Quantum Electrodynamics at Extremely Small Distances

The asymptotics of the Gell-Mann – Low function in QED can be determined exactly, β(g) = g at g → ∞, where g = e is the running fine structure constant. It solves the problem of pure QED at small distances L and gives the behavior g ∼ L. According to Landau, Abrikosov, Khalatnikov [1], relation of the bare charge e0 with the observable charge e in quantum electrodynamics (QED) is given by expression e = e0 1 + β2e0 lnΛ 2/m2 , (1) where m is the mass of the particle, and Λ is the momentum cut-off. For finite e0 and Λ → ∞ the ”zero charge” situation (e→ 0) takes place. The proper interpretation of Eq.1 consists in its inverting, so that e0 (related to the length scale Λ ) is chosen to give a correct value of e: e0 = e 1− β2e ln Λ2/m2 . (2) The growth of e0 with Λ invalidates Eqs.1,2 in the region e0 ∼ 1 and existence of ”the Landau pole” in Eq.2 has no physical sense. The actual behavior of the charge e as a function of the length scale L is determined by the Gell-Mann – Low equation [2] − dg d lnL2 = β(g) = β2g 2 + β3g 3 + . . . , g = e , (3) and depends on appearance of the function β(g). According to classification by Bogolyubov and Shirkov [3], the growth of g(L) is saturated, if β(g) has a zero for finite g, and continues to infinity, if β(g) is non-alternating and behaves as β(g) ∼ g with α ≤ 1 for large g; if, however, β(g) ∼ g with α > 1, then g(L) is divergent at finite L = L0 (the real Landau pole arises) and the theory is internally inconsistent due to indeterminacy of g(L) for L < L0. Landau and Pomeranchuk [4] tried to justify the latter possibility, arguing that Eq.1 is valid without restrictions; however, it is possible only for the strict equality β(g) = β2g , which is surely invalid due to finiteness of β3. One can see that solution of the problem of QED at small distances needs calculation of the Gell-Mann – Low function β(g) at arbitrary g, and in particular its asymptotic behavior for g → ∞.