The big picture search for a Theory of Everything

Photo by Aditya Chinchure on Unsplash

Turning my attention back to the theory of everything, I want to draw a “big picture” of the different paths taken in a search for a theory of everything and to plot a course forward. I still need to elaborate on all these disparate pieces so we can understand what has prevented us from writing down a working theory of everything. This is a long story that I am breaking into many bite sized parts. There are many threads to which I will return.

There are problems when building a theory of everything, as we have defined it. Many of these problems are often attributed to our lack of a quantum theory of gravity, which we will eventually learn is a huge part of the problem although not the only roadblock to a theory of everything. Even if we were able to quantize gravity successfully in exactly the same way we quantized the other forces, from where we sit today, there would still be large gaps in our understanding of the universe.

The search for a theory of everything has historically focused on either trying to quantize gravity as a traditional quantum theory or to find (or build) the quantum fields of the Standard Model inside an alternate version of general relativity (GR) that itself includes our usual version of GR.

My personal training in theoretical physics makes me much more familiar with the first approach which unfortunately leads to a dead end fairly quickly. There are many ways to see that gravity will be a problem in a quantum theory, but I will detail these issues in a later post. Today, I would like to see how the other half lives and describe some (but not all) of the details of the second approach.

One of the early success stories of the second approach came from a class of theories that have come to be known as Kaluza-Klein (KK) theories. In fact, string theory is a version of this class of theories. (As a friendly note, much of the original literature on KK theories is riddled with subtle sign errors. If you are working through these theories on your own and you are finding differences, you are not the first. Let computer algebra systems be your friend).

To give you some historical context, Albert Einstein introduced the special theory of relativity (SR) as part of his so-called annus mirabilis (or miracle year) in 1905 which included other foundational work on the photoelectric effect (which would eventually win Einstein his Nobel prize in 1921), Brownian motion, and the mass-energy relation. I am usually quick to point out to students that when Einstein first introduced the special theory of relativity, he did so in a way that is most likely unfamiliar and difficult for beginning students to follow. In his original paper, Einstein was trying very hard to show that SR was consistent with Maxwell’s well-known and understood classical theory of electromagnetism.

Working with what is already known is very important of course, but modern students are more likely to be introduced to SR via the frames of reference and the universal speed of light and worry about Maxwell later. This is the case for much of theoretical physics. As we come to understand a theory better and how it fits into the greater framework, we often uncover better ways of “seeing” what the theory was telling us all along and change the way it is taught.

The general theory of relativity (GR) is mathematically much more challenging than the special theory of relativity. Again, students are often surprised that you can learn special relativity with mathematics traditionally taught in high school (all you need is a little algebra and a mastery of square roots) having incorrectly conflated the two flavors of relativity together. And it’s true, the mathematics of GR is very challenging, however, that mathematics was already very well understood in Einstein’s day, however it was not typically taught to physicists at any level. For the most part, this is still true, although adding differential geometry into the standard vernacular of theoretical physics does seem to be well underway.

This has been the good fortune of many theoretical physicists who have wanted to apply a particular branch of mathematics to a physics problem only to find that all the heavy lifting was done years, decades, or even centuries earlier. Keep up the good work mathematicians, we will continue to do this as long as we are able!

For instance, on the advice of Max Born, matrices were introduced into quantum mechanics (QM) by Werner Heisenberg in a pivotal 1927 paper. At the time, Heisenberg had never heard of a matrix. Now, Heisenberg was no slouch. He was a brilliant scientist and literal wunderkind who had already completed his doctorate and his habilitation (a kind of super-PhD required by many European universities to become a full professor) by 1927, but he never learned matrices because it was not part of the usual scientific education of the day. Today, matrices are typically taught as early as middle school. Although some of the properties of matrices were known for a long time, the term “matrix” was not used in a mathematical sense until 1850 (as a play on words of the Latin word for a mother’s womb, which itself was a play on the idea of an expansion in minors. This is as good as it gets in math humor). Physicists were in a similar position with the introduction of groups into quantum field theory which we will discuss in a much later post.

General relativity was introduced to the world in 1915, but it took years and years of diligent study to learn how to test the theory and to find some of the more amazing features hidden within. In fact, several of scientists who made much of the early progress in GR already knew the “new” language of differential geometry including the Finnish mathematician Gunnar Nordström, who had a competing theory of gravitation that eventually fell to Einstein’s because Nordström’s theory did not correctly predict the bending of light, but was very similar to GR in many ways. There is currently a well-known exact black hole solution known as Reissner-Nordström metric named after him.

In an attempt to build a theory of everything for his day, Theodor Kaluza published a paper in 1921 in the Proceedings of the Prussian Academy of Sciences called “On the unity problem of physics” (or something like that, I am leaning on machine translated German for the title which I think gives the best flavor of the idea) on the advice and recommendation of Einstein himself. However, at the time his idea received very little attention (good old Nordström also published a similar idea, but in Swedish which probably drastically reduced his audience) until the idea was picked up by Oskar Klein in 1926 in an attempt to use these ideas within a quantum theory.

One final aside. At this time (1926), only two forces were known. This comes as a surprise to many students, but realize that we have had a theory of gravity since Newton and now Einstein’s GR to replace it. We also had a theory of the electromagnetic force in the form of Coulomb’s law, but really understood via Maxwell’s famous equations. But, we were still at the very birth of quantum theory and there was nothing but questions and confusions. The real structure of the nucleus was almost a mystery. Atomic theory was being elucidated by Rutherford in his famous Geiger-Marsden experiments, we had the (flawed) Bohr model, and the more correct Schrödinger equation by the time Klein published. However, the weak and strong forces are nuclear forces and the nucleus was still quite mysterious. In short, you couldn’t include weak and strong forces into a potential theory of everything in 1926 because we had no idea they existed. Sometimes there are unknown unknowns.

At their core, modern Kaluza-Klein theories are theories of higher-dimensions where the “extra” dimension(s) is/are thought of as being very small or “compactified.” Small can be a tricky and relative idea, but that’s another story for another post. In the original theory, Kaluza and eventually Klein, started simply by writing down what the theory of general relativity would look like in five dimensions. The details of what that would entail will have to come in a later post. We are still building a big picture. Then one of the dimensions was made to be very small, and in the case of Klein, periodic in the form of a simple circle. How did that one dimension become so small? We don’t know (yet), so we will have to come back to that question.

If you are having trouble visualizing this, just imagine that at each point in space there is a tiny, tiny little loop. So small that particle couldn’t travel around that loop. Perhaps a wave could move in there if it has the right wavelength, and that will eventually become important. So, a complete label of an event would need to be written as a point in (x, y, z, theta, t). That is, the usual position in (x, y, z) and then the angle around the tiny circle (or alternatively it’s position r/R where R is the tiny circular radius which is equivalent to the angle), and the time.

In doing this “compactification” the five-dimensional version of GR becomes a four-dimensional version of GR plus something very unexpected, we also find Maxwell’s equations (!) and one other thing a scalar-field. We don’t know what to do with the scalar field (yet) but at the time this sent a shock-wave through the world or theoretical physics. If we were actually living in a five-dimensional universe, with one very small “extra” circular dimension, then GR could be our theory of everything! It had everything that was known at the time inside it.

Well, we now know of two more forces, so this particular version of the theory is no longer tenable. But, one might add even more dimensions with a special shape to create everything we know about the standard model (which is how string theory works, in general) so we will have to leave this discussion for now. There are still missing pieces, but for a big picture, we are starting to see how this might or might not work.

Kaluza-Klein theories are an amazing idea and it gives us insight into how to build a candidate for a theory of everything. More importantly, perhaps, it highlights the difficulties in trying to build a complete theory with incomplete knowledge. On the other hand, this could be a feature if the theory predicts something you were unaware of which is then discovered. We will see this idea in action, too.

Dr. Bryan J Field is a theoretical high-energy physicist and Associate Professor. “A Theory of Everything” (MIT Press, Spring 2023). https://www.bryanjfield.com