So one day you look out at the world, or in at your mind and you see something beautiful. Now you could stop there and be an artist. Nothing wrong with seeing something beautiful and giving it a voice. But maybe you want to go one step further and give some sort of meaning to the beauty you just saw. If this is your passion, then you are a theorist. My field is theoretical high energy physics but there are universal threads to theory. Being a theorist is exciting and rewarding regardless of how good you are at it. Good let’s create a theory.
First off, you need to divine the “basic unit” of your theory. In number theory, the prime numbers are the basic unit. In chemistry, the atom is the basic unit. All the concepts your theory is concerned with can be constructed out of the basic unit through various operations. Chemical bonding in chemistry and multiplication in number theory build all the concepts in the theory’s universe from the basic unit. Physics is especially difficult because we are not totally sure what the basic unit is. In thermodynamics, it is the atom. Classical Mechanics uses conservation of energy and momentum (and their generalized symmetry based siblings such as angular momentum) along with space and time as conceptual building blocks. When we unify fields of physics such as classical mechanics (the mechanics of matter) with electromagnetism to form special relativity, everything changes. Space and time are a matter of perspective like the width and length of a container. Every time we look closer and unify disparate theories completely new theories become necessary. New theories with different basic units that make completely new deep philosophical statements about the universe. The central struggle in theoretical physics right now is unifying Quantum Field Theory (my area) with General Relativity. Quantum Field Theory nicely explains every phenomenon in the universe as long as we don’t consider gravity. Some theorists believe that they can be unified with string theory, so that the basic unit of the universe would be one dimensional strings.
Now why do theories give meaning to the beauty. What kind of meaning am I talking about here? Well, a well constructed and complete theory will make deep philosophical statements about the phenomenon it is concerned with. Physics is concerned with the universe, everything is under its domain, so its philosophical implications are the most vivid. The first proper physical theory was Newton’s Three Laws (1) An object in motions stays in motion, an object at rest stays at rest (2) F=ma (3) Every action has an Equal and Opposite Reaction. So these three laws seem simple and humble, but to paraphrase Richard Feynmann, just because you know how the pieces move in chess doesn’t mean you can predict a game played between Grandmasters. Let’s look to some of the more sophisticated consequences of these laws. The first law means that the velocity of an object is constant (velocity is the ratio of how much space a body traverses over how much time it traverses). The second law stipulates the only circumstance under which velocity can change. The third law is an extremely powerful philosophical statement. The actions referred to are forces. It says that in order for one object to modify the velocity of another object it must in turn pay the price of having its velocity changed in an equal but opposite amount. We take these things for granted, out intuitions are built upon them. However, the universe didn’t have to be this way. There is another interesting consequence of these laws. If they work on a given scale, they work on any given larger scale the same way. If you take lots of little things that F=ma applies to and momentum and energy is conserved for and clump them together into planets, lets say, all the same laws hold for the center of mass of those planets. The same is true when we clump those planets together into solar systems, galaxies, even the universe as a whole. These three laws also make an assumption that forces and distances in space are vectors. If you say something is a vector, it basically means that (1) if you walk x meters in any direction, or if you move the object x meters in any direction, the object is exactly the same (2) if you rotate the object through any angle, the object is exactly the same. That is a huge statement about the universe. It says, essentially, that space is homogeneous. Space looks the same all over and nature doesn’t care where in space you are standing, all her laws look the same there too. In fact, you cannot even define where in space you are standing, coordinate systems are always relative. In a deeper treatment of classical mechanics, thanks to Emmy Noether, we can mathematically prove that because space is homogeneous momentum must be conserved. Furthermore, because time is homogeneous (i.e. you wait 12 seconds and F still = ma) energy is conserved. Those simple three laws determine our most basic understandings of space, time, and the natural universe.
A good theory will make testable predictions which we then can verify to see if it’s a good theory. One could raise the objection that we are just jury rigging theories around empirical data. Well, firstly, that isn’t necessarily a bad thing. Organizing disparate sets of data into meaningful laws of nature has value in and of itself. Secondly, a good theory will make deep philosophical statements about the universe that make new predictions we have not yet tested. High energy physics is concerned with tiny objects (quantum mechanics is therefore needed), moving near the speed of light (special relativity is required). Thus we needed to reconcile special relativity and quantum mechanics into a single theory. Each of these theories make bizarre statements about our universe. Special relativity says space looks like time and time looks like space depending on what speed you are travelling at. Quantum mechanics says everything is really a wave. When we combine these two into a new conceptual basis for looking at the universe. I want to look at some of the new statements and predictions that Quantum Field Theory made about the universe. (1) Antiparticles exist (2) All electrons, photons, and other particles will be identical and indistinguishable from each other whether they are minted freshly in a particle collider or were created by the Big Bang. (3) Matter and energy will pop up out of nothing from the vacuum at tiny enough scales.
The first step to a good theory is modeling. You need to find a language in which you will write the rules of your model. Mathematics works miraculously well as the language of nature. There is no good reason why this should be the case, by the way. It really is a peculiar coincidence. Of course which branch of mathematics will be appropriate is the main challenge to modeling. If you are serious about theoretics and becoming a good theorist you need to learn modeling skills. I would recommend working through the Harvard Challenge Problems at the link below.