Well, your former ''this week's find in mathematical physics'' was far more attractive for me than your current blog. Online, I am now mainly active on PhysicsOverflow. But this term I have a sabbatical, which gives me more time to spend on discussions, so I became again a bit active here on PF.Nice to see you again!
In that regard the following is likely of interest:
One can avoid the divergences from the outset by using a carefully chosen mathematical setting. This is described in Scharf's book cited in the article, and in more detail in my insight article.
I replied:I’m a graduate student at UMass Boston and I really enjoyed your "Struggles with the Continuum” paper. I was wondering if you could give me a little more explanation for one part. I’m by no means a particle physicist, which is probably why I didn’t get what was probably a simple point. When we want to compute the magnetic dipole moment of an electron as predicted by QED, why should we consider a process where an electron exchanges a photon with a “massive charged particle”? Basically I want to know what intuition leads one to realize this is the calculation we want to do.
I'm no experimentalist, but to measure the magnetic dipole moment of an electron basically amounts to measuring its magnetic field. Like its electric field, the magnetic field of an electron consists of virtual photons. So when you "measure" its electric or magnetic field, it's exchanging a virtual photon with some charges in your detector apparatus.
The easiest way to model this is to imagine your detector apparatus is simply a single charged particle and compute the force on this particle as a function of its position and velocity, due to the virtual photon(s) being exchanged. (The magnetic force is velocity-dependent.)
But if that charged particle is light, quantum mechanics becomes important in describing its behavior: the uncertainty principle will make it impossible to specify both its position and velocity very accurately. So, it's better to take the limit of a very massive particle. In that limit, you can specify both position and momentum exactly.
"Imposing a cutoff" or "regularization" is one way to change the measure to get a convergent integral. This is an important first step. But in making this step, you are led to inaccurate answers to physics questions. This step amounts to pretending that virtual particles with large momenta are impossible, or less likely than we'd otherwise expect. That's not really true.
This step amounts to pretending that virtual particles with large momenta are impossible, or less likely than we'd otherwise expect. That's not really true.