I'm looking for a general overview of the various formulations of quantum electrodynamics and results, which can be be made completely mathematically rigorous.
From what I have learned so far, a full theory of interacting quantum electrodynamics has not been made mathematically precise yet. I would like to understand the shortcomings of the most recent theories towards that goal in more detail. I do not wish to consider gravitational effects and would rather not allow any arguments based on such effects, but rather consider just the interaction of the quantized electromagnetic field with the quantized Dirac field for electrons and positrons. Also I do not wish to consider path integrals here.
As far as I know the free quantum theories for both the Dirac field as well as the Photon field are well defined, that is: One has a Hilbert space (the fermionic and bosonic Fock space respectively) and a self-adjoint (in the precise sense of the term) Hamiltonian operator for the free evolution. In addition one has a unitary representation of the full (disconnected) Poincaré group (correct me if I'm wrong, precision is a big issue here). This allows one to treat the free theories just as nonrelativistic quantum theories. In some inertial frame of reference one can specify an initial state, specifying a field configuration, and will be able to calculate the state at a later time or as it would be seen from a different reference frame without the need for any perturbation theory. I think one can then meaningfully answer questions as "what is the probability of detecting exactly one photon within a given spacetime region". One would have to work in "position space" but it seems like a meaningfull question.
As soon as the two fields are made to interact via minimal coupling, one no longer has a self-adjoint Hamiltonian operator. I'm also not sure if one even still has a Hilbert space. Supposedly the only meaningfull questions one is allowed to ask from an interacting theory are the results of scattering experiments. I find this unsatisfactory and see no reason for it. All physical arguments I've seen for this end up talking about gravity and holographic principles, which don't appear anywhere in the theory. I see no reason why such an argument should be accepted as an explanation or excuse for the difficulties in non-perturbative treatment of the theory. I do not care about the fact that the theory is only "usefull" for describing scattering experiments and non-scattering problems can be explained very well with non-relativistic theories. I have found a paper that seems to be defining actual unitary time evolution from the non-self adjoint "Hamiltonian" of full interacting QED on Fock space. This seems too good to be true and I'm confused by it, especially since it is fairly technical. The question is:
- What is done in this paper (warning, pdf. I see no other way to link it)? Does it actually construct time evolution for interacting QED? If this paper doesn't do it, can a renormalized self-adjoint Hamiltonian, depending on some cutoff, be defined on Fock space?
As for perturbation theory, it seems to me like the standard approach (do expansion, organise it into Feynman diagrams, observe that everything is infinite, try to fix things), as effective as it may be, is basically due to not caring if the things one is dealing with are well defined. The more cumbersome approach, such as the Epstein and Glaser approach to Causal Perturbation Theory, as described in the book Finite Quantum Electrodynamics by Günter Scharf manages to get finite terms in the expansion from the very start by being more carefull with the manipulations. It still can't say anything about the convergence of the perturbation series (Scharf's book, 3rd edition, p. 160). There is the famous heuristic argument by Dyson, which states that the series must diverge for negative coupling and hence also diverges for positive coupling. I have a number of questions about this:
- Has it been proven that the (completely renormalized) perturbation series is an assymptotic series? Has it been proven that it diverges?
As far as I understand, presented with this perturbation expansion one has nothing to go on as to how precise the results may be, assuming one is capable of calculating the perturbations to arbitrary order. I believe currently we can do about 30 orders, which is a precision unlikely to be probed by experiment any time soon and also unlikely that we can get a lot more orders any time soon. In principle however, one would not know from what point onwards the results get worse again?! The heuristic approach to assymptotic series is to truncate at the smallest term (expected to be somewhere around order 137?), but I don't even see what "optimal truncation" for the QED perturbative expansion would mean in the first place. There is no "mathematically existing result that is being approximated by the series" to compare the expansion with. This opinion has been confirmed by a professor I briefly asked this before.
- Apparently lattice methods have something to say about this. What is the conceptual status of these methods? Assuming you have arbitrary computational power, does the theory commit to arbitrarily precise predictions for experiments? Can they in principle meaningfully answer questions such as specifying an allowed non-asymptotic initial quantum state and asking for photon detections in regions of spacetime? Does it specify what allowed quantum states are?