quantum electrodynamics - Which renormalisation techniques are available for 3+1 QED?

I hope my question is not too naive, but I would like to know what are the available renormalisation techniques for 3+1 QED. I have read a bit about Pauli-Villars, but I am wondering if there are others, and if possible a good detailed source to find them explained for a beginner.

Answer

There are many ways to regulate divergent quantities:

1. 
   

   Zeta regularization; nice review https://arxiv.org/pdf/1607.06493.pdf

   
   


2. 
   

   Heat kernel regularization; see discussion in above review as well

   
   


3. 
   

   Dimensional regularization; I highly highly recommend http://www.people.fas.harvard.edu/~hgeorgi/review.pdf both as a discussion of regularization and effective theory in general but also of dim reg specifically$^1$

   
   



4. 
   

   Hard cut-off; this is simply integrating up to some upper limit on the momentum, usually denoted capital lambda, here's another nice overview http://users.physik.fu-berlin.de/~kleinert/kleiner_reb8/psfiles/08.pdf

   
   


5. 
   

   Pauli--Villars is mentioned above: the idea is essentially the propagator is modified at high energies by adding extra factors of momentum in the denominator. That's a very heuristic description: more specifically what you're doing is adding fields to the action that have "the wrong sign kinetic term" (called ghosts)

   
   

When you calculate quantities in renormalizable theories, the whole idea is that physics should be independent of the UV details. Putting in a regulator alters what's happening at high energies, and so you should get the same answer for low energy physics independent of your choice of regulator. Some regulators are better than others: ideally, we may want to choose regulators that preserve symmetries or gauge invariance. Pauli--Villars looks like it violates unitarity, because of those negative kinetic terms. A hard cut--off is not gauge invariant -- dimensional regularization is nice for instance because it respects gauge invariance, and so usually is the regulator of choice in gauge theory$^2$. Using a regulator that breaks a symmetry of the theory is not "disallowed'' per se, it just means that you have to add counter terms to the action that also break this symmetry - this can be nasty since if you don't have symmetries constraining what you add to the action there may be many ugly terms you need to add in - and so making the symmetry manifest in your final answer is difficult.

For a great discussion of this topic generally I'd recommend Section 18 of Weinberg "The Quantum Theory of Fields" Vol 2.

$^1$_{There are subtleties to dim reg since it's modifying IR physics as well as UV physics -- it for instance can throw away IR divergences as well. The fact dim reg alters IR physics isn't a a problem when: 1) all particles are massive, so there are no dynamics in the infinitely long distance limit, or 2) we use effective field theory. The reason EFT solves this problem is that when we do renormalization in EFT, we are actually just calculating the change in the theory as we integrate out high energy modes. It doesn’t matter if dim reg affects the long distance behavior because any such effect cancels when we look at the change in the theory before and after we integrate out high energy modes.}

$^2$_{There are also subtlties in trying to use dim reg where there are objects in your theory that aren't defined in arbitrary dimensions, like the matrix $\gamma^5$. The prescription 't Hooft and Veltman came up with is to define $\gamma^5$ in a subspace of the $d$ dimensions; there's an example of this page 662 in Peskin and Schroeder where they use it to calculate the Adler-Bell-Jackiw anomaly. If you're new to QFT maybe all the details there are a little high powered, but the point is Equation (19.50) is just some integral involving gamma matrices, and they explain how to do that integral with dim reg.}