[Paper Review] Quantum systems in weak gravitational fields

G. Papini|ArXiv.org|Oct 11, 2001

Relativity and Gravitational Theory1 references4 citations

TL;DR

This paper derives fully covariant wave equations for quantum systems in weak gravitational and inertial fields, showing that quantum phases and spin-gravity coupling arise consistently with general relativity. It demonstrates that these effects—particularly spin-rotation coupling and gravitational phase shifts—can be tested in high-precision interferometry, with implications for neutrino oscillations and violations of the equivalence principle down to $10^{-13}$ cm scales.

ABSTRACT

Fully covariant wave equations predict the existence of a class of inertial-gravitational effects that can be tested experimentally. In these equations inertia and gravity appear as external classical fields, but, by conforming to general relativity, provide very valuable information on how Einstein's views carry through in the world of the quantum.

Motivation & Objective

• To derive quantum wave equations (Schrödinger, Klein-Gordon, Maxwell, Dirac) in weak gravitational fields using general covariance.
• To investigate how inertia and gravity induce quantum phases and spin-gravity coupling in a manifestly covariant framework.
• To assess the experimental feasibility of testing general relativity in quantum systems using high-precision interferometers.
• To explore the role of spin-rotation coupling in precision tests of fundamental symmetries and neutrino oscillations.
• To examine how violations of the weak equivalence principle may arise from non-universal coupling of rotation to matter, affecting particle helicity and neutrino behavior.

Proposed method

• Uses the weak field approximation (WFA) to expand the metric tensor as $g_{\mu\nu} \simeq \eta_{\mu\nu} + \gamma_{\mu\nu}$, enabling first-order perturbative treatment of gravitational and inertial fields.
• Applies the action principle with $\hbar = c = 1$ to derive the Hamiltonian for the Schrödinger equation, incorporating $\gamma_{\mu\nu}$ as perturbations.
• Derives exact first-order solutions for quantum phases in Klein-Gordon, Maxwell, and Dirac equations via covariant wave equations and path-ordered integrals.
• Introduces a phase factor $\Phi_g$ analogous to Berry’s phase, derived from curvature and torsion terms in the metric perturbation.
• Models spin-gravity coupling through the Dirac equation in curved spacetime, identifying the Mashhoon effect as a key mechanism.
• Analyzes neutrino oscillations in gravitational fields by solving coupled equations with effective potentials from $\gamma_{\mu\nu}$ and $\varphi$, including MSW-like terms.

Experimental results

Research questions

• RQ1 How do weak gravitational and inertial fields induce quantum phases in relativistic and non-relativistic wave functions?
• RQ2 To what extent do spin-gravity and spin-rotation couplings violate the weak equivalence principle and affect particle dynamics?
• RQ3 Can gravitational phase shifts in interferometers test general relativity at sub-10^{-11} cm scales?
• RQ4 What role does the Mashhoon effect play in the anomalous magnetic moment of muons and helicity oscillations?
• RQ5 Can small violations of the equivalence principle ($\Delta\alpha \sim 10^{-14}$) lead to observable neutrino oscillations even when $\Delta m^2 = 0$?

Key findings

• Quantum phases induced by gravity and inertia are derived exactly to first order in the WFA for Klein-Gordon, Maxwell, and Dirac equations, with solutions expressed as path-ordered phase factors.
• The Schrödinger equation in weak gravitational fields acquires effective terms $m\gamma_{0i}$ and $\gamma_{00}/2$, consistent with general covariance and experimental data down to $\sim 10^{-8}$ cm for neutrons.
• Spin-rotation coupling leads to the Mashhoon effect, which contributes to the anomalous magnetic moment of muons and extends the validity of the Dirac equation to $\sim 2 \times 10^{-13}$ cm.
• Neutrino oscillations can occur even when $\Delta m^2 = 0$ if $\alpha_1 \neq \alpha_2$, due to gravitational coupling, with oscillation frequency $\omega = \frac{\Delta m^2}{4E} - \frac{E\varphi \Delta\alpha}{2}$.
• A gravitational redshift correction and the Lense-Thirring effect are testable with atomic and molecular interferometers, potentially reaching $10^{-9} - 10^{-11}$ cm sensitivity.
• Violations of the weak equivalence principle with $\Delta\alpha \sim 10^{-14}$ can generate observable neutrino oscillations comparable in magnitude to the MSW effect, even without mass splitting.