How we can improve QED?
Let us now count arguments for and against the currently accepted form of the renormalized QED. We start with positive things:
- QED is relativistically invariant
- QED is cluster separable
- The S-matrix of QED perfectly agrees with experiment.
Now the negatives:
- The Hamiltonian of QED contains infinite counterterms
- QED interaction leads to unphysical processes of creation and absorption of extra particles from vacuum and 1-particle states
Is it possible to reformulate the theory so as to keep the positive stuff and get rid of the negative? Yes. The answer is here. The Hamiltonian of QED is apparently wrong, but according to Ekstein, we can modify the Hamiltonian without changing the S-matrix (i.e., without changing the agreement with experiment). This can be done by a unitary transformation. Moreover, if we are careful enough, we can choose this transformation such that both the relativistic invariance and the cluster separability are not affected. Now, the question is, whether one can simultaneously eliminate bad properties 1. and 2? And the answer is, again, 'yes'! One can find a unitary transformation of the Hamiltonian such that all interaction terms become finite, and all terms that act non-trivially on the vacuum and 1-particle states are eliminated. In one shot we kill two rabbits: with the new Hamiltonian the renormalization is not needed anymore and we have a clean physically satisfactory definitions of the vacuum and 1-particle states. This is called the "unitary dressing transformation". More about it in another post.
Update So, the conclusion is that the QED Hamiltonian is simply wrong and must be substituted by another Hamiltonian. Why did we use a wrong Hamiltonian for almost 80 years? One reason is in three "good" properties I listed above. This is a respectable reason. Another reason is that the Hamiltonian of QED was derived from correspondence with classical Maxwell's theory of electromagnetism. This is, actually, not a good reason. As we discussed here , Maxwellian description of radiation by a continuous wave is not accurate. There are numerous other flaws in Maxwell's theory, one of them being inadequate description of the "radiation reaction" force. So, derivation of quantum electrodynamics by "quantizing" classical theory is, at best, a heuristic trick.