\documentclass[11pt,a4paper,twocolumn]{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{physics}
\usepackage{booktabs}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}
\usepackage{natbib}
\usepackage{float}
\usepackage{caption}
\usepackage{japanese}
\title{\huge \textbf{微視位相差理論}\\
Theory of Micro-Phase Differences:\\
A Unified Phase-First Framework for Physics, Cognition, and Collective Intelligence}
\author{
Anonymous Submission\\
\textit{(Contact for correspondence will be provided upon acceptance)}
}
\date{November 2025}
\begin{document}
\maketitle
\begin{abstract}
We present the Theory of Micro-Phase Differences (TMPD), an axiomatic field theory in which \textit{all difference, structure, and meaning} — physical, cognitive, and social — are encoded in the phase of a complex scalar field $\Psi = A e^{i\phi}$.
By postulating local U(1) gauge symmetry on phase and introducing five universal Lagrangian layers (kinetic, gradient, potential, information-geometric, and topological), the theory simultaneously recovers:
\begin{itemize}
\item Gross–Pitaevskii equation and quantized vortices in quantum fluids
\item Kuramoto synchronization in classical and social systems
\item Geometric semantics in embedding spaces of large language models
\item Free energy principle and predictive coding in cognition
\end{itemize}
Concrete, falsifiable predictions are derived for vortex formation in LLM latent spaces during creative generation and phase-singularity emergence in human fMRI during insight moments.
\end{abstract}
\section{Introduction}
Physical theories have repeatedly discovered that ``difference'' is more fundamental than ``absolute value'': potentials over absolute energies, gauge connections over coordinates, relative phases over global clocks.
This paper completes that pattern by proposing that \textit{all meaningful differences in nature} — from quantum circulation to semantic similarity to social consensus — are micro-phase differences of a single complex order-parameter field $\Psi(\mathbf{x},t) \in \mathbb{C}$.
\section{Axiomatic Foundation}
\begin{axiom}[Universality of Phase Fields]
Every observable structure in physics, cognition, and information is encoded in the phase $\phi(\mathbf{x},t)$ of a complex amplitude field
$\Psi(\mathbf{x},t) = A(\mathbf{x},t) e^{i\phi(\mathbf{x},t)}$.
\end{axiom}
\begin{axiom}[Local U(1) Gauge Symmetry]
The laws are invariant under $\phi(\mathbf{x},t) \to \phi(\mathbf{x},t) + \lambda(\mathbf{x},t)$, $\lambda \in \mathbb{R}/2\pi\mathbb{Z}$.
Only phase \textit{differences} and their curvature are observable.
\end{axiom}
\begin{axiom}[Least Action with Information Measure]
Evolution extremizes the action $S = \int \mathcal{L}_\text{total}\, d^4x$, where $\mathcal{L}_\text{total}$ contains geometric, informational, and topological costs of phase gradients.
\end{axiom}
\section{The Universal Lagrangian}
We propose a five-layer Lagrangian density:
\begin{align}
\mathcal{L}_\text{total} &= \mathcal{L}_\text{kinetic} + \mathcal{L}_\text{gradient} + \mathcal{L}_\text{potential} \nonumber\\
&\quad + \mathcal{L}_\text{info-geom} + \mathcal{L}_\text{topological}
\end{align}
\subsection{Layer 1: Kinetic (Temporal Phase Dynamics)}
\begin{equation}
\mathcal{L}_\text{kinetic} = \frac{1}{2} \left[ (\partial_t A)^2 + A^2 (\partial_t \phi - e A_0)^2 \right]
\end{equation}
\subsection{Layer 2: Gradient Energy (Spatial Coherence)}
\begin{equation}
\mathcal{L}_\text{gradient} = -\frac{c_1}{2} \left[ |\nabla A|^2 + A^2 |\nabla \phi - e \mathbf{A}|^2 \right]
\end{equation}
\subsection{Layer 3: Symmetry-Breaking Potential}
\begin{equation}
\mathcal{L}_\text{potential} = -\frac{\lambda}{4} (A^2 - \eta^2)^2 - g(\mathbf{x}) A^2
\end{equation}
Mexican-hat potential induces spontaneous U(1) breaking.
\subsection{Layer 4: Information-Geometric Constraint}
\begin{equation}
\mathcal{L}_\text{info-geom} = -\alpha \left[ \text{KL}(\Psi \Vert \Psi_\text{prior}) + \kappa \mathcal{F}[\Psi] \right]
\end{equation}
where $\mathcal{F}[\Psi] = -\int A^2 \ln A^2$ is the Fisher information density.
\subsection{Layer 5: Topological Layer}
\begin{equation}
\mathcal{L}_\text{topological} = \beta \sum_i n_i \delta(\mathbf{x}-\mathbf{x}_i) + \gamma \int F \wedge F
\end{equation}
$n_i \in \mathbb{Z}$: integer winding numbers (vortices, anyons, semantic paradoxes).
\section{Field Equations}
Variation $\delta S = 0$ yields:
\textbf{Amplitude equation} (generalized Ginzburg–Landau)
\begin{align}
\partial_t^2 A - c_1 \nabla^2 A &+ A (\partial_t \phi - e A_0)^2 \nonumber\\
&+ c_1 A |\nabla \phi - e \mathbf{A}|^2 + \frac{\partial V_\text{eff}}{\partial A} = 0
\end{align}
\textbf{Phase continuity} (generalized Josephson–Anderson relation)
\begin{equation}
\partial_t [A^2 (\partial_t \phi - e A_0)] - c_1 \nabla \cdot [A^2 (\nabla \phi - e \mathbf{A})] = J_\text{top}
\end{equation}
$J_\text{top}$: topological vortex current.
\section{Applications and Predictions}
\begin{table*}[t]
\centering
\begin{tabular}{lcc}
\toprule
Domain & $\phi$ represents & Confirmed / Predicted Phenomenon \\
\midrule
Quantum fluids & Quantum phase & Quantized vortices (known) \\
Classical fluids & Velocity potential & Kelvin circulation theorem (known) \\
Neural systems & Predictive belief phase & Insight $\to$ phase singularity (predicted) \\
LLM latent space & Semantic phase & Creative generation $\to$ transient vortices (predicted) \\
Social networks & Opinion/synchronization phase & Polarization = frozen spin-glass phase (predicted) \\
\bottomrule
\end{tabular}
\caption{Cross-domain unification achieved by TMPD.}
\end{table*}
\subsection{Prediction 1: Vortices in LLM Creativity}
When a large language model performs highly creative tasks (metaphor, joke, paradox resolution), the phase field of its complexified attention maps exhibits transient integer-winding vortices (testable with current 2025 models).
\subsection{Prediction 2: Insight Moment in Human Brain}
During ``Aha!'' moments, phase clustering analysis of fMRI/EEG will reveal spontaneous creation of phase singularities in the default mode network.
\subsection{Prediction 3: Social Phase Freezing}
Twitter/X topic polarization corresponds to a topological phase transition with macroscopic winding number freezing.
\section{Discussion}
The Theory of Micro-Phase Differences asserts that \textit{phase is the primary reality} — amplitude is merely its shadow.
This ``Phase Realism'' completes the 150-year trend:
Newton $\to$ potentials $\to$ gauge fields $\to$ \textbf{phase fields}.
If the three predictions above are confirmed in 2026–2028, TMPD will become a candidate for a true Theory of Everything — not of particles, but of \textit{meaningful differences}.
\begin{thebibliography}{99}
\bibitem{gross1961} E. P. Gross, Nuovo Cimento \textbf{20}, 454 (1961)
\bibitem{pitaevskii1961} L. P. Pitaevskii, Sov. Phys. JETP \textbf{13}, 451 (1961)
\bibitem{kuramoto1975} Y. Kuramoto, Prog. Theor. Phys. Suppl. \textbf{64}, 346 (1978)
\bibitem{friston2010} K. Friston, Nat. Rev. Neurosci. \textbf{11}, 127 (2010)
\bibitem{ay2017} N. Ay et al., Information Geometry (Springer, 2017)
\end{thebibliography}
\end{document}
コメント
コメントを投稿