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Methodological notes

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

Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation

The holographic duality consisting in the functional coincidence of the spectra of the mean number of photons (or scalar quanta) emitted by a point 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 is discussed. Because they are functions of two variables and functionals of the common trajectory of the charge and the mirror, the spectra differ only by a factor $e^{2}/\hbar c$ (Heaviside units). The requirement $e^{2}/\hbar c$ =1 leads to unique values for the magnitude of the point 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 the finite bare charge. This requirement follows from the holographic bare charge quantization principle we propose here, according to which the charge and mirror radiations located correspondingly 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 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 close similarity of the values of the point bare charge $e_{0} = \sqrt {\hbar c}$, “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 vacuum by the neutral ideally conducting surfaces of a sphere of radius $a$ and a cube of side 2$a$, and the electron charge $e$ multiplied by $\sqrt {4\pi}$. The near equality $e_\mathrm{L} \approx \sqrt {4 \pi} e$ means that $\alpha_{0} \alpha_\mathrm{L} \approx \alpha$ — the fine structure constant.

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Fulltext is also available at DOI: 10.3367/UFNe.0183.201306c.0591
PACS: 03.70.+k, 12.20.−m, 41.60.−m (all)
DOI: 10.3367/UFNe.0183.201306c.0591
URL: https://ufn.ru/en/articles/2013/6/b/
000324296600002
2-s2.0-84888345421
2013PhyU...56..565R
Citation: Ritus V I "Duality of two-dimensional field theory and four-dimensional electrodynamics leading to finite value of the bare charge" Phys. Usp. 56 565–589 (2013)
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Received: 27th, July 2012, revised: 30th, April 2013, 7th, May 2013

Оригинал: Ритус В И «Дуальность двумерной теории поля и четырёхмерной электродинамики, приводящая к конечному значению затравочного заряда» УФН 183 591–615 (2013); DOI: 10.3367/UFNr.0183.201306c.0591

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