My Road to a Final Theory

I believe that there exists some … no, many students or scholars either studying or doing research in Physics have dreamed about or are chasing after a so-called final theory. And I am one of them. The pursuit of such a theory can be both exciting, miserable and dangerous, since it is so hard and there are full of suspects, distrusts and failures. Also, people who are still talking about a final theory after the famous “slap-face” in 1900 are thought to be “foolish” by some others. Despite all of these, the possibility of a final physical theory can be so enchanting that still, countless people have not yet lost their hope.

I myself was one of them. And I decided to stop this process a few days ago. The motivation of this article is nothing but to record some of the discoveries during my road towards the final theory, so that the follow-up people can have a reference.

What is a “Final Theory”?

We all heard of the powerfulness of a final physical theory. It is regarded as the most fundamental underlying rules of nature. However, things are not such simple when this illusion faces reality, just like the mercy faces the trolley problem. The first major problem coming up is nothing but the identification problem — what is a final theory?

To make things even worse, there are actually many explanations on a final theory, which all seems reasonable, but all with its shortcomings. Some may believe that the final theory should be about the motion of the most fundamental particles which our whole world consists of. They devote themselves to High Energy Physics, hoping to uncover the mystery of the basic particles. Some believe that the final theory should contain solely about the rules, without any involvement of the boundary conditions (including initial conditions as the temporal boundary) or information about the matter. Thus, they focus on Mathematical Physics and claim that the theories in High Energy Physics have so much involvement of the species of particles, which therefore lose their purity. Some even believe that each field has its own “final theory” and there are no ways to reduce one to another. Although seemed inaccurate, this view is referred as holism within this article.

Why not Holism?

This is apparently — one of an essential motivation for a final theory is its power to reconstruct the whole world. Thus, it is a must that the final theory should at least be able to reproduce all the other “final theories”.

What’s Wrong with High Energy Physics?

I know that the High Energy Physics is not only about the final theory. But be cool, there indeed exists some of our companions who put their hope on it. But I can clearly imagine the scene when they first dig into the frontier of HEP (especially those which connect with experiments). It is so much alike Chemistry — conducting collision and gather the info about the particles. And it is not about several particles, it is 61 (well, maybe not that much pragmatically). It is hardly believed that this is the “final layer” of nature. But, if it is not, then what? Supersymmetry? String Theory? Well, recent development reveals that there might have little hope. Now, what if String Theory is still not the final layer? What it should be? Please notice that the String Theory has already way out of our experimental ability. If it is still not the end, it looks like there is not going to be an end.

People are anxious. At least I was anxious. There is little hope. There is no even just signs which indicate that it can be successful. And moreover, as time goes by, it loses its charm.

So What?

I may never turn back and accept the holism. And I also decided to set aside High Energy Physics. Therefore, I finally turn to the philosophy of the so-called paradigm. This view claims that the final theory is in fact not a “theory”. Instead, it is only a “paradigm” which only gives the ways to generate the theory.

It looks like that this kind of “theory” provides nothing, since, before the arguments used to generate the theory are placed, the “theory” can output no predictions. However, as Noether’s theorem is uncovered in 1918, things had changed.

Noether’s theorem provides an astonishing fact of Lagrangian formalism — symmetry can lead to conservation. This is a very strong conclusion, since it provides a standard procedure to derive conservation laws using Lagrange formalism with a certain symmetry. Moreover, there are very few conditions for the theorem to hold, giving considerable freedom to the form of Lagrangian.

That’s why we call Lagrangian formalism a paradigm. It the theory is an abstraction of the real world, paradigm is the abstraction of theories. The paradigm represents the formulation that all theories share and connects with the basic principles of Physics. Unlike a “fundamental theory”, the paradigm cares nothing about scale or complexity. Instead, it provides a minimal guidance for how a theory should be built and what kind of basic property the theory will have.

Generally speaking, a paradigm is superfluous. However, the Noether’s theorem provides an exception. Actually, the procedure that creating a domain-specific theory under Lagrangian formalism and deriving the conserved current through Noether’s theorem has been a widely implemented standard. And it is not alone. λ-calculus exemplifies another pragmatically powerful paradigm. As a foundation of mathematical logic, λ-calculus is also widely used in the computability theory and the design of functional computer language. Therefore, we have every reason to believe that these sort of paradigm be even more pre-eminent than a simple “fundamental theory + first principle”.

New Insight

The investigation through the Lagrangian paradigm accidentally brings a new insight towards the construction of a final theory. Tracing back on the development of the High Energy Physics theory, we are forced to introduce more and more structures in order to describe Physics in smaller scales. Finally, it turns out that as the theory is getting “essential”, it is getting structurally complicated. This is exactly what should not happen, since intuitively, we are expecting a structurally simple theory as we approach the essence.

The above thinking leads to the distinguishment of two types of theory — fundamental theory and universal paradigm. A fundamental theory works at an elementary scale, and as the theory is getting elementary, it is getting specific. So there is more than normal that a fundamental theory will make itself more complicated during the construction process. However, this trait contradicts with our imagination of a final theory  —  conciseness.

The universal paradigm seems to be the only choice. Indeed, looking back on the history which physical theories were merged to form a more general one, we find that a major motivation for a final theory is to find a most general and universal theory that “contains” the others, just like how electromagnetic theory “contains” electric and magnetic theory. But a universal paradigm also has difficulties. And the most severe one is that it is clearly impossible to establish a “universal paradigm + first principle” scheme to reconstruct the world.

Thus, we’ve got two approaches towards a final theory: through digging deeper into the elementary layer of nature, we get a fundamental theory; through unification, we get paradigms. And moreover, each of them satisfies exactly one of our imaginations of a final theory and they contradict each other. Therefore, there are exactly no ways to meet both of our illusions.

Ultimate Settlement

Eventually, I gave up the fantasy of a final theory and set the Lagrangian formalism as the basis of all physical theories. After all, however, the very consolation of the disillusion is that the Lagrangian formalism can be sufficiently powerful to provide enough safety for Physics — there might be some time when all our theories would fail, but it is not an end. We will always have a theory generator anyway.

— April 13, 2017 at Manchester