– for the Indiegogo crowdfunding campaign
A THEORY OF EVERYTHING
welcome to the second newsletter of the Indiegogo crowdfunding campaign “A Theory of Everything”.
I intend to write this newsletter relatively short. The reason for this is that Johannes and I are at the moment involved in some technical analysis, which I prefer to write about only when it is completed. I hope that this will be the case within a couple of month — and until then I’ve thought it would be best to write simply a brief “part A” and then write a “part B” newsletter with a detailed account of our current work later, when I can fully estimate and evaluate its status.
So what I would like to do in this first part is to provide you with some background information on our work and to discuss some key questions in contemporary theoretical physics.
working with Johannes in Hannover, Germany.
Points in space
Let me start with a simple question: if I hold my two hands out in front of me, then how many points are there between them? Infinitely many? Or finitely many?
Well, if we take the mathematical point of view the answer is simple. There are infinitely many points, even uncountably many. The line segment from 0 to 1 contains infinitely many numbers and thus from a mathematical point of view we know the answer. But how many points are there, which can be physically distinguished? Still infinitely many? Or is the mathematical notion of a point too idealised, could it be that the number of physical points between my two hands is finite?
This question is closely related to the question of finding a theory of quantum gravity and the question whether the scientific project will eventually come to an end — questions, which I would like to discuss in the following. To do so I suggest we consider first the scientific project in its entirety.
The scientific project, which took its beginning in ancient Greece and has moved up through the centuries as an intellectual discipline passed on from generation to generation, is based on the credo ‘to make theories and hold them up against empirical evidence’. This project has been incredible successful and has by now produced a hierarchy of physical explanations, where biology and geology stands on chemistry, which stands on atomic physics, which again stands on nuclear physics and particle physics — and where cosmology provides an overall wrapping.
Now, this hierarchy of explanations raises the question whether it ends somewhere? Will we reach a point, where it is obvious that you cannot dig deeper into the fabric of Nature? Will we reach a bottom?
And what might that look like?
The ancient greeks asked this question too, but today we have knowledge, which they did not posses and which gives us a hint of an answer. This is what I would like to discuss now.
A key ingredient in science is measurement and to measure something we need to interact with the object of interest. And since physical scale — meters, centimeters, millimetres, … — is what orders the various physical theories we need to consider measurements at various scales.
How do we measure an object? Well, generally speaking we throw something at the object and observe how our probe might bounce back. For instance, we may shine a beam of light at the object and observe as some of that light reflects back. So a measurement involves a probe — throwing a rock into the dark or colliding beams of highly energised elementary particles — and the smaller the scale we wish to measure the more energy this probe must carry.
This means that if we wish to measure something at an extremely small scale we need to use probes with extremely high energies. But this inverse correspondence between physical scales and energy raises a curious issue, that involves Einsteins theory of relativity. The theory of general relativity tells us that it is the energy content of matter, that causes space and time to curve — what we experience as gravity — and that higher energies means higher curvature. This implies that when we want to measure something at a tiny scale and thus use a probe with a very high energy, then this probe will, at some point, begin to curve space and time around it.
And the shorter the scales the greater will the curvature of space and time caused by our probe be.
At the scales, which we operate with in our present physical theories, this effect is completely negligible, but there will be a point, where this effect becomes critical. That point is called the Planck scale. A measurement at the Planck scale will require energies so high that it will cause space and time to curve so strongly that it forms a black hole.
As I am sure you all know, a black hole is a region of space, which involves a curvature so great that nothing can escape it, not even light. That means that it is impossible to get a signal out from a black hole, and in particular it means that our probe, which we used for our hypothetical Planck scale measurement, will also not be able to escape!
This means that we simply cannot measure objects at scales shorter than the Planck scale. According to this simple argument distances shorter than the Planck scale are operational meaningless.
So what does this argument say about the number of physically distinguishable points between my two hands— the question I started out with? Well, if we are to believe that localisation below the Planck length is physically meaningless, then it must mean that the number of points between my hands is finite.
And what does this argument tell us about the scientific project? It tells us that the scientific project will end somewhere. Because each scientific reduction involves a jump to a shorter scale — think for instance of chemistry and atomic physics — and if there exist a shortest physical scale, then the process of scientific reduction must necessarily end there.
This implies that a final theory must exist.
Now, this argument is no proof, but in my opinion it is a strong hint that a final theory, that ends the scientific project, is really waiting for us to discover it.
What could such a theory possible look like? What theory could be immune to the process of scientific reduction? Is it even possible to imagine a theory, where the question “what causes its internal structure?” is rendered meaningless?
In my opinion these are the most fascinating questions imaginable — and since they are still unanswered we all share the privilege of having a chance to solve them.
The argument concerning Planck scale measurements, which I have just described, does not only suggest that a final theory exist, it also tells us something about what characteristics it might have. Let us take a look at this.
A key ingredient in all the theories, which we know today — Einsteins theory of relativity and the standard model of particle physics (which is a quantum theory) — is locality. What this means is that in the mathematics, which we use, we can in principle localise all quantities indefinitely, even to a single point.
In the case of Einsteins theory of relativity this is seen in its basic ingredient, the metric field. The metric field is an assignment of a collection of numbers to each point in space and time — numbers, which tell us something about the curvature of space and time — and there is nothing in Einsteins theory, which prevents this field from having highly localised extremities such as black holes (where the curvature becomes divergent, infinite) and the Big Bang itself (where everything is collapsed in a single point).
In the case of the standard model things are a little more complicated because we are not only dealing with various fields (similar to the metric field) but with fields that are quantised. This is what I discussed in my previous newsletter, namely quantum theory, where we are no longer dealing with numbers (think of the metric field, an assignment of numbers to each point in space and time) but instead with operators, which can be understood as the process of obtaining numbers. Now, this sounds complicated (and believe me, it is) but the point here is simply that in the standard model we are again dealing with local objects, where certain quantities — operators instead of number — are assigned to each point in space and time.
(As a side remark let me tell you that this locality, which we find in the standard model of particle physics, has some quite serious consequences. It turns out that when you begin to compute stuff in this quantum theory, then you immediately run into quantities, which are infinite. The occurrence of these infinities can in a certain sense be traced back to this local nature of the theory and in fact it means that the standard model — together with all non-trivial quantum field theories in 4 dimensions — does not exist in a strict mathematical sense. The standard model exist only in a perturbative sense, a rather surprising fact.)
But let me get back to my point. According to the argument concerning Planck scale measurements distances shorter than the Planck scale are operational meaningless. They don’t seem to make physical sense, at least not when you combine quantum theory and general relativity. So if this is true, shouldn’t we expect a final theory — a theory of quantum gravity — to be formulated in a mathematical system, that avoids locality? Shouldn’t we abandon the mathematical language employed in the theories, which we know today, of fields and ‘quantum fields’?
Yes, well, I believe this to be true, as do many others before me. But how do you do this? That is the question. It is not at all clear what such a mathematics should look like.
But I do have a suggestion. In fact, the question of locality is precisely what Johannes and I are working on at the moment. As I explained in the previous newsletter we have for some time now been working on a rigorous mathematical formulation of our theory, a task which for the past six month has thrown us into some rather involved analysis where the key technical step is precisely a letting go of locality. At the moment we are optimistic that this undertaking will be successful and I hope to be able to explain this to you in the second part of my newsletter.
I wish you a happy summer
With this I end this newsletter and wish you all the best till next time. I will write a “part B” as soon as I can report on our current work, which I hope will be the case within a few months.
Till then, may the force be with you!