Temporal Flows and Evolution

In this, I aimed to work on how my model predicts the formation of particles through the interactions of temporal flows. My goal was to refine the details and highlight the key insight that flows are linear. As you’ll see, this linearity is crucial because nonlinear effects don't significantly alter the outcome as traditional models might suggest. Essentially, we're dealing with very small values of flows that accumulate over time, and this process remains consistent across the simulations I ran, supporting the idea that the dynamics work out similarly despite the complexity.

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Core Principles and Fundamental Equations

1. Temporal Flows and Evolution:

Evolution Function

$$L(\phi, \nabla \phi):$$ $$\phi(t + \Delta t) = \phi(t) + L(\phi, \nabla \phi)$$

Quantization in Planck Time: Temporal flows are quantized in steps of Planck time $t_P$, ensuring discrete evolution steps. This highlights how fundamental flows evolve over time in a quantized manner.

Nonlinearities and Stability: Nonlinear terms in the evolution function $L(\phi, \nabla \phi)$ allow for complex interactions between flows, crucial for capturing realistic dynamics and the emergence of stable configurations.

Physical Implications of Nonlinearities: Different forms of the potential (e.g., cubic $\lambda \phi^3$ vs. quartic $\phi^4$) contribute to symmetry-breaking and particle formation. Cubic terms introduce asymmetry, which is essential for capturing realistic scenarios where interactions are not symmetric.

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2. Emergence of Space:

Spatial Coordinate

$$r(t) = \int F(t) \, dt$$

Space as an Emergent Property: Space arises from the accumulation of temporal flow differences, implying that spatial dimensions result from interactions and the evolution of these flows over time. This view ties the concept of distance to time, suggesting a dynamic and emergent nature of space.

Quantized Space: The discrete nature of flows (with minimum $t_P$ and maximum $c$) leads to a quantized yet continuous-appearing space at larger scales.

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3. Flow Interactions and Stability:

Interaction Term

$$I(\phi_i, \phi_j) = H(|\phi_i - \phi_j| - \epsilon)$$

Threshold Interaction: Interactions are significant when differences between flows exceed a threshold $\epsilon$, highlighting the role of phase and amplitude differences in governing flow dynamics.

Stability Function

$$S(\lambda, \mu, \theta) = A \exp\left[-\alpha_1(\lambda - \lambda_c)^2 - \alpha_2(\mu - \mu_c)^2 - \alpha_3(\lambda - \lambda_c)(\mu - \mu_c) - \alpha_4(\theta - \theta_{\text{opt}})^2\right]$$

Coherence Factor: Stability function depends on the coherence of flows, with factors like phase differences and amplitude contributing to the overall stability. Coherence decay and phase transitions are critical for understanding the dynamic behavior of flows.

Impact of Phase Differences: Phase differences between flows affect the formation of stable configurations, influencing the emergence of order and complexity. Constructive and destructive interference play key roles in these dynamics.

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4. Energy, Momentum, and Mass-Energy Relationship:

Energy Density Expression:

$$E(t) = \frac{1}{2} \left( |\partial_t \Psi(t, x)|^2 + c^2 |\nabla \Psi(t, x)|^2 \right)$$

Energy from Temporal Flows: Energy is defined in terms of temporal flow density, mirroring the classical energy-momentum tensor but adapted for temporal flows. This ensures a consistent energy definition within your framework.

Mass-Energy Equivalence:

$$m \sim \sum_i \left( \frac{dF_i}{dT} \right)^2 S(\lambda_i, \mu_i)$$

Mass from Flow Changes: Mass emerges from the rate of change in flow phases, linking mass to the evolution and stability of temporal fields. Mass is not an inherent property but arises from flow dynamics.

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5. Lorentz Invariance and Fundamental Constants:

Flow Transformation

$$\gamma_{\text{flows}}(v) = \frac{1}{1 - (f_{\text{flows}}(v) - t_P c - t_P)^2}$$

Respecting Special Relativity: Ensures that temporal flow transformations respect Lorentz invariance and principles of special relativity, crucial for the consistency and universality of your model.

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Most Recent Work

Representing Temporal Flows:

Multi-Component Field

$$\Phi(x, t) = (\phi_1(x, t), \phi_2(x, t), \phi_3(x, t))$$

Directional Components: Each component $\phi_i(x, t)$ measures the amplitude of flows in specific directions, capturing the multi-dimensional nature of temporal flows.

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Lagrangian Density and Evolution:

Lorentz-Invariant Lagrangian Density:

$$L = \frac{1}{2} |\partial_t \Phi|^2 - \frac{1}{2} c^2 |\nabla \Phi|^2 - V(\Phi)$$

Potential

$$V(\Phi) = \lambda_4 S(\lambda, \mu, \theta) |\Phi|^4$$

Nonlinear Interactions: Nonlinear potential encodes interactions among flows, with the stability function $S(\lambda, \mu, \theta)$ ensuring interactions are strongest near optimal configurations.

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Euler-Lagrange Equation:

$$\frac{\partial^2 \Phi}{\partial t^2} - c^2 \nabla^2 \Phi + \lambda S(\lambda, \mu, \theta) |\Phi|^2 \Phi = 0$$

Discrete Update Rule:

$$\Phi(t + t_P) = \Phi(t) + t_P \left[ \partial_t \Phi - \lambda S(\lambda, \mu, \theta) |\Phi|^2 \Phi + \mu \nabla^2 \Phi \right]$$

Time Evolution: This update rule incorporates the discrete, step-wise evolution of temporal fields, ensuring quantized evolution in Planck time steps.

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Effective Force:

$$F(x, t) = -\lambda S(\lambda, \mu, \theta) |\Phi(x, t)|^2 \Phi(x, t)$$

Force Density: Determines how temporal flows interact, driving the dynamics and formation of stable configurations.

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Coarse-Graining and Renormalization

Microscopic Evolution:

Finite-Difference Update:

$$\phi(t + \Delta t) = \phi(t) + \Delta t \left[ \lambda_0 \tanh(\phi^3) - \mu \nabla^2 \phi \right]$$

with $\Delta t = n t_P$ and additional rules (e.g., reflecting flows if $|\phi| > c$) to enforce physical limits.

Coarse-Graining: Every $N$ microscopic steps, the field is averaged:

$$\phi_{\text{block}}(x, T) = \frac{1}{\lambda_{\text{scale}}} \sum_{i=1}^{\lambda_{\text{scale}}} \phi(x, t_i)$$

producing a smoother, effective field.

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Renormalized Coupling:

Running Coupling with Scale:

$$\lambda_{\text{eff}} = \lambda_0 \left( 1 + A \lambda_0 \ln(N) \right)$$

For $N = 100$ and $A = 0.05$, obtained $\lambda_{\text{eff}} \approx 0.09775$, indicating a slight reduction from the bare coupling.

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Emergence of Force and Particle Formation

Force from Flow Interactions:

Potential Energy:

$$V(\Phi) = \lambda_0 4 S(\Phi) |\Phi|^4$$

Effective Force Density:

$$F(x, t) = -\lambda_{\text{eff}} S_{\text{eff}}(\Phi) |\Phi|^2 \Phi$$

This force drives flows to stabilize into bound states, interpreted as particles. Mass arises from the rate at which the phase of the flow changes:

$$m \sim \sum_i \left( \frac{dF_i}{dT} \right)^2 S(\lambda_i, \mu_i, \theta)$$

Particle Formation:

Eigenvalue Problem for Accumulation of Flows:

$$A \psi = \lambda_n \psi \, S(\lambda_n, \mu_n, \theta)$$

Particle State Representation: Here, $\psi$ represents a particle state, and the eigenvalue $\lambda_n$ is related to the energy and mass of that state. Particles form when flows "stick" together, meaning their amplitudes and phases align in a stable configuration.

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Emergent Electromagnetism and Relativity

Electromagnetic Fields from Temporal Flows:

Effective Fields:

$$E \sim -\partial_t \Phi, \quad B \sim \nabla \times \Phi$$

Wave Equation in Linear Regime: When nonlinearities are small, the evolution of $\Phi$ satisfies a wave equation:

$$\frac{\partial^2 \Phi}{\partial t^2} - c^2 \nabla^2 \Phi = 0$$

This equation is analogous to the electromagnetic wave equation in vacuum, demonstrating how electric and magnetic fields can emerge from temporal flows.

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Relativistic Effects:

Lorentz Invariance and Time Dilation: Since the maximum

velocity is $c$, all flows must obey relativistic effects like time dilation and length contraction. The modified Lorentz transformation for temporal flows:

$$\gamma_{\text{flows}}(v) = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}$$

describes the relativistic adjustment needed to preserve the invariance of the dynamics under transformations of velocity.

Reference Section:

Equations Used

1. Microscopic Evolution:
   The field evolution is governed by the following equation:

   $$\frac{\partial \phi}{\partial t} = - \mu \frac{\partial^2 \phi}{\partial x^2} + \lambda \tanh(\phi^3)$$
   

   where:

   • $\phi(x,t)$ is the field value at position $x$ and time $t$,
   • $\mu$ is the diffusion coefficient,
   • $\lambda$ is the coupling constant,
   • $\tanh(\phi^3)$ represents the nonlinear interaction term.

2. Coarse-Graining:
   The coarse-grained field is computed by averaging the field over blocks of size $N$:

   $$\phi_{\text{coarse}}(x) = \frac{1}{N} \sum_{i=0}^{N-1} \phi(x_i)$$
   

   where $N$ is the block size, and the summation is over the blocks.

3. Renormalized Effective Coupling:
   The effective coupling constant $\lambda_{\text{eff}}$ is given by the following equation:

   $$\lambda_{\text{eff}} = \frac{\lambda_0}{1 + A \lambda_0 \ln(N)}$$

   where:

   • $\lambda_0$ is the initial coupling constant,
   • $A$ is a constant factor,
   • $N$ is the block size.

Results for Simulation Set 1

1. Running Simulation with $N = 50, 100, 200, 400$ and Steps = 5,000,000

   • Effective Coupling and Coarse-Grained Information:

     • N = 50:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0981
       • Number of blocks: 20
       • Coarse-grained shape: (20, 200)

     • N = 100:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0977
       • Number of blocks: 10
       • Coarse-grained shape: (10, 200)

     • N = 200:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0974
       • Number of blocks: 5
       • Coarse-grained shape: (5, 200)

     • N = 400:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0971
       • Number of blocks: 2
       • Coarse-grained shape: (2, 200)

Results for Simulation Set 2

1. Running Simulation with $N = 50, 75, 150, 300$ and Steps = 5,000,000

   • Effective Coupling and Coarse-Grained Information:

     • N = 50:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0981
       • Number of blocks: 20
       • Coarse-grained shape: (20, 200)

     • N = 75:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0979
       • Number of blocks: 13
       • Coarse-grained shape: (13, 200)

     • N = 150:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0976
       • Number of blocks: 6
       • Coarse-grained shape: (6, 200)

     • N = 300:

       • Effective Coupling ($\lambda_{\text{eff}}$): 0.0972
       • Number of blocks: 3
       • Coarse-grained shape: (3, 200)

Key Observations

• Effective Coupling ($\lambda_{\text{eff}}$) decreases as $N$ increases, reflecting the renormalization effect in the model. This trend is observed in both simulation sets.
• The number of blocks decreases as $N$ increases, which is expected due to the larger block size reducing the number of coarse-graining blocks.
• The coarse-grained shape is determined by the number of blocks and the spatial grid size, showing how the field is averaged over larger scales.