Pair Production Problem and Canonical Quantization of Nonlinear Scalar Field in Terms of World Lines

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Science
101 Vernadskii Ave., Moscow, 117526, Russia.
Updated August 22, 2001


A new quantization scheme (WL-scheme), using world lines as objects of quantization, is proposed. Applying to nonlinear scalar field, the WL -scheme is investigated and compared with the conventional PA-scheme of quantization. In the PA-scheme objects of quantization are particles and antiparticles, which are fragments of the total physical object -- world line (WL). Applying to the nonlinear field, the PA-scheme of quantization leads to such difficulties as nonstationary vacuum, obligatory use of perturbation theory technique, normal ordering and cut-off  in the scattering problem. These difficulties are corollaries of inconsistency of PA-scheme. The WL-scheme is free of these difficulties. These difficulties are connected with the reconstruction problem of the total world lines from their fragments (particles and antiparticles). In the case, when these fragments interact between themselves, such a reconstruction is very complicated problem. The new WL-scheme of quantization is free of all these problems, because it does not cut the total world line into fragments (particles and antiparticles). Formally appearance of fragments in the conventional quantization PA-scheme is a corollary of identification of the energy with the Hamiltonian. In fact such an identification is not necessary. It leads only to difficulties. The new  WL-scheme of quantization does not use this identification and enables to go around all these problems. The WL-scheme enables not to use additional (to nonrelativistic QM) quantization rules, used in the relativistic QFT (normal ordering, perturbation technique, renormalization).
English version of the paper and figure in Postscript.
Russian  version of the paper and figure in Postscript.