Nonperturbative QED on the Hopf bundle

Quantum electrodynamics (QED) and the electroweak theory are very successful in explaining quantum phenomena for electromagnetic and weak interactions. Their predictions agree with experimental data to great precision. Attempts to use this technique for strong interactions do not lead to a full success. This suggests that at present we do not clearly understand the nature of quantization. It is reasonable to assume that there should exist some well-defined mathematical procedure of quantization which can be applied to any field theory.

Taking all this into account, it is of great interest to construct nonperturbative QED for some simple case in order to compare perturbative and nonperturbative QED. It would allow one to understand the physical essence of such phenomena like the renormalization, convergence of the Feynman integral, etc. One can assume that such a possibility may arise in constructing QED on some compact manifold, since in this case spectra of eigenvalues of the Dirac and Maxwell equations will presumably be discrete. Here we suggest a scheme of nonperturbative quantization of coupled Dirac and Maxwell fields on the Hopf bundle.

The most important part of our study is the procedure of nonperturbative quantization suggested for the interacting Dirac and Maxwell fields. Following Heisenberg, we have replaced the classical equations by equations for operators of the corresponding fields. Since the operator equation can scarcely be solved somehow, it is replaced by an infinite set of equations for Green's functions.

To illustrate the suggested scheme of nonperturbative quantization, we have considered some physical system possessing perfectly definite Ansatze for the spinor and electromagnetic fields. This gives the much simpler set of equations, for which we have written out the first few equations for 1- and 2-point Green's functions.