In the following I will discuss what I consider the most fascinating and intriguing development in theoretical high-energy physics in the past some forty years, namely the research field known as noncommutative geometry.
The core business of noncommutative geometry, which is pioneered by the french mathematician and field medalist Alain Connes, is is to produce a dual formulation of Riemannian geometry (the geometrical framework, that we use in Einsteins theory of general relativity), that has a straight-forward generalisation to a much wider class of exotic geometries known as noncommutative geometries.
Noncommutative geometry turns our usual notions about geometry upside down and shows us that there exist other types of geometries, which we previously had no access to — at least not within a geometrical formalism — and where one of the first examples of such an exotic geometry turns out to be the standard model of particle physics coupled to general relativity . That is, essentially everything we know about the universe can be formulated in terms of a single geometrical framework.
What I am going to do is to first give you a sketch of what noncommutative geometry is all about. I’ll leave out all details and technical noise and just give you a rundown the basic concepts and ideas. The aim is to get as far as the connection to the standard model — and in particular to discuss what this connections means. What questions it raises and in which new directions it sends us searching for answers.
Distances between points
As the name suggests noncommutative geometry is about geometry. So let us begin by looking at ordinary geometry or what we call Riemannian geometry, which is the framework that is used in general relativity. Basically geometry is about distances between points in various spaces, so let us begin with a space and consider its geometry, which is usually encoded in a metric. A metric is a field, that assigns a set of number to each point and which tells us someting about the curvature in that point.
Think of a straight line. If we wish to measure the distance between two points A and B on the line, then what do we do? Well, we get out the good old ruler and place it on the line and read off the distance. Now, the ruler is here the equivalent of a metric. The metric is so to say a ruler generalised to more complicated spaces with nontrivial geometries.
But now I would like to propose a second way of measuring the distance between the points A and B. What I would like to do is to pick a function on the line — a function is simply an assignment of numbers to each of the points on the line — and then I will read off its values in each of the two points and subtract them. Voila! That gives us a distance on the line.
“But wait a minute, that can give us anything” I imagine you might object. And you are of course right, so what we need to do is to restrict our choice of function. If we for instance choose a function that has a gradient of exactly one, i.e. that the function increases by one every time we take a step along the line, then it is clear that when we subtract its values at the points A and B then that number will be equal to the distance between these points.
If you think that this sounds rather trivial then you are completely right, but hang on a moment, this will soon develop into something very far from trivial.
Let us now look at the generalisation of this second way of measuring distances to the case where we have a space with a more complicated geometry. So we pick a function on our space and then we need a way to ensure that this function has what amounts to a gradient of one (the precise meaning of this statement is expressed in Connes so-called distance formula). The way to do this is to introduce what is called a Dirac operator, which is a mathematical entity that gives us the gradient of functions.
So when we generalise the second way of measuring distances we end up with a framework, where we have functions combined with a Dirac operator. These two ingredients need, however, a third ingredient, which is called a Hilbert space and which is roughly speaking a place where we can evaluate the difference between two functions. Finally, we organise the functions in what we call an algebra, which is a basic mathematical framework that involves addition and multiplication.
To summarise, we have now these three ingredients: 1) an algebra of functions, 2) a Dirac operator and 3) a Hilbert space. This is what we call a spectral triple.
Now, we can lift this second approach to a more abstract level. What I mean by this is that we can forget about the space and just consider the spectral triple as an abstract object together with the mathematical rules — axioms — it is required to satisfy. So we have an abstract algebra that interacts with a Dirac operator and where it all lives in a Hilbert space. And note this: the space itself has now become a secondary quantity, we do not need a space to define an abstract algebra, it can be derived from the algebra in a secondary step. Also, we should note that it is the Dirac operator, that encodes geometrical data. The Dirac operator corresponds in a certain sense to a metric.
In 2008 Alain Connes proved that his framework is equivalent to the framework with a metric field. The so-called equivalence theorem states that it makes no difference which framework we choose, they amount to one and the same geometry.
So these two ways of formulating geometry are equivalent, that is nice. But there is one very important difference between these two frameworks and that difference is the very reason why this is interesting.
The key to understand this difference is the algebra of functions. I’ve already reminded you that an algebra is a collection of elements, which can be added and multiplied. Numbers form an algebra and so do functions on a space. You can add or multiply two numbers to get a third number and you can add or multiply two functions to get a third function. And note this: when it comes to multiplication then it does not matter in which order you do this. Two times three equals six as does three times two. Numbers commute, they don’t care in which order we multiply them.
Now, it turns out that the mathematical conditions, the axioms, that we require a spectral triple to satisfy in order to give us a geometry, can be straightforwardly generalised to algebras that do not commute. That is, algebra where it matters in which order their elements are multiplied. These are called noncommutative algebras.
And with this we finally arrive at what we call noncommutative geometries. A noncommutative geometry is given by a spectral triple, which involves a noncommutative algebra.
This is where this story gets really interesting. The point is that this generalisation to noncommutative algebras opens the door to a completely new type of geometrical spaces and it turns out — surprisingly — that the standard model of particle physics coupled to Einsteins theory of general relativity gives us an example of such a geometry.
If you think this sounds a little intriguing, perhaps even f§!€ incredible, then I agree with you. So let us jump into a couple of details to understand this better.
An almost commutative geometry
Let us consider a very simple example of a noncommutative algebra, namely one given by rotations in a three-dimensional space. We can rotate around the three axes and we can combine two rotations to get a third one, which means that we can multiply them. But note this: it matters in which order we execute these rotations (try it out!), they do not commute, and hence an algebra build from such rotations will be noncommutative. This gives us what we call a matrix algebra. Now, we can combine such a matrix algebra with the algebra that we already have worked with, namely the algebra of functions, and when we do that we obtain what is called an almost commutative algebra.
This type of algebra is very close to the algebraic setup that corresponds to general relativity alone. All we have done is to add a little bit of noncommutativity, we have added a matrix factor. And it is this type of algebra, that leads us to the standard model of particle physics. If we add just the right type of matrix algebra and turn all the cranks on the machinery of noncommutative geometry, then the standard model coupled to general relativity comes out in the other end.
I think that this is amazingly cool.
With this machinery we have a completely new way of understanding the standard model. Things, which in the old formulation looked odd, suddenly make more sense. The best example is probably the Higgs mechanics, which in this new formulation emerges as an integrated part of the gauge sector. The Higgs is here not some slightly weird add-on, its an integral part of the geometrical machinery. It has to be there.
In what direction does this lead us?
There is a tonne of details, which I could write about here. There are caveats — for instance the important restriction that this formulation really only works in the Euclidian setting, i.e. when the time-direction is complex — and other limitations. But although these details are all important, essential even, I believe that its wise to leave them aside and instead look at the bigger picture. What does all this tell us and where might it lead us?
There are two points, which I would like to emphasise first:
- The noncommutative formulation of the standard model is a purely gravitational formulation. The standard model and general relativity are here combined into an integrated whole, a single gravitational theory over an odd noncommutative space. This is in my opinion remarkable.
- The fact that this is possible is nontrivial. Not all field theories fit into the framework of noncommutative geometry. Had the standard model looked a little different it could not have been formulated within this framework. This means that the standard model is special. The framework of noncommutative geometry singles out the standard model as exceptional.
This raises two key questions:
- Where does the almost-commutative algebra originate from? Why this specific structure? What might explain this particular choice of mathematics? I believe that this is one of the most interesting questions in modern theoretical physics (and, oddly, one that is barely being asked).
- What role does quantum field theory play in all this?
The second question requires some explanations. The standard model is a quantum field theory, which means that it is written in the language of quantum mechanics (operators and Hilbert spaces). But the noncommutative formulation of the standard model only captures the classical theory. It does not involve the quantum aspect.
It is not hard to understand why this has to be the case. Since the noncommutative formulation is essentially a gravitational formulation it would, if it were to involve quantised fields, involve some kind of quantisation of gravity too. And that is something we don’t know how to do. So what Chamseddine and Connes have done is to apply the procedure of quantisation in a secondary step and only to the standard model — not to gravity.
Now, this is clearly unsatisfactory. If this formulation — which I assure you is truly beautiful — should be fundamental then ought it not involve quantum theory as a primary ingredient and not as a more or less arbitrary add-on? Hence my question: what role does quantum field theory play in the noncommutative formulation of the standard model? I think that this question is essential.
It is a strange situation. On the one hand we have the old formulation, where the standard model is a quantum field theory and where gravity is a separate theory, and on the other hand we have the new formulation, where the standard model and gravity at a classical level are unified but where the quantum is left out. What does this mean? Nobody knows.
Let me end this with my own point of view. I believe that Chamseddine and Connes formulation of the standard model is a roadsign. I think it points us in the direction of a more fundamental theory and I think that it is our job to read this roadsign very carefully.
The key point is, in my opinion, that with the formalism of noncommutative geometry it is possible to boil down the structure of both the standard model of particle physics and general relativity to something very simple, namely an almost commutative geometry.
I believe that this points us in the direction of a theory of quantum gravity, which should do two things: first it should involve the language of noncommutative geometry and second it should produce the matrix factor, which is needed to generate the standard model, in a classical limit. A corollary of this opinion is that quantum field theory as we know is in fact be a low-energy limit of a quantum theory of gravity.
These considerations are in fact what initially motivated the mathematician Johannes Aastrup and myself to begin our work on our own approach to a theory of quantum gravity. In our work we have proposed a setup, which is essentially purely quantum gravitational and which is capable of delivering an almost commutative algebra in a classical limit. We are, however, not yet certain that this almost commutative algebra suffices to get us all the way to the standard model itself — more work is required — but we are optimistic that we’ll know the answer relatively soon. I will write more about that later.
And with that I end this piece.
Best wishes, Jesper