Crazy stuff happens when you try to measure the mass and charge and momentum of particles—stuff beyond the measurement problem in QM.

Suppose you step on a scale and weigh yourself, call that your bare mass m₀. You then dive into a pool and try to measure your momentum.

The mass term that shows up in your momentum calculation is not your bare mass. It will instead be some mass, m, that differs experimentally from m₀. Why?

Because of interactions. There‘s an added mass m’ associated with your interaction with the pool such that:

m = m₀ + m’

m is what we call the renormalized mass, but let’s put that off for a second. The quantity m₀ makes sense to us because we can remove ourselves from the pool and step on a scale.

But what about in the case of an electron, where the analogue of the pool is its field?

In the electron’s case, the bare mass m₀ is a quantity in a fictional theory where interactions have been turned off. But it’s not physical. It’s not what we measure.

Interestingly, the physicists who tried to calculate the self-interaction of the electron faced trouble!

Lorentz and others tried to, using the classical theory of electromagnetism, calculate the self-energy/self-force of the electron. Feynman has a great chapter on this where he gives the self-force calculation: http://feynmanlectures.caltech.edu/II_28.html 

The leading term is the electromagnetic mass

What do you think happens when a, the electron size, goes to zero? That is, what do you think will happen when we consider the electron as a point particle? Take a guess!

The term becomes infinite! This should not happen. We do not want our calculations of physical quantities to diverge.

That sucks, but there’s a new quantum theory on the table, quantum electrodynamics (QED). Can it treat the infinities that arose in our classical theory?

Dirac, Oppenheimer, and others set out to do just this. A quote from Oppenheimer:

A brief, simplified answer: the divergences (infinities) also show up in the quantum theory, though they’re somewhat less scary (mostly logarithmic rather than quadratic). Still, we want to get rid of them. What do we do?

“Why not simply toss them out?” the physicists thought.

Putting things roughly, we go back to the earlier toy equation and

m = m₀ - m_{self}

we subtract the divergent term from the bare mass, adjusting the bare mass to absorb it, and reproduce the measured, physical (renormalized) mass m.

Ok, things are getting spooky. The physicists noticed that the integrals they were taking had an upper limit of infinity, and that they could impose a cutoff Λ

∫^∞ k dk —> ∫^Λ k dk

where k is momentum. They were integrating over arbitrarily high momentum scales until then.

Obvious problem: physics can’t depend on our epistemic position. We can’t just impose an arbitrary cutoff and expect things to be alright. We have to remove the cutoff-dependence from our physics.

But that’s what the above infinity subtraction is for!

After we impose the cutoff, we can adjust the values of the bare parameters in the theory (bare charge, bare mass, etc.) in order to cancel out the cutoff-dependence! It’s a mathematical trick, but at least we’re reproducing experimental results, eh?

A popular way to do this is: https://twitter.com/wickrotate/status/1286044683030769667

This is the pre-Wilsonian picture of renormalization, but it’s still used. I’ll show how it’s done in 𝜙^4 theory in a bit!