LOCALITY OF QUANTIZED SCALAR FIELDS FOR GENERATIONS OF PARTICLES
Keywords:
convergence of integrals, differential equations, generations of particles, microcausality principle, indefinite metrics
Abstract
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.Downloads
Download data is not yet available.
References
1. Bogolubov N.N., Shirkov D.V. Introduction to theory of quantized fields. – Moscow: Nauka, 1967. – 465p. (in Russian)
2. 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)
3. 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)
4. Itzykson C., Zuber J.-B. Quantum field theory. Vol. 1. – N.Y.: Mc Graw. – Hill Book Company; Moscow: Mir, 1984. – 448 p. (in Russian)
5. 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)
6. Gaziorowicz S. Elementary particle physics. – New York-London-Sydney: John Wilej & Sons Inc.; Moscow: Nauka, 1969. ‑ 743 p. (in Russian)
7. Akhiezer A.I., Peletminskij S.V. Theory of fundamental interactions. – Kiev: Naukova Dumka, 1993. – 570p. (in Russian)
8. Fikhtengolts M. G. Course of differential and integral calculus. Vol. 3. – Moskow: Nauka, 1966. – P.221. (in Russian)
9. Budak B.M., Fomin S.V. Multiple integrals and series. – Moskow: Nauka, 1967. – P.387 – 401. (in Russian)
10. 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.
11. 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.
12. 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.
13. Whittaker E.T., Watson G.N. A course of modern analysis. Vol. 2. – Cambridge: University Press, 1927; Moscow: Gosizdat phys-math. Lit., 1963. – 215p.
14. Berezhnoj Yu.A., Gakh A.G. Functions of theoretical physics. – Kharkov: Karazin V.N. Kharkov National University, 2011. ‑ 124p. (In Russian)
15. Akhiezer A.I., Peletminskij S.V. Fields and fundamental interactions. – Kiev: Naukova Dumka, 1986. – 552p. (in Russian)
16. Kulish Yu.V. Classification of particles at arbitrary quantity of generations. I. Hadrons // EEJP. – 2016. – Vol. 3. – No. 4. – P. 22-33.
2. 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)
3. 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)
4. Itzykson C., Zuber J.-B. Quantum field theory. Vol. 1. – N.Y.: Mc Graw. – Hill Book Company; Moscow: Mir, 1984. – 448 p. (in Russian)
5. 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)
6. Gaziorowicz S. Elementary particle physics. – New York-London-Sydney: John Wilej & Sons Inc.; Moscow: Nauka, 1969. ‑ 743 p. (in Russian)
7. Akhiezer A.I., Peletminskij S.V. Theory of fundamental interactions. – Kiev: Naukova Dumka, 1993. – 570p. (in Russian)
8. Fikhtengolts M. G. Course of differential and integral calculus. Vol. 3. – Moskow: Nauka, 1966. – P.221. (in Russian)
9. Budak B.M., Fomin S.V. Multiple integrals and series. – Moskow: Nauka, 1967. – P.387 – 401. (in Russian)
10. 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.
11. 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.
12. 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.
13. Whittaker E.T., Watson G.N. A course of modern analysis. Vol. 2. – Cambridge: University Press, 1927; Moscow: Gosizdat phys-math. Lit., 1963. – 215p.
14. Berezhnoj Yu.A., Gakh A.G. Functions of theoretical physics. – Kharkov: Karazin V.N. Kharkov National University, 2011. ‑ 124p. (In Russian)
15. Akhiezer A.I., Peletminskij S.V. Fields and fundamental interactions. – Kiev: Naukova Dumka, 1986. – 552p. (in Russian)
16. Kulish Yu.V. Classification of particles at arbitrary quantity of generations. I. Hadrons // EEJP. – 2016. – Vol. 3. – No. 4. – P. 22-33.
Published
2017-12-15
Cited
How to Cite
Kulish, Y. V., & Rybachuk, E. V. (2017). LOCALITY OF QUANTIZED SCALAR FIELDS FOR GENERATIONS OF PARTICLES. East European Journal of Physics, 4(4), 4-11. https://doi.org/10.26565/2312-4334-2017-4-01
Section
Original Papers
Copyright (c) 2017 Yu. V. Kulish, E. V. Rybachuk
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
