Duality of two-dimensional field theory and four-dimensional electrodynamics leading to a finite value of the bare charge

We dicuss the holographic duality consisting in the functional coincidence of the spectra of the mean number of photons (or scalar quanta) emitted by a point-like electric (scalar) charge in $(3 + 1)$-space with the spectra of the mean number of pairs of scalar (spinor) quanta emitted by a point mirror in $(1 + 1)$-space. Being functions of two variables and functionals of the common trajectory of the charge and the mirror, the spectra differ only by the factor $e^{2}/\hbar c$ (in Heaviside units). The requirement $e^{2}/\hbar c$ =1 leads to unique values of the point-like charge and its fine structure constant, $e_{0} = \pm \sqrt {\hbar c}$, $\alpha_{0} = 1/4 \pi$, all their properties being as stated by Gell-Mann and Low for a finite bare charge. This requirement follows from the holographic bare charge quantization principle we propose here, according to which the charge and mirror radiations respectively located in four-dimensional space and on its internal two-dimensional surface must have identically coincident spectra. The duality is due to the integral connection of the causal Green's functions for $(3 + 1)$- and $(1 + 1)$-spaces and to connections of the current and charge densities in $(3 + 1)$-space with the scalar products of scalar and spinor massless fields in $(1 + 1)$-space. We discuss the closeness of the values of the point-like bare charge $e_{0} = \sqrt {\hbar c}$, the ‘charges’ $e_\mathrm{B} = 1.077 \sqrt {\hbar c}$ and $e_\mathrm{L} = 1.073 \sqrt {\hbar c}$ characterizing the shifts $e^{2}_\mathrm{B,L} /8\pi a$ of the energy of zero-point electromagnetic oscillations in the vacuum by neutral ideally conducting surfaces of a sphere of radius $a$ and a cube of side 2$a$, and the electron charge $e$ times $\sqrt {4\pi}$. The approximate equality $e_\mathrm{L} \approx \sqrt {4 \pi} e$ means that $\alpha_{0} \alpha_\mathrm{L} \approx \alpha$ is the fine structure constant.