Finite value of the bare charge and the relation of the fine structure constant ratio for physical and bare charges to zero-point oscillations of the electromagnetic field in a vacuum

The duality of four-dimensional electrodynamics and the theory of a two-dimensional massless scalar field leads to a functional coincidence of the spectra of the mean number of photons emitted by a point-like electric charge in 3+1 dimensions and the spectra of the mean number of scalar quanta pairs emitted by a point mirror in 1+1 dimensions. The spectra differ only by the factor $e^2/\hbar c$ (in Heaviside units). The requirement that the spectra be identical determines unique values of the point-like charge $e_0=\pm \sqrt {\hbar c}$ and its fine structure constant $\alpha _0=1/4\pi \alpha$, which have all the properties required by Gell-Mann and Low for a finite bare charge. The Dyson renormalization constant $Z_3\equiv \alpha /\alpha _0= 4\pi \alpha$ is finite and lies in the range $0 < Z_3 < 1$, in agreement with the K$\ddot {\rm a}$ll$\acute {\rm e}$n–Lehmann spectral representation sum rule for the exact Green's function of the photon. The value of $Z_3$ also lies in a very narrow interval $\alpha _{\rm L} < Z_3 \equiv \alpha /\alpha _0 = 4\pi \alpha < \alpha _{\rm B}$ between the values $\alpha _{\rm L} = 0.0916$ and $\alpha _{\rm B} = 0.0923$ of the parameters defining the shifts $E_{\rm L, \,B} = \alpha _{\rm L, \,B}\hbar c/2r$ of the energy of zero-point fluctuations of the electromagnetic field in cubic and spherical resonators with the cube edge length equal to the sphere diameter, $L = 2r$. In this case, the cube is circumscribed about the sphere. That the difference between the coefficients $\alpha _{\rm L,\, B}$ is very small can be explained by the general property of all polyhedra circumscribed about a sphere: despite the difference between their shapes, they share a topological invariant, the surface-to-volume ratio $S/V = 3/r$, the same as for the sphere itself. Shifts of the energy of zero-point oscillations in such resonators are also proportional to this invariant: $E_{\rm L, \,B} = \alpha _{\rm L, \,B} \hbar c S/6V$. On the other hand, the shifts $E_{\rm L, \,B} = \alpha _{\rm L, \,B}\hbar c/2r$ of the energy of zero-point oscillations of the electromagnetic field essentially coincide with the energy of the mean squared fluctuations of the volume-averaged electric and magnetic fields in resonators, equal to $Z_3\hbar c/2r$ in order of magnitude. It hence follows that $\alpha _{\rm L, \,B}\approx Z_3$, as it should for the coefficients $\alpha _\gamma $ of the shifts $E_\gamma = \alpha _\gamma \hbar c/2r$ in other resonators $\gamma $ circumscribed about a sphere. The closeness of $\alpha _{\rm L}$ and $\alpha _{\rm B}$ to the $Z_3$ factor is confirmed by the K$\ddot {\rm a}$ll$\acute {\rm e}$n–Lehmann spectral representation and agrees with asymptotic conditions relating the photon creation amplitudes for free and interacting vector fields.