p-Adic mathematical physics and B. Dragovich research

  • I. Ya. Aref’eva
  • G. S. Djordjevic
  • A. Yu. Khrennikov
  • S. V. Kozyrev
  • Z. Rakic
  • I. V. Volovich
Short Communications


We present a brief review of some parts of p-adic mathematical physics related to the scientific work of Branko Dragovich on the occasion of his 70th birthday.

Key words

theoretical physics mathematical physics p-adic mathematical physics p-adic analysis p-adic genetic code adelic models ultrametrics 


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  1. 1.
    I. V. Volovich, “p-Adic string,” Class. Quant. Grav. 4, L83–L87 (1987).MathSciNetCrossRefGoogle Scholar
  2. 2.
    L. Brekke and P. G. O. Freund, “p-Adic numbers in physics,” Phys. Rep. 233, 1–66 (1993).MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Sci. Publ., Singapore, 1994).CrossRefMATHGoogle Scholar
  4. 4.
    A. Khrennikov, Information Dynamics in Cognitive, Psychological, Social and Anomalous Phenomena (Kluwer Acad. Publishers, Dordrecht, 2004).CrossRefMATHGoogle Scholar
  5. 5.
    A. Yu. Khrennikov and M. Nilsson, p-Adic Deterministic and Random Dynamics (Kluwer Acad. Publishers, Dordrecht, 2004).CrossRefGoogle Scholar
  6. 6.
    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V.Volovich, “On p-adicmathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 1–17 (2009) [arXiv:0904.4205 [math-ph]].MathSciNetCrossRefGoogle Scholar
  7. 7.
    I. Ya. Aref’eva, B. G. Dragovic and I. V. Volovich, “On the adelic string amplitudes,” Phys. Lett. B 209, 445–450 (1988).MathSciNetCrossRefGoogle Scholar
  8. 8.
    I. Ya. Aref’eva, B. G. Dragovic and I. V. Volovich, “Open and closed p-adic strings and quadratic extensions of number fields,” Phys. Lett. B 212, 283–291 (1988).MathSciNetCrossRefGoogle Scholar
  9. 9.
    I. Ya. Aref’eva, B. G. Dragovic and I. V. Volovich: “p-Adic superstrings,” Phys. Lett. B 214, 339–349 (1988).Google Scholar
  10. 10.
    B. Dragovich, “Zeta strings,” in Proc. SQS’07, Dubna, Russia (2007) [arXiv:hep-th/0703008].Google Scholar
  11. 11.
    B. Dragovich, “Zeta nonlocal scalar fields,” Theor. Math. Phys. 157 (3), 1671–1677 (2008) [arXiv:0804.4114 [hep-th]].MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    I. Ya. Aref’eva, B. G. Dragovic and I. V. Volovich, “On the p-adic summability of the anharmonic oscillator,” Phys. Lett. B 200, 512–514 (1988).MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. G. Dragovic, “p-Adic perturbation series and adelic summability,” Phys. Lett. B 256 (3,4), 392–39 (1991).MathSciNetCrossRefGoogle Scholar
  14. 14.
    B. G. Dragovic, “Power series everywhere convergent on R and Qp,” J. Math. Phys. 34 (3), 1143–1148 (1992) [arXiv:math-ph/0402037].CrossRefMATHGoogle Scholar
  15. 15.
    B. G. Dragovic, “On p-adic aspects of some perturbation series,” Theor. Math. Phys. 93 (2), 1225–1231 (1993).MathSciNetCrossRefGoogle Scholar
  16. 16.
    B. G. Dragovic, “Rational summation of p-adic series,” Theor. Math. Phys. 100 (3), 1055–1064 (1994).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    B. Dragovich and N. Z. Misic, “p-Adic invariant summation of some p-adic functional series,” p-Adic Numbers Ultrametric Anal. Appl. 6 (4), 275–283 (2014), [arXiv:1411.4195v1 [math.NT]].MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    B. Dragovich, A. Yu. Khrennikov and N. Z.Misic, “Summation of p-adic functional series in integer points,” accepted for publ. in Filomat, [arXiv:1508.05079v1 [math.NT]] (2015).Google Scholar
  19. 19.
    B. Dragovich, “Adelic model of harmonic oscillator,” Theor. Math. Phys. 101, 1404–1412 (1994) [arXiv:hepth/ 0402193].MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    B. Dragovich, “Adelic harmonic oscillator,” Int. J. Mod. Phys. A 10, 2349–2365 (1995) [arXiv:hepth/ 0404160].MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    B. Dragovich, Ya. Radyno and A. Khrennikov, “Distributions on adeles,” J. Math. Sci. 142 (3), 2105–2112 (2007).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    B. Dragovich, “On generalized functions in adelic quantum mechanics,” Integr. Transf. Spec. Func. 6, 197–203 (1998).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    G. S. Djordjevic and B. Dragovich, “p-Adic path integrals for quadratic actions,” Mod. Phys. Lett. A 12, 1455–1463 (1997) [arXiv:math-ph/0005026].MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    G. S. Djordjevic, B. Dragovich and Lj. Nesic, “Adelic path integrals for quadratic Lagrangians,” Infin. Dim. Anal. Quant. Prob. Relat. Top. 6, 179–195 (2003) [arXiv:hep-th/0105030].MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    I. V. Volovich, “Number theory as ultimate physical theory,” p-Adic Numbers Ultrametric Anal. Appl. 2 (1), 77–87 (2010).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    G. S. Djordjevic and B. Dragovich, “p-Adic and adelic harmonic oscillator with time-dependent frequency,” Theor. Math. Phys. 124, 1059–1067 (2000) [arXiv:quant-ph/0005027].MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    G. S. Djordjevic, B. Dragovich and Lj. Nesic, “p-Adic and adelic free relativistic particle,” Mod. Phys. Lett. A 14, 317–325 (1999) [arXiv:hep-th/0005216].MathSciNetCrossRefGoogle Scholar
  28. 28.
    I. Ya. Aref’eva, B. Dragovich, P. H. Frampton and I. V. Volovich, “Wave function of the Universe and p-adic gravity,” Int. J. Mod. Phys. A 6, 4341–4358 (1991).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    B. Dragovich, “Adelic wave function of the Universe,” Proc. Third A. Friedmann Int. Seminar on Gravitation and Cosmology, pp. 311–321 (Friedman Laboratory Publishing, St. Petersburg, 1995).Google Scholar
  30. 30.
    G. S. Djordjevic, B. Dragovich, Lj. Nesic and I. V. Volovich, “p-Adic and adelic minisuperspace quantum cosmology,” Int. J. Mod. Phys. A 17, 1413–1434 (2002) [arXiv:gr-qc/0105050].MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    B. Dragovich and A. Yu. Dragovich, “A p-adic model of DNA sequence and genetic code,” p-Adic Numbers Ultrametric Anal. Appl. 1, 34–41 (2009) [arXiv:q-bio.GN/0607018].MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    B. Dragovich and A. Dragovich, “p-Adic modelling of the genome and the genetic code,” Computer J. 53 (4), 432–441 (2010) [arXiv:0707.30.43 [q-bio.OT]].CrossRefMATHGoogle Scholar
  33. 33.
    I. Ya. Aref’eva, B. G. Dragovic and I.V. Volovich, “Extra time-like dimensions lead to a vanishing cosmological constant,” Phys. Lett. B 177, 357–360 (1986).MathSciNetCrossRefGoogle Scholar
  34. 34.
    I. Ya. Aref’eva, I. V. Volovich, B. G. Dragovic, “Spontaneous reduction in multidimensional (D=10, 11) supergravity theories with arbitrary signature” Theor. Math. Phys. 70 (3), 297–304 (1987).CrossRefGoogle Scholar
  35. 35.
    B. G. Dragovic, “On signature change in p-adic space-times,” Mod. Phys. Lett. A 6, 2301–2307 (1991).MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    B. G. Dragovic, P. H. Frampton and B. V. Urosević, “Classical p-adic space-time,” Mod. Phys. Lett. A 5, 1521–1528 (1990).MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    B. Dragović and B. Sazdović, “On the structure of ultraviolet divergences in the vacuum region of quantum electrodynamics,” J. Phys. A: Math. Gen. 14, 915–920 (1981).CrossRefGoogle Scholar
  38. 38.
    B. G. Dragovic, D. P. Mavlo and A. T. Filippov, “Investigation of solutions of the Dyson-Schwinger equation for electron propagator in quantum electrodynamics,” Fizika 10, 51–74 (1978).Google Scholar
  39. 39.
    B. Dragović, “On dynamical mass generation in quantum electrodynamics,” J. Phys. G: Nucl. Phys. 9, L1–L5 (1983).CrossRefGoogle Scholar
  40. 40.
    B. Dragović and B. Sazdović, “On the possibility of dynamical mass generation in axial electrodynamics,” J. Phys. G: Nucl. Phys. 8, 1637–1640 (1982).CrossRefGoogle Scholar
  41. 41.
    B. Dragovich and Z. Rakic, “Path integrals in noncommutative quantum mechanics,” Theor. Math. Phys. 140, 1299–1308 (2004) [arXiv:hep-th/0309204].MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    B. Dragovich and M. Dugić, “On decoherence in noncommutative plane with perpendicular magnetic field,” J. Phys. A:Math. Gen. 38, 6603–6611 (2006) [arXiv:quant-ph/0503163].MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    B. Dragovich, A. Khrennikov and D. Mihajlović, “Linear fractional p-adic and adelic dynamical systems,” Rep. Math. Phys. 60 (1), 55–68 (2007) [arXiv:math-ph/0612058].MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    B. Dragovich and A. Khrennikov, “p-Adic and adelic superanalysis,” Bulgarian J. Phys. 33 (s2), 159–173 (2006) [arXiv:hep-th/0512318].MathSciNetMATHGoogle Scholar
  45. 45.
    B. Dragovich and D. Joksimović, “On possible uses of p-adic analysis in econometrics,” Megatrend Rev. 4 (2), 5–16 (2007).Google Scholar
  46. 46.
    I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic, “Some cosmological solutions in a nonlocal modified gravity,” Filomat 29 (3), 619–628 (2013) [arXiv:1508.05583 [hep-th]].CrossRefMATHGoogle Scholar
  47. 47.
    B. Dragovich, “On nonlocal modified gravity and cosmology,” Springer Proc. Math. Stat. 111, 251–262 (2014) [arXiv:1508.06584 [hep-th]].CrossRefMATHGoogle Scholar
  48. 48.
    I. Dimitrijevic, B. Dragovich, J. Grujic, A. S. Koshelev and Z. Rakic, “Cosmology of modified gravity with non-local f(R),” [arXiv:1509.04254 [hep-th]] (2015).Google Scholar
  49. 49.
    I. Dimitrijevic, B. Dragovich, J. Stankovic, A. S. Koshelev and Z. Rakic, “On nonlocal modified gravity and its cosmological solutions,” Springer Proc. Math. Stat. 191, 35–51 (2016) [arXiv:1701.02090 [hep-th]].CrossRefGoogle Scholar
  50. 50.
    B. Dragovich, “p-Adic and adelic cosmology: p-adic origin of dark energy and dark matter,” in p-Adic Mathematical Physics, AIP Conf. Proc. 826 (2006) [arXiv:hep-th/0602044].Google Scholar
  51. 51.
    B. Dragovich, “Towards p-adic matter in the Universe,” Springer Proc. Math. Stat. 36, 13–24 (2013) [arXiv:1205.4409 [hep-th]].MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • I. Ya. Aref’eva
    • 1
  • G. S. Djordjevic
    • 2
  • A. Yu. Khrennikov
    • 3
  • S. V. Kozyrev
    • 1
  • Z. Rakic
    • 4
  • I. V. Volovich
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Faculty of Science and MathematicsNišSerbia
  3. 3.International Center forMathematical Modeling in Physics, Engineering, Economics, and Cognitive ScienceLinnaeus UniversityVäxjö-KalmarSweden
  4. 4.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

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