Abstract
The regularization of quantum electrodynamics in the space of functions $\psi_a(x)$, which depend on both the position $x$ and the scale $a$, is presented. The scale-dependent functions are defined in terms of the continuous wavelet transform in $\mathbb{R}^4$ Euclidean space, with the derivatives of Gaussian served as basic wavelets. The vacuum polarization and the dependence of the effective coupling constant on the scale parameters are calculated in one-loop approximation in the limit $p^2 \gg 4m^2$.
Highlights
This paper was initially conceived as an erratum to the paper [1], where we have found technical errors in the evaluation of one-loop diagrams [Eqs. (34) and (36)] in wavelet-based quantum electrodynamics (QED)
II, we summarize the scale-dependent approach to QED, described in the previous paper [1], and present the results of one-loop calculations performed in Euclidean R4 space, with two different wavelets, viz. the first and the second derivatives of the Gaussian
Quantum electrodynamics was the first quantum field theory model to face the problem of deriving finite observable quantities—physical charge and physical mass of the electron—from formally divergent Feynman integrals
Summary
This paper was initially conceived as an erratum to the paper [1], where we have found technical errors in the evaluation of one-loop diagrams [Eqs. (34) and (36)] in wavelet-based quantum electrodynamics (QED). QED, briefly described in the aforementioned paper, can shed some new light on the scale dependence of the coupling constant on the observation scale in an Abelian gauge theory—starting from completely finite quantum field theory model with no need of renormalization. We make a simplifying assumption that all measurable quantitiesPcan be determined in terms of effective fields ψAðxÞ ∼ A≤a≤∞ ψaðxÞ (with the meaning of the sum clarified later in the text), which are the sums of all fluctuations larger than the observation scale A This approach allows us to start with a standard QED Lagrangian at large scales, with the “bare”. The remainder of this paper is organized as follows: In Sec. II, we summarize the scale-dependent approach to QED, described in the previous paper [1], and present the results of one-loop calculations performed in Euclidean R4 space, with two different wavelets, viz. In the Conclusions, we summarize the reasons for violation of a locally defined gauge invariance by finite-scale wavelet calculations and propose to substitute it by the scale-dependent gauge invariance, which has been already considered by different authors [2,13]
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