The Flying Fish model:
Quantum spinor dynamics and gravity
The purpose of the Flying Fish model is to investigate the interactions between quantum spinor dynamics and gravity. He investigates how quantum fields can affect space-time through their interactions, creating localized temporal variations and gravitational effects. These fields obey the Dirac equation:
\[(i \gamma^\mu \nabla_\mu – m) \psi = 0\]
where \(\gamma^\mu\) represents the gamma matrices for spinors in curved spacetime, \(\nabla_\mu\) is the covariant derivative, and \(m\) is the spinor field mass. This equation is the basis for describing the interactions in the model. Spinor fields can create coherent oscillations between different spin states. This oscillatory behavior, similar to the Josephson effect observed in superconductors [14], enables complex dynamics in quantum systems that respond to initial conditions and external magnetic fields. Under certain conditions, these systems can spontaneously transition into new quantum states [2].
Notation and symbols
– \(\mathcal{G}_t\): Local time gradient, representing variations in the flow of time caused by spinor transitions.
– \(\mathcal{T}(x)\): Dynamic topological invariant that stabilizes quantum knots and responds to local time gradients.
– \(\rho_{\text{spinor}}(r)\): Local energy density of the spinor field.
– \(T_0\): Reference time flow in the absence of quantum transitions.
– \(P^2\): Spacetime curvature function, describing geodesic distances in the modified spacetime.
– \(\alpha_s\): coefficient quantifying the effect of the spinor field energy density on the flow of time.
The time gradient
The Flying Fish model introduces the concept of space-time gradients created by quantum transitions in spinor fields. The time gradient, represented by the symbol \(\mathcal{G}_t\), involves interactions between the spinor field energy and topological stabilization [13]. It is defined as:
\[\mathcal{G}_t = T_0 \left( 1 + \frac{\alpha_s \rho_{\text{spinor}}(r) \cdot L_P}{c^2 (1 + \beta \mathcal{T}(x))} \cdot \left( 1 + \eta \frac{E}{c^2} \right) \right)\]
The topological invariant \(\mathcal{T}(x)\) adapts based on local time gradients:
\[\mathcal{T}(x) = \mathcal{T}_0 \cdot \exp\left(-\frac{\mathcal{G}_t}{T_0} + \kappa \cdot \Delta \phi^2\right)\]
where \(\kappa\) is an amplification coefficient caused by quantum fluctuations.
The model extends the equations of general relativity by incorporating dynamical contributions from spinor fields and topological invariants. The energy density of spinor fields and local time gradients contribute to the curvature of spacetime. This interaction is expressed by the expression:
\[f(\rho_{\text{spinor}}, \mathcal{G}_t) = \alpha_s \cdot \rho_{\text{spinor}} \cdot \left( 1 + \frac{\mathcal{G}_t} {T_0} \right) \cdot \exp\left(-\frac{\mathcal{G}_t}{T_0}\right)\]
\[G_{\mu \nu} + f(\rho_{\text{spinor}}, \mathcal{G}_t) + \mathcal{T}(x) = \frac{8 \pi G}{c^4} T_{\mu \nu},\]
where \(f(\rho_{\text{spinor}}, \mathcal{G}_t)\) quantifies the effect of spinor fields on curvature, while \(\mathcal{T}(x)\) provides topological stabilization.
This modification minimizes the impact of quantum transitions while ensuring compatibility with the strong equivalence principle [4]. After this modification, it is possible to analyze the spatial geometric properties of spacetime affected by these phenomena. The function for the curvature of spacetime, defined as:
\[P^2 = \int \left(G_{\mu\nu} + f(\rho_{\text{spinor}}, \mathcal{G}_t) + \mathcal{T}(x)\right) dx^\mu dx^\nu,\]
allows the calculation of the resulting geodesic distance between points in this modified spacetime. This relation captures the contributions to the curvature resulting from the energy density of the spinor fields and the effect of quantum knots. The function \(P^2\) provides a geometric interpretation of the modified spacetime and can be used to analyze the local or global effects of the spinor field dynamics on the structure of spacetime. This approach allows for the explicit study of the spatial aspects of the interactions within the model. To verify, we can use NMR spectroscopy to detect changes in the precessional frequencies of protons.
Energy transfer between states of spinor fields
The energy density \(\rho_{\text{spinor}}(r)\) of the spinor field is critical in generating spacetime gradients. It is expressed as:
\[\rho_{\text{spinor}}(r) = \langle \bar{\psi}(r) \gamma^0 \psi(r) \rangle\]
– \(\bar{\psi} = \psi^\dagger \gamma^0\) is the spinor conjugate,
– \(\langle \cdots \rangle\) denotes the quantum expectation value [13].
Energy transfer occurs due to transitions between accelerated and decelerated states of the spinor field, as it adjusts to changes in the local flow of time. This energy exchange is quantified by:
\[\Delta E = \hbar \omega \cdot \left( \frac{\mathcal{G}_t – T_0}{T_0} \right)\]
– \(\hbar \omega\) characterizes the energy associated with the quantum transition
This formulation describes how the energy of a spinor field dynamically adapts to local space-time conditions. In extreme conditions, the energy transferred between states of the spinor field increases with the time gradient (\(\mathcal{G}_t\)). The relation for \(\Delta E\) is defined in the basic part of the model. The energy density of the spinor fields (\(\rho_{\text{spinor}}(r)\)) and the time gradients (\(\mathcal{G}_t\)) contribute to the accumulation of energy:
\[E_{\text{accum}} = \int_{r_{\text{horizon}}} \rho_{\text{spinor}}(r) \cdot \mathcal{G}_t \cdot \left( 1 + \frac{\mathcal{T}(x)}{T_0} \right) dr\]
The energy density of a spinor field can be analyzed calorimetrically.
Equations for gravitational and inertial mass
In the model, gravitational and inertial mass are emergent properties that depend on local spatiotemporal conditions and the stabilizing effects of quantum knots. These weights are expressed as:
\[M_{\text{grav}} = \frac{1}{2 \alpha_s \cdot \rho_{\text{spinor}}} \left( 1 – \frac{\mathcal{G}_t}{T_0} \right) \cdot (1 + \delta \mathcal{T}(x))\]
\[M_{\text{inert}} = \frac{r c^4}{\rho_{\text{spinor}}} \left( \frac{\mathcal{G}_t}{T_0} – 1 \right) \cdot (1 + \delta \mathcal{T}(x))\]
– \(\delta\) represents the stabilizing influence of quantum knots on gravitational and inertial mass,
– \(r\) is a spatial factor.
These equations show that mass is not an intrinsic property but a dynamical result of interactions between spinor fields. Quantum nodes stabilize these interactions by reducing fluctuations and maintaining coherence in the system. This relation can be formalized using the effective gravitational Hamiltonian:
\[\hat{H}_{\text{grav}} = \int d^3x \, \left( \rho_{\text{spinor}} \cdot \mathcal{G}_t \cdot \mathcal{T} (x) \right)\]
– \(\hat{H}_{\text{grav}}\) is the effective gravitational Hamiltonian associated with the energy density of the spinor field.
The \(\alpha_s\) parameter quantifies how the energy density from quantum transitions in spinor fields affects the gravitational acceleration. This relationship is given by:
\[\alpha_s = \frac{g_{\text{spinor}}}{\rho_{\text{spinor}}}\]
– \(g_{\text{spinor}}\) is the gravitational acceleration influenced by the energy density of the spinor field,
– \(\rho_{\text{spinor}}\) is the local energy density of the spinor field.
Higher \(\rho_{\text{spinor}}\) energy densities amplify gravitational effects in specific regions, while \(\alpha_s\) provides a direct measure of how quantum transitions and spinor fields affect their dynamics. This ensures that the model accurately reflects the interplay between quantum effects and gravitational phenomena.
Planck scale in the model
In this model, the Planck length is the natural limit where quantum effects create an extreme time gradient [10]. The high energy density of spinor fields in this scale leads to a slowing down of the flow of time, effectively freezing the dynamics of quantum processes and making them unobservable by classical methods. Consequently, quantum processes in this region appear to be macroscopically static. This extreme time dilation is not stable. The high energy density at the Planck scale induces quantum fluctuations that can lead to the creation and annihilation of virtual particles in accordance with the Heisenberg uncertainty principle. These fluctuations are described by the expression [12]:
\[\langle (\Delta \phi)^2 \rangle = \frac{\rho_{\text{spinor}}}{E} \cdot \left(1 + \frac{\mathcal{G}_t}{T_0}\right).\]
For extreme conditions where nonlinear effects are needed:
\[\langle (\Delta \phi)^2 \rangle = \frac{\rho_{\text{spinor}}}{E} \cdot \exp\left(\frac{\mathcal{G}_t}{T_0}\right).\]
where \(\delta \phi\) represents the quantum fluctuation of the field.
These quantum effects affect space-time at the smallest scales and form the basis for the quantum nature of gravitational phenomena. Adding energy increases their stability, this stabilization is reflected in the lifetime of quantum structures:
\[\tau = \tau_0 \cdot \left( 1 + \frac{\rho_{\text{spinor}} \cdot E \cdot L_P^4}{\hbar \cdot c \cdot M_P} \cdot \frac{\mathcal{G}_t}{T_0} \right)\]
where:
– \(\tau\) is the lifetime of the structure,
– \(\tau_0\) is the base lifetime without added energy,
– \(E\) is the added energy.
Design of the experimental part of the Flying Fish model
The aim of the experiment is to determine whether coherent magnetic oscillations, generated by closely spaced harmonic frequencies, can accelerate quantum precession in spinor fields. By generating pulse frequencies, the experiment could induce time dilation effects similar to weak gravitational fields [11]. Helmholtz coils will generate precise magnetic oscillations to modulate spinor fields, while NMR spectroscopy will measure changes in proton precession. The selected frequencies are:
– First frequency: \( f_1 = 127.740000 \, \text{MHz} \),
– Second frequency: \( f_2 = 127.74000563 \, \text{MHz} \),
– Third frequency: \( f_3 = 127.74001126 \, \text{MHz} \).
These frequencies are selected to produce interference (beat) frequencies within the Hz range, specifically around the desired value of 5.63 Hz. The differences between these close frequencies generate low-frequency components that control the quantum transitions of the spinor fields:
\[f_{\text{beat1}} = f_2 – f_1 = 5.63 \, \text{Hz}, \quad f_{\text{beat2}} = f_3 – f_2 = 5.63 \, \text{Hz}, \quad f_{\text{beat3}} = f_3 – f_1 = 11.26 \, \text{Hz}.\]
These low-frequency beat components are critical for achieving the desired modulation of the spinor field dynamics, facilitating quantum transitions that align with the local time gradients, leading to time dilation effects similar to those experienced in Earth’s gravitational field. The coherent manipulation of these frequencies is essential to exploring the relationship between quantum transitions in spinor fields and their impact on local gravitational behavior.
Generating magnetic pulses
Three superconducting magnets and Helmholtz coils will generate harmonic frequencies \( f_1 \), \( f_2 \), and \( f_3 \), creating coherent magnetic oscillations crucial for influencing the dynamics of the spinor fields. The magnetic pulses will be synchronized with a precision of ±0.1 radians to ensure accurate modulation of the spinor fields. The pulse settings are as follows:
– Pulse duration: 10 microseconds,
– Interval between pulses: 20 microseconds,
– Number of cycles per pulse: 100 cycles per pulse.
The use of superconducting magnets with a field intensity of 2 T will enhance the interaction of spinor fields and increase the precision of the measured data. By creating interference between the frequencies \( f_1 \), \( f_2 \), and \( f_3 \), low-frequency beat components will be generated, which are essential for achieving the desired spinor state transition. The differences between these frequencies will produce beat frequencies in the Hz range, specifically 5.63 Hz, enabling controlled modulation of spinor flipping and the associated quantum transitions. These low-frequency components are tuned to a target frequency, calculated based on experimental conditions, to match the gravitational effects induced by the spinor fields, analogous to the weak gravitational acceleration observed in orbit (about 8.7 m/s²).
Preparation of test material and environment
The test material will be titanium hydride (TiH₂) or palladium hydride (PdHx), as these materials effectively bind hydrogen atoms and provide a sufficient number of protons. The experiment will be conducted in a cryogenic environment with the temperature maintained at 77 K using liquid nitrogen. This will ensure the superconducting state of the magnets and provide sufficient stability for the quantum states of hydrogen protons while minimizing thermal noise. The sample will be placed at the center of the experimental setup, between three Helmholtz coils that will generate a homogeneous magnetic field.
To verify the predictions of the Flying Fish model, precise measurements of the temporal gradients and their effects on local dynamics are needed. One possible approach involves detecting changes in proton precession frequencies due to changes in the temporal gradient. The relationship is expressed as:
\[\Delta \omega = \omega_{\text{prec}} – \omega_L \propto \mathcal{G}_t\]
where:
– \(\Delta \omega\): Change in the precession frequency caused by the spinor field dynamics,
– \(\omega_{\text{prec}}\): Adjusted precession frequency affected by the time gradient,
– \(\omega_L\): Larmor frequency, which represents the fundamental precession frequency without external influence.
This proportional relationship determines that changes in \(\mathcal{G}_t\) directly affect the proton precession frequencies. By generating controlled magnetic fields via superconducting magnets, it is possible to induce specific time gradients. Using NMR spectroscopy [7], these frequency shifts can be precisely measured, providing experimental confirmation of the influence of spinor field dynamics on the properties of local spacetime. After the magnetic fields are turned off, the gravitational field should return to its original state, confirming that the observed gravitational changes are directly related to the modulation. A successful experiment would confirm the model’s assumption that quantum transitions in spinor fields are responsible for the observed gravitational variations.
Discussion
The model assumptions are subject to certain limitations:
- Planck Scale: The model assumes that spinor fields and their time gradients remain coherent up to the Planck scale. At smaller scales, unpredictable quantum fluctuations dominate.
- Extreme Curvature: The model effectively describes weak to moderate curvature of spacetime. Cases such as black holes or singularities are only marginally addressed by this model.
- Stabilization of Quantum Knots: The model assumes well-defined topological invariants for stabilization. In chaotic or extreme conditions, these assumptions may change.
Conclusion and future research
The Flying Fish model opens new avenues for understanding the interactions between quantum fields and gravitational phenomena. Its predictions could influence the following areas:
- Gravity Control Technologies: Stabilizing quantum knots could form the basis for the development of devices for manipulating gravitational fields with applications in space exploration (satellites that could control their orbits more efficiently, or propulsion systems of the future), energy optimization, and quantum technologies.
- Quantum-Gravitational Transitions: The model facilitates the study of spinor field behavior under extreme conditions, offering insights into singularities and quantum black holes.
- Experimental Validation: The predicted effects of time gradients and spinor transitions can be verified through spectroscopic techniques, potentially confirming key aspects of the model.
Author
Madala Roman I madala@modifiedplanck.com I December, 2024
References
1. Chang, MS., Qin, Q., Zhang, W. et al. Coherent spinor dynamics in a spin-1 Bose condensate. Nature Phys 1, 111–116 (2005). https://doi.org/10.1038/nphys153. Available: https://www.nature.com/articles/nphys153
2. Prüfer, M., Kunkel, P., Strobel, H. et al. Observation of universal dynamics in a spinor Bose gas far from equilibrium. Nature 563, 217–220 (2018). https://doi.org/10.1038/s41586-018-0659-0. Available: https://www.nature.com/articles/s41586-018-0659-0
3. Renato M. Angelo and K. Furuya, Semiclassical limit of the entanglement in closed pure systems, Phys. Rev. A 71, 042321 – Published 13 April 2005. Available: https://arxiv.org/abs/quant-ph/0410103
4. Sabín, C. Mapping curved spacetimes into Dirac spinors. Sci Rep 7, 40346 (2017). https://doi.org/10.1038/srep40346. Available: https://www.nature.com/articles/srep40346
5. Naderi, F., Hasanirokh, K. Quantum transport of massless Dirac fermions through wormhole-shaped curved graphene in presence of constant axial magnetic flux. Sci Rep 14, 7763 (2024). https://doi.org/10.1038/s41598-024-57718-3. Available: https://www.nature.com/articles/s41598-024-57718-3
6. Shi, YH., Yang, RQ., Xiang, Z. et al. Quantum simulation of Hawking radiation and curved spacetime with a superconducting on-chip black hole. Nat Commun 14, 3263 (2023). https://doi.org/10.1038/s41467-023-39064-6. Available: https://www.nature.com/articles/s41467-023-39064-6
7. Gooth, J., Niemann, A., Meng, T. et al. Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP. Nature 547, 324–327 (2017). https://doi.org/10.1038/nature23005. Available: https://www.nature.com/articles/nature23005
8. Verresen, R., Moessner, R. & Pollmann, F. Avoided quasiparticle decay from strong quantum interactions. Nat. Phys. 15, 750–753 (2019). https://doi.org/10.1038/s41567-019-0535-3. Available: https://www.nature.com/articles/s41567-019-0535-3
9. Hall, D., Ray, M., Tiurev, K. et al. Tying quantum knots. Nature Phys 12, 478–483 (2016). https://doi.org/10.1038/nphys3624. Available: https://www.nature.com/articles/nphys3624
10. Toedt, JN., Hansen, W. Dynamic control of spin-wave propagation. Sci Rep 11, 7821 (2021). https://doi.org/10.1038/s41598-021-86887-8. Available: https://www.nature.com/articles/s41598-021-86887-8
11. Wu, X., Wang, Z., Yang, F. et al. Dissipative time crystal in a strongly interacting Rydberg gas. Nat. Phys. 20, 1389–1394 (2024). https://doi.org/10.1038/s41567-024-02542-9. Available: https://www.nature.com/articles/s41567-024-02542-9
12. S Torquato et al J. Stat. Uncovering Multiscale Order in the Prime Numbers via Scattering, Mech. (2018) 093401. Available: https://arxiv.org/abs/1802.10498
13. P. Debray, Spinors, [Online]. Available: https://adebray.github.io/lecture_notes/m392c_Spin_notes.pdf
14. P. Blair Blakie, Barbara Capogrosso-Sansone, Magnetic atoms push interactions to new lengths for quantum simulation, Nature 622, 704-705 (2023). Available: https://www.nature.com/articles/d41586-023-03152-w
15. Pita-Vidal, M., Bargerbos, A., Žitko, R. et al. Direct manipulation of a superconducting spin qubit strongly coupled to a transmon qubit. Nat. Phys. 19, 1110–1115 (2023). https://doi.org/10.1038/s41567-023-02071-x. Available: https://www.nature.com/articles/s41567-023-02071-x
16. Casotti, E., Poli, E., Klaus, L. et al. Observation of vortices in a dipolar supersolid. Nature 635, 327–331 (2024). DOI: https://doi.org/10.1038/s41586-024-08149-7
17. Yuen and Demetriadou. Exact Quantum Electrodynamics of Radiative Photonic Environments. Phys. Rev. Lett. 133, 203604 – Published 14 November, 2024. DOI: https://doi.org/10.1103/PhysRevLett.133.203604