– for the Indiegogo crowdfunding campaign
A THEORY OF EVERYTHING
Johannes and I have just posted a new preprint on the archive and therefore I would like to write a newsletter and explain to you what we have found.
Things are looking very promising at the moment. We have spent a long time pondering about some rather fundamental and critical questions — problems, that I wrote about in my last newsletter and which at one point made me think that our project might finally have come to a dead end. But in our new paper we have found a number of solutions, which make us feel much more optimistic.
What I will do in the following is to first explain what we have found and what I believe it means and secondly I will briefly discuss the main obstacles ahead of us.
This is all somewhat technical, but I will do my best to keep it at a level where most of you should be able to understand roughly what is going on — and please hang in there, because what we have found is very surprising.
THE OLD PROOF
Two years ago Johannes and I published a proof that our theory exist — technically, what we proved was that we had found a pairing between our basic algebra and a so-called Hilbert space, which is the scene on which a quantum theory “lives”. But our proof had certain deficits, it had a couple of problems, which our reviewers were more than willing to point out to us when we submitted it to a journal of mathematical physics.
What were these deficits? Well, they are both linked to one and the same thing, which is what we call a gauge symmetry. All the fundamental theories, which describe the forces in Nature — the electro-weak and strong nuclear forces; gravity — are formulated in a mathematical language that involves a certain redundancy, which means that there is an invariance under a specific symmetry. And that symmetry is called a gauge symmetry.
Normally this symmetry is critical for a quantum theory to make sense. If it is broken then the theory will mostly be considered meaningless. And that is where the first problem with the proof, that Johannes and I published, shows up. It turns out that our proof involved a broken gauge symmetry.
The second problem with the proof has to do with how our theory behaves on extremely short scales. Perhaps some of you remember an argument, which I gave you some time ago, that combines Einsteins theory of relativity with quantum mechanics and which strongly suggests that distances shorter than the so-called Planck length are physically meaningless. The argument basically goes like this: if you wish to measure distances shorter than the Planck length, then your test particle (whichever object you use for your measurement, photons for instance) will have such a short wavelength (high energy) that it creates a black hole — extreme curvature of space and time — which means that no information can escape, alas no measurement.
Now, the proof that Johannes and I published involved a certain mechanism, that ensure that our theory does not involve distances shorter than a given length — presumably the Planck length. Its a tricky business how precisely this is done, but essentially our theory comes with a built-in ultra-violet cut-off (this would be the technical term). Now, this is problem number two, because such a cut-off would normally not be considered a part of a physical theory but rather a computational tool, an unphysical artefact. But in our framework it had to be part of the physical theory.
As I said, these two problems are actually one and the same since the ultra-violet cut-off is precisely the mechanism, that broke the gauge symmetry.
Now, this all sounds terrible (and some of our reviewers definitely used some strong words), but the reason why we believed (and still believe) that our result was significant is that the type of mathematical existence, which we proved, is rather rare. In our point of view the fact that we could prove existence was a strong hint that the two problems, which I have just told you about, probably could be solved.
THE NEW FORMULATION
In our new paper we have presented a formulation of our theory, that indeed solves these two problems. We now have a formulation of our theory, that does not break the gauge symmetry. This is very, very good news.
The way this is done is that we have managed to formulate the mechanism, that removes short distances from our construction, in a way that does not violate the gauge symmetry.
This probably sounds rather technical to most of you, but what comes out of all this is highly interesting. It turns out that our new formulation amounts to a geometrical structure on a huge infinite-dimensional space.
Let me try to explain this. Imagine you have a stick and you would like to move it from a point A to a point B along a certain path in space. How do you do this? Well, you can twist the stick to the right while you move it along the path, or alternatively you can tilt it forwards while you move it, or you can rotate it 37655 and a half times to the left — there are infinitely many different ways to move the stick along the path from A to B. Now, imagine the abstract space where each point is one particular recipe for moving your stick along various paths in space. This is the infinite-dimensional space we are talking about — the technical term for this is a configuration space — and it is on this space that we find a geometrical structure. Broadly speaking, what we find is an ‘Einsteins theory of general relativity’ on this huge, infinite-dimensional space.
If you think think this sounds a little crazy then I kind of agree with you. But I assure you, the math is completely solid. We know this with certainty.
And by the way, the idea to dress up this so-called configuration space with a geometrical structure is not new. A few people worked on similar ideas years ago — Richard Feynman, the famous US physicist who won the 1967 Nobel Prize together with Julian Schwinger and Sin-Itiro Tomonaga, is one of them — but the way we are doing it is new and has a broader scope.
So what we find is that when we fix the two problems, that I mentioned before, then we end up with a geometrical construction over this huge, infinite-dimensional space, which does not mess up the gauge symmetry. This is a very interesting and to the best of my knowledge completely novel mathematical object.
Now, we know that the geometrical structure, that I just talked about, exist, but we do not yet have a proof that our theory as a whole exist. We are confident that it does, but we haven’t yet extended our proof to this more general case. In our new paper we formulate it as a conjecture and provide arguments why we believe it must hold true.
QUANTUM FIELD THEORY
But let me now talk about what all this really means. It means that a geometrical theory of the kind that we have found will, in a certain low-energy limit, always look like a Yang-Mills type of quantum field theory.
What is a Yang-Mills type of quantum field theory, you may ask. Well, Yang-Mills theory is the general framework that is used to describe for instance the three fundamental forces known as the weak, strong and electro-magnetic forces. And quantum field theory is the mathematical framework, that describes how fields are quantised. That is, quantum mechanics extended to fields such as the electro-magnetic field.
This means that we obtain completely new conceptual explanations for key elements in quantum field theories. For instance, the canonical commutation relations, which are normally taken as the starting point of a quantum field theory, emerge from our theory from a very natural algebraic relation in a certain local limit (the canonical commutation relations are closely related to the Heisenberg uncertainty relations). Also, the Hamilton operator, which is the mathematical object that gives us the time evolution in a quantum theory, has in our framework a purely geometrical origin.
What we find is sort of a new dictionary, which translates some of the objects, which we have worked with for decades, into a new, geometrical language.
Now, the question that I imagine you’d like to ask here is whether this ‘dictionary’ also translates physically realistic quantum field theories into this new language? That is, can we construct a geometrical theory of the kind, that I have talked about, that gives rise to, say, the standard model of particles physics?
Or put differently: can we find such a geometrical theory, that explains the structure of the standard model?
We don’t know the answer to this question. Of course, it is our hope that this will be the case and we do have a couple of ideas, which we are investigating right now, but we do not know whether this could be so. I am hopeful that I will be able to write more about this question in my next newsletter.
One particularly striking aspect of the framework that we have presented, is that it provides a completely new conceptual role for the fermions.
Fermions, what are they? Well, basically there are two types of particles in Nature. There are bosons and there are fermions. Bosons are related to the forces and fermions constitute the matter. STUFF. Electrons, for instance, are fermions. Photons, on the other hand, are bosons.
And the fermions, that emerge from our framework have a particularly intersting role to play. It is somewhat technical, but roughly speaking they emerge as a key technical component in the metric structure that I talked about before. The fermions are not some more or less arbitrary “add-on”, which for various reasons we give a particular statistics (Fermi statistics, as it is called); no, they are there in order for us to construct our geometrical theory on this huge, infinite-dimensional space of “all possible ways to move a stick around”.
There is one more aspect of the quantum field theories, that we obtain, that I would like to briefly mention. Normally a quantum field theory is constructed on a space that is flat. That means, it does not have curvature, it does not involve Einsteins theory of general relativity. It is possible to construct quantum field theories on curved backgrounds, but its involved and difficult.
Now, it turns out that the quantum field theories, that we find, live on arbitrarily curved backgrounds. It appears that we have found a general framework for quantum field theories on curved backgrounds, curved spaces. This is interesting.
QUANTUM GRAVITY OR NOT?
But this raises another question. When we talk about quantum field theory and Einsteins theory of general relativity, one is almost immediately lead to the question of quantum gravity. Does a quantum theory of gravity exist? Can our framework say anything about this?
Well, it seems that it can. And it seems to tell us that gravity is not quantum. That a quantum theory of gravity does not exist.
Or rather, that a quantum theory of gravity need not exist.
Now, this is not a definite statement, we are still trying to work out what our framework contains and what applications it has. What it is telling us. But our framework does appear to offer a way to combine quantum theory with general relativity in which Einsteins theory remains, essentially, classical.
I find this fascinating. It appeals to me that the explanation why it has been so hard, so very, very hard, to find a quantum theory of gravity could simply be that it does not exist. Theoretical physicists and mathematicians have searched for this mythical theory for decades, the unicorn of theoretical physics. Perhaps it really is a myth, a fairytale? I have come to suspect that this is the case.
This is surprising too. Johannes and I have worked for more than 15 years in the hope that we would be able to produce a candidate for a theory of quantum gravity but now we have discovered that what we are working is not that at all. This is not quantum gravity, it is something different. This insight has in fact been screaming at us for some time, we have been aware of the possibility for a few years, but we didn’t want to see it, we were determined that what we wanted was a theory of quantum gravity. But in the end it is not us, who decide, but the math. We decide where we dig our hole, but we do not decide what we will find.
As I said, I am fascinated by this. At first it was a little confusing, unsettling, to shift our perspective, but somehow we are used to tectonic intellectual shifts — is our project bogus? Have we hit a gold ore? Are we completely lost? — but now I find it exiting and interesting. It is a new scenery. We are not playing Shakespeares Hamlet; it is a play by Ibsen. The question is which one.
THE ROAD AHEAD
Basically, there are two key questions that we need to work on now. The first concerns the fermions; the second concerns existence.
The problem with the fermions is that they have the wrong form. They are what is called vectors; they should be what is called spinors (this is related to spin, i.e. how they behave when they are rotated in space. Vectors have spin one; spinors have spin half). This is a very serious problem, it meddles with Einsteins special theory of relativity, what is called the Lorentz symmetry, and specifically what is called the spin-statistics theorem. It is a very naughty business — this problem must be solved if our framework is to have any application to the real world.
Luckily we believe that we see a solution to the fermion-problem. Sometimes the places, that look the darkest, the most scary, turn out to be the places that hide the greatest treasures. I think that this is the case with the fermion-problem — this is what we are working on right now; hopefully I can tell you more about this soon.
The second problem is to complete our proof of existence. This is a technical problem. It may be hard, we are not sure yet, but we are pretty optimistic that our basic conjecture holds true. This problem is not as scary as the fermion-problem. At least not to me. Johannes might see this differently, but he is also a mathematician. They are the ones who worry about such seemingly insignificant issues as whether things exist or not.
MY BOOK “SHELL BEACH”
As I told you in my previous newsletter my book “SHELL BEACH — the search for a final theory” has now been published in Danish. This is the book that I have written as a reward to those of you, who ordered it in my crowdfunding campaign.
The book has received good reviews. The science editor on the Danish newspaper Ingeniøren wrote (my translation):
“Jesper Møller Grimstrup has written a formidable book … throughout the years I have read many books both in Danish and English about physics specifically and about science in general, but none, which compares to Grimstrup’s book.”(Jens Ramskov)
I am still working on finding an English/US publisher for the english version of my book. Those of you who live in Denmark and who have ordered the book will soon receive it in the mail.
HAVE A NICE WINTER
With this I end my newsletter. I expect to write the next newsletter in the spring — depending on how things are going and whether I’ve got something interesting to tell you about.
I wish you all a nice autumn and winter — and to those of you who live in the southern hemisphere; have a nice summer, I hope it will not be too hot.
Best wishes, take good care,