Re: Hydrogen in Relativistic QM

Chris H. Fleming wrote:

> How do I know there is a ground state in the fully relativistic and
> electrodynamic, quantum problem. I only know there is a bound state in
> the nonrelativistic problem, or the relativistic problem with an
> electrostatic field. But neither of these cases had pathology in the
> classical regime. I was never in doubt of these cases.

That's a good question. Relativistic QED has a trouble in describing
the bound state of the hydrogen atom. This trouble is explained by
Weinberg in the beginning of chapter 14 of his "The quantum theory of
fields" vol. 1. All we can do in QED is to calculate the S-matrix.
It is known that the bound states of the hydrogen atom must show up
as poles of the electron-proton scattering amplitudes. So, in principle,
the renormalized QED has a chance to give us the energies of
the stable bound states which correspond to the positions of the poles
(note that getting the corresponding wave functions is a much more
difficult task).
However, in order to reproduce these poles one needs to sum up a very
large (infinite?) number of Feynman diagrams. Weinberg proposes a
solution that does not look elegant: approximate the contribution of
some (infinite number) of these diagrams by a solution of the
Dirac equation with fixed 1/r potential and treat other diagrams as
perturbations leading to the Lamb shifts. This seems to be working fine,
but it doesn't answer directly your question, whether the bound state
exist in the full relativistic renormalized theory. We are assuming
their existence instead of deriving it ab initio.

Fortunately, there is another approach to the bound state problem
in QED that moves us a bit closer to the answer. This is the
"dressed particle" approach described (among other places) in chapter
12 of my book http://arxiv.org/abs/physics/0504062 .
In this approach, the Hamiltonian of QED is rewritten in terms of
physical particles (electrons and protons). In the position
representation, this Hamiltonian has explicit 1/r interaction
term (plus relativistic correction) already in the 2nd perturbation
order. So, by simply diagonalizing this 2nd order Hamiltonian
in the electron-proton sector of the Fock space you can get 99%
complete description of the bound state of the hydrogen atom. Additional
corrections (line widths or Lamb shifts) appear from higher order
contributions. So, in the "dressed particle" approach, the existence
and properties (including the wave functions) of the bound states
follow directly from the Hamiltonian.

>>Conservation laws (e.g., the conservation of the baryonic and leptonic
>>charges) prevents its decay into lighter particles. Note that these
>>decay laws do not apply to the positronium (all charges are zero),
>>so it is allowed to decay into photons, as it does.
>
>
> That is more in line with what I was thinking.
>
> So is it true that if you modeled hydrogen simply with the electron and
> proton as different mass, spin-1/2 particles interacting via photon
> exchange, and ignored all other forces, then hydrogen would not be
> stable. Is it true that forces other than electrodynamics keep hydrogen
> and all other elements electronically stable.

I think you misunderstood. The situation is quite opposite, in my view.
Electromagnetic interactions obey the laws
of conservation of the baryonic and leptonic charges, so the hydrogen
atom cannot decay if only EM interactions are at play.
There are hypotheses about additional forces of
nature that do not obey the law of conservation of the baryonic
charge. If these forces existed, then the proton would be unstable
and the hydrogen atom would decay as well. Needless to say, that
such interactions have not been observed yet.

Eugene.

.