A couple of months ago, I finally completed a project I embarked upon many years back and defended (successfully, in case you were wondering) my Master’s thesis. I was working on mathematical physics, and more specifically in an arcane subdomain that studies quantum groups, a concept that even most of my fellow theorists usually don’t come across. As a consequences, whenever somebody asked my what my research was actually about, I’d usually shrug my shoulders and reply “It’s hard-to-impossible to explain when you’re not familiar with the framework”. Ultimately though, I grew dissatisfied with this answers and started thinking about how to render the whole affaire more comprehensible, without restoring to a lot of technical jargon and formulae. What follows is my attempt to cast the thesis into terms that require nothing but elementary knowledge of maths and physics, but hopefully provide a small window onto the fascinating world of theoretical physics.
Physics is often thought of as an extremely difficult subject that only those rare geniuses with a natural feel for numbers can master. And yet, in a sense, it is also a very simple science. The objects we deal with — quarks, atoms, fluids, planets, galaxies and so on — are nowhere near as complicated as the most primitive mammals, let alone human societies. It is this fact that makes it possible to test our predictions with astonishing precision, leading to a body of knowledge that seems to grow at an ever-accelerating rate.
The payoff, of course, is that it’s rather hard to find instances of these ideal models in nature. And even when we do, there are hardly any cases where we can calculate the value of some quantity we are interested in — say, the lowest possible energy of the system under consideration — exactly, without restoring to approximations and numerical methods. However, there are exceptions. One important class that defies these restrictions, known as integrable models, will be the most relevant for us. Let me explain what this means.
An illustrative example that helps to understand the concept was introduced by Werner Heisenberg, one of the pioneers of quantum mechanics, and originator of the famous Heisenberg uncertainty principle. He suggested a model of particles placed on a line at constant distance that interact with one another through an intrinsic property called the spin: a (one-dimensional) quantum spin chain . If you wish, think of these spins as needles that can point in two directions only, either up or down . Their pinpoints are connected by an elastic rubber band, so that flipping one spin around will have an effect on its neighbors. Continuing with our analogy, the tension of the rubber band would be a measure for the overall energy of the system. In the simplest configuration, all spins pointing in one direction correspond to the lowest possible energy.
A peculiar feature of quantum physics is that the energy of the system cannot just take any value, but comes in discrete portions, or levels. What is quite extraordinary in Heisenberg’s model is that it is actually possible to calculate all the ‘allowed’ energy levels exactly, without sweeping something under the carpet midway through, and no matter how many particles the system contains! That this works at all has something to do with the mathematical structure of the model, so let us make this a bit more explicit.
We all love symmetries, in arts, in architecture, and even in faces; but physicists hold them especially dear. And, of course, they have a rather particular understanding of the word. Whereas most people, when thinking of symmetries, would probably picture a mandala that looks the same when reflected or rotated in a certain way, physicists use it todescribe transformations of a system that does not make it behave ‘differently’. Specifically in quantum mechanics, this often relates to changes on the microscopic level (which we cannot observe) that leave macroscopic quantities (which we can measure) the same. For example, in our spin chain, it doesn’t actually matter which direction we call ‘up’, and which we call ‘down’: If all spins point in the same direction, the energy will be the same! In the same vein, one can show that configurations with the same overall spin (number of needles pointing up minus of needles pointing down) are indistinguishable: When we do measurements, we will get the same results for two ‘up’ spins on the very left, in the middle, or anywhere else in the chain as long as all other needles point down .
There are several mathematical concepts that describe these and similar and symmetries very precisely. The most important one for us, as it’s the central theme of the thesis, is that of an algebra . There is a very technical definition of it, but for our purposes, we can imagine it as a set of (abstract) objects, together with a prescription how to combine these elements with one another. Take vectors on a plane, for example. Our set would be all the arrows pointing from one spot to another. And we understand what it means to add them: The starting point of one vector is moved to the tip of the other, as you can see it in the illustration below. Moreover, it doesn’t actually matter in which way we add them (v+w or w+v), the outcome will be the same. Note also that we can build any vector whatsoever from v and w if we are allowed to stretch them (multiply with a number) before adding them. We say that these two vectors form a (two-dimensional) basis for the plane.
In the most general case, algebras can be rather unintuitive. For one thing, we typically don’t require a fixed number of basis elements — in fact, we can even have infinitely many of them (it’s just not longer possible to draw this). And second, it’s not always true that the order in which we connect our objects doesn’t matter, contrary to the example of the two-dimensional vector space above.
Consider an analogy: Assume you had three friends, Albert (A), Berta (B) and Cecilia (C). Albert doesn’t know the two ladies, but Berta and Cecilia are in touch regularly. And in fact, their friendship is a little bit strained because Berta The Jealous always suspects you like Cecilia more than her.
Now let’s say you have an exciting piece of information — maybe a new job — that you want to share with your friends. With Albert, there is no problem: If you let him know ahead of Berta (which we will denote by A*B) or afterwards (B*A) makes no difference, since the two don’t talk with each other. The same is true for him and Cecilia: A*C = C*A. However, things are different for your two female friends: Telling Cecilia first will not go unnoticed and make Berta furious, whereas the reverse will have less dramatic consequences. Mathematicians, who feed on precision, even have a tool to measure how important that order is: It’s called a commutator, and the formula for it is [B,C] = B*C — C*B. They say that elements commute (such as Albert and Berta) if [A,B] = 0.
As it turns out, this commutator is very relevant for the concept of algebras I introduced above. In fact, what we typically do to define an algebra is to write down how the results of this commutator look for all the basis elements of the algebra (it’s possible to have two algebras made up of exactly the same elements and yet possess very different structures). Additionally, the commutator can tell us something about the center of an algebra, which is a label given to elements that commute with all other elements. In our fictitious example, Albert was the center of our ‘friend algebra’ (though not necessarily the center of attention).
This was, of course, a very simple case, and things can get a lot more involved. What I did in my thesis was to look at a complicated algebra that can be viewed as a generalization of structures attributed to Heisenberg’s model. And despite the fact that this object is infinite-dimensional and has painfully involved commutators, we were able to show exactly how its center looks like. If you dare, you’re invited to have a look at the technical demonstration here.
The upshot is this: Physical symmetries can be encoded algebraically, and by studying these abstract structures, we can hope to become better at solving our models, and ultimately to gain a deeper understanding of the empirical world that surrounds us. Make no mistake: there is no guarantee this strategy always succeeds. But history is ripe with examples where ivory-towery theories found their way into concrete applications, and the present work might enjoy the same fate one day.
 Lest you argue that this hardly sounds like a ‘realistic’ model, let me assure you that modern laser-cooling techniques can indeed recreate this set-up and make the theory experimentally accessible.
 Note that this doesn’t represent the most general case. Depending on the particles that are being used, there may be more than just two different spin states, but this fact is not essential for the rest of the article.
 In the absence of external (magnetic) fields, that is. When such fields exists, they will favor one direction — remember playing with that horseshoe magnet to orient metallic needles? Physicists call this procedure symmetry breaking, and it plays a fundamental role in particle physics.
 To avoid confusion, I should add that this is not the way the word is used in school, where it describes rules for manipulating mathematical symbols and solving equations.