We examine the origin of the Newton–Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the 'Moller–Rosenfeld equation') based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the 'collapse of the wave function', we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.We examine the origin of the Newton–Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the 'Moller–Rosenfeld equation') based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the 'collapse of the wave function', we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.

arXiv:1403.4921New Journal of PhysicsAug 19, 201490 citations

Higher conservation laws for the quantum non-linear Schrödinger equation

Brian Davies

Physica A Statistical Mechanics and its ApplicationsAug 1, 199042 citations

"The Schrödinger Equation from Gravitational Null-Zone Geometry

Rob Facer

This is the third paper in the Eigenfield series. Building directly on the original framework (Zenodo 18646417, v11) and its companion note on the derivation of ℏ, we show that the time-symmetric nature of gravity at null zones (saddle points where net gravitational force vanishes) naturally forces the Schrödinger equation to appear — without any external quantum postulate. The central insight is gentle and geometric: for the retarded + advanced balance condition $\vec{g}_{\rm ret} + \vec{g}_{\rm adv} = \vec{0}$ to hold at every instant, the residual tidal field can only vibrate in closed orbits on the eigenvector sphere $S^2$ . Open or fractional orbits eventually break the saddle condition. Closed orbits require the action to come in exact integer multiples of a fundamental unit — the same unit we previously identified as Planck’s reduced constant $\hbar = c^5 / (16\pi G \delta\lambda_{\rm min})$ . From this single requirement we derive the full harmonic-oscillator Schrödinger equation for a test particle near a null zone. Every term (potential, frequency $\omega = \sqrt{\delta\lambda_{\rm min}}$ , and $\hbar$ ) arises directly from the tidal residual and time-symmetric propagation at $c$ . Written at a level accessible to advanced undergraduates while remaining mathematically precise, the paper uses only the eigenvalue algebra, interference diagnostics, and torque/precession machinery already present in the earlier work. No new postulates are introduced — quantisation emerges as a boundary condition imposed by the geometry itself. This provides a concrete, pedagogical bridge from classical multi-body gravitational geometry to the form of quantum mechanics.This is the third paper in the Eigenfield series. Building directly on the original framework (Zenodo 18646417, v11) and its companion note on the derivation of ℏ, we show that the time-symmetric nature of gravity at null zones (saddle points where net gravitational force vanishes) naturally forces the Schrödinger equation to appear — without any external quantum postulate. The central insight is gentle and geometric: for the retarded + advanced balance condition $\vec{g}_{\rm ret} + \vec{g}_{\rm adv} = \vec{0}$ to hold at every instant, the residual tidal field can only vibrate in closed orbits on the eigenvector sphere $S^2$. Open or fractional orbits eventually break the saddle condition. Closed orbits require the action to come in exact integer multiples of a fundamental unit — the same unit we previously identified as Planck’s reduced constant $\hbar = c^5 / (16\pi G \delta\lambda_{\rm min})$. From this single requirement we derive the full harmonic-oscillator Schrödinger equation for a test particle near a null zone. Every term (potential, frequency $\omega = \sqrt{\delta\lambda_{\rm min}}$, and $\hbar$) arises directly from the tidal residual and time-symmetric propagation at $c$. Written at a level accessible to advanced undergraduates while remaining mathematically precise, the paper uses only the eigenvalue algebra, interference diagnostics, and torque/precession machinery already present in the earlier work. No new postulates are introduced — quantisation emerges as a boundary condition imposed by the geometry itself. This provides a concrete, pedagogical bridge from classical multi-body gravitational geometry to the form of quantum mechanics.

Mar 3, 20260 citations

On the Non-Linear Schrödinger Equation

A.B. Datzeff

Springer eBooksJan 1, 19850 citations

Relating Madelung–Bohm trajectories

Gilberto Silva-Ortigoza, Ramón Silva‐Ortigoza, Israel Julián-Macías +1 more

In this work, we use the quantum potential approach to quantum mechanics to show that the Madelung–Bohm trajectories for a particle in a constant gravitational field can be related to those of a free particle by means of a quantum point transformation defined in the extended configuration space. We find that the point transformation also gives a connection between the corresponding quantum Hamiltonians determined by the solutions of the corresponding Schrödinger equations for these two problems. We show that the Madelung–Bohm trajectories determined by the stationary solutions to the Schrödinger equation for the particle in a constant gravitational field are straight ones, while the corresponding ones for the free particle are parabolic trajectories. The Airy beam is one example of this type of solution. We study the properties of a solution to the Schrödinger equation for a free particle with phase singularities (zeroes), and we find that the corresponding solution to the Schrödinger equation for the particle in a constant gravitational field also has zeroes at the same spacetime points. However, the Madelung–Bohm trajectories determined by the two solutions are totally different. Furthermore, we remark that similar results can be directly obtained for the paraxial wave equation.In this work, we use the quantum potential approach to quantum mechanics to show that the Madelung–Bohm trajectories for a particle in a constant gravitational field can be related to those of a free particle by means of a quantum point transformation defined in the extended configuration space. We find that the point transformation also gives a connection between the corresponding quantum Hamiltonians determined by the solutions of the corresponding Schrödinger equations for these two problems. We show that the Madelung–Bohm trajectories determined by the stationary solutions to the Schrödinger equation for the particle in a constant gravitational field are straight ones, while the corresponding ones for the free particle are parabolic trajectories. The Airy beam is one example of this type of solution. We study the properties of a solution to the Schrödinger equation for a free particle with phase singularities (zeroes), and we find that the corresponding solution to the Schrödinger equation for the particle in a constant gravitational field also has zeroes at the same spacetime points. However, the Madelung–Bohm trajectories determined by the two solutions are totally different. Furthermore, we remark that similar results can be directly obtained for the paraxial wave equation.

Journal of the Optical Society of America BDec 10, 20241 citation

\vec{g}_{\rm ret} + \vec{g}_{\rm adv} = \vec{0}

to hold at every instant, the residual tidal field can only vibrate in closed orbits on the eigenvector sphere

S^2

. Open or fractional orbits eventually break the saddle condition. Closed orbits require the action to come in exact integer multiples of a fundamental unit — the same unit we previously identified as Planck’s reduced constant

\hbar = c^5 / (16\pi G \delta\lambda_{\rm min})

. From this single requirement we derive the full harmonic-oscillator Schrödinger equation for a test particle near a null zone. Every term (potential, frequency

\omega = \sqrt{\delta\lambda_{\rm min}}

, and

\hbar

) arises directly from the tidal residual and time-symmetric propagation at

c

. Written at a level accessible to advanced undergraduates while remaining mathematically precise, the paper uses only the eigenvalue algebra, interference diagnostics, and torque/precession machinery already present in the earlier work. No new postulates are introduced — quantisation emerges as a boundary condition imposed by the geometry itself. This provides a concrete, pedagogical bridge from classical multi-body gravitational geometry to the form of quantum mechanics.