In the following I am going to discuss the theory, which I have developed with the mathematician Johannes Aastrup over the past some 15 years. This work is rather technical and involves several different fields in contemporary theoretical physics as well as new conceptual ideas rooted in modern mathematics. In this piece I am going to give a rough sketch of our work and leave the details aside. I expect to write more pieces about our work and there I will dig a little deeper. But since its monday and this is the first post I write about this, I’m going to keep it fairly light.
With this preamble aside let us throw ourselves into the fight. I would like to begin with the notion of a final theory. What do we mean by that?
In his book Dreams of a final theory the theoretical physicist and nobel laureate Steven Weinberg writes about what he calls the arrows of explanations and how they appear to converge at a single point just out of sight. An explanation has a direction — we can explain something by means of something else. We can explain data by means of a principle, for instance. And we can explain one principle by means of another principle. In physics we have uncovered a large number of such arrows of explanations — the laws of Nature — and what Weinberg writes about is that these arrows appear to converge in a single point. The deeper we dig into the fabric of reality the simpler things look. The arrows of explanations do not form isolated blobs or separate islands of logical structures, no, they appear to converge in a single point. That point is what Weinberg refers to as the final theory.
The final theory is a theory that cannot itself be reduced to another, deeper theory. It is it is where the onion ends, the end of the line, the final word, it is the last turtle, if you like.
Okay, so we’ve got that, but how can we possible imagine what such a theory might look like? Is that even conceivable? It seems like a very tall order to cook up a theory, which on the one hand cannot be reduced to another, deeper theory and on the other hand must involve sufficiently complicated mathematics to explain, well, everything.
This consideration is where my research project with Johannes Aastrup begins. The starting point of our work is the idea that such a theory, a final theory, must be so simple that it borders a triviality. From a conceptual point of view it must be almost empty. What else could it be? If a final theory involves complex structure it will open the door to further scientific explanations, to further scientific reductions, which means that it can’t be final. We would want to know where that structure originate from. Why is it this way and not that way? A truly final theory must be imune to such questions and therefore it must be close to being conceptually empty. The only structure it can involve must be self-evident and obvious to such a degree that it makes no sense asking where it originates from.
The idea we are working on is to construct a candidate for a final theory by starting with empty space.
Empty space. Just that. And, well, add a tiny touch of salt too.
So, what can we say about empty space? Well, one aspect of space is that we can move stuff around. We can consider the action of moving an object from one point A to another point B. And don’t worry about the object itself, this is empty space, there are no objects, not yet, we just consider the action itself of moving stuff around.
This is where we begin. We consider an algebra that includes all the possible ways you can move stuff around in empty space.
Why is that a cool idea? Well, think about this — if it really turns out to be possible to uncover sufficiently complex mathematical structures from an algebra, that simply encodes how you move stuff around in space, if it is possible to generate a theory from that, which explains everything we know, then such a theory must be final. How would you be able reduce a theory like that to something deeper? You cannot, it does not make sense to ask what “moving stuff around” is made of. There is noting to reduce here. Such a theory would be conceptually empty.
Now, it turns out that the story, which we derive from this starting point, naturally involves the mathematical research field of non-commutative geometry as well as certain elements of what is known as loop quantum gravity, which is an attempt to formulate a quantum theory of general relativity. It is a story, which we are still busy writing — we’re not finished yet, far from it! — but things do look promising so far. We begin to see mathematical structures, which we recognise from the standard model of particle physics, emerge.
But let us throw ourselves into the deep end and take a better look at the theory, which Johannes and I have found.
So, we start with an algebra that involves all the ways you can move stuff around in empty space. Such an algebra naturally involves what is known as gauge fields — which is a type of fields, that in the standard model of particle physics describe the electro-weak and the strong nuclear forces and which are also related to Einsteins theory of relativity via the so-called Ashtekar variables. A gauge field encodes information about how you move various degrees of freedom between neighbouring points in space and thus it is natural that this type of fields turn up when we discuss an algebra that describes how stuff is moved between arbitrary points in space.
In fact, it turns out that our algebra — we call it the Quantum Holonomy-diffeomorphisms algebra — naturally encodes the basic algebraic relations of a quantum field theory of gauge fields. These relations are the so-called canonical commutation relations, which are the starting point of a quantum theory of gauge fields — or, if we work with Ashtekar variables, a quantum theory of gravity.
So let us sum this up. We started out with an algebra that encodes information about how stuff is moved around in space and we quickly reach the conclusion that this algebra includes the basic algebraic machinery of a quantum theory of gauge fields.
Thats nice. If you know a little about mathematics its not extremely surprising but its nevertheless a nice construction, and its new, nobody has looked at this algebra before we stumbled upon it.
A question about time
There are a number of technical steps, which one should work through when given an algebra like the one I have just described. The most important step is to analyse whether it has what is called a corresponding Hilbert space (in math lingo: a Hilbert space representation). An algebra does by itself not give us a theory, to have a theory we need a mechanism that can translate various algebraic structures into numbers, which then, hopefully, can be measured in experiments. That mechanism is called a Hilbert space, it is so to say the place, where the algebra lives, it is its second half, together they form a theory.
Or rather, the algebra and the Hilbert space give us what we call the kinematical part of the theory. It is the foundation of the theory. When we have the kinematics in place we can begin the show, we just need some time.
Time. Dynamics. Stuff happening, time going by, the tick-tock of the passing of time, this is what we need.
But time does not just happen in a quantum theory. We need something to define the passing of time, a mathematical mechanism, that adds this aspect to our construction.
The name of that mechanism is called the Hamilton operator. It is a quantity, which once defined determines how states in the Hilbert space evolves with time.
But how do we know what Hamilton operator to chose? This is an absolutely crucial question. When we chose the Hamilton operator we essentially pin down what theory we are working with.
But if our aspiration is to find a final theory, then the amount of extra mathematical structure which we add and which could potentially call for further explanations, should be as little as possible and preferably zero. If we seek to find a theory, that stands by itself, then we need some mathematical mechanism, that gives us the Hamilton operator more or less as a consequence of the algebra that we have chosen. It must come for free, we cannot allow ourselves to pay for it by adding further mathematical structures. Or at least we cannot pay very much.
Usually in a quantum theory one picks the Hamilton operator by looking at the corresponding classical theory and then modeling the Hamilton operator in a way so that it produces its classical counterpart in a classical limit. But that method will not do if our aspiration is to find a theory that explains everything, including the classical theory.
What we do is something completely different. And that something is — perhaps not so surprising — a little complicated. So what I will do here is to just give you a brief taste of it and then I will write about it in more details later. But basically we employ the machinery of non-commutative geometry to the quantum holonomy-diffeomorphism algebra and obtain a Hamilton operator from that.
Within the formalism of non-commutative geometry a key concept is what is called a Dirac operator. Ordinary geometry is about curvature of space (and possibly time) and that curvature is encoded mathematically in a metric. A metric is a field, that tells us something about the curvature in space. Now, in non-commutative geometry this picture is turned around and the geometry is encoded in the Dirac operator.
This is a very long story but to cut it short I will just tell you that when we formulate a Dirac operator, that interacts with our algebra (the algebra containing all the “moving stuff around”), then it turns out that it on the one hand has a canonical structure — i.e. there is very little choice in how this Dirac operator can be formulated — and on the other hand it gives us a time-evolution. It gives us the tick-tock that we were looking for.
And with that we have a theory. It is a quantum theory that is build over the concept of moving stuff around and it is well defined and completely consistent. We know that. What we don’t yet know is exactly what this theory describes. We think we know but we are not yet certain. We think it tells us something about quantum gravity and we think that there is a link to the standard model of particle physics hidden within its mathematical structures. We think that this is a candidate for a final theory. But thinking that is not enough, what we need is proof. And that is what we are working on at the moment.
So, I hope that I with this cliffhanger have given you the intellectual equivalent of a New York cheese cake: it tastes incredible and certainly calls for more!
Take good care, may the horse be with you,