Gravitation and quantummechanical localization of macroobjects (1412.0201v1)

Abstract: We propose nonlinear Schr\"odinger equation with gravitational self-interacting term. The separability conditions of Bialynicki-Birula are satisfied in asymptotic sense. Solitonlike solutions were found.

• The paper demonstrates that integrating gravitational interactions into the Schrödinger framework yields solitonlike solutions for stable macroscopic localization.
• It develops a nonlinear integro-differential equation that meets the Bialynicki-Birula separability condition under asymptotic gravitational limits.
• Results include quantitative predictions for wavepacket spread, offering a potential bridge between quantum mechanics and gravitational physics.

An Analysis of Nonlinear Schrödinger Equations and Gravitational Localization of Macroobjects

This paper presents a theoretical investigation aiming to address the localization of macroscopic objects within the framework of quantum mechanics. The primary focus is the incorporation of gravitational interactions into a nonlinear Schrödinger equation to potentially offer stable, solitonlike solutions that maintain the localization of macroscopic entities. The paper addresses the classical observation that a macroscopic object's center of mass cannot possess a localized stationary wavefunction due to quantum mechanical spreading.

Key Contributions

The author introduces a nonlinear interaction term within the Schrödinger equation using Newtonian gravity to tackle the challenges of wavepacket spreading for macroobjects. The nonlinear integro-differential equation derived in this framework shows properties conducive to stationary, localized solutions, identified as solitonlike, under certain conditions.

1. Nonlinear Quantum Mechanics: The paper extends quantum mechanics by integrating gravitational interaction, inherently breaking the conventional linear framework. This approach attempts to reconcile quantum mechanics and gravity — domains not yet unified satisfactorily in contemporary physics.
2. Derivation and Analysis: The integration of gravitational effects within Schrödinger's equation leads to a form that respects the Bialynicki-Birula separability condition in an asymptotic sense, showing promise for consistent theoretical behavior under conditions where gravitational interactions between components are negligible.
3. Soliton Solutions: A significant result is the formulation of solitonlike solutions for the equation derived. These solutions suggest the possibility of describing macroscopic objects as having stable localized wavepackets, a hypothesis that deviates from traditional quantum expectations.
4. Energy Minimization and Localization Width: The analysis provides quantitative expressions for the width of the wavefunction's spread. For a pointlike macroscopic object, the relation $a \approx \hslash/(GM)$ is derived, specifying the wavepacket's natural uncertainty in position.
5. Extended Objects: Extending the model to non-pointlike objects, the paper elucidates how the interaction potential modifies the energy functional, thereby adjusting the spatial spread. The derived expression $a_0(R) \approx a_0 \cdot R^{3/4}$ stands in contrast to traditional scaling laws, marking a significant prediction requiring further exploration.

Implications and Future Directions

The theoretical framework proposed challenges the prevailing delineation between micro- and macroscopic domains by suggesting that gravitational nonlinearity introduces a natural localization mechanism potentially observable in high-precision experiments. Conservation laws for momentum and energy within this framework corroborate its theoretical consistency.

The implications of this paper hint at new avenues for exploring the intersection of quantum mechanics and gravitational physics. It invites future investigations that could substantiate or refute the hypothesis empirically, potentially refining the understanding of quantum phenomena at macroscopic scales. The broader implications for fields such as quantum gravity, cosmology, and even technological applications in quantum computing warrant significant attention.

Conclusion

This research contributes a novel perspective on macroscopic localization by introducing gravitational influences into a nonlinear quantum framework. As the paper anchors its theoretical advances in mathematically robust formulations and produces predictions contrasting with traditional quantum mechanics, it lays conceptual groundwork for future explorations into unified physical theories. The soliton solutions derived within this framework provide a basis for the anticipated demarcation between micro and macro realms — a division that continues to provoke rigorous inquiry.