A final theory

Imagine if there exists a single mathematical principle, that explains all of reality — in a reductionist sense — and which cannot itself be reduced to other, deeper principles. The belief that such a principle exists and the quest to find it lies at the heart of modern physics. It is the search for a final theory.

Why do we believe that such a principle exist?

The rationale behind this belief comes from the experience of scientists during the last centuries, who saw that there is a convergence of explanations, an increase of simplicity in the physical principles that have been found to explain the physical reality, that we observe.

Take Maxwell’s equations as an example. Here the electric and magnetic fields are combined in a set of differential equations that form classical electromagnetism, which describes phenomena such as light and electricity. These equations combine two different phenomena — the electric and magnetic fields — and link them together in a single framework, which immediately opens up doors to completely different and much broader aspects of physics. The mathematical structure of Einstein’s special theory of relativity lies dormant in Maxwell’s equations and with it the seeds for general relativity, which explains the structure of the universe, black holes, and the big bang. Equally important are the natural generalisations of Maxwell’s equations known as non-Abelian gauge theory, which we today understand as the correct framework to describe both the strong and electro-weak nuclear forces and even gravity. All this and more come out of the unification of the electric and magnetic fields, insights that were not previously available.

Another example of the convergence of explanations is the periodic table of chemical elements, where the complex properties of the various chemical elements are explained by atomic and nuclear physics.

What we see is a fascinating convergence of fundamental principles, where apparently arbitrary structures are suddenly understood in terms of simpler mathematical principles. Science is abundant with such examples of the convergence of explanations and thus the idea is born that eventually, it will be possible to reduce all of it, everything we know about the physical reality, to a single, final mathematical principle.

This is a grand aspiration.

What might a final principle look like?

There is no consensus as to what a final principle will look like. There are a number of general directions of research, a few ideas, and no agreement among scientists.

There is, however, no question as to what a final theory must achieve. It must explain the fundamental physical theories that we know today, which are

• Einsteins theory of general relativity, and
• the standard model of particle physics.

General relativity formulates gravitation in terms of space-time curvature and explains the large scale structure of the universe, such as the big bang and the expansion of the universe. The standard model of particle physics explains on the other hand phenomena at the scale of elementary particles and involves the strong, weak, and electromagnetic forces. A final theory must include both of these theories as limiting cases.

But this is not enough. A final theory must also explain several open questions such as

• what is dark matter?
• what is dark energy (or why is the cosmological constant so small?)
• what happened at the big bang?
• what happens at the singularity of a black hole?

Note that when we discuss a final theory there are in fact two different issues, which might be good to hold apart. First of all, general relativity is what is known as a classical theory, which means that it does not involve the principles of quantum mechanics. The standard model, on the other hand, is built on the principles of quantum mechanics. Thus there is firstly the problem of reconciling gravity with quantum theory — this problem is mostly viewed as a problem of constructing a quantum theory of general relativity  — and secondly, there is the problem of unifying all the four fundamental forces, namely the three described in the standard model and gravity.

It is not clear if a final principle should be based primarily on the principles of quantum mechanics or if it should be based primarily on principles of general relativity – or both (in which case: how?!) or neither. It is not even clear that a final theory must involve a quantization of gravity or a unification of the four fundamental forces. There are, however, a few very general characteristics, which most physicists (but not all!) would agree that a final principle will have. These are:

• Minimal structure. Clearly, if there is structure it raises the possibility for further scientific reductions. If the principle is “it is all turtles stacked on each other” it raises the question “why turtles?”. Thus: a minimal amount of structure.
• A minimal number of free parameters. Similarly, there must be a minimal number of free parameters as too much freedom leaves room for deeper levels of explanations.
• A non-zero “aha!”-score. Conservation of momentum can be understood in terms of a principle of invariance under spatial translation. Conservation of angular momentum can be understood in terms of rotational invariance. General relativity is based on the principle of invariance under coordinate changes. These principles all have in common that they make sense. In quantum mechanics, the basic principles are more opaque, but there is nevertheless a sense of deep meaning to them. One might expect a final principle to score relativity well on the “aha!”-scale.
• Non-trivial and rigorous mathematics. Obviously, if a complex theory is to emerge there must be a mathematical richness involved.

With quantum holonomy theory we have put forward a candidate for a final theory. This theory is based on a simple and novel mathematical principle that meets the aforementioned requirements. The principle comes in the form of an algebra, which simply encodes information on how objects are moved in space. This algebra combines key elements of both general relativity and quantum mechanics and involves very few free variables. In fact, it is so elementary that it seems completely immune to further reduction and at the same time it involves a mathematical richness, which we suspect suffices to generate a serious candidate for a fundamental theory.

Outline of quantum holonomy theory.