Foundations of Physics

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

On Gravitational Effects in the Schrödinger Equation

  • M. D. PollockEmail author


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 \(8 \le \mathcal {D} \le 10\).


Superconductivity Schrödinger equation Curved space–time Quantum cosmology 



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


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© Springer Science+Business Media New York 2014

Authors and Affiliations

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

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