- #36

meopemuk

- 1,761

- 53

The non-relativistic Hamiltonian is well defined and self-adjoint. In QED there isn't really a Hamiltonian for hydrogen, just the QED Hamiltonian.

I am OK with that. In the dressed theory there is also a full Hamiltonian ##H= H_0+V## in the entire Fock space. This Hamiltonian is obtained by a unitary transformation from the "QED Hamiltonian". ##H## is a normally-ordered polynomial in particle creation and annihilation operators, like in eq. (4.4.1) of Weinberg's vol. 1. This polynomial is Poincare invariant, cluster separable and has huge (infinite) number of different terms. However, if we are interested only in the hydrogen atom (which I define as a bound state of two particles -- an electron and a proton), we don't need the full interaction ##V##, we can choose only few terms that can be relevant to this physical system.

Let me introduce the following creation/annihilation operators: ##a^{\dagger}, a## for electrons, ##d^{\dagger}, d## for protons, ##c^{\dagger}, c## for photons. Then in a pretty decent approximation, the part of the polynomial ##V## that is relevant to the description of the hydrogen atom is

##V = d^{\dagger}a^{\dagger}da + d^{\dagger}a^{\dagger}dac + d^{\dagger}a^{\dagger}c^{\dagger}da + \ldots##

For brevity, I left only operator symbols and omitted coefficient functions, integral signs and other paraphernalia. If necessary, I can supply the missing info. These three terms have clear physical meanings:

## d^{\dagger}a^{\dagger}da## is the direct electron-proton interaction potential. Its leading part is the usual Coulomb potential, plus there are relativistic (contact, spin-orbit, spin-spin) corrections, plus -- starting from the 4th perturbation order -- there are corrections responsible for the Lamb shift. If we diagonalize ##H_0 + d^{\dagger}a^{\dagger}da## in the electron+proton sector of the Fock space, we will get a decent energy spectrum of hydrogen.

## d^{\dagger}a^{\dagger}dac## is interaction by which the atom can absorb a free photon and jump to a higher energy level

## d^{\dagger}a^{\dagger}c^{\dagger}da## is interaction by which photons are emitted from excited atomic levels. So, the energy widths of the excited levels can be properly described.

In brief, the dressed approach provides a clear physical picture of the hydrogen atom with all important interactions. The relevant part of the Hamiltonian is resembling the familiar non-relativistic hydrogen Hamiltonian. However, there are also relativistic and radiative corrections, which ensure that the whole theory is Poincare invariant and cluster separable.

Now, you are saying that the "proper" field-based QED does not use the ideas of particles, their creation/annihilation operators and their Fock space, when interacting systems (such as the hydrogen atom) are involved. On the other hand, you claim that this theory possesses a finite well-defined cutoff-independent Hamiltonian. I am really interested in learning how this Hamiltonian looks like and how one can extract from it useful info, like the energy spectrum, lifetimes, wave functions, etc?

Eugene.