Applications of "finite mathematics" to physics

Question

Disclaimer: I know that what follows is a biased view on applications, one of the points of the question is to eliminate some of that bias.

When I think of applications of maths outside of itself, I have the impression that applications in physics are mostly related to "continuous/smooth mathematics" : representation theory of Lie groups, PDEs, functional analysis, different kinds of differential geometry (I don't know if they technically fit into that, but I include, say, symplectic geometry and riemannian geometry in that word), probability theory, and a bunch of other stuff, but all somehow related to $\mathbb{R,C}$ and the differential or topological or measurable structure on these (or related structures);

while "discrete/finite mathematics" (here I'm almost sure I'm not using the right terminology - what I mean by that is stuff like finite group theory, representation theory of abstract groups, ring theory, linear algebra over finite fields, algebraic geometry, combinatorics, finite probabilities, number theory, graph theory and again a bunch of stuff that somehow fits the intuitive meaning one could put behind "finite" or "discrete" mathematics) seems to have applications mainly in computer science and related fields.

Now this view is probably very biased, and that's because I don't know that many applications of maths/much applied maths. The point of this question is to, if possible, get rid of some of that bias. Since asking "what are applications of mathematics ?" would be way too broad, I'll ask something more specific and more related to my personal interests.

What, if any, are some applications of "finite/discrete mathematics" to physics ? More specifically of "finite/discrete" algebra ?

(Note that here I use words "finite/discrete mathematics" in the sense that I tried to describe vaguely above, not in the common sense, if it is different)

Answer 1

This answer is (intentionally) partial in order to present a special topic, concerning an application of graph theory in quantum physics.

Recently at MathOverflow Mario Krenn asked “a purely graph-theoretic question motivated by quantum mechanics” (and a special case of the questions asked in a less than a month old arXiv paper "Questions on the Structure of Perfect Matchings inspired by Quantum Physics” by Mario Krenn, Xuemei Gu and Daniel Soltész). I allow myself to quote here fragments from the beginning and the conclusion of the paper:

A bridge between quantum physics and graph theory has been uncovered recently [1, 2, 3]. [These are fresh papers, among others, of the first two authors and Anton Zeilinger, a famous specialist in quantum physics. AR] It allows to translate questions from quantum physics – in particular about photonic quantum physical experiments – into a purely graph theoretical language. The question can then be analysed using tools from graph theory and the results can be translated back and interpreted in terms of quantum physics. The purpose of this manuscript is to collect and formulate a large class of questions that concern the generation of pure quantum states with photons with modern technology. This will hopefully allow and motivate experts in the field to think about these issues. ...

Every progress in any of these purely graph theoretical questions can be immediately translated to new understandings in quantum physics. Apart from the intrinsic beauty of answering purely mathematical questions, we hope that the link to natural science gives additional motivation for having a deeper look on the questions raised above.

So now we start a joint project between quantum physicists and graph theorists to deal with these problems. Yesterday Mario Krenn wrote to me:

I was thinking that it might potentially interesting to think about a project along the lines of "Quantum Physics for Graph Theoretists", kind of an educational text that helps Graph theoretists to understand the concepts in quantum physics. Our connection here with ivc is a special type of quantum experimental construction, but it encodes many nontrivial essential quantum phenomena, such as quantum superposition, quantum entanglement, quantum interference (!), special purpose quantum computation, entanglement swapping [for all of these effects, i know how they look in graphs]. probably quantum teleportation (as a special case of entanglement swapping), probably many others. maybe we can go through quantum textsbooks and see what other ideas we could translate and explain in a language that graph theoretists would understand and apprechiate.

as a benefit, it would give me a different way to think about these topics also :) maybe it could even help to see things in quantum physics that i was not aware of, but that you see in graph theory.

Answer 2

Crystallography is full of discrete/lattice results.

Crystallographic restriction theorem

Crystallographic point groups

Bieberbach's theorems

FKT Algorithm

Answer 3

• Rep Theory. Granted what physicists calls group theory tends to really mean representation (rep) theory. There’s an inordinate amount of physics related to reps of Lie algebras but I’m going to assume you wanted only reps of finite groups. The preceding (rep theory for finite groups) applies to rigid body rotations and reflections for certain cases, abstract fourier series on the torus, the concept was also heavily used for FFT algorithms whose use is insanely widespread well beyond physics, reps of point groups/space groups (in crystallography/quasicrystals).

• Ring Theory. This is heavily used in certain areas of crystallography as you’re looking at $\mathbb{Z}$ lattices.

• Finite Groups. Braid groups have been very popular in physics lately, also - such as in quantum computers and anyons, field theories, and the Potts model of statistical mechanics.

• Probability & Combinatorics. If you want physical applications of probability and combinatorics, look into statistical mechanics. Aside of the Potts model, one can discuss random walks on lattices and a bunch of different Monte Carlo simulations; there are many many more applications here; the ones I've worked with closely are in condensed matter theory.

• Difference Equations & Dynamical Systems. Difference equations are also important. There are a few in quantum mechanics I’m aware of like studying the evolution of discrete states which can lead to cool topics in dynamical systems and sometimes quantum chaos.

In addition, physicists almost always assume their Hilbert space of quantum states is separable (has a countable basis) in which case it is algebraically isomorphic to little $\ell^2(\mathbb{Z})$. As in this example, the discrete and the continuous are not always such repelling concepts. The advisor I work under did some work using cohomology to study a discrete (quasicrystal) lattice.