Conversation

Mathematicians get annoyed at how physicists take beautiful formulas and clutter them up with 'useless' constants like

𝑐 - the speed of light
ℏ - Planck's constant
𝑘 - Boltzmann's constant
𝐺 - the gravitational constant

making it harder to see the essence of things. Mathematicians prefer units where all these constants are set equal to 1.

I used to be like that too - but right now I'm doing a project where I 𝑛𝑒𝑒𝑑 these constants to see the essence of things!

(Of course it's good to keep these constants around to avoid making mistakes where you equate quantities with different dimensions: this is what computer scientists call a 'type discipline'. That's important, but it's NOT what I'm talking about now.)

When you're studying just one physical theory at a time, you can set dimensionful constants equal to 1 to simplify things. But often we like to study a whole 𝑓𝑎𝑚𝑖𝑙𝑦 of physical theories at once - a family where those constants take different values! We can't set them to 1 if we're interested in what happens when they approach 0. For example:

As 1/𝑐 → 0, special relativity reduces to Newtonian physics.
As ℏ → 0, quantum mechanics reduces to classical mechanics.
As 𝑘 → 0, statistical mechanics reduces to classical mechanics.
As 𝐺 → 0, general relativity reduces to special relativity.

And this is just the beginning of the story: various collections of constants can approach 0 at different rates, and so on.

When we do this, we're studying what mathematicians would call a 'moduli space' of theories - or even better, a 'moduli stack'. We may want to do 'deformation theory', where we expand answers in powers of some constant. And so on.

So don't scorn those constants!