Advertisement

Foundations of Physics

, Volume 44, Issue 4, pp 368–388 | Cite as

On Gravitational Effects in the Schrödinger Equation

  • M. D. PollockEmail author
Article

Abstract

The Schrödinger  equation for a particle of rest mass \(m\) and electrical charge \(ne\) interacting with a four-vector potential \(A_i\) can be derived as the non-relativistic limit of the Klein–Gordon  equation \(\left( \Box '+m^2\right) \varPsi =0\) for the wave function \(\varPsi \), where \(\Box '=\eta ^{jk}\partial '_j\partial '_k\) and \(\partial '_j=\partial _j -\mathrm {i}n e A_j\), or equivalently from the one-dimensional  action \(S_1=-\int m ds +\int neA_i dx^i\) for the corresponding point particle in the semi-classical approximation \(\varPsi \sim \exp {(\mathrm {i}S_1)}\), both methods yielding the equation \(\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi \) in Minkowski  space–time  , where \(\alpha ,\beta =1,2,3\) and \(\phi =-A_0\). We show that these two methods generally yield equations  that differ in a curved background  space–time  \(g_{ij}\), although they coincide when \(g_{0\alpha }=0\) if \(m\) is replaced by the effective mass \(\mathcal{M}\equiv \sqrt{m^2-\xi R}\) in both the Klein–Gordon  action \(S\) and \(S_1\), allowing for non-minimal coupling to the gravitational  field, where \(R\) is the Ricci scalar and \(\xi \) is a constant. In this case \(\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi \), where \(\phi ^{(\mathrm g)} =\sqrt{g_{00}}\) and \(\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} \), the correctness of the gravitational  contribution to the potential having been verified to linear order \(m\phi ^{(\mathrm g)} \) in the thermal-neutron beam interferometry experiment due to Colella et al. Setting \(n=2\) and regarding \(\varPsi \) as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time  coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div\({{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0\), where \({{\varvec{A}}}^{\alpha }=-A^{\alpha }\) and \({{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }\). The quantum-cosmological Schrödinger  (Wheeler–DeWitt) equation is also discussed in the \(\mathcal{D}\)-dimensional  mini-superspace idealization, with particular regard to the vacuum potential \(\mathcal V\) and the characteristics of the ground state, assuming a gravitational  Lagrangian  \(L_\mathcal{D}\) which contains higher-derivative  terms up to order \(\mathcal{R}^4\). For the heterotic superstring theory  , \(L_\mathcal{D}\) consists of an infinite series in \(\alpha '\mathcal{R}\), where \(\alpha '\) is the Regge slope parameter, and in the perturbative approximation \(\alpha '|\mathcal{R}| \ll 1\), \(\mathcal V\) is positive semi-definite for \(\mathcal{D} \ge 4\). The maximally symmetric ground state satisfying the field equations is Minkowski  space for \(3\le {\mathcal {D}}\le 7\) and anti-de Sitter  space for \(8 \le \mathcal {D} \le 10\).

Keywords

Superconductivity Schrödinger equation Curved space–time Quantum cosmology 

Notes

Acknowledgments

This paper was written at the University of Cambridge, Cambridge, England.

References

  1. 1.
    Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity (In Russian). Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)Google Scholar
  2. 2.
    Pollock, M.D.: On vacuum fluctuations and particle masses. Found. Phys. 42, 1300–1328 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Higgs, P.W.: Spontaneous symmetry breakdown without massless bosons. Phys. Rev. 145, 1156–1163 (1966)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Grib, A.A., Mostepanenko, V.M., Frolov, V.M.: Spontaneous breaking of gauge symmetry in a nonstationary isotropic metric. Teor. Mat. Fiz. 33, 42–53 (1977). [Theor. Math. Phys. 33, 869–876 (1977)]Google Scholar
  5. 5.
    Klein, O.: Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Phys. 37, 895–906 (1926)Google Scholar
  6. 6.
    Gordon, W.: Der Comptoneffekt nach der Schrödingerschen Theorie. Z. Phys. 40, 117–133 (1926)Google Scholar
  7. 7.
    Lämmerzahl, C.: A Hamilton operator for quantum optics in gravitational fields. Phys. Lett. A203, 12–17 (1995)ADSCrossRefGoogle Scholar
  8. 8.
    von Weyssenhoff, J.: Anschauliches zur Relativitätstheorie. Z. Phys. 95, 391–408 (1935)Google Scholar
  9. 9.
    Zel’manov, A.L.: Chronometric invariants and frames of reference in the general theory of relativity. Dokl. Akad. Nauk SSSR 107, 815–818 (1956). [Sov. Phys. Dokl. 1, 227 (1956)]Google Scholar
  10. 10.
    Pollock, M.D.: On the gravito-electromagnetic analogy. Acta Phys. Pol. B42, 1767–1796 (2011)CrossRefGoogle Scholar
  11. 11.
    Kramers, H.A.: On the application of Einstein’s theory of gravitation to a stationary field of gravitation. Proc. Kon. Ned. Akad. Wet. Amsterdam 23, 1052–1073 (1922)ADSGoogle Scholar
  12. 12.
    De Donder, Th.: La gravifique einsteinienne. Ann. de l’Obs. Royal de Belgique, 3. sér., t. I, Bruxelles: M. Hayez, 1922, pp. 73–268; Premiers compléments de la gravifique einsteinienne, pp. 317–355 (1922)Google Scholar
  13. 13.
    Lanczos, K.: Ein vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen. Phys. Zeit. 23, 537–539 (1922)Google Scholar
  14. 14.
    Lancius, K.: Zur Theorie der Einsteinschen Gravitationsgleichungen. Z. Phys. 13, 7–16 (1923)Google Scholar
  15. 15.
    Fock, V.: The Theory of Space, Time and Gravitation, 2nd edn. Pergamon Press, Oxford (1964)Google Scholar
  16. 16.
    Colella, R., Overhauser, A.W., Werner, S.A.: Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472–1474 (1975)ADSCrossRefGoogle Scholar
  17. 17.
    Greenberger, D.M., Overhauser, A.W.: Coherence effects in neutron diffraction and gravity experiments. Rev. Mod. Phys. 51, 43–78 (1979)ADSCrossRefGoogle Scholar
  18. 18.
    Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. Lond. A117, 610–624 (1928)ADSCrossRefGoogle Scholar
  19. 19.
    Dirac, P.A.M.: The quantum theory of the electron. Part II. Proc. R. Soc. Lond. A118, 351–361 (1928)ADSCrossRefGoogle Scholar
  20. 20.
    Fock, V.: Geometrisierung der Diracschen Theorie des Elektrons. Z. Phys. 57, 261–277 (1929)Google Scholar
  21. 21.
    Schrödinger, E.: Diracsches Elektron im Schwerefeld I. Sitz. Preuss. Akad. Wiss. Berlin, pp. 105–128 (1932)Google Scholar
  22. 22.
    Pollock, M.D.: On the Dirac equation in curved space–time. Acta Phys. Pol. B41, 1827–1846 (2010)MathSciNetGoogle Scholar
  23. 23.
    Weyl, H.: Elektron und Gravitation I. Z. Phys. 56, 330–352 (1929)Google Scholar
  24. 24.
    Pauli, W.: Über die Invarianz der Dirac’schen Wellengleichungen gegenüber Ähnlichkeitstransformationen des Linienelementes im Fall verschwindender Ruhmasse. Helv. Phys. Acta 13, 204–208 (1940)Google Scholar
  25. 25.
    Noether, E.: Invariante Variationsprobleme. Nachr. Kgl. Gesell. Wiss. Göttingen, Math-Phys. Kl., pp. 235–257 (1918)Google Scholar
  26. 26.
    Weinberg, S.: From BCS to the LHC. Int. J. Mod. Phys. A23, 1627–1635 (2008)Google Scholar
  27. 27.
    Duff, M.J.: In: Isham, C.J., Penrose, R., Sciama, D.W. (eds.) Inconsistency of Quantum Field Theory in Curved Space-Time, in Quantum Gravity 2, pp. 81–105. Clarendon Press, Oxford (1981)Google Scholar
  28. 28.
    Møller, C.: The Theory of Relativity, 2nd edn. Clarendon Press, Oxford (1972)Google Scholar
  29. 29.
    Wheeler, J.A.: Superspace and the nature of quantum geometrodynamics. In: DeWitt, C.M., Wheeler, J.A. (eds.) Battelle Rencontres, 1967 Lectures in Mathematics and Physics, pp. 242–307. Benjamin, New York (1968)Google Scholar
  30. 30.
    DeWitt, B.S.: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113–1148 (1967)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Pollock, M.D.: On the derivation of the Wheeler–DeWitt equation in the heterotic superstring theory. Int. J. Mod. Phys. A7, 4149–4165 (1992); Erratum: On the derivation of the Wheeler–DeWitt equation in the heterotic superstring theory A27, 1292005(E) (2012)Google Scholar
  32. 32.
    Pollock, M.D.: Chronometric invariance and string theory. Mod. Phys. Lett. A23, 797–813 (2008)ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    Pollock, M.D.: On the quartic higher-derivative gravitational terms in the heterotic superstring theory. Int. J. Mod. Phys. A21, 373–404 (2006); Erratum: On the quartic higher-derivative gravitational terms in the heterotic superstring theory A28, 1392001(E) (2013)Google Scholar
  34. 34.
    Bennett, C.L., et al.: First-year Wilkinson microwave anisotropy probe (WMAP) observations: preliminary maps and basic results. Astrophys. J. Suppl. Ser. 148, 1–27 (2003)ADSCrossRefGoogle Scholar
  35. 35.
    Komatsu, E., et al.: Five-year Wilkinson microwave anisotropy probe observations: cosmological interpretation. Astrophys. J. Suppl. Ser. 180, 330–376 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Pollock, M.D.: On the positivity of the gravitational potential in the quantum-cosmological Schrödinger equation. Int. J. Mod. Phys. D3, 569–578 (1994)ADSCrossRefMathSciNetGoogle Scholar
  37. 37.
    Pollock, M.D.: On the horizon hypothesis in quantum cosmology. Int. J. Mod. Phys. D5, 193–208 (1996)ADSCrossRefMathSciNetGoogle Scholar
  38. 38.
    Pollock, M.D.: On the thermodynamics of cosmic dust. Acta Phys. Pol. B42, 195–207 (2011); Erratum: On the thermodynamics of cosmic dust B43, 121(E) (2012)Google Scholar
  39. 39.
    Cai, R.-G., Kim, S.P.: First law of thermodynamics and Friedmann equations of Friedmann Robertson Walker universe. J. High Energy Phys. 0502, 050 (2005)Google Scholar
  40. 40.
    Cai, R.-G., Cao, L.-M., Hu, Y.-P.: Hawking radiation of an apparent horizon in a FRW universe. Class. Quantum Grav. 26, 155018 (2009)Google Scholar
  41. 41.
    Pollock, M.D.: The Wheeler–DeWitt equation for the heterotic superstring theory including terms quartic in the Riemann tensor. Int. J. Mod. Phys. D4, 305–326 (1995); Erratum: The Wheeler–DeWitt equation for the heterotic superstring theory including terms quartic in the Riemann tensor D21, 1292002(E) (2012)Google Scholar
  42. 42.
    Pollock, M.D.: On the superstring Hamiltonian in the Friedmann space–time. Int. J. Mod. Phys. D15, 845–868 (2006); Erratum: On the superstring Hamiltonian in the Friedmann space–time D22, 1392001(E)(2013)Google Scholar
  43. 43.
    Pollock, M.D.: On the quantum cosmology of the superstring theory including the effects of higher-derivative terms. Nucl. Phys. B324, 187–204 (1989)ADSCrossRefMathSciNetGoogle Scholar
  44. 44.
    Pollock, M.D.: Maximally symmetric superstring vacua. Acta Phys. Pol. B40, 2689–2701 (2009)Google Scholar
  45. 45.
    Boulware, D.G., Deser, S.: String-generated gravity models. Phys. Rev. Lett. 55, 2656–2660 (1985)ADSCrossRefGoogle Scholar
  46. 46.
    Pollock, M.D.: On the semi-classical approximation to the superstring theory. Int. J. Mod. Phys. A7, 6421–6430 (1992)ADSCrossRefMathSciNetGoogle Scholar
  47. 47.
    Lapchinsky, V.G., Rubakov, V.A.: Canonical quantization of gravity and quantum field theory in curved space-time. Acta Phys. Pol. B10, 1041–1048 (1979)ADSMathSciNetGoogle Scholar
  48. 48.
    Halliwell, J.J., Hawking, S.W.: Origin of structure in the Universe. Phys. Rev. D31, 1777–1791 (1985)ADSMathSciNetGoogle Scholar
  49. 49.
    Kiefer, C.: Quantum Gravity, 3rd edn. Oxford University Press, Oxford (2012)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.V. A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations