I build tools, mathematical tools to be specific, and I want those tools to be useful. I want them to be used to study the real world. But when I build those tools, most of the time, I don’t test them on the real world. I use toy models, simpler cases, theories that don’t describe reality and weren’t intended to.
I do this, in part, because it lets me stay one step ahead. I can do more with those toy models, answer more complicated questions with greater precision, than I can for the real world. I can do more ambitious calculations, and still get an answer. And by doing those calculations, I can start to anticipate problems that will crop up for the real world too. Even if we can’t do a calculation yet for the real world, if it requires too much precision or too many particles, we can still study it in a toy model. Then when we’re ready to do those calculations in the real world, we know better what to expect. The toy model will have shown us some of the key challenges, and how to tackle them.
There’s a risk, working with simpler toy models. The risk is that their simplicity misleads you. When you solve a problem in a toy model, could you solve it only because the toy model is easy? Or would a similar solution work in the real world? What features of the toy model did you need, and which are extra?
The only way around this risk is to be careful. You have to keep track of how your toy model differs from the real world. You must keep in mind difficulties that come up on the road to reality: the twists and turns and potholes that real-world theories will give you. You can’t plan around all of them, that’s why you’re working with a toy model in the first place. But for a few key, important ones, you should keep your eye on the horizon. You should keep in mind that, eventually, the simplifications of the toy model will go away. And you should have ideas, perhaps not full plans but at least ideas, for how to handle some of those difficulties. If you put the work in, you stand a good chance of building something that’s useful, not just for toy models, but for explaining the real world.
Great thinking! I really like your posts.
Which begs the question of what the most important differences between your favorite toy models and reality are.
One of the biggest differences has to do with dimensions.
Four is a special number. Special things happen in four dimensions that don’t happen elsewhere. You’ve probably heard them mentioned in other contexts.
When calculating in the real world, people often use something called Dimensional Regularization. You let the dimension vary, slightly away from four dimensions, in order to avoid divergences.
Doing this spoils a lot of the magic of four dimensions. That magic is quite powerful, and it’s at the heart of a lot of advances in amplitudes. In particular, it’s been very useful in N=4 super Yang-Mills, where for a variety of reasons you can either avoid thinking about Dimensional Regularization, or avoid doing it altogether.
Thus, the tricky part, and the issue to keep in mind: to generalize from the toy model, you must either find ways to keep using four dimensions despite not using Dimensional Regularization, or find a different regularization you can use in the real world.
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FWIW, I love your use of the term “magic” in the non-magical sense here, which is something that I do quite frequently when talking about physics, math and law.