Spin Field Dynamics
Spin dynamics play a crucial role in understanding the fundamental interactions in quantum systems. This study explores how accelerated spin states can influence spacetime curvature, a concept that bridges quantum mechanics and general relativity. We present a theoretical model and an experimental proposal to investigate these effects using highly polarized particles in a controlled environment.
Theoretical framework
The model emphasizes the importance of interference patterns in quantum states and their influence on system dynamics. These patterns affect the formation of quantum knots, the behavior of particles and the transfer of energy within the system. The efficiency of energy transfer depends on the overlap of the wave functions and the interaction energy between the points. These interactions are described using the extended Schrödinger equation, which accounts for the influence of interference nodes on the state of the particle. The feedback between a particle and the interference nodes it creates reveals the interactive nature of quantum systems. Instead of the classical concept of curvature used in general relativity, the model uses the concept of time points and their gradients. The model uses the dimensions of the Planck scale for normalization and dimensional consistency, allowing a detailed study of interactions between quantum states and fundamental forces.
1. Quantum interference nodes and energy transfers
1.1 Dynamics of quantum interference nodes
\[ n(x, t) = n_0 \exp\left(\min\left(\lambda(E, S), \lambda_{max}\right) t\right) \]
Describes the evolution of quantum interference nodes, which affect the quantum structure of space.
1.2 Particle energy dynamics
\[ E^2 = (mc^2)^2 + \left(\hbar k \lambda S \right)^2 + (\Lambda c^4)^2 – \min(E_{fluc}(\Delta T), E_{fluc_{max}}) \]
Includes contributions from spin to the total energy of particles.
1.3 Dynamic interactions
\[ i\hbar\frac{\partial \psi(x,t)}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + \Phi\left(\frac{T(x, t)}{t_P}\right) + \lambda \int \min\left(\Phi(x – x’, t), \Phi_{max}\right) |\psi(x’,t)|^2 dx’\right]\psi(x,t) \]
This extended Schrödinger equation models the dynamical interactions between quantum particles and interference nodes, contributing to the completeness and consistency of the model.
2. Modified local time and spin
2.1 Modified local time
\[ T_s(x) = T_0 \left(1 – \min\left(\alpha_s \frac{S(x)}{S_P}, 1 – \epsilon\right)\right) \]
Shows how local time depends on spin interactions, impacting particle dynamics.
2.2 Gradient of local time
\[ \nabla T_s(x) = -\frac{\alpha_s T_0}{S_P} \min\left(\nabla S(x), \nabla S_{max}\right) \]
It describes how local time in space changes depending on rotation changes.
2.3 Spin dynamics
The dynamics of particle spin can be influenced by the local curvature of spacetime:
\[ \frac{dS(x, t)}{dt} = -\beta \left(1 + \delta \frac{n(x)}{n_0}\right) S(x, t) \]
3. Temporal gradients and modified Einstein equation
3.1 Hamiltonian function
\[ \mathcal{H}\left(\frac{m}{m_P}, \frac{S}{S_P}\right) = \mathcal{H}_0 \left(1 + \delta \min\left(\frac{S}{S_P}, k\right)\right) \]
Determines the energy of the system including spin. This energy is fundamental for understanding the dynamics of particles and their interactions in the gravitational field.
3.2 Modified Einstein equation with quantum corrections
Einstein’s equation with a modified metric tensor can be written as:
\[ R_{\mu\nu} – \frac{1}{2} R g_{\mu\nu}(x) = \frac{8 \pi G}{c^4} T_{\mu\nu}(x) \]
By substituting the modified metric tensor:
\[ R_{\mu\nu} – \frac{1}{2} R \left( g_{\mu\nu}^{(0)} \left(1 + \delta \frac{n(x)}{n_0}\right) \right) = \frac{8 \pi G}{c^4} T_{\mu\nu}(x) \]
where \( T_{\mu\nu}(x) \) includes the classical energy-momentum tensor as well as contributions from quantum effects and spin dynamics:
\[ T_{\mu\nu}(x) = T_{\mu\nu}^{(classical)} + \eta \left(\frac{S_{\mu} S_{\nu}}{S_P^2}\right) + \lambda \left(\frac{n_{\mu} n_{\nu}}{n_P^2}\right) \]
Experimental design with calculations to demonstrate changes in spacetime curvature
Experiment objective:
Demonstrate changes in spacetime curvature through the manipulation of spin in a system of highly polarized particles, utilizing quantum and relativistic effects to modify the physical properties of matter.
Materials and system
1. Superconductor – HgMnTe (HgTe with added Mn):
– Role: Initiates spin in a material composed of dysprosium or holmium through magnetic pulses.
– Preparation: Cooling to temperatures between 20 mK and 50 mK to minimize decoherence.
– Lattice structure development: Optimized through annealing processes and precise control of Mn concentration (2% to 5%).
2. High spin atoms – Dysprosium (Dy) or Holmium (Ho):
– Particle selection: Atoms with high spin quantum numbers (\(J = 8.5\) for Dy and \(J = 7.5\) for Ho).
– Arrangement: Atoms arranged in a regular cubic lattice with a spacing of 50 nm to 100 nm.
Experimental setup
1. Spin initiation and acceleration:
– Mechanism: Superconductor HgMnTe through magnetic pulses initiates spin in dysprosium or holmium atoms.
– Magnetic pulses: Duration 10 ns to 100 ns and amplitude 1 mT to 10 mT.
– Spin propagation: Spin acceleration in individual atoms through particle entanglement in the lattice.
2. Enhancement of spin-orbit interaction:
– Implementation: Use of strong local electric fields (~10^6 – 10^8 V/m).
– Impact on spacetime: Local spacetime deformations caused by strong spin-orbit interaction.
Measurement devices
1. Sensitive gravimeters: Sensitivity \(10^{-18} \, g\).
2. Interferometers: Sensitivity \(10^{-21} \, m\).
3. Atomic clocks: Optical lattice clocks with accuracy up to \(10^{-18}\).
Simulation of calculations
Step 1: Calculate the density of quantum nodes
Using the equation:
\[ n(x, t) = n_0 \exp\left(\min\left(\lambda(E, S), \lambda_{max}\right) t\right) \]
For values:
\[ n(x, t) = 10^3 \exp\left(\min(0.01, 0.02) \times 1\right) \]
\[ n(x, t) \approx 10^3 \times 1.01005 \approx 1010.05 \]
Step 2: Calculate spin change
Using the equation:
\[ \Delta S(x, t) \approx -\beta \left(1 + \delta \frac{n(x)}{n_0}\right) S(x, t) \Delta t \]
For values:
\[ \Delta S(x, t) \approx -1 \left(1 + 0.1 \frac{1010.05}{10^3}\right) 0.5 \times 0.1 \]
The change in spin over the time interval \( \Delta t \) is approximately \( -0.05505 \).
Impact on spacetime curvature
Using the modified Einstein equation:
\[ R_{\mu\nu} – \frac{1}{2} R \left( g_{\mu\nu}^{(0)} \left(1 + \delta \frac{n(x)}{n_0}\right) \right) = \frac{8 \pi G}{c^4} T_{\mu\nu}(x) \]
The change in spin \( \Delta S(x, t) \) affects \( T_{\mu\nu}(x) \), which in turn affects the curvature of spacetime \( R_{\mu\nu} \).
Conclusion
The dynamics of rotation is described by a differential equation that includes coefficients affecting the change of rotation over time. Quantum nodes behave according to an exponential function that depends on the parameters \(\lambda\) and \(\lambda_{max}\). These nodes in turn affect the metrics in Einstein’s equations, leading to changes in the curvature of spacetime. Based on the spin change, we can modify the tensor \( T_{\mu\nu} \) and then solve the equations using numerical methods to determine the specific effect on the curvature of spacetime.
Author
Madala Roman I 2024 I madalar2@gmail.com
References
Avoided quasiparticle decay from strong quantum interactions
Tying quantum knots
Magnetically tunable supercurrent in dilute magnetic topological insulator-based Josephson junctions
Crystallization of bosonic quantum Hall states in a rotating quantum gas