Quantization of Non-standard Hamiltonians and the Riemann Zeros

• Rami Ahmad El-Nabulsi ^[1]
  1. [1] Athens Institute for Education and Research
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 18, Nº 1, 2019, págs. 69-84
• Idioma: inglés
• DOI: 10.1007/s12346-018-0277-0
• Enlaces
  • Texto completo
• Resumen

  • 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+ξ/p) where ξ 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+ξ/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.

• Referencias bibliográficas
  • 1. Alonso, M.A.: Second order differential operators and their eigenfunctions, Talk given at Winter College on Fibre Optics, Fibre Lasers...
  • 2. Aschheim, R., Castro, C., Irwin, K.: The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes. Int. J....
  • 3. Bender, C.M., Boettcher, S.: Real spectra in non-hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)
  • 4. Berry, M.V., Keating, J.P.: The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41(2), 236–266 (1998)
  • 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...
  • 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....
  • 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)
  • 8. Carinena, J.F., Ranada, M.F., Santander, M.: Lagrangian formalism for nonlinear second-order Riccati Systems: one-dimensional integrability...
  • 9. Carinena, J.F., Nunez, J.F.: Geometric approach to dynamics obtained by deformation of Lagrangians. Nonlinear Dyn. 83(1), 457–461 (2016)
  • 10. Carinena, J.F., Nunez, J.F.: Geometric approach to dynamics obtained by deformation of timedependent Lagrangians. Nonlinear Dyn. 86(2),...
  • 11. Carinera, J.F.: Theory of singular Lagrangians. Fortschr. Phys. 38(9), 641–679 (1990)
  • 12. Cieslinski, J.L., Nikiciuk, T.: A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical...
  • 13. Cisneros-Parra, J.U.: On singular Lagrangians and Dirac’s method. Rev. Mex. Fis. 58, 61–68 (2012)
  • 14. Connes, A.: Formule de trace en géométrie non-commutative et hypothèse de Riemann. C R Acad. Sci. Paris 323, 1231–1236 (1996)
  • 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...
  • 16. EL-Nabulsi, R.A.: Non-linear dynamics with non-standard Lagrangians. Qual. Theory Dyn. Syst. 12(2), 273–291 (2013)
  • 17. El-Nabulsi, R.A.: Non-standard non-local-in-time Lagrangians in classical mechanics. Qual. Theory Dyn. Syst. 13(1), 149–160 (2014)
  • 18. El-Nabulsi, R.A.: Non-standard power-law Lagrangians in classical and quantum dynamics. Appl. Math. Lett. 43, 120–127 (2015)
  • 19. Edwards, H.M.: Riemann’s Zeta Function. Academic Press, New York (1974)
  • 20. Faria, C.F.M., Fring, A.: Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the...
  • 21. Figueira de Morisson Faria, C., Fring, A.: Time evolution of non-Hermitian Hamiltonian systems. J. Phys. A 39, 9269–9289 (2006)
  • 22. Figotin, A., Schenker, J.H.: Hamiltonian treatment of time dispersive and dissipative media within the linear response theory. J. Comput....
  • 23. Figotin, A., Schenker, J.H.: Hamiltonian structure for dispersive and dissipative dynamical systems. J. Stat. Phys. 128(4), 969–1056 (2007)
  • 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)
  • 25. Hardy, G.H., Littlewood, J.E.: The zeros of Riemann’s zeta-function on the critical line. Math. Zeitschrift 10, 283–317 (1921)
  • 26. Hojman, S., Urrutia, L.F.: On the inverse problem of the calculus of variations. J. Math. Phys. 22, 1896–1903 (1981)
  • 27. Knauf, A.: Number theory, dynamical systems and statistical mechanics. Rev. Math. Phys. 11(8), 1027–1060 (1999)
  • 28. Kurokawa, N.: Multiple zeta functions: an example. In: Zeta Functions in Geometry (Tokyo, 1990). Advanced Studies in Pure Mathematics,...
  • 29. Lapidus, M.L.: In search of the Riemann zeros, Strings, fractal membranes and noncommutative spacetimes. American Mathematical Society,...
  • 30. Liu, S., Guan, F., Wang, Y., Liu, C., Guo, Y.: The nonlinear dynamics based on the nonstandard Hamiltonians. Nonlinear Dyn. 88, 1229–1236...
  • 31. Montgomery, H.: Analytic Number Theory, vol. 24, pp. 181–193. American Mathematical Society, Providence, RI (1973)
  • 32. Musielak, Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A Math. Theor....
  • 33. Musielak, Z.E.: General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos, Solitons Fractals...
  • 34. Nucci, M.C.: Spectral realization of the Riemann zeros by quantizing H w(x)(p + l 2 p p): the Lie-Noether symmetry approach. J....
  • 35. Rajeev, S.: A canonical formulation of dissipative mechanics using complex-valued Hamiltonians. Ann. Phys. 322(3), 1541–1555 (2007)
  • 36. Riemann, B.: Uber die Anzahl der Primzahlen unter einer gegebenen Große, Monatsberichte der Berliner Akademie 1859, pp. 671–680. Berlin...
  • 37. de Rittis, R., Marmo, G., Platania, G., Scudellaro, P.: Inverse problem in classical mechanics: dissipative systems. Int. J. Theor. Phys....
  • 38. Saha, A., Talukdar, B.: Inverse variational problem for non-standard Lagrangians. Rep. Math. Phys. 3(3), 299–309 (2014)
  • 39. Sierra, G., Rodriguez-Laguna, J.: The H = xp model revisited and the Riemann zeros. Phys. Rev. Lett. 106, 200201–200204 (2011)
  • 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)
  • 42. Zhang, Y., Zhou, X.S.: Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians. Nonlinear Dyn. 84(2),...