What's wrong with the Hamiltonian of QED?

I mentioned several times on this blog that the non-trivial action of the QED interaction V on the vacuum and one-particle states is deeply disturbing. What's wrong with it?

The interaction Hamiltonian V written in terms of particle creation and annihilation operators contains so-called tri-linear terms. An example of such a term is the product of three particle operators a*c*a, where I denoted 'a*' an electron creation operator, 'a' an electron annihilation operator, and 'c*' a photon creation operator. Let us focus just on this term and write the full Hamiltonian H in the truncated form (all momentum, spin, and polarization labels are omitted as well as numerical factors, integration signs, etc.)

H = H_0 + a*c*a + ... (1)

Having the full Hamiltonian H one can form the time evolution operator U = exp(iHt) and study the time dependence of states and observables. Let us consider the time evolution of a state a*|0> which at time t=0 had only one electron (here |0> denotes the vacuum state). We would expect (according to observations) that a single electron always remain in the one-electron state. Let us now find out whether this expectation is fulfilled with our Hamiltonian. We are going to find out how this state will look like after short time t. Let us take t small enough, so that the time evolution operator U can be approximated by linear terms in the series over t. Then

exp(iHt) a*|0> = exp(it(H_0 + a*c*a + ...)) a*|0>
= (1 + it H_0 + it a*c*a + ...) a*|0>
= a*|0> +it H_0 a*|0> + it a*c*a a*|0> + ... (2)

The first two terms in this expansion are harmless. They both correspond to states with one electron, as expected. However, the third state is different. After bringing the particle operators to the normal order and omitting, again, all numerical factors, this term will look like

a*c*|0> (3)

and this is now a state with one electron and one photon. So, the time evolution with the QED Hamiltonian (1) leads to the "emission" of photons by a single free electron. Such an effect has never been seen experimentally. The problem becomes even more severe if we take more terms in the expansion of the operator U and consider additional terms in the interaction V. It is easy to show that the vacuum state |0> will also become unstable with respect to the decay into multiparticle states containing photons and electron-positron pairs. This is also unphysical.

The cure for this desease is known for many years. It is often said that creation and annihilation operators, like a* and a, do not correspond to real physical particles. They describe creation and annihilation of fictitious "bare" particles. Real electrons are complex linear combinations of multiparticle "bare" states. Although, this has been known for decades, there were suprisingly few attempts to reformulate QED in terms of physical particles that can be observed. In view of the complete failure of QED to describe the time evolution, such reformulation of the theory seems to be a reasonable step. This step is exactly the essence of the "dressed article" program initiated in 1958 by the work of Greenberg and Schweber [1].

I would like to mention just one more thing. It appears that in a properly dressed theory all ultraviolet divergences disappear from both the S-matrix and the Hamiltonian. More about the "dressed particle" approach in later posts.

[1] O. W. Greenberg and S. S. Schweber, "Clothed particle operators in simple models of quantum field theory", Nuovo Cim., 8, (1958) 378.