Hi everyone,

It’s finally time for a newsletter. Johannes and myself have just posted two new papers on the archive (you can find one here, the other one will appear in a few days. I’ll attach both of them to this newsletter). In these two papers we present a very simple and somewhat surprising result concerning Yang-Mills quantum field theory. One of the most important unsolved problems in contemporary theoretical high-energy physics is the rigorous formulation of Yang-Mills quantum field theory. This is one of the famous Millenium Prize Problems put forward by the Clay Institute of Mathematics. In our two papers we show that there exist a very simple connection between Yang-Mills quantum field theory and an elementary geometrical construction on a so-called configuration space, a result that we suspect might point towards a solution of the millenium problem. 

In this newsletter I will explain to you what this means and put our result into context. We begin with Yang-Mills quantum field theory and after that we will discuss Dirac operators. As always I will try to keep the discussion as simple as possible and focus primarily on the big picture. But in order to give you an idea about just how simple a Dirac operator really is I will in this newsletter digress a little into some details and use a small amount of mathematics. I will, however, try to provide those of you, who are not too keen on mathematics, with an easier bypass route.

Yang-Mills quantum field theory

As I already said, one of the most important unsolved problems in contemporary theoretical high-energy is the rigorous formulation of a Yang-Mills quantum field theory. Since Yang-Mills theory is one of the central pillars of the standard model of particle physics, where it describes the force fields (the strong and electro-weak forces), it is vital to understand how a quantum theoretical formulation of Yang-Mills theory works. This problem has been known for more than 50 years and yet it remains unsolved. The Millenium Prize that the Clay Institute put forward has never been claimed.

This is truly remarkable. We know exceedingly well how the mathematics of quantum mechanics works, i.e. how to rigorously formulate a quantum theory that involves particles. We have known this for many decades. But what is not known is how to rigorously formulate a quantum theory of fields. 

The fact that this problem has remained unsolved for so long suggests that the correct framework to address it in has not yet been found. Some of the best minds have worked unsuccessfully on this problem for decades, and thus it seems likely that some of our basic assumptions must be wrong. In other words, that we need a completely new idea. This is where Johannes and my work enters the picture.

What is a Dirac operator?

But before I can get to our work I need to explain to you what a Dirac operator is, and to do that I am first going to introduce the concept of a Clifford algebra. To do that I will use a minimum amount of mathematics. If you don’t feel comfortable with that I invite you to just skip over those parts; you will be able to get the gist of the argumentation without understanding all the details. But for those readers, who are comfortable with a small dose of math, I would like to show you just how simple a Dirac operator is.

And to keep the discussion as simple as possible I will first work in two dimensions. That is, we consider points in a two-dimensional plane. 

Now, many students (and maybe also some mathematicians and physicists), would probably prefer that the following equation was true:

                               

This is of course not the case, but what if it was?

A common praxis in mathematics and physics is to define new objects that satisfy a desired equation or property. In the example here we could for instance consider something like , where and are some new “mysterious” objects, and try to get to be equal to . A small computation gives us: 

If we want this to be equal to then we need the following two conditions to be true:

(1)  

(2)

The question is if such “mysterious” objects and exist and what they might look like? The answer is that they do exist; the two conditions listed above describes what is known as a Clifford algebra associated to a two-dimensional plane. For those of you who are familiar with matrices you can check that the matrices

,

satisfy the two conditions above. So one can think of Clifford algebras in terms of matrices.

Clifford algebras is the type of algebra that the British mathematical physicist P.A.M. Dirac was lead to consider in 1928 when he was searching for a version of the Schrödinger equation that is compatible with Einsteins theory of special relativity.

But why did Dirac need a Clifford algebra? 

The answer is that he was searching for a differential operator with some very particular characteristics. Dirac needed a first order differential operator, whose square would give him the Laplace operator, which is a second order differential operator. 

Let me explain. First of all, a differential operator is a mathematical entity that involves taking the derivative of functions (i.e. computing its gradient in some direction). For instance, in two dimensions we write the derivatives of a function   in the x and y-directions as

.

If a differential operator is first order it simply means that it involves only one derivative; if it is second order it involves two derivatives. And a Laplace operator, which we write as ., is the operator that in two dimensions is the sum of two terms: one term with two derivatives in the x-direction and one term with two derivatives in the y-direction:

.

The Laplace operator is found everywhere in physics, for instance in the Schrödinger equation that Dirac was working on. The problem that Dirac was trying to solve is that the Schrödinger equation is first order in time (i.e. it involves one derivation in time) but second order in space (because it involves the Laplace operator). But Einsteins theory of special relativity tells us that space and time should be treated on an equal footing, which is not the case in the Schrödinger equation. So Dirac was trying to find an equation that was first order in both space and time.

Now, what Dirac realised was that the solution to this problem involves a Clifford algebra. The first order differential operator, that he was searching for — and which is today known as a Dirac operator — involves elements like and that we discussed above. In two dimensions the Dirac operator looks like

(here the ‘i’ is the imaginary unit. If you don’t know what that is it does not matter).

You can check that the square of gives us the Laplace operator , which is what Dirac needed in order to write down his famous equation. Dirac’s equation lead to the discovery of the anti-particles and earned him a Nobel prize in physics.

Now, the Dirac operator that Dirac used was four-dimensional (i.e. it involves three spatial dimensions and one time dimension), but let us take a closer look at the two-dimensional Dirac operator that I wrote down above. What does this operator actually do? Well, it takes the derivative of a function in the x-direction and throws the result into the -slot of the Clifford algebra and then it takes the derivative of a function in the y-direction and throws the result into the -slot of the Clifford algebra. And it does this in a way that guarantees that  the square of gives us the Laplace operator . 

What I want to convey to you here is just how simple a Dirac operator is. Once you have the Clifford algebra the Dirac operator is the simples differential operator you can write down.

Today the Dirac operator plays a pivotal role in the standard model of particle physics, where it tells us how the matter fields interact with the force fields, which are described by Yang-Mills theory. Thus, Dirac operators are primarily understood to play an important role in particle physics and therefore they are mostly associated with 3 and 4-dimensional spaces. But Dirac operators can be formulated in any space-time dimensions. Also, in mathematics it has long been known that the Dirac operator plays an intrinsically geometrical role. They can straightforwardly be formulated on spaces, which have a non-trivial geometry, and on such spaces the Dirac operator encodes information about that geometry. 

Dirac operators on configuration spaces

The reason why I spend so much time on Dirac operators here is that they play a pivotal role in my work with Johannes. As I have discussed previously we are interested in the geometry of what is known as a configuration space.

Let us briefly recall what a configuration space is (and in particular, a configuration space of gauge fields). When you move an object around in your living room you can do that in many different ways: you can rotate it twice to the right, you can rotate it 4576 times to the left, or you can rotate it as many times as you wish around some other arbitrary axis. There are infinitely many ways to move an object between two points in space and all this information is stored in what is known as a gauge field (which, incidentally, is the type of field that one encounters in a Yang-Mills theory). A configuration space is then the enormous, infinite-dimensional space that includes all possible gauge fields. So one point in a configuration space is one gauge field, which tells us how to move objects between arbitrary points in your living room.

— I know, it’s all very abstract, but stay with me.

By the way, configuration spaces of this kind are well known in physics. For instance, they play a central role in quantum field theory. What is completely new is, however, the idea to consider their geometry and in particular, to do that by constructing a Dirac operator on them.

So this is where we come full circle. Because the way that Johannes and myself formulate a geometry on a configuration space is by constructing a Dirac operator on that space.

And since the configuration space is infinite-dimensional it means that the Clifford algebra, that one needs in order to construct a Dirac operator on that space, must also be infinite-dimensional. Thus we are now dealing with a very different scenario than the two-dimensional case that we discussed before, but much of the mathematics is the same.

Back to Yang-Mills quantum field theory

But what does all this have to do with Yang-Mills quantum field theory, which is where we started? Well, once we have constructed a Dirac operator on a configuration space we can begin to play around with it. And it turns out that if we rotate this Dirac operator in this infinite-dimensional space (a rotation in such a space is called a unitary transformation), and if we take the square of the rotated Dirac operator then we get an expression that is identical to a sector of Yang-Mills quantum field theory known as the self-dual sector. 

Let me explain. A Yang-Mills theory can be divided into two sectors, a self-dual and an anti-self-dual sector, and the sum of these two sectors gives you the full Yang-Mills theory. And what we find is that we obtain one of these sectors if we rotate our infinite-dimensional Dirac operator and takes its square. Which sector we get (self-dual or anti-self-dual) depends on which direction we rotate the Dirac operator in.

What does all this mean?

That is a good question. To be honest we are a little perplexed by this result. On the one hand it is extremely simple, the computations are almost trivial. For this reason we have been doubtful that this result is actually new; because it is so simple we first thought that it must be known. But the truth is that we haven’t known about it before and since we are the first researchers to construct Dirac operators on configuration spaces we don’t think anyone else have thought about this. 

Now, I should mention that one can get a similar result by rotating a Laplace operator on a configuration space, so we cannot completely rule out that someone somewhere already considered this idea, but again, we are not aware of it.

But leaving that issue aside, the question remains what it means? 

We think that this result tells us that there is a very simple connection between Yang-Mills quantum field theory and Dirac operators on configuration spaces and the research field of noncommutative geometry, which is based on Dirac operators. That these two frameworks are closely connected.

And the reason why we think this is important is that it might be the missing ingredient needed to solve the Millenium Prize Problem of Yang-Mills theory. Perhaps the reason why this problem has remained unsolved for so long is that people have worked on it in the wrong setting; perhaps the right setting for solving this problem is one based on Dirac operators on configuration spaces?

And this, incidentally, is what Johannes and myself have been working on for the past two decades. We have already shown that a geometrical construction on a configuration space that involves Dirac operators naturally leads to a framework that involves many of the basic building blocks of contemporary theoretical high-energy physics. And now we find a simple roadsign that points in the direction of Yang-Mills quantum field theory.

In other words, perhaps a quantum theoretical description of Yang-Mills theory will turn out to be much more than just Yang-Mills theory quantised. 

Other bits and pieces

The two papers that we have just published includes more material than what I have discussed here. In those papers (and in particular the longer one) we continue the development of the geometrical construction on the configuration space. One key new result is the inclusion of fermions with half-integer spin. This is a problem that we have worked on before. A few years ago we found a solution to this problem, but we have not been completely satisfied with that solution and thus we have now revisited this issue and found what we believe is a more natural solution. We have also been developing the mathematics around the Dirac operator on the configuration space. This work is, however, very technical and thus not suitable for a discussion here.

There is a lot of interesting things going on in my camp at the moment, both with respect to my work with Johannes but also on other fronts that I am working on. As soon as some of these activities are ready to be made public I will write another newsletter to you about them; I expect this to happen fairly soon, probably within a few months time. So stay tuned, I look forward to telling you about it all.

Have a nice autumn 

With this I will end this newsletter. I hope that you are all doing well and that life is treating you kindly. 

Take good care,

Kind regards, Jesper