英語論文用 LaTeX 形式で出力 Unified Formalization of the Theory of Micro-Phase Differences

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\title{Unified Formalization of the Theory of Micro-Phase Differences \\ \large A Complete Mathematical Framework}
\author{Yoshirou Shinkawa}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
We present a unified framework for the theory of micro-phase differences, combining physical, cognitive, and informational phenomena into a single axiomatic and variational formalism. The approach defines a complex phase field whose amplitude and phase encode state, meaning, and structure, integrates gradient, potential, information-geometric, and topological contributions in the Lagrangian, and provides testable predictions across domains. Numerical schemes and experimental protocols are also outlined.
\end{abstract}

\tableofcontents

\section{Axiomatic Foundations}

\subsection*{Axiom 1 (Universality of Phase Fields)}
All physical, cognitive, and informational states are represented as a complex amplitude field
\[
\Psi: M \times \mathbb{R} \rightarrow \mathbb{C},
\]
whose phase component \(\phi\) encodes difference, structure, and meaning.

\subsection*{Axiom 2 (Local \(U(1)\) Gauge Symmetry)}
The phase field obeys local gauge transformations
\[
\phi(x) \rightarrow \phi(x) + \lambda(x), \quad \lambda: M \rightarrow \mathbb{R}/2\pi\mathbb{Z},
\]
and physical laws are invariant. Observables depend only on phase differences, connections, and curvature.

\subsection*{Axiom 3 (Principle of Least Action)}
The system evolves according to the stationary condition of the action functional \(S[\Psi]\):
\[
\delta S = 0,
\]
with \(S[\Psi]\) integrating amplitude dynamics, phase gradients, topological invariants, and information-geometric terms.

\section{Full Lagrangian Density}

The proposed five-layer Lagrangian reads:
\[
\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{kinetic}} + \mathcal{L}_{\text{gradient}} + \mathcal{L}_{\text{potential}} + \mathcal{L}_{\text{info-geom}} + \mathcal{L}_{\text{topological}}.
\]

\subsection*{Layer 1: Kinetic Term}
\[
\mathcal{L}_{\text{kinetic}} = \frac{1}{2} \left[ (\partial_t A)^2 + A^2 (\partial_t \phi - e A_0)^2 \right],
\]
where \(A_0\) is the temporal connection (external potential), and \(e\) is the coupling constant.

\subsection*{Layer 2: Gradient Energy}
\[
\mathcal{L}_{\text{gradient}} = -\frac{c_1}{2} \left( |\nabla A|^2 + A^2 |\nabla \phi - e A|^2 \right),
\]
with spatial connection \(A\) and phase stiffness \(c_1\).

\subsection*{Layer 3: Self-Interaction Potential}
\[
\mathcal{L}_{\text{potential}} = -\frac{\lambda}{4} (A^2 - \eta^2)^2 - g(x) A^2,
\]
allowing spontaneous symmetry breaking for \(\eta \neq 0\), and \(g(x)\) represents external driving fields.

\subsection*{Layer 4: Information-Geometric Term}
\[
\mathcal{L}_{\text{info-geom}} = -\alpha \left[ KL(\Psi \| \Psi_{\text{prior}}) + \kappa F[\Psi] \right],
\]
where \(KL\) is Kullback–Leibler divergence, \(F[\Psi] = -\int A^2 \log A^2\) is a Fisher information penalty, and \(\alpha, \kappa\) are information constraint strengths.

\subsection*{Layer 5: Topological Term}
\[
\mathcal{L}_{\text{topological}} = \beta \sum_i n_i \delta(x - x_i) + \gamma \int F \wedge F,
\]
with vortex cores \(x_i\) and winding numbers \(n_i \in \mathbb{Z}\), and a Chern–Simons term for 4D topology.

\section{Field Equations}

\subsection*{Amplitude Field}
\[
\partial_t^2 A - c_1 \nabla^2 A + A(\partial_t \phi - e A_0)^2 + c_1 A |\nabla \phi - e A|^2 + \frac{\partial V_{\text{eff}}}{\partial A} = 0,
\]
\[
V_{\text{eff}} = \frac{\lambda}{4} (A^2 - \eta^2)^2 + g(x) A^2 + \alpha I_{\text{info}}.
\]

\subsection*{Phase Field (Continuity Equation)}
\[
\partial_t \left[ A^2 (\partial_t \phi - e A_0) \right] - c_1 \nabla \cdot \left[ A^2 (\nabla \phi - e A) \right] + (\text{topological source}) = 0.
\]

\section{Applications Across Domains}

\begin{itemize}
    \item \textbf{Quantum systems:} \(A = \rho\), \(\phi\) = quantum phase. Recovers Gross–Pitaevskii, quantized vortices.
    \item \textbf{Classical fields / fluids:} \(\nabla \phi\) = velocity potential. Predicts quantized circulation, Kelvin-wave dispersion.
    \item \textbf{Cognition / semantics:} \(\phi\) encodes meaning. Phase gradients correlate with embedding geometry.
    \item \textbf{Collective/social phenomena:} \(\phi\) = synchronization phase. Kuramoto model emerges in continuum limit.
\end{itemize}

\section{Topological Invariants}

\subsection*{Winding Number}
\[
W = \frac{1}{2\pi} \oint_{\partial D} \nabla \phi \cdot d\ell \in \mathbb{Z}.
\]

\subsection*{Chern Number (2D)}
\[
C = \frac{1}{2\pi} \int_M F \in \mathbb{Z}.
\]

\subsection*{Berry Phase}
\[
\gamma = \oint_C \mathbf{A} \cdot d\mathbf{R} = \int_\Sigma F.
\]

\section{Numerical Implementation}

\subsection*{Crank–Nicolson Time Evolution}
\[
\frac{\Psi^{n+1} - \Psi^n}{\Delta t} = \frac{1}{2} \left[ H[\Psi^{n+1}] + H[\Psi^n] \right].
\]

\subsection*{Finite Differences}
\[
\nabla \phi(x_i) \approx \frac{\phi(x_{i+1}) - \phi(x_i)}{\Delta x}.
\]

\subsection*{Vortex Tracking}
\begin{itemize}
    \item Compute $\nabla \times (\nabla \phi)$.
    \item Identify singular points.
    \item Track trajectories over time.
\end{itemize}

\section{Experimental Protocols}

\begin{itemize}
    \item \textbf{BEC experiments:} Vortex lattice formation in rotating trap; $\Omega_c \propto \lambda \eta^2 / c_1$.
    \item \textbf{Cognitive tasks:} fMRI phase synchrony correlates with task difficulty; harder tasks → higher $\int A^2 |\nabla \phi|^2$.
    \item \textbf{Societal dynamics:} Phase reconstruction from SNS diffusion; influencers = high-amplitude hubs.
\end{itemize}

\section{Open Problems}

\begin{enumerate}
    \item Quantum-classical transition under stochastic phase noise.
    \item Determination of information functional $I_{\text{info}}$ from cognitive data.
    \item Classification and stability of 3D topological solitons (skyrmions, hopfions).
    \item Interpreting neural network training as phase-field gradient flow.
\end{enumerate}

\section{Minimal Python Implementation}

\begin{verbatim}
import numpy as np

class PhaseFieldUnified:
    def __init__(self, grid_size, c1, lam, eta, alpha, beta):
        self.N = grid_size
        self.c1, self.lam, self.eta = c1, lam, eta
        self.alpha, self.beta = alpha, beta
        self.A = eta * np.ones((self.N, self.N))
        self.phi = np.random.uniform(0, 2*np.pi, (self.N, self.N))

    def laplacian(self, field):
        return (np.roll(field,1,0) + np.roll(field,-1,0) +
                np.roll(field,1,1) + np.roll(field,-1,1) - 4*field)

    def step(self, dt):
        dA_dt = (self.c1 * self.laplacian(self.A) -
                 self.lam * self.A * (self.A**2 - self.eta**2))
        grad_phi_x = np.roll(self.phi, -1, axis=1) - self.phi
        dphi_dt = self.c1 * self.laplacian(self.phi) / (self.A**2)
        self.A += dt * dA_dt
        self.phi = (self.phi + dt * dphi_dt) % (2*np.pi)
\end{verbatim}

\section{Conclusion}

This framework provides a mathematically rigorous, physically realizable, cognitively interpretable, and numerically testable unified theory of micro-phase differences. Future work includes domain-specific calibration, empirical determination of information terms, analysis of topological phase transitions, and embedding in machine learning architectures.

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