Johannes and I have just uploaded a new paper to the archive. This is without any doubt the most important paper that we have published in our careers; I feel very strongly about it. It is clear to me that our project has reached a milestone: what was hitherto an ambitious project with new ideas and partial results is now a more or less complete mathematical structure that ties together essentially all the basic ingredients of modern high-energy physics. I am not aware of any other mathematical framework in theoretical high-energy physics today that compares in scope, simplicity and mathematical completeness to what we have discovered.
In this newsletter I will explain to you what we have found. This newsletter will be a little different from the previous ones as I will permit myself to use some technical language now and again. I will still do my best to keep the exposition at a level where everyone can follow me, but once in a while I am going to shift gears. Also, I expect this newsletter to be longer than usual.
So hang on and lets begin!
The basics of our construction
The starting point for our construction is a very simple ansatz, namely an algebra that contains information about how objects are moved around in a three-dimensional space. We have named this algebra the QHD algebra.
When you move, say, a pencil from A to B then you might think that there are no ambiguities involved in that, but in fact there is: if the pencil points “up” at A then where should it point at B? Should it also point “up” or perhaps “down”? The mathematics, that encodes this information, is called a gauge theory and the central object is called a gauge field. A gauge field is like a huge recipe for moving sticks around in space, from A to B to C to wherever.
Now, there are very many ways to move sticks around in space, that is, there are countless ways such a gauge field can be configured. Imagine the huge, infinite-dimensional space where each point is such a gauge field. At one point in this space is the (trivial) gauge field that tells you that all sticks end up pointing in the same direction as they did when you started no matter where you move them. And at another point is the gauge field that tells you to rotate the stick wildly when you move it just a tiny bit. This enormous space is called a configuration space of gauge fields.
Here is a question: does it make sense to talk about distances on this configuration space? If we have one gauge field, that doesn’t rotate the stick and another one that only rotates it a tiny, tiny bit, are they then in some sense closer to each other compared to a gauge field that rotates the stick wildly? Is such a mathematical concept meaningful?
And note this: if a concept of distances on a configuration space is meaningful then it will imply that such spaces have curvature. That is, a geometry.
It turns out that such a concept is indeed meaningful — this is precisely what Johannes and I have worked on for the past 15 years. And it turns out that it leads to a mathematical machinery that has a very rich structure which involves almost all the basic ingredients of modern theoretical high-energy physics.
But before I begin to explain this to you let us just pause for a moment and catch our breath. So we start out with something very, very simple, just an algebra of moving stuff around in space and we combine it with a metric principle. That is all.
And once we begin to turn the cranks on this seemingly elementary piece of mathematical machinery a surprising amount of structure starts to fall out of it; structure that we have seen before but which now appears in a new setting. I find this fascinating.
The treasures we find
Let me begin by simply making a list of all the things, that this construction delivers to us. Then we can discuss the details later.
- the canonical commutation relations for a bosonic quantum gauge theory.
- the canonical commutation relations for a fermionic quantum field theory.
- the Hamilton operator for a Yang-Mills theory.
- the Dirac Hamilton operator for a fermionic field.
- all the above on a curved background.
- the lapse and shift fields.
- higher-order derivative terms.
- a theta-term or topological Yang-Mills.
- a metric invariant.
- an almost commutative algebra very similar to the one given by the standard model of particle physics.
- Hilbert space representations of the above elements(1)
All of these items and more come out of the basic ansatz, that I described to you before.
Now, I imagine that most of you do not recognise these items or know what they mean. So let me tell you that the standard model of particle physics and general relativity — the two theories that contain essential everything we know about the Universe (understood in a reductionistic sense) — are built from these building blocks.
Moreover, what we have is a non-perturbative framework. There is no perturbation theory here. The standard model of particle physics is a perturbative quantum field theory. This is not (and that is a good thing!).
We do not yet see the Hamiltonian of General Relativity — that is, Einsteins full theory — but we have very good reasons to believe that this item is stored away in some of the boxes, which we have not yet opened. We believe that this must be the case, as a mathematical necessity.
But I ask you to pause for a moment and take a look at this list: It all comes out of the basic ansatz of moving stuff around. I find this remarkable, it is hard to believe that this is merely a coincidence.
No Big Bang and no Quantum Gravity
Now, before we begin on the dirty business of explaining some of the details let me list three of the most interesting consequences of our construction. These are:
- There was no Big Bang.
- There are no black hole singularities.
- Gravity is a classical theory; a theory of quantum gravity does not exist.
These three points are all related. One of the most convincing arguments for the existence of a theory of quantum gravity goes like this: if you want to probe reality at ever smaller distances then your test particle will carry more and more energy that will curve space and time and eventually create a black hole horizon, which makes observations impossible. Distances shorter than what is known as the Planck length appear to be operational meaningless.
It is generally believed that a theory of quantum gravity will generate such a short-distance screening. If the geometry of space and time is somehow quantised, then it seems natural that the distance between objects will be a quantum operator and thus have a discrete spectrum: that is, there will be a shortest distance, which is different from zero.
But what if this short-distance screening comes from some other source? What if there exist another mechanism that prevents us from peeking beyond the lowest distances of Nature? That is precisely what we find and that seems to tell us the three points listed above: if there is a limit to what distances can appear in Nature then there will be no gravitational singularities — that is, no Big Bang and no black hole singularities — and also no need for a quantum theory of gravity, since such a theory would never be probed.
Perhaps what we see now — the standard model of particle physics, quantum theory of fields, various gauge groups, general relativity — is essentially all there is? Perhaps we have discovered almost all of it already? This seems to be what our construction is telling us. Yes, there are some novelties, but not that many, just a very rigorous mathematical machinery, which puts together stuff, that we have known for decades. Of course there will be surprises — the origin of dark matter and energy, for instance — but nothing that blows our minds out. No extra dimensions, no crazy stuff. I find this thought rather interesting.
Two conceptual novelties
Let me now dive into some details. I would like to explain to you two key ingredients in our construction; two mathematical objects, which are completely new. This part of my newsletter will be more techical, but I hope that you will hang on.
1. Dynamical gauge-covariant regularisation
A central ingredients in our paper is what we call a dynamical gauge-covariant regularisation. To the best of my knowledge this is a completely new concept. So lets take a look at it.
First: a regularisation? What is that?
One of the main challenges when you work with fields instead of particles is that they involve infinitely many degrees of freedom. Every point in space and time is a degree of freedom — and there are uncountable many points in space. Or think of fields in terms of waves: you have waves with different wave length and the wave length can be as large or as small as you wish. Again: there are infinitely many.
Infinite is a difficult number to work with and thus one usually employs what is called a regularisation. And as the major problems usually arise from short-scale degrees of freedom it is usually an ultra-violet regularisation.
Now, a regularisation can be very crude — if we for instance decide that we shall only consider waves with wavelength larger than a meter and if we assume that our space is finite, then we have cut down the degrees of freedom to a finite number — and it can be more sophisticated, like simply weighting degrees of freedom with a smaller and smaller weight as we go to shorter distances. But a regularisation always means that we reduce the degrees of freedom to a manageable level.
Here is an important point: a regularisation is almost always a technical artefact. That is, its a computational tool that cannot be part of the physical data. In perturbative quantum field theory, for instance, regularisations are used all the time and its always a critical test of ones computations that the result does not depend on the regularisation.
Could the regularisation be physical? Is that even possible?
Well, there are some very major problems that make such an idea look like a no-starter. First of all, there are countless ways you can regularise so how do you pick a particular one? Even worse, your regularisation will break important symmetries, such as the gauge symmetry (a defining concept in a gauge theory), which is beyond bad.
Johannes and I started thinking about regularisations when we were looking for what is called a Hilbert space representation of the QHD algebra (the algebra of moving stuff around in space). A Hilbert space is the mathematical stage on which an algebra acts, its a necessity to have such a space for a quantum theory to make sense, and thus we put a lot of thought into finding one.
And we did eventually find a Hilbert space for the QHD algebra, but the representations we found turned out to non-local. That is, they involved quantities that cannot be indefinitely localised in space. The reason for this is that our representations involved an ultra-violet regularisation.
Now, a Hilbert space representation is very important, so we didn’t want to throw it away once we had found it, but on the other hand it seemed problematic that it depended on a regularisation — and on top of that it broke the gauge invariance: double-bad. This clash of concepts, however, lead us down a path, that turned out to be extremely interesting.
The first problem we had to deal with was the gauge symmetry. As I said before, a broken gauge symmetry must be avoided at all cost and yet a regularisation will apparently always break it.
But we found a way out. If you permit the regularisation itself to depend on the gauge field then you can formulate it in a way that does not break the gauge symmetry.
Technically, what we do is to replace all derivatives in our regularisation with covariant derivatives. In this way the regularisation will be gauge covariant.
But this changes everything! If the regularisation depends on the gauge field, then it will, in a quantum theory of gauge fields, be time-dependent. It will be dynamical.
We didn’t realise this until a week ago. It is completely obvious but we didn’t see it until just before we were ready to publish our paper.
Here is a fascinating fact: if the regularisation is dynamical, that is, if it evolves in time, then there is no problem with choosing a particular one: we don’t have to, the construction will automatically move between different regularisations.
It reminds me of Baron von Münchhausen, who could lift himself and his horse out of a swamp by pulling his own hair. In a way this is what our theory does. It is standing over the abyss of every larger energy-scales, there appear to be nothing that prevents it from tumbling into this pit of mathematical divergencies, and yet our theory just floats there in mid-air, seemingly oblivious to the harrowing void below it.
And that Baron von Münchhausen trick appears to leave no room for a theory of quantum gravity. A dynamical gauge-covariant regularisation solves the problem of ultra-violet screening in a way that leaves no necessity for gravity to be quantised.
What I find intersting here is that once you have decided to interpret your ultra-violet regularisation as physical, then you will invariably be lead to the construction that we have found. A physical regularisation must be gauge covariant, which means dynamical. It is very obvious — in hindsight.
Let me end this part with a technical remark: from the perspective of perturbative quantum field theory you will probably never see this ultra-violet regularisation. The dynamical regularisation will simply look like an infinite series of interactions. In fact, I think it will be hard to see this regularisation directly. I am not certain, but it seems to me that the evidence will at best be indirect.
2. An infinite-dimensional Bott-Dirac operator
Another key technical ingredient in our construction is the Bott-Dirac operator. The geometry of the infinite-dimensional configuration space is encoded in this object, so lets take a look at this monster.
Some of you probably know a Dirac operator on a finite-dimensional space. Famously, P.A.M. Dirac discovered this operator as he was searching for a relativistic Schrödinger equation and soon realised that the structure of this new operator lead to the anti-particles.
Now, roughly speaking a Dirac operator is a mathematical entity that probes how rapidly a function on a space changes. Think of a wave. If a wave has a long wavelength, then it changes slowly; if it has a short wavelength it changes quickly. The Dirac operator picks up this information and stores it in something called a Clifford algebra. You can think of the Clifford algebra as a big cupboard with lots of shelves.
Now, a Bott-Dirac operator is just a slight variation of a Dirac operator and an infinite-dimensional one is an operator that probes functions with infinitely many variables. You can think of its Clifford algebra as an infinitely big cupboard with infinitely many shelves.
So what is the point of all this, you might ask? Well, the point is that this operator, when it is given a specific gauge field, will register how rapidly the field changes in its different variables and store this information in the many shelves of its Clifford algebra. The nature of this information is metric.
It is not a straight forward matter to construct such an operator in a meaningful way. There is a great deal of technicalities here but I shall gladly spare you; the important point is that it is possible.
And basically that is it! Our construction consist of the algebra, that I described to you before, and this Bott-Dirac operator. An algebra of moving stuff around combined with a metric principle. And all of it wrapped in a Hilbert space.
Generally speaking this construction constitutes a geometry on the configuration space of connections — but what does that actually mean? Well, there are several issues here, but one thing, which we analysed in our paper, is that the square of the Bott-Dirac operator gives us the time-evolution with the correct Hamilton operators, that I mentioned in my list before, both the bosonic and the fermionic parts.
The hunt for a key ingredient
Some time in early March this year Johannes and I were working on a formulation of the Bott-Dirac operator. But then Johannes checked out, he was going on paternity leave, and at the same time the Corona lock-down hit Denmark.
I had actually decided to take a vacation during Johannes paternity leave; I was tired and needed a break; but with the lock-down it seemed safer to just stay at home. And then I started thinking about a technical detail that was troubling me.
In our paper it is formula 45 on page 20. I just knew that this solution existed but I couldn’t find it. It had to satisfy a number of requirements, and the more I dug myself into the techical details the more certain I became that this could be done. But I still couldn’t find the solution.
It took me six weeks to find it. During those six weeks I didn’t see anyone. I was staying in my cabin in the countryside and I worked basically 24/7 on this problem. I tried to relax, but once I start a hunt like this I find it very difficult to check out again. My living room was filled with notes with formulas, I often woke up at night thinking that finally I had the solution and raced to my desk to write it down, but the next morning when I read my notes I saw no divine intervention but mostly minor misunderstandings. It was deeply frustrating.
Every second day I would convince myself that now I had it, finally, the solution, and I would celebrate without daring to check all the details: I wanted to enjoy the calm waters for a moment. And sure enough, when I checked my computations the day after I immediately saw my mistake and the hunt continued. Until some day when I ran out of mistakes and finally wrote down the solution.
This paper hasn’t come for free. I don’t recall ever having fought so hard in my life.
It all fits onto one tiny matchbox
I want to emphasise again how remarkable I find it that so much structure comes out of such a simple starting point. As we have worked our way into this construction Johannes and I have again and again been surprised at how much it gives us.
Take for instance the lapse and shift fields, which encode information about the time-component of the metric field. In the construction of the Bott-Dirac operator it is necessary to add a certain auxiliary spin structure, which entail a dependency, which we didn’t like. We were, however, able to tie this dependency back into the theory but at the cost of adding more structure. Now, it turns out that this structure delivered us the lapse and shift fields.
Now, for most of you this will sound like gibberish and I only include it to exemplify how this construction keeps giving. We shake the tree a little and treasures fall into our hands.
Another example is the metric invariant, that we find. It emerges from the computation of the Dirac Hamilton operator and it related to what is known as the eta-invariant first introduced by Atiyah, Patodi and Singer in 1975. I contacted two experts on the eta-invariant who reported back to us that they had not seen this new invariant analysed before. We have been able to establish its existence as well as some of its basic features but we do not yet know its physical meaning. A first guess is that it could be related to a time-dependent cosmological constant. A third example is the theta-term, which comes out of the same computation and which is related to what is known as instantons.
All of this just fell out of the tree like ripe apples in autumn.
The fermions as a storehouse of geometrical data
I want to write about an aspect, which isn’t easy to explain but which I find highly interesting. The title of our paper, The Metric Nature of Matter, refers to this, which is the intrinsic geometrical role that matter plays in our construction.
To explain this let me begin with a question: what is matter?
Well, we all know what matter is. Stuff, right? The chair I sit in, trees, highways, planets, solar systems and books about solar systems. Matter. its all matter.
Good, that all makes sense, but from a mathematical and physical perspective it is less clear what exactly matter is. Or rather: where the mathematical structure of matter actually originates.
This is a technical story (which is why it is hard to explain), but fermionic quantum field theory — and fermions is a fancy word for matter fields — basically consist of quantum fields with a very peculiar algebraic property, namely that they anti-commute.
Anti-commute? What does that mean?
It means that it matters in which order you multiply them and that they pick up a minus sign when you interchange them.
Why is it like that? Well, ask any theoretical physicist and she will give you a long and complicated explanation that nobody understands but which sounds convincing. The truth is that we don’t know.
But our framework actually gives us an explanation. This is what I want to tell you about.
Any Dirac operator comes with what is called a Clifford algebra. As I mentioned earlier, you can think of a Clifford algebra as a huge cupboard which the operator uses to store geometrical data about the underlying space. And the notable thing is that the Clifford algebra has precisely this anti-commuting feature that I just mentioned.
Now, the Bott-Dirac operator, which we have constructed, has an infinite-dimensional Clifford algebra — that is, an insanely large cupboard! — and that is precisely where our fermions come from.
The infinite-dimensional Clifford algebra entails mathematical structures that are identical to those of a fermionic quantum field theory. That is: matter. Stuff. The solar system and all the books about it.
This means that in our mathematical framework the fermions play the role of an infinite-dimensional storehouse for geometrical data about the underlying configuration space. That is, data about all the possible ways reality can be arranged. I am absolutely certain that Spock would have loved this idea!
The road ahead
Traveller, your footprints Are the path and nothing more; Traveller, there is no path, The path is made by walking. By walking the path is made And when you look back You'll see a road Never to be trodden again. Traveller, there is no path, Only trails across the sea…
Our work is like solving a really hard puzzle.
First you cannot see the hidden image, you have ideas but you cannot know what it is; then you manage to finish the entire edge and isolated islands of the puzzle, which gives you clues about the image but still you cannot see it.
And then, suddenly, you manage to put together so much of the puzzle that you finally know what it is. Work still needs to be done, islands of unfinished business remains, areas, which may contains surprises, big surprises too, but basically you feel very confident that the puzzle you are solving shows the image of ‘Earthrise’ taken by William Anders on the Apollo 8 mission in 1968.
Now, we are at a point where we believe that we know what our puzzle is hiding. But there are still large areas of unfinished business and there can still be great surprises. And of course, there is always the possibility that things do not add up. We do not have an insurance policy against that possibility, not yet.
Ahead lies a tonnes of work, I believe it will take decades to fully analyse the mathematical machinery that we have uncovered. And it will take much more manpower. Two people cannot possible do this work by themselves.
The two most important tasks ahead are the following:
1. we need a proof that this construction exist in a rigorous mathematical sense. We have a proof of existence in a particular case but we do not have a general proof. We are, however, almost 100% certain that it exists. But almost certain is no proof and we need proof. Alas, job number one.
2. we need to analyse the possible connection to the standard model of particle physics that we have found. The algebra, that we end up with is tantalisingly similar to what Chamseddine and Connes have identified as the algebraic backbone of the standard model, but several questions remain as does a clear definition of the limit in which the standard model might emerge. There will almost certainly be some Higgs-mechanisms hiding here (they are kind of the hallmark of non-commutative geometry), which will likely play an important role. We need to work all this out. Alas, job number two.
The end of the crowdfunding campaign
The goal of our 2016 crowdfunding campaign was to find either a consistent theory or to prove a link between our work and that of Alain Connes’, i.e. to prove that our framework involves unification. With our new paper I believe that we have reached this goal. Pockets of uncertainty remain, but we have come much farther than I had ever imagined possible. Thus I believe that we have come to the end of this road and I thus declare mission accomplished and the scientific part of the crowdfunding campaign for ended.
Of course, those of you who ordered an english version of my book will receive this; that part of the campaign is still open. But with our recent paper I consider the scientific part closed. This means that henceforth I will no longer acknowledge the crowdfunding campaign in our papers.
I am eternally grateful for the support you have offered to Johannes and myself! It has been very important to us, not only financially but also morally. Thank you so very much! I treasure your support and trust and I will never forget it.
I will continue to write newsletters and give you updates on our progress. If you wish to receive these newsletters please subscribe to my blog.
An invitation to participate
If you wish to continue your association with our research project you are very welcome to sign up as sponsors. You can do that via my homepage or by writing to me. I am grateful for any contribution, small or large. Everything helps!
In fact, recently a handful of you signed up for a fixed monthly contribution. Such fixed contributions are very helpful as they give me a steady income. At the moment I have one larger sponsor and a handful of smaller ones, which all adds up to a around 350 Euros per month. This is, however, not enough and thus I welcome any help and encourage you to pass this on to others who might be interested in sponsoring what I without hesitation will call the most promising research project in contemporary theoretical high-energy physics.
If you wish to read about why I conduct my research on a basis of private sponsors I have written a blog post about this.
Have a enjoyable and safe autumn
With this I end my newsletter. I wish you all the best, take good care of yourselves and stay safe!
Best wishes, kind regards,
(1) We only have a proof that a Hilbert space representation of the QHD algebra exist in the non-covariant case. We have, however, strong reasons to expect that this result also applies to the covariant case.