Yu. V. Kulish, E. V. Rybachuk


It is shown that an integral corresponding to the contribution of one particle to equal-time commutator of quantized scalar fields diverges in a reality, contrary to usual assumption that this integral vanishes. It means that commutator of scalar fields does not vanish for space-like intervals between the field coordinates. In relation with this divergence the generalization of the Klein-Gordon equation is considered. The generalized equation is presented as products of the operators for the Klein-Gordon equation with different masses. The solutions of derived homogeneous equations are sums of fields, corresponding to particles with the same values of the spin, the electric charge, the parities, but with different masses. Such particles are grouped into the kinds (or families, or dynasties) with members which are the particle generations. The commutator of fields for the kinds of particles can be presented as sum of the products of the commutators for one particle and the definite coefficients. The sums of these coefficients for all the generation equal zero. The sums of the products of these coefficients and the particle masses to some powers equal zero too, i.e., for these coefficients some relations exist. In consequence of these relations the commutators of the fields for the particle generations vanish on space-like intervals. Thus, the locality (the microcausality) is valid for the fields of the particle kinds. It is possible if the number of the generations is greater than two.


convergence of integrals; differential equations; generations of particles; microcausality principle; generations of particles; indefinite metrics

Full Text:



Bogolubov N.N., Shirkov D.V. Introduction to theory of quantized fields. – Moscow: Nauka, 1967. – 465p. (in Russian)

Schweber S.S. An introduction to relativistic quantum field theory. – N.Y.: Brandeis Univ. – Row, Peterson and Co. Evanston, Ill., 1961; Moscow: Izlftel’stvo Inostrannoj Literatury, 1963. – 843 p. (in Russian)

Bjorken J.D., Drell S.D. Relativistic quantum fields. Relativistic quantized fields. – Vol.2. – N.Y.: Mc Graw. Hill Book Company, 1965; Moscow: Nauka, 1978. – 408 p. (In Russian)

Itzykson C., Zuber J.-B. Quantum field theory. Vol. 1. – N.Y.: Mc Graw. – Hill Book Company; Moscow: Mir, 1984. – 448 p. (in Russian)

Barton G. Introduction to dispersion techniques in field theory. – New York, Amsterdam: Univ. of Sussex. W.A. Benjamin. Inc., 1965; Moscow: Atomizdat, 1968. – 392 p. (In Russian)

Gaziorowicz S. Elementary particle physics. – New York-London-Sydney: John Wilej & Sons Inc.; Moscow: Nauka, 1969. ‑ 743 p. (in Russian)

Akhiezer A.I., Peletminskij S.V. Theory of fundamental interactions. – Kiev: Naukova Dumka, 1993. – 570p. (in Russian)

Fikhtengolts M. G. Course of differential and integral calculus. Vol. 3. – Moskow: Nauka, 1966. – P.221. (in Russian)

Budak B.M., Fomin S.V. Multiple integrals and series. – Moskow: Nauka, 1967. – P.387 – 401. (in Russian)

Kulish Yu., Rybachuk E.V. Necessary generalization of Klein-Gordon and Dirac equations and existence of particle generations // Problems of Atomic Science and Technology. – 2012. – No.1 (77). – P. 16–20.

Kulish Yu. V., Rybachuk E. V. Divergences of integrals for Green functions and necessary existence of particle generations // The Journal of Kharkiv National University, physical series “Nuclei, Particles, Fields”. – 2011. – No. 955. – Iss.2(50). – P.4-14.

Kulish Yu.V. Elimination of singularities in causal Green functions for generalized Klein-Gordon and Dirac equations on light cone // EEJP. – 2016. – Vol. 3. – No. 3. – P. 73-83.

Whittaker E.T., Watson G.N. A course of modern analysis. Vol. 2. – Cambridge: University Press, 1927; Moscow: Gosizdat phys-math. Lit., 1963. – 215p.

Berezhnoj Yu.A., Gakh A.G. Functions of theoretical physics. – Kharkov: Karazin V.N. Kharkov National University, 2011. ‑ 124p. (In Russian)

Akhiezer A.I., Peletminskij S.V. Fields and fundamental interactions. – Kiev: Naukova Dumka, 1986. – 552p. (in Russian)

Kulish Yu.V. Classification of particles at arbitrary quantity of generations. I. Hadrons // EEJP. – 2016. – Vol. 3. – No. 4. – P. 22-33.


  • There are currently no refbacks.

Copyright (c) 2017 Yu. V. Kulish, E. V. Rybachuk