Theoretical physics isn’t easy. Everyone who has seen a documentary about physics or taken a graduate level physics course knows this. Much of theoretical physics is steeped in complicated math with talk about symmetries and conservation laws. Even understanding where basic kinematic equations comes from requires a bit of calculus. But it’s not hard for no good reason. Physicists want certain things out of their theories and they use abstract ideas to get them.
Take Lie algebras, for example. These are abstract algebraic objects defined by weird things like Lie brackets and vector spaces over fields. Most math or physics students won’t formally encounter these things until graduate school because they’re pretty abstract. But everyone who has taken high school level physics intuitively knows what a Lie algebra is, they just don’t know that they know it1.
But why do we even need to deal with something so abstract, like Lie algebras? Why can’t we just work with real numbers and tangible things? To someone who hasn’t studied any math, talking about things like continuous symmetries and group orbits seems needlessly complicated. It’s hard to even get to a point where talking about these things seems like a discussion worth having. But abstract mathematical objects, such as Lie algebras and symmetries, are so ubiquitous in physics for a simple reason: they come directly from our ideas on what good scientific theories look like.
A naive universal theory of physics
Let’s describe, in very general terms, what a scientific theory looks like. At it’s most basic, a scientific theory is a consistent set of ideas that makes accurate predictions about what the world will look like in the future. For physics, this is a set of rules that give answers to questions that often sound like “where will this ball be in the future if I throw it this fast at this angle?” or “how hot will this wire get if I run a certain amount of electricity through it?” But questions like these come from hundreds of years of studing physics, so we know a thing or two about which questions to ask. What if we knew nothing about physics and just wanted to think about what a good physical theory would look like? Where should we start?
Theories, being about predictability, should predict as broad of a picture as it can while still being accurate. Saying a certain set of rules work in one context, but a different set of rules work in another seems like some patchwork notion of what a theory should look like. Even if that worked we’d still need some way of figuring out which rules work in which scenarios. At the very least, a “good” universal physical theory, then, would be a theory that is the same everywhere. The rules that tell you how things work on Earth should also work in space, and if they don’t we should know why they don’t and be able to explain it. Rules that tell me how things work where I stand should also tell you how things work where you stand, and we both should agree on each other’s descriptions of how things work, generally. What’s worth noting is that this isn’t a necessity, prescribed by the universe about how things should operate. It’s just a desire from people who want to understand the world about what that understanding should look like.
If you’ve heard about Einstein’s theory of relativity, you may be aware that the first fundamental postulate of that theory is that the laws of physics take the same form in all intertial frames of reference. This first postulate states exactly what I’ve described above as a good physical theory2. There’s no logical dictum stating why this must be true. But without knowing any physics you can understand why you’d want this notion as a part of your theory of the universe.
How does this relate to symmetries?
An important part about science is thinking about the consequences of your theory. Since we have this idea about a “good universal physical theory” should look like, what does that mean for how we describe it? What consequences does this definition of “good” have on our theories, without making further assumptions about it? Do these consequences give us any clue about where to start looking and what tools we should bring with us?
Let’s start by defining the space we’re working in as \(X\). This can be something like space, or time, or spacetime, but without making specific claims yet, we can say that our goal for studying physics is to try to find a set of physical rules that govern what happens at each point in our physical space, \(X\). We don’t know what these physical rules are, or what they may even look like. But let’s define the set \(A\) as the set of all sets of physical rules. Formally, then, our goal can be re-stated as finding the element(s) in \(A\) that describe the world we live in.
From our desired universal theory, we can say that we’re actually only concerned with a subset of \(A\). That is, we only care about sets of physical rules that include the “physical theories are the same everywhere in \(X\)” rule. Let me stress that up until this point, we haven’t really made any assumptions about how the universe works. We are just thinking about what studying physics means.
Let’s build on this idea. Let’s consider some mapping, \(\pi: X \rightarrow A\), where \(\pi(x)\) is the set of physical rules that govern what happens at point \(x\). Then, because of our desire for a “good” theory, if there is some total list of physical rules that we call \(\alpha\), we can immediately say something about \(\pi\):\[\pi(x) = \alpha \forall x \in X\]
Stated another way, \(\pi(x)\) is invariant under transformations in \(X\).
What does this tell us? It tells us that we should start thinking about invariant transformations and symmetries. What math do we know about that is concerned with invariant transformations and symmetries? Groups. Rings. Vector spaces. Lie algebras. A whole plethora of mathematical tools immediately become available to us simply because we want something in our description of the universe. This isn’t some statement from on high that we need to obey, it results directly from what we think good scientific theories are. It tells us that, if nothing else, there is a fundamental conservation law at the heart of our physical theory.
Theoretical physics isn’t easy, but the mathematical tools that physicists use don’t show up for no reason. What we want our physical theories to look like points us to things we should think about, like symmetries and invariant transformations. This naturally leads to thinking about Lie algebras, representation theory, vector spaces, and the like.
The reason these tools are so abstract is that we don’t want to predetermine what the universe should look like if we don’t have to. Using 3-dimensional real vectors presupposes that we live in three spatial dimensions and that our universe is based on real numbers. If we don’t have to make that assumption, it’s probably safer not to, even if it seems reasonable. Lots of things that seems reasonable in day-to-day life don’t end up producing reasonable predictions in extreme circumstances, so we have to be careful about each and every assumption we make. But keeping these assumptions in mind and taking them to their logical consequences can help inform you of what tools you may need and where you will need to look to find something new.
A simple example of a Lie algebra is the set of 3-dimensional real vectors with the cross product. ↩
Including “inertial reference frames” in the first postulate extends what I’ve mentioned above about a good scientific theory. This little bit of physics sprinkled into the first postulate essentially make the theory a statement about a good physical theory instead of just a good scientific theory. ↩