Why Gauge Theory Emerges in Temporal Flow Physics

By John Gavel

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Abstract

In Temporal Flow Physics (TFP), quantized 1D temporal flows are fundamental, and space emerges from their relational dynamics. This post demonstrates how gauge theory—a cornerstone of the Standard Model—arises naturally from TFP’s internal symmetries, flow comparisons, and entropy gradients. I derive the mathematical structures of gauge fields, symmetries, and couplings from first principles, showing how TFP emulates QED’s fine-structure constant ($\alpha \approx 1/137$). Using recent simulation results ($N_1 \approx 10.90$, $\alpha_{\text{eff}}^{-1} \approx 136.97$), I reveal a deep unification of information geometry, emergent spacetime, and gauge interactions—addressing criticisms about parameter derivations and offering testable predictions.

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1. Core Principle: Temporal Flow as Fundamental

TFP posits that the universe’s fundamental degrees of freedom are quantized 1D temporal flows, $F_i(t)$, indexed by discrete nodes $i = 1, 2, \ldots, N$ in a network. Unlike traditional physics, where fields exist in a pre-existing spacetime, TFP treats time as fundamental and space as emergent. Spatial structure arises from comparisons between flows, such as differences in their rates or phases.

Consider a set of complex-valued flows:

$$\{ F_i(t) \}, \quad F_i \in \mathbb{C}, \quad i = 1, 2, \ldots, N$$

These evolve under a Lagrangian action, with interactions defined by local comparisons, e.g., flow differences:

$$\Delta F_{ij} = F_j - F_i$$

In simulations ($N = 100$), flows take forms like:

• Sinusoidal: $F_i = A \sin(2 \pi i / N)$
  

• Random: $F_i = A (\text{normal}(0, 0.1) + i \, \text{normal}(0, 0.1))$
  

• Oscillatory: $F_i=A{e}^{i0.1i}$

Space emerges from an entropy field, $S_i$, which quantifies flow disorder:

$$S_i = \exp\left( -\frac{(i - i_0)^2}{2 \sigma^2} \right), \quad i_0 = N/2$$

The gradient $\Delta S_{ij} = S_i - S_j$ acts as a pseudo-distance, defining emergent spatial separation—analogous to a metric in general relativity.

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2. Internal Symmetries from Flow Phase Redundancy

Each flow $F_i$ has an internal phase, due to its complex nature. A local phase transformation:

$$F_i(t) \rightarrow e^{i \theta_i(t)} F_i(t)$$

leaves the magnitude $|F_i|^2$ invariant, forming a U(1) symmetry group. This redundancy reflects unobservable reparametrizations of flow orientation, akin to gauge symmetry in QED.

For multicomponent flows (e.g., doublets $F_i = (F_i^1, F_i^2)$), internal rotations yield non-abelian symmetries like SU(2) or SU(3). For simplicity, we focus on U(1), as it directly emulates QED’s electromagnetic gauge symmetry, producing the fine-structure constant:

$$\alpha \approx \frac{1}{137.036}$$

In TFP, the coupling is:

$$\alpha_1 = \frac{\lambda_1}{4 \pi N_1}, \quad \lambda_1 = 1, \quad N_1 = \sum_j W_{ij}$$

Recent simulations yield $N_1 \approx 10.90$, giving:

$$\alpha_1^{-1} = 4 \pi \cdot 10.90 \approx 136.97$$

This symmetry is intrinsic, emerging from the quantization of flows—not imposed externally.

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3. Gauge Fields from Flow Comparisons

Comparing flows at nodes $i$ and $j$ requires accounting for phase differences:

$$\Delta F_{ij} = F_j - F_i = |F_j| e^{i \theta_j} - |F_i| e^{i \theta_i}$$

To ensure invariance under $F_i \rightarrow e^{i \theta_i} F_i$, we introduce a connection field $A_{ij}$, transforming as:

$$A_{ij} \rightarrow A_{ij} + \theta_j - \theta_i$$

This defines a covariant difference:

$$D_{ij} F_i = F_j - F_i - i A_{ij} F_i$$

In the continuum limit, this becomes the gauge covariant derivative:

$$D_\mu F(x) = (\partial_\mu - i A_\mu(x)) F(x)$$

with U(1) gauge symmetry. The field strength:

$$F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$

emerges naturally, as in QED’s electromagnetic field.

In simulations, $A_{ij}$ is implicit in the causal kernel:

$$W_{ij} = \exp\left( -\frac{|\Delta F_{ij}|^2}{\epsilon + (\Delta S_{ij})^2} \right), \quad \epsilon = 10^{-6}$$

Here, $\Delta F_{ij}$ encodes phase and amplitude differences, mediated by gauge-like interactions.

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4. Gauge Invariance in the TFP Lagrangian

The TFP Lagrangian encapsulates flow dynamics:

$$\mathcal{L}_{\text{TFP}} = \frac{1}{2} (D_\mu \Phi_G)^2 - \frac{k}{2} |\Phi_G|^2 - \xi (\partial_\mu S)^2 + \eta S |\Phi_G|^2 - \frac{1}{4} F_{\mu \nu}^a F^{a \mu \nu}$$

• $\Phi_G = \bar{F} + \delta F$: Flow field, with $\bar{F}$ as the mean flow (computed as $\langle F_i \rangle$) and $\delta F$ as fluctuations.

• $D_\mu \Phi_G$: Ensures gauge invariance.

• $F_{\mu \nu}^a$: Gauge field strength, supporting U(1) or non-abelian symmetries.

• $S$: Entropy field, shaping emergent space.

• Parameters: Set $k = \xi = \eta = 1$ for simplicity, ensuring intrinsic derivations.

The kinetic term $\frac{1}{2} (D_\mu \Phi_G)^2$ requires covariance to maintain phase coherence, while $F_{\mu \nu}^a$ arises from the relational consistency of flow comparisons, as derived in Section 3.

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5. Entropy Gradients and Gauge Couplings

Entropy $S_i$ drives emergent spatial structure and modulates gauge couplings. The Lagrangian terms:

$$-\xi (\partial_\mu S)^2 + \eta S |\Phi_G|^2$$

couple entropy to flows, yielding an effective mass:

$$m_{\text{eff}}^2 = -k + 2 \eta S(x)$$

In simulations, $S_i$ is Gaussian:

$$S_i = \exp\left( -\frac{(i - i_0)^2}{2 \sigma^2} \right), \quad \sigma \approx 4.3$$

The kernel $W_{ij}$ depends on:

$$\Delta S_{ij} = S_i - S_j$$

which sets the interaction range. The coupling strength:

$$\alpha_1 = \frac{1}{4 \pi N_1}, \quad N_1 = \sum_j W_{ij}$$

is modulated by $\sigma$. Deriving $N_1$ yields:

$$W_{ij} \approx \exp\left( -\frac{A^2 (\pi (i-j)/N)^2}{\epsilon + \frac{(j-i)^4}{4 \sigma^4} S_i^2} \right)$$

This quantifies the number of effectively interacting flows, linking entropy gradients to gauge couplings—a key feature of TFP’s emergent structure.

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Conclusion

Gauge symmetry in Temporal Flow Physics is not postulated—it emerges naturally from the quantized dynamics of 1D temporal flows. Phase redundancy gives rise to local gauge invariance, flow comparisons produce gauge fields, and entropy gradients shape coupling strengths. The fine-structure constant $\alpha \approx 1/137$ arises intrinsically from the interaction count $N_1$, modulated by flow amplitude and entropy width.

This unification of flow coherence, entropy geometry, and gauge dynamics shows that spacetime and forces may both emerge from the same foundational principle: temporal flow. As simulations become more refined, TFP offers a promising route toward reconciling quantum field theory, thermodynamics, and gravity—from first principles.