Skip to main content
SpringerLink
Log in

Quantization of Non-standard Hamiltonians and the Riemann Zeros

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Relations between number theory, quantum mechanics and statistical mechanics are of interest to mathematicians and physicists since it was suggested that the zeros of the Riemann zeta function might be related to the spectrum of a self-adjoint quantum mechanical operator related to a one-dimensional Hamiltonian \( H = xp \) known as Berry–Keating–Connes Hamiltonian. However, this type of Hamiltonian is integrable and the classical trajectories of particles are not closed leading to a continuum spectrum. Recently, Sierra and Rodriguez-Laguna conjectured that the Hamiltonian \( H = x(p \,+\, {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) \) where \( \xi \) is a coupling constant with dimensions of momentum square is characterized by a quantum spectrum which coincides with the average Riemann zeros and contains closed periodic orbits. In this paper, we show first that the Sierra–Rodriguez-Laguna Hamiltonian may be obtained by means of non-standard singular Lagrangians and besides the Hamiltonians \( H = x(p + {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) \) and \( H(x,p) = px \) are not the only semiclassical Hamiltonians connected to the average Riemann zeros. We show the presence of a new Hamiltonian where its quantization revealed a number of interesting properties, in particular, the sign of a trace of the Riemann zeros.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Alonso, M.A.: Second order differential operators and their eigenfunctions, Talk given at Winter College on Fibre Optics, Fibre Lasers and Sensors, Abdus Salam Center for Theoretical Physics, 5–9 Feb, 2007

  2. Aschheim, R., Castro, C., Irwin, K.: The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes. Int. J. Geom. Meth. Mod. Phys. 14(6), 1750109–1750137 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bender, C.M., Boettcher, S.: Real spectra in non-hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berry, M.V., Keating, J.P.: The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41(2), 236–266 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berry, M.V., Keating, J.P.: H = xp and the Riemann zeros. In: Keating, J.P., Khmelnitskii, D.E, Lerner, I.V. (eds.) Supersymmetry and Trace Formulae: Chaos and Disorders, pp. 355–367. Plenum, New York (1998)

    Google Scholar 

  6. Berry, M.V., Keating, J.P.: A compact Hamiltonian with the same asymptotic mean spectral density as the Riemann zeros. J. Phys. A Math. Theor. 44, 285203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhaduri, R.K., Khare, A., Law, J.: Phase of the Riemann zeta function and the inverted harmonic oscillator. Phys. Rev. E 52, 486 (1995)

    Article  Google Scholar 

  8. Carinena, J.F., Ranada, M.F., Santander, M.: Lagrangian formalism for nonlinear second-order Riccati Systems: one-dimensional integrability and two-dimensional superintegrability. J. Math. Phys. 46, 062703–062721 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carinena, J.F., Nunez, J.F.: Geometric approach to dynamics obtained by deformation of Lagrangians. Nonlinear Dyn. 83(1), 457–461 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carinena, J.F., Nunez, J.F.: Geometric approach to dynamics obtained by deformation of time-dependent Lagrangians. Nonlinear Dyn. 86(2), 1285–1291 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carinera, J.F.: Theory of singular Lagrangians. Fortschr. Phys. 38(9), 641–679 (1990)

    Article  MathSciNet  Google Scholar 

  12. Cieslinski, J.L., Nikiciuk, T.: A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients. J. Phys. A Math. Gen. 43, 175205–1752222 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cisneros-Parra, J.U.: On singular Lagrangians and Dirac’s method. Rev. Mex. Fis. 58, 61–68 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Connes, A.: Formule de trace en géométrie non-commutative et hypothèse de Riemann. C R Acad. Sci. Paris 323, 1231–1236 (1996)

    MathSciNet  MATH  Google Scholar 

  15. Conrey, J.B.: More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. Reine Angew. Math. 399, 1–26 (1989)

    MathSciNet  MATH  Google Scholar 

  16. EL-Nabulsi, R.A.: Non-linear dynamics with non-standard Lagrangians. Qual. Theory Dyn. Syst. 12(2), 273–291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. El-Nabulsi, R.A.: Non-standard non-local-in-time Lagrangians in classical mechanics. Qual. Theory Dyn. Syst. 13(1), 149–160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. El-Nabulsi, R.A.: Non-standard power-law Lagrangians in classical and quantum dynamics. Appl. Math. Lett. 43, 120–127 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Edwards, H.M.: Riemann’s Zeta Function. Academic Press, New York (1974)

    MATH  Google Scholar 

  20. Faria, C.F.M., Fring, A.: Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time-dependent quantum mechanical formulation. Laser Phys. 17, 424–437 (2007)

    Article  Google Scholar 

  21. Figueira de Morisson Faria, C., Fring, A.: Time evolution of non-Hermitian Hamiltonian systems. J. Phys. A 39, 9269–9289 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Figotin, A., Schenker, J.H.: Hamiltonian treatment of time dispersive and dissipative media within the linear response theory. J. Comput. Appl. Math. 204, 199–208 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Figotin, A., Schenker, J.H.: Hamiltonian structure for dispersive and dissipative dynamical systems. J. Stat. Phys. 128(4), 969–1056 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gupta, K.S., Harikumar, E., de Queiroz, A.R.: A Dirac type xp-Model and the Riemann Zeros. Eur. Phys. Lett. 102, 10006 (2013)

    Article  Google Scholar 

  25. Hardy, G.H., Littlewood, J.E.: The zeros of Riemann’s zeta-function on the critical line. Math. Zeitschrift 10, 283–317 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hojman, S., Urrutia, L.F.: On the inverse problem of the calculus of variations. J. Math. Phys. 22, 1896–1903 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  27. Knauf, A.: Number theory, dynamical systems and statistical mechanics. Rev. Math. Phys. 11(8), 1027–1060 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kurokawa, N.: Multiple zeta functions: an example. In: Zeta Functions in Geometry (Tokyo, 1990). Advanced Studies in Pure Mathematics, vol. 21, pp 219–226. Kinokuniya, Tokyo (1992)

    Chapter  MATH  Google Scholar 

  29. Lapidus, M.L.: In search of the Riemann zeros, Strings, fractal membranes and noncommutative spacetimes. American Mathematical Society, Providence (2008)

    Book  MATH  Google Scholar 

  30. Liu, S., Guan, F., Wang, Y., Liu, C., Guo, Y.: The nonlinear dynamics based on the nonstandard Hamiltonians. Nonlinear Dyn. 88, 1229–1236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Montgomery, H.: Analytic Number Theory, vol. 24, pp. 181–193. American Mathematical Society, Providence, RI (1973)

    Book  Google Scholar 

  32. Musielak, Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A Math. Theor. 41, 055205–055222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Musielak, Z.E.: General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos, Solitons Fractals 42(15), 2645–2652 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nucci, M.C.: Spectral realization of the Riemann zeros by quantizing \( H = w(x)(p + {{l_{p}^{2} } \mathord{\left/ {\vphantom {{l_{p}^{2} } p}} \right. \kern-0pt} p}) \): the Lie-Noether symmetry approach. J. Phys. Conf. Ser. 482, 012032 (2014)

  35. Rajeev, S.: A canonical formulation of dissipative mechanics using complex-valued Hamiltonians. Ann. Phys. 322(3), 1541–1555 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Riemann, B.: Uber die Anzahl der Primzahlen unter einer gegebenen Große, Monatsberichte der Berliner Akademie 1859, pp. 671–680. Berlin (1860)

  37. de Rittis, R., Marmo, G., Platania, G., Scudellaro, P.: Inverse problem in classical mechanics: dissipative systems. Int. J. Theor. Phys. 22(10), 931–946 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Saha, A., Talukdar, B.: Inverse variational problem for non-standard Lagrangians. Rep. Math. Phys. 3(3), 299–309 (2014)

    Article  MATH  Google Scholar 

  39. Sierra, G., Rodriguez-Laguna, J.: The H = xp model revisited and the Riemann zeros. Phys. Rev. Lett. 106, 200201–200204 (2011)

    Article  Google Scholar 

  40. Sierra, G.: The Riemann zeros as spectrum and the Riemann hypothesis, arXiv:1601.01797

  41. Sierra, G.: A quantum mechanical model of the Riemann zeros. New J. Phys. 10, 033016 (2008)

    Article  MathSciNet  Google Scholar 

  42. Zhang, Y., Zhou, X.S.: Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians. Nonlinear Dyn. 84(2), 1867–1876 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous referees for their useful comments and valuable suggestions.

Author information

Authors and Affiliations

  1. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

Authors
  1. Rami Ahmad El-Nabulsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rami Ahmad El-Nabulsi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Nabulsi, R.A. Quantization of Non-standard Hamiltonians and the Riemann Zeros. Qual. Theory Dyn. Syst. 18, 69–84 (2019). https://doi.org/10.1007/s12346-018-0277-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-018-0277-0

Keywords

Search

Navigation