A true theory of everything must explain itself
Why Grand Unified Theories are not theories of everything
Why Grand Unified Theories are not theories of everything
Physicists typically think about what the laws of physics are or make measurements to determine whether what they think the laws are actually match what we observe. Yet, very little is said about what determines the laws of physics themselves.
The typical physicist on the street might say, “symmetry”. Symmetry does determine what conservation laws exist which, in turn, does determine how things move, flow, or change. Yet, symmetry itself is simply a product of the description of the phenomena of interest.
I think it important to distinguish between these two levels of questioning. On the one hand, there are questions of “how does X lead to Y” or “why is Y? Well, because of X of course!”. These are the bread and butter of physics. Why does a ball roll? Because it is round. That is level 1 questioning: the how of things.
Level 2 questioning is to ask not how X leads to Y but why does X exist. For example, why does our universe allow round things?
You might think you could answer level 2 questions in the same way as level 1 questions. After all, if you can have a level 1 question, “why does the ball roll?” You could surely have a question “why is the ball round?” There are two problems with this. The first has to do with the word “does” versus the word “is”. Physics, as it is ordinarily considered, is about “doing” not “being”. It is interested in causes and effects and the rules that connect them. The second problem is that the statement is tautological, meaning that it implies itself. A ball is, by definition, a round thing, and so there is no cause and effect here. It is simply a statement of something that is allowed to exist: namely, round balls. A square ball is a contradiction in terms. Thus, the answer to the question is simply a linguistic one that balls are all round.
Maybe the question was just put poorly and we should remove the concept of ball and simply ask: “why roundness?” This gets closer to the problem. Roundness is a concept. We could define it very precisely using mathematics, but we could never get closer to answering why the universe allows it. If we tried to tie it to some concept like rotational symmetry, we would simply be back at the problem of asking why are balls round because of course rotational symmetry depends on the concept of roundness itself. We could ask why rotational symmetry? But that is in some sense just a restatement of the same question. We have no answer to this. It is simply built into the universe we know.
These are metaphysical questions in the sense that they are meta-rules about physical things but they are not necessarily philosophical. We have already made great progress in addressing such questions in physics by looking at the concept of spontaneous symmetry breaking. Spontaneous symmetry breaking is where you have a physical law or rule and, under certain conditions, a symmetry that it apparently has is broken. For example, when water is frozen, the water molecules are arranged in regular formations, but when the temperature is high enough that symmetry breaks and the water becomes liquid.
In high energy physics, likewise, we know that the electromagnetic force and the weak force, which governs radioactive decay, combine into one force, the electroweak force, when the energy is high enough. While the weak force doesn’t behave like a force at all at low energies, being neither attractive nor repulsive, it does when at high energies. The electroweak force has a particular symmetry to it that is broken at ordinary energies.
This is a small step towards answering a level 2 question because when you ask “why the weak force?” you can say “it is a broken symmetry of the electroweak force”. In this case, we aren’t talking about cause and effect. We are talking about why the weak force is allowed to exist in our universe and we can describe the precise mechanism and energy level at which it comes into being.
This kind of symmetry breaking leads one to imagine that all the forces might combine together at a high enough energy. This is an example of a Grand Unification Theory or GUT. A GUT is not concerned with cause and effect. Rather, it attempts to answer the level 2 question of why the forces we measure exist at all.
I think, however, that GUTs do not go far enough to answering the level 2 questions about our universe. At most they describe the potential processes by which forces and matter come into being from a single overarching theory. But they stop there and that is not good enough.
Whether you describe a GUT as a God equation or a universal force or what have you, there is still the problem of why that equation exists. It seems never ending. Nevertheless, we may find an end in sight if we press on. Perhaps we simply need a level 3 type of question which is how do laws of physics come into being in the first place?
Laws of physics in general have a description which is given by some mathematical object called a Hamiltonian or equivalently a Lagrangian. I am going to focus on the Hamiltonian description. The Hamiltonian describes a landscape called a phase space. This is the space in which any physical system, be it a spring on a table or a black hole, resides in. A single point in a phase space is a complete description of a system both in terms of where it is and where it is headed. Therefore, given the Hamiltonian, you can determine not only the shape of the phase space but also what paths a system is allowed to take within it.
Thus, the phase space is like a meta-description of the physical system under consideration. Yet, given the Hamiltonian, you know exactly how any point in the phase space leads to any other point.
So, in that case, you might ask how the Hamiltonian comes about. And that is the problem: we don’t really know. In some sense, a Hamiltonian is a description of the physical behavior of the system. Therefore, it somehow describes how all the parts of a system interact with each other and how they interact with the outside world. The Hamiltonian can almost be thought of as the rule book in a game of chess. It tells you, given the state of the board, what moves are allowed. The phase space meanwhile is the entire space of allowable board configurations as well as all the possible moves for each one so you know how one configuration gets to another.
The Hamiltonian creates the shape of the phase space because without it we would not know how the different phase points are connected. With the Hamiltonian, we know how the phase space curves into a shape.
Since, however, the Hamiltonian creates the phase space, you might wonder if we have it backwards. Perhaps it is the phase space that creates the Hamiltonian.
My thesis is that it is a little of both. The phase space is like the surface of a little world and the Hamiltonian is how we describe the topology of that world — its mountains and valleys, rivers and oceans.
In mathematical physics this is known as symplectic geometry and can describe all dynamic systems. It has even extended to quantum theory using an uncommon approach called the phase-space formulation of quantum mechanics.
That suggests that the Hamiltonian is simply a way of expressing the phase space and allowed pathways through it rather than the other way around.
The level 3 question applied to phase spaces is to ask if something determines the shape of the phase space and therefore the Hamiltonian for a given particle or field. That is, again, what creates the laws of physics? If there is such a thing I image it would play a similar role that matter and energy play in Einstein’s theory of gravity. Matter and energy tell space and time how to curve. Space and time tell matter and energy how to move. If you carry this analogy back to phase spaces it suggests that particles and fields themselves (or more correctly their phase space trajectories) tell phase spaces how to curve and the phase spaces tell them how to move. In that case the laws of physics might be self-determining like a form of unsupervised machine learning.
Such a theory could rightly be called a theory of everything since it explains where the laws that govern everything come from.
I would hardly refer to most GUTs on the market today as theories of everything. Even string theory fails to propose where strings come from or why they obey the laws that they do. It is a way of unifying without explaining and so falls into the level 2 category.
Perhaps a level 4 question would be: where do phase spaces even come from? At this point we might be getting into metaphysics since everything in order to be a thing existing has a phase space. I will leave that to the philosophers.