— what are we looking for?
One of the main reasons for me to write this blog is that I would like to write about my work with the mathematician Johannes Aastrup. Over the past some 15 years we have developed a candidate for a fundamental theory, a final theory. But before I write about our own work there are two questions, which first need to be asked:
- Why do we believe that a final theory exist?
- To what questions do we hope that a final theory will provide the answers?
In the following I will discuss these two questions — and explain why I think that we should expect a final theory to tell us both more than is normally expected and at the same time much less than what we hope for.
Why do we believe that a final theory exist?
Modern science has given us a tower of scientific explanations — biology, chemistry, atomic, nuclear and particle physics — where each level can be understood in terms of a deeper level. The periodic system can, for instance, be understood in terms of atomic physics and the knowledge of electron orbits; much of biology can be understood in terms of chemistry. But this tower of explanations leads us to the obvious question whether it ends somewhere or whether reductionism will continue forever. Does this onion of a Universe, which we are peeling, have a finite number of layers?
There are good reasons to believe that the onion is finite. There exist a simple argument, that combines quantum mechanics with Einsteins theory of general relativity, which strongly suggest that distances below the Planck length are operational meaningless — and if this is the case, then the tower of explanations will be finite. And the lowest level of explanations, the bottom of the ladder of reductionism, is what we call the final theory.
The argument that I just mentioned? Well, the argument simply says that when you measure shorter and shorter distances you will need test particles with shorter and shorter wavelengths, which means increasingly high energies. But energy causes space and time to curve — according to Einstein — and when the energy of the test particle is sufficiently high it will form a black hole! And we all know that nothing escapes a black hole, not even light, which means that there exist a distance below which measurements do not make sense. That distance is the Planck length.
So this is the reason why we believe that a final theory exist. This argument is no proof but it is a very strong hint that this onion of a Universe, that we inhabit, is finite in depth. And if that is the case we sure as #§&$ would like to know what the last onion layer looks like.
What would we like a final theory to tell us?
Well, the way this onion works is that each layer explains why the previous layer looks the way it does. Particle physics tells us something about nuclear physics, atomic physics tells us something about chemistry, chemistry tells us something about biology and so on. So we would like the next layer, which we believe could be the final layer (but this belief might very well be wrong!), to explain to us why the deepest layer that we have discovered so far looks the way it does. We would like a final theory to explain to us why the theories, which we have already found, have their particular structures.
So before we plunge ourselves into the second question we need to look at where modern physics stands today.
Basically modern theoretical physics operates with two fundamental theories. There is Einsteins theory of general relativity and there is the standard model of particle physics. The first tells us about the large scale structure of the Universe, the latter tells us about the behaviour of elementary particles, their interactions with the electro-weak and the strong nuclear forces and the Higgs mechanism.
Let me add a few more details about these two theories. Firstly, with Einsteins theory one should distinguish between the special and the general theory. The special theory, which can be understood as a local approximation of the general theory, is essentially a consequence of light having the same speed in all systems. The general theory then links the geometry of space and time to its energy content, i.e. it is matter that causes the curvature of space and time. Secondly, the standard model of particle physics is a relatively complicated mathematical machinery that describes how the three fundamental forces — the electro-magnetic, the weak and the strong forces — interact with matter. These forces are formulated in terms of what is called gauge theory, which is a mathematical machinery that describes rotations in certain internal spaces of different dimensions. Finally, the standard model is a quantum field theory, which means that the machinery of quantum mechanics is applied to its basic constituents, namely fields.
So, what we would like a final theory (or at least the next onion layer) to explain to us is why these two theories — Einsteins theory of relativity and the standard model — look the way they do. Nothing less, we would like to understand the basic structure of these theories as properties emerging from a deeper theory.
This statement can be expanded into a number of specific questions:
- Why does Einsteins theory look the way it does? In particular, why is the speed of light the same in all reference systems? Also, why does energy and momentum cause space and time to curve? Or put differently, why does Einsteins theory of general relativity couple to matter the way it does?
- Why does the standard model have its particular structure? It is a rather complicated mathematical machinery — it looks almost arbitrary and yet its involves a mathematical finesse, which seems anything but arbitrary. Why? Specifically, why this particular choice of gauge groups, why gauge theory, why three particle generations, why the Higgs, etc.
- Why quantum? Why do we have quantum mechanics — and quantum field theory, which is the quantum formalism applied to the fields of the standard model? And what about a quantum theory of gravity? As it is now we do not understand whether Einsteins theory can or should be formulated in the language of quantum mechanics.
The third question can in fact be expanded somewhat. The point is that quantum field theory comes in two variants depending on the type of fields involved. There is bosonic and fermionic quantum field theory — bosons describes the forces and fermions describes matter (electrons, for instance) — and these two variants of quantum field theory look different. I would therefore like to expand question 3 with another question:
3.2. Why does fermionic quantum field theory look the way it does?
So, basically we are asking a final theory to explain to us the entire mathematical structure of what we already know. All of it.
More and less
When physicists discuss the question of a final theory and the candidates, which have already been proposed — the most prominent one being string theory — then the list of questions, which one would like to have answered, is usually somewhat less ambitious than the list that I have just given you. What is often discussed is for instance the ratio of different coupling constants (why are they so different?), it is the question about the singularities within black holes, it is dark matter and dark energy, the cosmological constant, it is the question of wether the three forces in the standard model can be unified and so on.
In fact, it is often said that high-energy physics is in a crisis because we have no new experimental data to explain. The Higgs particle was found at the LHC at CERN but no new particles were found, the standard model seems to account for everything we have observed and thus theoretical physicists do not know in which direction to search. The standard model is too perfect, it explains too much. I think that we with this point of view are missing a crucial point. The standard model is by itself experimental data that we must explain. General relativity is by itself experimental data that we must explain. Thus my list. What we need a final theory to explain to us is essentially why these theories look the way they do — everything included, hair and skin and all!
I therefore suspect that a final theory — once it is found! — will tell us much more than we ever hoped for. At the same time I think that it will tell us much less than we hoped for too. What I mean by this is that I suspect that a final theory will be so simple that it will not tell us much more than why our present theories look the way they do (and a few more things such as dark matter for instance) — but not much more than that.
Why is that? Well, how can a theory be final if it is not incredible simple? Structure calls for explanation, which means more scientific reductions and thus not final. I therefore believe that a final theory must be so simple that it is close to being conceptually empty. And if it is empty then it will not tell us much of interest — except why our universe looks the way it does.
My work with Johannes Aastrup
What might a final theory, that is close to empty, actually look like? Well, obviously there is no consensus on this, but the reason why I write about these things here is that I, together with the mathematician Johannes Aastrup, have proposed a candidate for such a theory. Our theory is based on a very simple mathematical principle that provides us with an example of something that is close to empty while it still involves sufficiently complex mathematical structure to make it conceivable that it could be a fundamental theory. That theory will be the subject of another blog posting.
But let me tell you this before I end my post. Although our theory is not fully analysed we do see a possibility that the list of questions, which I gave you in the above, could be explained by it. Why general relativity, why the standard model and why quantum. And that possibility is rather interesting. From the get-go the aim of our work was to explain the structure of the standard model, which is a story that involves the mathematical research field of non-commutative geometry and something as trivial as the mathematics of empty space. I will write about all this in a later post.
Till then take good care, Jesper