CHRONOLOGICAL PRODUCTS, ENERGY-MOMENTUM TENSORS OF SCALAR FIELDS FOR PARTICLE GENERATIONS, AND INDEFINITE METRICS
The solutions of generalized Klein-Gordon equations are considered. The generalizations of the Klein-Gordon equation allow one to derive convergent integrals for the Green functions of these equations. The generalized equations are presented as products of the operators for the Klein-Gordon equation with different masses. The solutions of derived homogeneous equations (total fields) 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 (families, dynasties) with members which are the generations. The chronological products of the scalar fields for kinds of particles are obtained at arbitrary quantity of the generations. These chronological products are the causal Green functions of generalized Klein-Gordon equations. The Lagrangians for the generalized Klein-Gordon equations of arbitrary order are derived. These Lagrangians are used to obtain the energy-momentum tensors for the particle kinds at arbitrary quantity of generations. It is shown that the generalized Hamiltonians (for total fields) have got positive eigenvalues for all the generations. These results are derived with the use of the indefinite metrics.
2. Schweber S.S. An introduction to relativistic quantum field theory. – Brandeis Univ. – Row, Peterson and Co. Evanston, Ill., N.Y., 1961; Izlftel’stvo Inostrannoj Literatury. – Moscow, 1963. – 843p. (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. – 408p. (in Russian)
4. Itzykson C, Zuber J.-B. Quantum field theory. Vol. 1. – N.Y.: Mc Graw-Hill Book Company; Moscow: Mir, 1984. – 448p. (in Russion)
5. Barton G. Introduction to dispersion techniques in field theory. – New York, Amsterdam: Univ. of Sussex. W.A. Benjamin. Inc., 1965; Moscow: Atomizdat, 1968. – 392p. (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., Peletminskii S.V. Fields and fundamental interactions. – Kiev: Naukova Dumka, 1986. – 552p. (in Russian)
8. Akhiezer A.I., Peletminskii S.V. Theory of fundamental interactions. – Kiev: Naukova Dumka, 1993. – 570p. (in Russian)
9. Kulish Yu. V., Rybachuk E.V. Divergences of integrals for Green functions and necessary existence of particle generations // Journal of Kharkiv National Univ. – 2011. – No.955. – Iss. 2(50). – P.4-14.
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. Mathews J, Walker R.L. Mathematical methods of physics. - Calif. Inst. of Tech. – NewYork-Amsterdam: W.A. Benjamin Inc, 1964; Moscow: Anomizdat, 1972. – 400p. (in Russian)
12. Kulish Yu. V., Rybachuk E.V. Locality of quantized scalar fields for generations of particles // EEJP. – 2017. – Vol. 4. – No. 4. – P. 4-11.
13. Kulish Yu.V. Classification of particles at arbitrary quantity of generations. I. Hadrons // EEJP. – 2016. – Vol. 3. – No. 4. – P. 22-33.
14. Korn G.A., Korn T.M. Mathematical handbook for scientists and engineers. Definitions, theorems and formulas for reference and review. – New York, San Francisco, Toronto, London, Sydney: Mc Graw Book Company, 1968; Moscow: Nauka, 1978. – P. 34. (in Russian)
15. Esl’sgol’ts L.E. Variational calculus. – Moscow: Gostechizdat, 1952. (in Russian)
16. Myshkis A.D. Mathematics. Special courses for higher technical schools. – Moscow: Nauka, 1971. – 632p. (in Russian)
17. Nagy K.L. State vector spaces with indefinite metric in quantum field theory. – Budapest: Akademial Kiado, 1966; State vector spaces with indefinite metric in quantum field theory. Library of “Mathematics” Collection. – Moscow: Mir, 1969. – 136p. (in Russian)
18. Whittaker E.T., Watson G.N. A course of modern analysis. Vol. 2. – Cambridge: University Press, 1927; Moscow: Gosudarstvennoe izdatel’stvo fisiko-matematicheskoj literatuty, 1963. – P.215.
19. Berezhnoj Yu.A., Gakh A.G. Functions of theoretical physics. – V.N. Karazin Kharkov National University. – Kharkov. – 2011. – 124p. (in Russian)
20. 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.
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 acknowledgement 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 acknowledgement 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).