTs obey some kind of positivity constraint. QFTs have a specific mathematical structure, and mathematics is overflowing with examples where the basic definition of a mathematical structure, turns out to have far-reaching further implications for what's possible within that structure. (A modern example: the long path from the definition of a Diophantine equation, to the proof of Fermat's last theorem, which says that a certain infinite class of Diophantine equations has no solutions.)

It's also possible that this positivity hypothesis is wrong! Maybe there are valid QFTs that don't obey it, just as there are solutions to a^n + b^n = c^n when n = 1 or 2.

mitchell porter said:

It's not going to be "all possible theories of everything" because you always have e.g. cellular automata theories, or other discrete computational models, that just aren't QFTs.

Couldn't there be Quantum Cellular Automata models that would be QFTs or equivalent to them?

Suekdccia said:

Couldn't there be Quantum Cellular Automata models that would be QFTs or equivalent to them?

Yes, that would be lattice qft. Every qft can be approximated by lattice qft AFAIK.