Geodesic
Developing a non-Hermitian algebraic theory with the $\gamma_5$-extension of mass
Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 1, pp. 124-139 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a non-Hermitian fermionic model with the $\gamma_5$-extension of mass: $m\to m_1+\gamma_5m_2$. We establish the relation between this theory and a geometric theory with the maximum mass in the quantum mechanical approximation. We propose a more detailed condition of $PT$-symmetry preservation in the theory. It implies segregating the initial domain of $PT$-symmetry preservation into subdomains corresponding to descriptions of standard and exotic particles. We calculate the operator $\mathcal C$ in the new scalar product in such a theory with a non-Hermitian Hamiltonian and describe some consequences of introducing this operator. We find the eigenvalues and eigenvectors of this Hamiltonian.
Mots-clés : maximum mass
Keywords: ferminonic sector, $PT$ symmetry, pseudo-Hermitian theory. 
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V. N. Rodionov; G. A. Kravtsova. Developing a non-Hermitian algebraic theory with the $\gamma_5$-extension of mass. Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 1, pp. 124-139. http://geodesic.mathdoc.fr/item/TMF_2015_182_1_a7/

[1] C. M. Bender, S. Boettcher, Phys. Rev. Lett., 80:24 (1998), 5243–5246, arXiv: physics/9712001 | DOI | MR | Zbl

[2] C. M. Bender, S. Boettcher, P. N. Meisinger, J. Math. Phys., 40:5 (1999), 2201–2229, arXiv: quant-ph/9809072 | DOI | MR | Zbl

[3] A. Mostafazadeh, Internat. J. Geom. Meth. Modern Phys., 7:7 (2010), 1191–1306, arXiv: 0810.5643 | DOI | MR | Zbl

[4] A. Khare, B. P. Mandal, Phys. Lett. A, 272:1–2 (2000), 53–56, arXiv: quant-ph/0006126 | DOI | MR | Zbl

[5] G. Lévai, M. Znojil, Modern Phys. Lett. A, 16:30 (2001), 2273, arXiv: hep-th/0111213 | DOI | MR | Zbl

[6] A. Mostafazadeh, J. Phys. A: Math. Gen., 38:29 (2005), 6557–6569 | DOI | MR | Zbl

[7] C. M. Bender, D. C. Brody, J.-H. Chen, H. F. Jones, K. A. Milton, M. C. Ogilvie, Phys. Rev. D, 74:2 (2006), 025016, 10 pp., arXiv: hep-th/0605066 | DOI

[8] C. M. Bender, K. Besseghir, H. F. Jones, X. Yin, J. Phys. A: Math. Theor., 42:35 (2009), 355301, 10 pp., arXiv: 0906.1291 | DOI | MR | Zbl

[9] A. Khare, B. P. Mandal, Pramana J. of Physics, 73:2 (2009), 387–395 | DOI

[10] P. Dorey, C. Dunning, R. Tateo, J. Phys. A: Math. Gen., 34:28 (2001), 5679–5704, arXiv: hep-th/0103051 | DOI | MR | Zbl

[11] C. M. Bender, D. C. Brody, H. F. Jones, Phys. Rev. D, 70:2 (2004), 025001, 19 pp., arXiv: ; Erratum 71:4 (2005), 049901, 1 pp. hep-th/0402183 | DOI | DOI

[12] C. M. Bender, S. F. Brandt, J.-H. Chen, Q. Wang, Phys. Rev. D, 71:6 (2005), 065010, 7 pp., arXiv: hep-th/0412316 | DOI

[13] C. M. Bender, H. F. Jones, R. J. Rivers, Phys. Lett. B, 625:3–4 (2005), 333–340, arXiv: hep-th/0508105 | DOI | MR | Zbl

[14] C. M. Bender, D. C. Brody, H. F. Jones, Phys. Rev. Lett., 89:27 (2002), 270401, 4 pp. ; Erratum 92:11 (2004), 119902, 1 pp. | DOI | MR | Zbl | DOI

[15] C. M. Bender, J. Brod, A. Refig, M. E. Reuter, Phys. A: Math. Gen., 37:43 (2004), 10139–10165, arXiv: quant-ph/0402026 | DOI | MR | Zbl

[16] A. Mostafazadeh, J. Math Phys., 43:1 (2002), 205–214, arXiv: math-ph/0107001 | DOI | MR | Zbl

[17] A. Mostafazadeh, A. Batal, J. Phys. A: Math. Gen., 37:48 (2004), 11645–11679, arXiv: quant-ph/0408132 | DOI | MR | Zbl

[18] A. Mostafazadeh, J. Phys. A: Math. Theor., 36:25 (2003), 7081–7091, arXiv: quant-ph/0304080 | DOI | MR | Zbl

[19] M. Mostafazadeh, Class. Quantum Grav., 20:1 (2003), 155–171, arXiv: math-ph/0209014 | DOI | MR | Zbl

[20] M. Mostafazadeh, Ann. Phys., 309:1 (2004), 1–48, arXiv: gr-qc/0306003 | DOI | MR | Zbl

[21] M. Mostafazadeh, F. Zamani, Ann. Phys., 321:9 (2006), 2183–2209, arXiv: ; 2210–2241, arXiv: quant-ph/0602151quant-ph/0602161 | DOI | MR | Zbl | DOI | Zbl

[22] M. Mostafazadeh, Internat. J. Modern Phys. A, 21:12 (2006), 2553–2572 | DOI | MR | Zbl

[23] F. Zamani, M. Mostafazadeh, J. Math. Phys., 50:5 (2009), 052302, 35 pp., arXiv: 0805.1651 | DOI | MR | Zbl

[24] V. G. Kadyshevsky, Nucl. Phys. B, 141:4 (1978), 477–496 ; “Fundamental length hypothesis in a Gauge theory context”, Proceedings of International Integrative Conference on Group Theory and Mathematical Physics (Austin, Texas, September 11–16, 1978), Lecture Notes in Physics, 94, eds. W. Beiglbock, A. Bohm, E. Takasugi, Springer, Berlin, 1979, 114–124 ; В. Г. Кадышевский, ЭЧАЯ, 11:1 (1980), 5–39 | DOI | MR | DOI | MR

[25] M. A. Markov, Progr. Theoret. Phys. Suppl., Extra Number (1965), 85–95; М. А. Марков, ЖЭТФ, 51:3 (1967), 878–890

[26] V. G. Kadyshevsky, M. D. Mateev, V. N. Rodionov, A. S. Sorin, Towards a maximal mass model, arXiv: 0708.4205

[27] V. G. Kadyshevskii, M. D. Mateev, V. N. Rodionov, A. S. Sorin, Dokl. RAN, 408:4 (2006), 465–468, arXiv: hep-ph/0512332 | DOI | MR

[28] V. G. Kadyshevskii, EChAYa, 29:3 (1998), 563–576 | DOI

[29] V. G. Kadyshevsky, M. D. Mateev, Phys. Lett. B, 106:1–2 (1981), 139–142 | DOI

[30] V. G. Kadyshevsky, M. D. Mateev, Nuovo Cimento A, 87:3 (1985), 324–349 | DOI

[31] M. V. Chizhov, A. D. Donkov, V. G. Kadyshevsky, M. D. Mateev, Nuovo Cimento A, 87 (1985), 350–372 | DOI

[32] M. V. Chizhov, A. D. Donkov, V. G. Kadyshevsky, M. D. Mateev, Nuovo Cimento A, 87:4 (1985), 373–396 | DOI

[33] V. G. Kadyshevsky, D. V. Fursaev, JINR Rapid Commun., 1992, no. 6, 5–8

[34] R. M. Ibadov, V. G. Kadyshevskii, O preobrazovaniyakh supersimmetrii v teorii polya s fundamentalnoi massoi, preprint OIYaI R2-86-835, OIYaI, Dubna, 1986

[35] V. G. Kadyshevskii, D. V. Fursaev, O kiralnykh fermionnykh polyakh pri vysokikh energiyakh, preprint OIYaI R2-87-913, OIYaI, Dubna, 1987; Докл. АН СССР, 306:4 (1989), 856–858 | MR

[36] V. G. Kadyshevsky, V. N. Rodionov, Phys. Part. Nucl., 36:1 (2005), S34–S37

[37] V. N. Rodionov, PT-symmetric pseudo-Hermitian relativistic quantum mechanics with maximal mass, arXiv: 1207.5463

[38] V. N. Rodionov, Exactsolutions for non-Hermitian Dirac–Pauli equation in an intensive magnetic field, arXiv: 1406.0383

[39] I. M. Ternov, V. R. Khalilov, V. N. Rodionov, Vzaimodeistvie zaryazhennykh chastits s silnym elektromagnitnym polem, Izd-vo Mosk. un-ta, M., 1982

[40] I. P. Volobuev, V. G. Kadyshevskii, M. D. Mateev, M. R. Mir-Kasimov, TMF, 40:3 (1979), 363–372 | DOI | MR