ytuab said:
Did you only transfer the infinity to e^{i\Phi}?
The idea that the infinity is removed by the infinity is the same as that of the traditional methods?
Does it have any more merits in comparison to the traditional methods?
In traditional QED you cannot use the usual Hamiltonian obtained from naive analogy with classical electrodynamics. High order perturbation contributions to scattering amplitudes come out divergent, i.e., useless. Tomonaga-Schwinger-Feynman renormalization approach "fixes" this problem by adding divergent counterterms to the above Hamiltonian. If the counterterms are chosen correctly, then the original divergences get canceled by the divergences from counterterms, and the full renormalized S-matrix is finite and agrees with experiment very well.
Unfortunately, this great result is achieved at the expense of screwing up the Hamiltonian completely. The Hamiltonian of the renormalized theory contains divergent counterterms, so this Hamiltonian H is useless for any type of calculation except the S-matrix calculation (where all divergences cancel out, as I said before). For example, you cannot form the time evolution operator \exp(iHt) and therefore you cannot study the interacting time evolution of states and observables. Usually, this problem is not regarded as a big deal, because it is almost impossible to measure the time evolution of particles in scattering experiments. However, I believe that without a well-defined finite Hamiltonian and time evolution operator a theory (QFT) cannot be considered successful and complete.
The dressed particle approach suggests to fix the above problem by choosing a new Hamiltonian H', which is obtained from the (divergent) Hamiltonian H of QED by means of a unitary (dressing) transformation
H' = e^{i\Phi} H e^{-i \Phi}
It can be proven that the Hermitian operator \Phi (which generates the unitary transformation) can be chosen in such a way that
1. The S-matrix calculated with H' is the same as the (accurate) S-matrix calculated with H
2. The relativistic invariance of the theory is preserved.
3. All divergences contained in H get "absorbed" in \Phi, so that new "dressed particle" Hamiltonian H' is divergence-free.
After this dressing transformation is done we can simply forget about divergent quantities H and \Phi, and perform all calculations (S-matrix, bound states, time evolution, etc.) with our new finite Hamiltonian H'. Remarkably, in these calculations we will never meet divergences, and we will never need to perform renormalization. So, in this approach, we have managed not only sweep divergences under a rug (as before), we also throw away the rug and all divergences under it.
There is a well-defined prescriprion of how to evaluate the dressed particle Hamiltonian H' in each perturbation order. In the lowest (2nd) order the Hamiltonian H' for charged particles coincides with the well-known Darwin-Breit Hamiltonian. For 2 particles in the non-relativistic approximation, it takes the usual form
H' = H_0 + e^2/r
Eugene.
