13.09.2023

Hi everyone,

it’s finally time for a newsletter. I spent the last two weeks in Hannover, Germany, working with Johannes, and today we posted a new paper on the archive. We have made some significant progress and in this newsletter I would like to explain in a non-technical manner what we have found. 

Our new paper has been underway for a long time. One year ago we believed that we were ready to publish our results, but then some technical complications turned up, which took us a year to work out. But now we have finally reached a point where it can all be published. 

This newsletter will be a little longer and at times slightly more involved than what I usually write. Thus, it might be a good idea to get yourself a strong cup of coffee (or a big martini and a snack, depending on your temperament) before you jump into it all. And remember: if you get stuck somewhere then just jump a bit ahead. It is not necessary to understand all the details to get the overall flavour.

The search for a fundamental theory: what is required

Let us begin with the overall goal. We are searching for a fundamental theory and thus we should begin by stating what such a theory must accomplish. Basically, there are four requirements:

1. It must have an extremely simple foundation. If a fundamental theory is indeed fundamental (and perhaps even final), then it must be simple. Simplicity means immunity against further scientific reductions, just as simplicity reduces the number of new questions that the theory raises. 
2. It must be able to reproduce the physics that we already know. That is: the standard model of particle physics (which includes quantum field theory) and general relativity.
3. It must be able to explain the physical phenomena that we are at present unable to explain. That means dark matter and dark energy. Furthermore, it should be able to make further predictions.
4. It must be formulated in rigorous mathematics. 

Note that hidden within the second requirement lies also the question of how to reconcile general relativity with quantum theory. If a single theory can reproduce both general relativity and the standard model of particle physics, then it must have solved the almost ancient question of how to reconcile Einsteins theory of gravity with quantum theory. This problem has been around for almost a century without any decisive breakthrough.

Now, the idea that Johannes and myself have been working on for almost two decades is that it should be possible to build a fundamental theory out of the mathematics of empty space. This idea lead us to consider an elementary algebra, that basically encodes how stuff is moved around in empty space, and with that we were lead to consider the geometry of what is called a configuration space, which is a space where each point represents a configuration of a certain type of fields known as gauge fields. 

What we found is that a lot of the mathematics, which we already known from physics today, emerge from a geometrical construction over the configuration space. We found that key elements of quantum field theory emerge, alongside elements of general relativity. And in our new paper we show that it is possible to formulate all this rigorously and to prove that it exists mathematically. 

That is: we can now show that our construction exist in terms of rigorous mathematics. Previously we had some partial results concerning existence, but they all came with some significant limitations. In our new paper we are able to expand these previous results and widen their applicability considerably.  Thus, we can now prove that we are standing on solid ground.

This means that we have a candidate for a fundamental theory that meets points number 1) and 4) in the above list, and which shows promising signs in the domain of point number 2). This is where we stand today.

Now, it is not really possible to explain in a non-technical manner how we prove that our theory exist mathematically, so what I am going to do in the following is to give you a fairly detailed summary of the theory as it stands today and along the way I will then also touch upon the mathematical mechanism, that secures its existence. 

A geometry on a configuration space

So the basic idea is to begin with an algebra that simply encodes how stuff is moved around in empty space. And note this: there is no stuff here, not yet, just the mathematical notion of moving something around in a three-dimensional space. This algebra is interesting for two reasons: 

1. it has a very high level of canonicity (in physics we use the term canonical in situations where there is no room for variation), the algebra only depends on the dimension of space1. One could also call it universal.
2. the algebra gives rise to some very rich mathematics, as I shall discuss in the following.

Now, this algebra, which we call the holonomy-diffeomorphism algebra, or just the HD-algebra, naturally leads one to consider the configuration space that I mentioned a moment ago. The configuration space is a huge space where each point is one recipe for how to move an object between arbitrary points. There are infinitely many ways you can move, say, a stick between two points in your living room. You can turn it to the right, you can turn it 45998 times to the left, or you can flip it twice forwards – and so on. All these different possibilities correspond to different recipes, which we also call gauge fields. And the configuration space is the space of all these recipes. 

By the way, such configuration spaces are well known in physics, especially in quantum field theory. What is new here is what comes next.

So here is our central idea: we want to consider the geometry of this space. You can think of this as a meta-geometry. 

This idea turns out to be interesting for several reasons:

• it may not at first be obvious, but this idea constitutes a completely new take on the old problem of how to reconcile general relativity with quantum theory. Usually the strategy has been to apply the framework of the latter to the former, that is, to construct a theory of quantum gravity. But there exist of course also the other possibility: to apply the machinery of general relativity to quantum theory. This is what we are proposing. The central mathematical playing field in a quantum theory is precisely the configuration space. In a quantum theory you can have superpositions between different classical configurations (for instance, an electron being on the moon and right here, simultaneously), which in mathematical terms means that you need to work with the configuration space.

• Now, what is truly interesting about this idea is that the configuration space, which we consider, has one very important feature, which is called a gauge symmetry. Gauge symmetries are very important in theoretical high-energy physics, where they for instance play a key role in the standard model of particle physics. In our case, the presence of the gauge symmetry means that the geometry of the configuration space (that is, the meta-geometry) has to be compatible with this symmetry. This is a necessity, if the geometry is not compatible with the gauge symmetry it will make little mathematical sense. But here is the crux: a geometry of the configuration space that is compatible with the gauge symmetry must be dynamical. It is too technical to explain precisely why this is so, but the mathematics is quite strict when it comes to gauge symmetries. The important point here is simply that the geometry of the configuration space must have a time-evolution. The meta-geometry will be dynamical, just as we know it from Einsteins theory of general relativity.

What does that mean? It means that what we are proposing is in fact a natural generalisation of Einsteins two theories of relativity: first there is special relativity, which does not include gravity, secondly, there is general relativity, which does include gravity, and now we propose a third theory of relativity, which we tentatively call configurational relativity², and which includes quantum theory too.

Several layers of unification

Now, the interesting thing about this idea is that it naturally involves several layers of unification. We already know the concept of unification from high-energy physics, where for instance supersymmetry were supposed to unify bosons and fermions through a certain extension of space-time symmetries. There are also the various types of GUT’s, grand unified theories, where the idea is to unify the three forces in the standard model in a single force at a higher energy. And then there is also string theory, where the fundamental particles including a graviton (which is the postulated quantised gravitational field; similar to the photon, which comes from the quantisation of the electro-magnetic field) supposedly emerge from tiny vibrating strings living in 10 dimensions. 

Here is the reason why unification is important: if you want to go from a simple starting point to a relatively messy and complex end point like the standard model of particle physics, which has several types of fields, which obey various symmetries, then you need some mathematical mechanism that can generate all that complexity. That is what unification does for you. The aim is to find mechanisms that can generate rich mathematical structures or, equivalently, explain how the complex structures we observe originate from something simpler, without adding more assumptions to the starting point.

In our framework there are three different types of unification:

1. First, there is a unification between bosons and fermions (bosons are forces and fermions are matter fields). This unification turns up when we construct the geometry on the configuration space. Why is that? Well, it is somewhat technical,  but basically we use the mathematics of noncommutative geometry, where the central object is called a Dirac operator. It turns out than when you construct a Dirac operator on a configuration space then you automatically get fermionic degrees of freedom. 
2. Secondly, there is an intrinsic type of unification that comes from the HD-algebra being noncommutative. That an algebra is noncommutative simply means that the ordering of the elements in the algebra is important. When you multiply numbers it does not matter in what order you do that: 2 times 3 equals 3 times 2 (equals 6). So numbers form a commutative algebra. But the HD-algebra is noncommutative, and what this means is that there exists an additional mathematical structure, which is very important. In the case of the standard model of particle physics it is known that this effect gives rise to the entire bosonic sector, including both the forces and the Higgs field. In our case we do not yet know what the effect of this additional structure will be, only that it must be there.
3. Finally, we believe that the geometry of the underlying three-dimensional space must also be encoded into our construction, and we believe this geometry must be dynamical. Right now we are not able to prove this, but it is clear that the geometry of the configuration space does involve information about the geometry of the three-dimensional space (the space in which we live), and since the meta-geometry is dynamical, so must the three-dimensional geometry be.

Now, as I have already said, what this means is that our construction on the one hand emerges from a very simple and canonical starting point (the mathematics of empty space) and at the same time gives rise to some very rich mathematics, which we recognise as the mathematics that we already know from high-energy physics. That is, key elements of quantum field theory and gravity and several mechanisms of unification. 

There is one thing that I want you to note here, and that is how the fermions emerge. I told you right in the beginning that there is no stuff at the outset of our construction, we simply start with the mathematical notion of moving stuff around in space. But fermions (which represent matter like electrons and quarks etc.) naturally emerge in our construction, and the place where that happens is when we construct the geometry. They are sort of baked into the mathematics. 

Another important point to note is that there is significant difference between the different layers of unification, which turn up in our framework, and the ones I mentioned at the beginning of this section, which were the different types of unification already known in high-energy physics. The unification found in supersymmetry, in GUT’s, and in string theory all take place after quantum field theory has been applied. Thus, they are post-QFT unifications. In contrast to this the unification that takes place in our framework are all pre-QFT. That is, they take place at a point that is deeper than quantum field theory. In our framework quantum field theory is something that emerges, quantum theory is an output, not an input. And the different elements of unification, which we encounter, all take place before quantum field theory emerges. 

This all sounds very complicated — and believe me, it is! — but what it really means is that we have uncovered a piece of mathematical machinery, which has an almost universal foundation, namely the HD-algebra, and which gives rise to a rich jungle of mathematical structures, that all look so tantalisingly similar to the mathematics we already know from high-energy physics. I keep repeating this, but that is because its important. 

Mathematical existence

Now, as I already said, what we have accomplished in our new paper is to prove that all this can be formulated rigorously. It really exists in mathematical terms.

This is not a trivial thing. The problem of formulating quantum field theory in three spatial and one temporal dimension is still unsolved and as far as I know there is no proof that string theory exists outside of perturbation theory. So the fact that we have obtained a rigorous result is significant.

There are several reasons why we are able to prove this and they all have to do with one basic assumption that we make. This assumption is rather abstract: we assume that the volume of the configuration space is finite. 

To explain what this means let us first ask what geometry actually is? Well, geometry has to do with the shape of things, it has to do with distances between points. But once you have distances between points you can also compute the volume associated to a given geometry. For instance, in three dimensions the volume of, say, a donut, may be 300 cubic centimeters (I have no idea if that is realistic). In two dimensions the ‘volume’ is called the area, and the area of a sheet of paper may be, say, 500 square centimeters.

So what is the volume of the configuration space? Remember that the configuration space is infinite dimensional. It is the space that includes all possible recipes for moving stuff around in space, or (equivalently) all the different ways a gauge field can be configured. Such a space has infinite many dimensions and that means that the volume of this space is a more abstract concept. But it can still be formulated mathematically, and what we assume is that the volume of this space is a finite number. 

This finite-volume assumption has several consequences. One of them is that the theory that we obtain must be non-local. That is, it will not be possible to localise physical entities arbitrarily in this theory. 

Is this a good thing? Is non-locality something we should be happy about? The answer is yes, and to understand why that is so we need to look at an old argument that combines general relativity and quantum theory. 

The argument basically tells us that it is impossible to measure anything beyond the Planck length, and it goes like this: if you want to measure something very tiny then you need a test particle that is equally tiny. But in terms of its Compton wavelength this means that as you measure smaller and smaller objects, the energy of your test particle will get larger and larger, and eventually it will be so large that it generates an event horizon. That is, your test particle will generate a black hole. And since nothing can escape a black hole this means that your measurement becomes impossible. The scale where this happens is the Planck scale.

What this means is that non-locality seems to emerge naturally when we combine general relativity and quantum theory. So we should expect a fundamental theory to be non-local. This is why our assumption that the volume of the configuration space is finite makes sense.

There are other reasons why this initial assumption is sensible, but they are more technical and less easy to explain. Basically, the problem is that we must construct what is called a Hilbert space, which is the stage on which all our mathematical actors perform. In our case such a Hilbert space will involve what is called a path integral, which is something that is known from quantum field theory, and which normally does not exist in rigorous mathematical terms. The reason why it exist in our case is precisely our finite-volume assumption.

Those of you who are familiar with theoretical high-energy physics might recognise the finite-volume assumption to be what is called an ultra-violet regularisation. An ultra-violet regularisation is something that we use in quantum field theory to make sense of quantities that are in fact infinite. An ultra-violet regularisation cuts down such quantities in a way so that they become finite and thus easier to handle mathematically. But here is the bug: in quantum field theory such ultra-violet regularisations are always computational artefacts, they are never physical features. So what is going on here, is what Johannes and myself are saying really meaningful?

This is a very good question and the answer is yes. The reason why this makes sense is the regularisation that we work with is dynamical. We call it a dynamical ultra-violet regularisation. This is a completely new concept in theoretical physics, as far as we know it has never been studied before. The point is that once the regularisation is dynamical it is feasible to interpret it as being physical. 

Anti-gravity?

There is one aspect of this finite-volume assumption, which is particular intersting, and that is related to a kind of anti-gravity forcing.

The point is that the non-locality, which I just discussed, will have a strong impact in physical situations, where a lot of matter is highly localised, and we know of two situations, where this is the case: at the beginning of the big bang and in black holes. 

As far as we know today the big bang was a space-time singularity, where the entire universe was compressed into a single point. But if a fundamental theory is non-local that will mean that there could not have been such an initial singularity. The non-locality would have prevented this from happening and thus it will represent an anti-gravitational forcing. It will be something that repels, where gravity attracts.

The same can be said about black holes. As far as we know today there is nothing that can stop the gravitational collapse that takes place in a black hole and thus we assume that it is a singularity, where the entire mass of the black hole is compressed in a single point with infinite space-time curvature. But again, if we have a fundamental theory that is non-local, such a singularity will not be permitted. This means that there must be something in it that pushes against the gravitational collapse, and that something will therefore look like an anti-gravitational forcing. 

I do not think that this anti-gravitational forcing will be a force in the same way that gravity or the strong or electro-weak force is it. What I mean is that it will not be represented by a field. But what it exactly looks like we do not yet know. All we know is that the theory that we have found is non-local and that fact prohibits these highly localised configurations.

Emergence of general relativity

One of the problems, which we have not yet solved, is concerned with general relativity. Here is the idea: for various technical reasons it is clear that a geometry on a configuration space (the meta-geometry) must encode metric information — that is, information of a geometrical nature — about the underlying three-dimensional space (the space we all live in). This is simply how the mathematics work, it is very clear that the meta-geometry involves information about the geometry of the physical three-dimensional space. The question is how to extract that information, and the question is in what form it might come. We do not know the answer to these questions but we can make some guesses.

First of all, the question is whether this geometry will be dynamical. That is: will it have a time-evolution. The meta-geometry is dynamical, as I explained to you a little while ago, but does that imply that the geometry of the underlying three-dimensional space will be dynamical too? Clearly, this is what we want. What we are hoping is that it is Einsteins theory of general relativity that emerges from our construction, and in Einsteins theory the geometry is dynamical.

Secondly, the question is whether the time-evolution of the three-dimensional space will be generally covariant? General covariance is one of the key features in Einsteins theory of relativity, it means that the theory does not depend on a given coordinate system. It seems plausible that the answer here is yes, it will be generally covariant, at least with respect to the three-dimensional space. There is simply nothing in our construction that could give rise to something that is not. 

The interesting thing about these questions is that it is no longer a question about finding the right framework and conjuring up some new theory. The theory is already there, we have found it, now it is simply a matter of analysing the mathematics. The answers are there, they must be, it is just a matter of time until we or someone else figure out what they are.

Physical implications

Some of you have asked me about the physical implications of our theory. What does it say about the real world? 

The truth is that we do not yet know the answer to this question. In fact, we do not know if it has anything to do with the real world. 

Our approach is to work top-down. We have conjured up a consistent piece of mathematical machinery based on a very simple ansatz, and we have shown that it for instance leads to the mathematics of quantum field theory. What we need to do next is to map out in greater detail precisely what this machinery contains and whether it has any physical implications. This will require a lot of analysis. 

Having said that I believe that it is nevertheless possible to say a few things:

• the anti-gravitational forcing, which I mentioned above, must have implications in cosmology. If black holes have a bottom, that is, if there is something in them that prevents them from forming a singularity, then that must have an impact on their total mass, which in turn should impact the overall curvature of the universe. That is, it must impact the cosmological constant.
• our theory already has a lot to say about the origin of quantum theory. What we find is that key elements of both bosonic and fermionic quantum field theory (the canonical commutation and anti-commutation relations, the Hamilton operators, alongside Hilbert space representations, Fock spaces etc.) emerge from our construction. This means that our theory has strong implications for the interpretation of quantum theory.

(concerning point number two, then we do not, however, have anything to say about the measurement problem)

Open problems

Let me give you a list of the most important open problems that we are faced with right now:

1. The emergence of general relativity. I already discussed this point, but one thing worth mentioning is the question about the Lorentz symmetry. The Lorentz symmetry is the local rotational symmetry in Einsteins theory (it is the basis of special relativity) and what we would like to understand is how this symmetry might emerge from our framework. 
2. A connection to the standard model of particle physics. This is a key question, which was what first got us interested in the HD-algebra. The point is that there exist a very intersting formulation of the standard model in terms of noncommutative geometry, which is due to the physicist Ali Chamseddine and the mathematician Alain Connes. This formulation is very algebraic and some of the structures that Chamseddine and Connes identified within the standard model are similar to what the HD-algebra produces in a certain limit. But so far this is just a hunch, we need much more analysis to be able to answer this question.
3. To understand how half-integer spin emerges in our theory. This is a question that has bugged me for some years. The point is that fermions must have half-integer spin, but within our framework they will a priori have integer spin. In our last publication we came up with a solution to this problem, but I am no longer certain that this solution is the right one. Today I suspect that the answer is much more simply and has to do with the type of configuration space that we work with. It is too technical to explain here, but this problem is probably not as hard as I previously thought it was.
4. There is a technical detail that has to do with quantisation of gauge theories and which is known as the Gribov ambiguity. The Gribov ambiguity is a headache in theoretical high-energy physics that has lasted almost 50 years. It has to do with the non-perturbative formulation of gauge theories (such as the standard model of particle physics). The point is that there is a generic obstruction that one will always encounter when one attempts to formulate such a theory. It is a rather technical matter, but in our new paper Johannes and I suggest a very simple solution to this problem. It may be that within our particular setup the Gribov ambiguity is immaterial. Whether our solution is correct remains to be seen, but in principle it is something that one can check (although I think it will require some mathematical heavy weightlifting). 

Call for sponsors

I would like to end this newsletter with a huge thank you to my sponsors. I have a small but very faithful circle of sponsors, who has kept me financially afloat during the past years. In particular I would like to mention the generous support of the danish entrepreneur Kasper Gevaldig. On my homepage you can find a full list of my sponsors. I am deeply grateful to you all, just as I am awed and inspired by your trust in me and my work. The truth is that without your support, financially and morally, I would have given up long ago. 

However, the truth is also that my ship is slowly sinking. I need some more help and therefore I would like to encourage anyone, who is interested in sponsoring one of the most innovative, coolest, and most independent research projects in theoretical high-energy physics, to support me. Everything counts, large and small donations. You can support me via PayPal or you can contact me (for larger donations please write me an email). I will recognise all my sponsors on my homepage and in my publications. I will recognise large donations (more than 2000 US$) on the title page of future publications.

Johannes and myself publish our work in world-leading journals in mathematical physics. You can check my publications on my homepage and verify that our work is of the highest international standard. The problem in theoretical physics today is that with the present sociological setup dissenting voices and alternative ideas — which used to be the essence of science — are not supported or rewarded, and thus we have been forced to seek support elsewhere.

My book “Shell Beach” is available as a hardcover

My book Shell Beach – the search for the final theory, which I wrote as a reward to the backers in my Indiegogo crowdfunding campaign, is now out in a hardcover. The book offers a non-technical introduction to the ideas behind my research project with Johannes Aastrup, just as it gives a peek behind the scenes in the research community of high-energy physics. 

The science editor Jens Ramskov at the danish journal ‘Ingeniøren’ write about Shell Beach:

“Jesper has written a formidable book. I have read many books about physics and science but none that compares to his book”

Although the book was written a few years ago it is still a good introduction to my work and the general philosophy behind our research project. 

Have a nice fall!

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

Take good care of yourselves, kind regards,

Jesper

Footnotes:

 1. The HD-algebra depends a priori on the dimension of the manifold (space) on which it is defined as well as a choice of gauge group. But if one choses the setup where the HD-algebra encodes how extended objects in space are moved around (i.e. tensor degrees of freedom) then the choice of gauge group is essentially fixed to be either SO(3) or SU(2). In principle one can, however, define the HD-algebra for any (compact) gauge group and thereby consider rotations in some internal space.

 2. We have previously used the name “quantum holonomy theory”, but we feel that configurational relativity is a better term. We are thankful to Jarl Sidelmann for suggesting this name to us.