  We consider the Dirac equation and Maxwell's electrodynamics in $\mathbb{R}
\times S^3$ spacetime, where a three-dimensional sphere is the Hopf bundle $S^3
\rightarrow S^2$. In both cases, discrete spectra of classical solutions are
obtained. Based on the solutions obtained, the quantization of free,
noninteracting Dirac and Maxwell fields is carried out. The method of
nonperturbative quantization of interacting Dirac and Maxwell fields is
suggested. The corresponding operator equations and the infinite set of the
Schwinger-Dyson equations for Green's functions is written down. To illustrate
the suggested scheme of nonperturbative quantization, we write a simplified set
of equations describing some physical situation. Also, we discuss the
properties of quantum states and operators of interacting fields.
