On Gravitational Effects in the Schrödinger Equation

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 $8 \le \mathcal {D} \le 10$.