Simulating physics: the current status of lattice field theories

[ErikZorkin]

I recently watched this video by David Tong on computer simulation of quantum fields on lattices, fermionic fields in particular. He said it was impossible to simulate a fermionic field on a lattice so that the action be local, Hermitian and translation-invariant unless extra fermions get introduced. This is known as the Nielsen–Ninomiya theorem.
David Tong mentioned that simulating physics (to be precise, quantum field theories) remains one the most challenging problems of physics and just a handful of people are currently working on it.

Question: what is currently the most accepted method of simulating a lattice field theory? In particular, what conditions of the said theorem does the Susskind's staggered fermion model discard and what is the physical implications?

 

[Demystifier]

He said it was impossible to simulate a fermionic field on a lattice so that the action be local, Hermitian and translation-invariant unless extra fermions get introduced. This is known as the Nielsen–Ninomiya theorem.

I don't think that's exactly true. What is impossible is to make the action chiral, but locality and hermiticity should not be a problem. Translation invariance is violated on lattice by definition, I think no one even tries to avoid it. My favored method for dealing with the problem of lattice fermions is the Wilson fermions, which requires fine tuning but I don't think it's such a big problem.

For more details see also Tong's lectures https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html Sec. 4.

 

[WernerQH]

Thanks for the pointers. I enjoy Tong's videos, but I find the particular emphasis on the Nielsen-Ninomiya theorem misplaced. I don't consider any fields of the Standard Model fundamental; the Standard Model is an effective field theory sharing "long" wavelength features with the true theory (which we do not yet have). There's no problem at all with Fermion doubling, since the lattice itself is an approximation. All that matters is that the numerical simulations produce reasonable "long" range correlation functions.