Advertisement

The 3D Perturbed Schrödinger Hamiltonian in a Friedmann Flat Spacetime Testing the Primordial Universe in a Non Commutative Spacetime

  • S. Fassari
  • F. Rinaldi
  • S. ViaggiuEmail author
Article

Abstract

In this paper we adapt the mathematical machinery presented in Albeverio et al. (Nanosystem 8(2), 153, 2017) to get, by means of the Laplace-Beltrami operator, the discrete spectrum of the Hamiltonian of the Schrödinger operator perturbed by an attractive 3D delta interaction in a Friedmann flat universe. In particular, as a consequence of the treatment in Albeverio et al. (Nanosystem 8(2), 153, 2017) suitable for a Minkowski spacetime, the discrete spectrum consisting only of the ground state and the first excited state in the above-mentioned cosmic framework can be regained. Thus, the coupling constant λ must be chosen as a function of the cosmic comoving time t as λ/a2(t), with λ being the one of the Hamiltonian studied in the aforementioned article. In this way we can introduce a time dependent delta interaction which is relevant in a primordial universe, where a(t) → 0 and becomes negligible at late times, with a(t) >> 1. We investigate, with such a model, quantum effects provided by point interactions in a strong gravitational regime near the big bang. In particular, as a physically interesting application, we present a method to depict, in a semi-classical approximation, a test particle in a (non commutative) quantum spacetime where, thanks to Planckian effects, the initial classical singularity disappears and, as a consequnce, a ground state with negative energy emerges. Conversely, in a scenario where the scale factor a(t) follows the classical trajectory, this ground state is unstable and thus must be physically ruled out.

Keywords

Schrödinger operators Friedmann universe Point interactions Non commutative quantum spacetime 

Mathematical Subject Classification (2010)

83C65 83F05 58B34 34L10 34L40 35J10 35P15 47A10 

Notes

References

  1. 1.
    Albeverio, S., Fassari, S., Rinaldi, F.: The behaviour of the three-dimensional Hamiltonian −Δ + λ[δ(x + x 0) + δ(xx 0)] as the distance between the two centres vanishes. Nanosystem 8(2), 153 (2017)Google Scholar
  2. 2.
    Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge University Press, Cambridge (1982)CrossRefGoogle Scholar
  3. 3.
    Shestakova, T.P.: Is the Wheeler-DeWitt equation more fundamental than the Schrödinger equation? Int. J. Mod. Phys. D 27, 1841004 (2018)ADSCrossRefGoogle Scholar
  4. 4.
    Fassari, S., Rinaldi, F., Viaggiu, S.: The spectrum of the Schrödinger Hamiltonian for trapped particles in a cylinder with a topological defect perturbed by two attractive delta interactions. Int. J. Geom. Meth. Mod. Phys. 15(8), 1850135 (2018)CrossRefGoogle Scholar
  5. 5.
    Albeverio, S., Fassari, S., Rinaldi, F.: The discrete spectrum of the spinless Salpeter Hamiltonian perturbed by δ-interactions. J. Phys. A: Math. Theor. 48, 185301 (2015)ADSCrossRefGoogle Scholar
  6. 6.
    Albeverio, S., Fassari, S., Rinaldi, F.: Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II. Nanosyst.: Phys. Chem. Math. 7(5), 803 (2016)zbMATHGoogle Scholar
  7. 7.
    Albeverio, S., Fassari, S., Rinaldi, F.: Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin. Nanosyst.: Phys. Chem. Math. 7(2), 268 (2016)zbMATHGoogle Scholar
  8. 8.
    Albeverio, S., Fassari, S., Rinaldi, F.: A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ-interaction centred at the origin: double degeneracy and level crossing. J. Phys. A: Math. Theor. 46, 385305 (2013)ADSCrossRefGoogle Scholar
  9. 9.
    Belloni, M., Robinett, R.W.: The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics. Phys. Rep. 540, 25 (2014)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Pethick, J., Smith, H.: Bose Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  11. 11.
    Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187 (1995)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Doplicher, S.: Space-time and fields: a quantum texture. In: Karpacz, New Developments in Fundamental Interaction Theories, pp. 204–213 (2001). arXiv:hep-th/0105251
  13. 13.
    Doplicher, S., Fredenhagen, K., Roberts, J.E.: Spacetime quantization induced by classical gravity. Phys. Lett. B 331, 39 (1994)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bahns, D., Doplicher, S., Morsella, G., Piacitelli, G.: Advances in Algebraic Quantum Field Theory. In: Quantum spacetime and algebraic quantum field theory, pp. 289–330. Springer, Berlin (2015)Google Scholar
  15. 15.
    Tomassini, L., Viaggiu, S.: Physically motivated uncertainty relations at the Planck length for an emergent noncommutative spacetime. Class. Quantum Grav. 28, 075001 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Tomassini, L., Viaggiu, S.: Building noncommutative spacetimes at the Planck length for Friedmann flat cosmologies. Class. Quantum Grav. 31, 185001 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Viaggiu, S.: Entropy, energy and temperature-length inequality for Friedmann universes. Int. J. Mod. Phys. D 3, 1650033 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Viaggiu, S.: Planckian corrections to the Friedmann flat equations from thermodynamics at the apparent horizon. Mod. Phys. Lett A 31(4), 1650016 (2016)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Bilic, N.: Randall-Sundrum versus holographic cosmology. Phys. Rev. D 93, 066010 (2016)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Piacitelli, G.: Twisted covariance as a non-invariant restriction of the fully covariant DFR model. Commun. Math. Phys. 295, 701 (2009)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Ashtekar, A.: Singularity resolution in loop quantum cosmology: a brief overview. J. Phys. Conf. Ser. 189, 012003 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di Fisica Nucleare, Subnucleare e delle RadiazioniUniversitá degli Studi Guglielmo MarconiRomeItaly
  2. 2.CERFIMLocarnoSwitzerland
  3. 3.Università di Roma “Tor Vergata”RomaItaly
  4. 4.INFN, Sezione di NapoliComplesso Universitario di Monte S. AngeloNapoliItaly

Personalised recommendations