# On the existence of dynamics in QED

This is an attempt to ask separately about aspects of my previous question, which was closed as too broad. Note that I strongly prefer results that are or can be made mathematically completely rigorous. The first part of the linked question, dealing with divergence of perturbation series, was discussed here.

Recently I came across a japanese Paper from 2014, published online in Journal of Mathematical Physics:

Shinichiro Futakuchi and Kouta Usui, "Construction of dynamics and time-ordered exponential for unbounded non-symmetric Hamiltonians", Journal of Mathematical Physics 55, 062303 (2014); doi: http://dx.doi.org/10.1063/1.4878737 or arxiv.org/abs/1309.5194v1

This paper is fairly long and very technical; in particular I don't understand most of it. Furthermore it does not appear to be cited by anyone other than the authors themselves.

The content of the paper is the mathematically rigorous construction of the Dyson series for non-normal unbounded "Hamiltonians" and application to Quantum Electrodynamics. It is well known, that the interacting theory, even after any regularization or renormalization procedures, currently does not have a mathematically rigorous description, including observables and states. To me it seems that this paper achieves this! (although it certainly sounds way to good to be true and I don't seriously believe it.)

Let me try to summarize what I think are the main points made in the application of their theory to QED (last 3rd of the paper): The free, non-interacting theories both of the photon field (in Lorentz gauge), as well as of the Dirac spinor field for the electron-positron field are first defined on their Fock spaces. The authors provide second quantized Hamiltonians for both fields, which are unbounded self adjoint operators and have spectrum bounded from below (they just take the positive root in the dispersion relation for the Spinor field). Then, on page 24, they define the interaction Hamiltonian: $$H_{\text{int}}\Psi:=e \int_{\mathbb{R}^3}d^3x \ \chi(\mathbf{x})j^{\mu}(\mathbf{x})\otimes A_{\mu}(\mathbf{x})\Psi$$ in the sense of a strong Bochner integral, where $\chi\in L^1(\mathbb{R}^3)$ implements a spacial cutoff. The field operator $A_{\mu}$ is the usual expression from the free theory and the current is formed from the dirac spinor field in the usual manner. The domain of definition for $H_{\text{int}}$ is specified in the paper. This interaction turns out to be densely defined but not normal. The total Hamiltonian is obtained by adding this interaction to the free Hamiltonians, which in total gives a Hamiltonian operator defined on the tensor product of the two Fock spaces.

That is, the interaction is given by minimal coupling, which is just what is usually written down and what perturbation theory is applied to. The perturbation theory, renormalized in this (or a similar) way, has finite terms in the series, however we expect that the series itself does not converge (Dyson's argument). In the paper the interaction Hamiltonian, as well as the total Hamiltonian (not being self-adjoint), are proven to be $\eta$-self adjoint, which to me seems like a mathematically rigorous implementation of the Gubta-Bleuler method, although I certainly do not understand the details there.

The final result is constructing what they call the "Time Evolution", which is a well-defined isometry defined on a subset of the total Hilbert space. It is given by the Dyson series, which is claimed to converge for suitable states. I have been told numerous times that this is not possible to achive, supposedly there are even results suggesting that this is impossible in principle. What am I missing in that paper?

Another thing, which puzzles me somewhat, is that the authors make absolutely no comment as to applications of their results. For example whether their time evolution can be used to compute (approximately) the same amplitudes as obtained by perturbation theory. I'm very thankfull indeed to anyone who would briefly look over it and tell me, which possibly trivial detail I'm missing, that would render the discussion irrelevant to physics, which would have to be the case judging from the lack of attention this paper received.

They are using a momentum cutoff (p. 24) and a space cutoff (p. 26) on $H_{int}$. $H_{int}$ is then $N_b^{1/2}$-bounded, so they can show it is self-adjoint by Nelson's Theorem and that the exponential series converges (Eq. 6.74, p. 30). The hard part is removing the cutoffs.