More important or relevant progress in discretizing hard problems in physics in last decade

Question

This is a reference request, and soft question as companion.

I'm curious to ask, from an informative point of view, what about the more important progress in the goal to discretize hard problems in physics and that were in the literature recently, let's say in this decade (2009-2020), as remarkable advances.

For the following question I was inspired in the problems explained in [1] (if I understand/interpret well the words of the professor in slides of his section What About Future Laws of Physics?, my understanding is that these discussions are related to hard problems in physics).

Question. I would like to ask as reference request, or soft question, for the more important or relevant recent progress in discretizing hard problems in physics.

I'm asking it as a reference request for the more recent advances, then I'm going to try to search and read those references from the literature. If in your discussion as soft question you want to refer about other past articles feel free to do it.

My knowledges in physics or mathematical physics and discretization methods aren't the best, but I think that this could be an interesting post for your colleagues, and reference for all us. Feel free to add your feedback about the post in comments.

References:

[1] David Tong, Physics and the Integers, University of Cambridge, Trinity Maths Society (2010).

Answer 1

I understand the question as a request for pointers in the literature to research in the discretization of spacetime. There are two reasons why this is an active research topic, a fundamental and a practical reason: Fundamentally, spacetime might be discrete at the smallest levels (Planck scale); practically, to simulate relativistic quantum field theories (either on a classical computer, or, eventually, on a quantum computer), we need to discretize continuous degrees of freedom.

For an overview that I found instructive, and you might also like, I point to Causal Fermions in Discrete Spacetime by Farrelly and Short:

We consider fermionic systems in discrete spacetime evolving with a strict notion of causality, meaning they evolve unitarily and with a bounded propagation speed. First, we show that the evolution of these systems has a natural decomposition into a product of local unitaries, which also holds if we include bosons. Next, we show that causal evolution of fermions in discrete spacetime can also be viewed as the causal evolution of a lattice of qubits, meaning these systems can be viewed as quantum cellular automata. Following this, we discuss some examples of causal fermionic models in discrete spacetime that become interesting physical systems in the continuum limit: Dirac fermions in one and three spatial dimensions, Dirac fields and briefly the Thirring model. Finally, we show that the dynamics of causal fermions in discrete spacetime can be efficiently simulated on a quantum computer.