2025/11正式 微小位相差理論ーー"Theory of Micro-Phase Differences: A Unified Mathematical Framework"（微小位相差理論:統一的数学的枠組み）

 

Theory of Micro-Phase Differences: A Unified Mathematical Framework

Complete Formalization and Cross-Domain Applications

Yoshiro Shinkawa
Graduate School of Arts, Kyoto University of the Arts
Photography and Moving Images Field (写真映像領域)

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Abstract

We present a unified theoretical framework—the Theory of Micro-Phase Differences (TMPD)—that describes physical, cognitive, and informational phenomena through a single mathematical structure based on complex phase fields and U(1) gauge symmetry. The theory proposes that all observable phenomena arise from microscopic phase differences in an underlying field, with observable quantities determined exclusively by phase gradients, gauge connections, and topological invariants rather than absolute phase values.

This framework emerges from direct phenomenological experience—including precognitive dreams, extended déjà vu episodes, and photographic practice—subsequently formalized into rigorous mathematical language. The theory provides testable predictions across quantum systems, classical fluids, cognitive processes, and social dynamics, while offering a novel perspective on consciousness, perception, and the nature of reality itself.

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I. Foundational Axioms

Axiom 1: Universality of Phase Fields

All physical, cognitive, and informational states are represented by a complex amplitude field:

$$\Psi: M \times \mathbb{R} \to \mathbb{C}$$

whose phase component

$$\phi = \arg(\Psi)$$

encodes difference, structure, and meaning. The amplitude $A = |\Psi|$ represents the intensity or probability density of the field configuration.

**Ontological Commitment**: The phase $\phi$ is considered ontologically primary; amplitude is derivative. This inversion of the usual quantum mechanical interpretation reflects the phenomenological insight that *relationships and differences* (encoded in phase gradients) are more fundamental than *absolute values* (amplitude).

Axiom 2: Local U(1) Gauge Symmetry

The phase field is invariant under local gauge transformations:

$$\phi(x) \to \phi(x) + \lambda(x), \quad \lambda: M \to \mathbb{R}/2\pi\mathbb{Z}$$

Observable quantities depend only on:

• Phase differences: $\Delta\phi = \phi_2 - \phi_1$
• Gauge-covariant derivatives: $D_\mu \phi = \partial_\mu \phi - eA_\mu$
• Field strength (curvature): $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$

This symmetry principle ensures that absolute phase is unobservable—only relative phases and their gradients carry physical meaning.

Axiom 3: Principle of Least Action

System evolution follows from the stationary condition:

$$\delta S[\Psi] = 0$$

where the action $S$ incorporates contributions from amplitude dynamics, phase gradients, topological invariants, and information-geometric terms.

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II. Five-Layer Lagrangian Density

We propose a comprehensive Lagrangian structure:

$$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{kinetic}} + \mathcal{L}_{\text{gradient}} + \mathcal{L}_{\text{potential}} + \mathcal{L}_{\text{info}} + \mathcal{L}_{\text{topo}}$$

Layer 1: Kinetic Term (Temporal Dynamics)

$$\mathcal{L}_{\text{kinetic}} = \frac{1}{2}\left[(\partial_t A)^2 + A^2(\partial_t \phi - eA_0)^2\right]$$

• $A_0$: temporal gauge connection (external potential)

  Theory of Micro-Phase Differences: A Unified Mathematical Framework

  Complete Formalization and Cross-Domain Applications

  Yoshiro Shinkawa
  Graduate School of Arts, Kyoto University of the Arts
  Photography and Moving Images Field (写真映像領域)

  ---

  Abstract

  We present a unified theoretical framework—the Theory of Micro-Phase Differences (TMPD)—that describes physical, cognitive, and informational phenomena through a single mathematical structure based on complex phase fields and U(1) gauge symmetry. The theory proposes that all observable phenomena arise from microscopic phase differences in an underlying field, with observable quantities determined exclusively by phase gradients, gauge connections, and topological invariants rather than absolute phase values.

  This framework emerges from direct phenomenological experience—including precognitive dreams, extended déjà vu episodes, and photographic practice—subsequently formalized into rigorous mathematical language. The theory provides testable predictions across quantum systems, classical fluids, cognitive processes, and social dynamics, while offering a novel perspective on consciousness, perception, and the nature of reality itself.

  ---

  I. Foundational Axioms

  Axiom 1: Universality of Phase Fields

  All physical, cognitive, and informational states are represented by a complex amplitude field:

  $$\Psi: M \times \mathbb{R} \to \mathbb{C}$$

  whose phase component

  $$\phi = \arg(\Psi)$$

  encodes difference, structure, and meaning. The amplitude $A = |\Psi|$ represents the intensity or probability density of the field configuration.

  **Ontological Commitment**: The phase $\phi$ is considered ontologically primary; amplitude is derivative. This inversion of the usual quantum mechanical interpretation reflects the phenomenological insight that *relationships and differences* (encoded in phase gradients) are more fundamental than *absolute values* (amplitude).

  Axiom 2: Local U(1) Gauge Symmetry

  The phase field is invariant under local gauge transformations:

  $$\phi(x) \to \phi(x) + \lambda(x), \quad \lambda: M \to \mathbb{R}/2\pi\mathbb{Z}$$

  Observable quantities depend only on:

• Phase differences: $\Delta\phi = \phi_2 - \phi_1$
• Gauge-covariant derivatives: $D_\mu \phi = \partial_\mu \phi - eA_\mu$
• Field strength (curvature): $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$

This symmetry principle ensures that absolute phase is unobservable—only relative phases and their gradients carry physical meaning.

Axiom 3: Principle of Least Action

System evolution follows from the stationary condition:

$$\delta S[\Psi] = 0$$

where the action $S$ incorporates contributions from amplitude dynamics, phase gradients, topological invariants, and information-geometric terms.

---

II. Five-Layer Lagrangian Density

We propose a comprehensive Lagrangian structure:

$$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{kinetic}} + \mathcal{L}_{\text{gradient}} + \mathcal{L}_{\text{potential}} + \mathcal{L}_{\text{info}} + \mathcal{L}_{\text{topo}}$$

Layer 1: Kinetic Term (Temporal Dynamics)

$$\mathcal{L}_{\text{kinetic}} = \frac{1}{2}\left[(\partial_t A)^2 + A^2(\partial_t \phi - eA_0)^2\right]$$

• $A_0$: temporal gauge connection (external potential)
• $e$: coupling constant (electric charge in physics; attention weight in cognitive systems)

This term governs temporal evolution and couples amplitude and phase dynamics.

Layer 2: Gradient Energy (Spatial Structure)

$$\mathcal{L}_{\text{gradient}} = -\frac{c_1}{2}\left(|\nabla A|^2 + A^2|\nabla \phi - e\mathbf{A}|^2\right)$$

• $\mathbf{A}$: spatial gauge connection (vector potential)
• $c_1$: phase stiffness constant (controls correlation length and coherence)

The gauge-covariant gradient $\nabla \phi - e\mathbf{A}$ ensures gauge invariance while allowing for vortex solutions.

Layer 3: Self-Interaction Potential

$$\mathcal{L}_{\text{potential}} = -\frac{\lambda}{4}(A^2 - \eta^2)^2 - g(x)A^2$$

• Mexican-hat potential with $\eta \neq 0$ induces spontaneous symmetry breaking
• $g(x)$: external driving field (context or observational bias)
• $\lambda$: self-interaction strength

This term generates non-trivial vacuum structure and enables phase transitions.

Layer 4: Information-Geometric Term

$$\mathcal{L}_{\text{info}} = -\alpha\left[D_{\text{KL}}(\Psi \| \Psi_{\text{prior}}) + \kappa F[\Psi]\right]$$

where:

• $D_{\text{KL}}$: Kullback-Leibler divergence
• $F[\Psi] = -\int A^2 \log A^2$: Fisher information functional
• $\alpha, \kappa$: information constraint strengths

This term embeds Bayesian inference, variational free energy, and belief updating directly into the phase-field dynamics—a crucial innovation for cognitive applications.

Layer 5: Topological Term

$$\mathcal{L}_{\text{topo}} = \beta \sum_i n_i \delta(x - x_i) + \gamma \int F \wedge F$$

• First term: vortex singularities with winding number $n_i \in \mathbb{Z}$
• Second term: Chern-Simons or $\theta$-term (4D topological curvature)

This accounts for quantized defects, topological invariants, and emergent discrete structure.

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III. Field Equations

From $\delta S = 0$, we derive coupled equations for amplitude and phase:

Amplitude Equation

$$\partial_t^2 A - c_1 \nabla^2 A + A(\partial_t \phi - eA_0)^2 + c_1 A|\nabla \phi - e\mathbf{A}|^2 + \frac{\partial V_{\text{eff}}}{\partial A} = 0$$

where the effective potential is:

$$V_{\text{eff}} = \frac{\lambda}{4}(A^2 - \eta^2)^2 + g(x)A^2 + \alpha I_{\text{info}}$$

Phase Equation (Continuity-like)

$$\partial_t[A^2(\partial_t \phi - eA_0)] - c_1 \nabla \cdot [A^2(\nabla \phi - e\mathbf{A})] + J_{\text{topo}} = 0$$

• $J_{\text{topo}}$: topological source term (vortex creation/annihilation)

This equation exhibits the structure of a conservation law for the phase current $j^\mu = A^2 D^\mu \phi$.

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IV. Cross-Domain Applications and Testable Predictions

Quantum Systems

• Interpretation: $A = \sqrt{\rho}$

• $e$: coupling constant (electric charge in physics; attention weight in cognitive systems)

This term governs temporal evolution and couples amplitude and phase dynamics.

Layer 2: Gradient Energy (Spatial Structure)

$$\mathcal{L}_{\text{gradient}} = -\frac{c_1}{2}\left(|\nabla A|^2 + A^2|\nabla \phi - e\mathbf{A}|^2\right)$$

• $\mathbf{A}$: spatial gauge connection (vector potential)
• $c_1$: phase stiffness constant (controls correlation length and coherence)

The gauge-covariant gradient $\nabla \phi - e\mathbf{A}$ ensures gauge invariance while allowing for vortex solutions.

Layer 3: Self-Interaction Potential

$$\mathcal{L}_{\text{potential}} = -\frac{\lambda}{4}(A^2 - \eta^2)^2 - g(x)A^2$$

• Mexican-hat potential with $\eta \neq 0$ induces spontaneous symmetry breaking
• $g(x)$: external driving field (context or observational bias)
• $\lambda$: self-interaction strength

This term generates non-trivial vacuum structure and enables phase transitions.

Layer 4: Information-Geometric Term

$$\mathcal{L}_{\text{info}} = -\alpha\left[D_{\text{KL}}(\Psi \| \Psi_{\text{prior}}) + \kappa F[\Psi]\right]$$

where:

• $D_{\text{KL}}$: Kullback-Leibler divergence
• $F[\Psi] = -\int A^2 \log A^2$: Fisher information functional
• $\alpha, \kappa$: information constraint strengths

This term embeds Bayesian inference, variational free energy, and belief updating directly into the phase-field dynamics—a crucial innovation for cognitive applications.

Layer 5: Topological Term

$$\mathcal{L}_{\text{topo}} = \beta \sum_i n_i \delta(x - x_i) + \gamma \int F \wedge F$$

• First term: vortex singularities with winding number $n_i \in \mathbb{Z}$
• Second term: Chern-Simons or $\theta$-term (4D topological curvature)

This accounts for quantized defects, topological invariants, and emergent discrete structure.

---

III. Field Equations

From $\delta S = 0$, we derive coupled equations for amplitude and phase:

Amplitude Equation

$$\partial_t^2 A - c_1 \nabla^2 A + A(\partial_t \phi - eA_0)^2 + c_1 A|\nabla \phi - e\mathbf{A}|^2 + \frac{\partial V_{\text{eff}}}{\partial A} = 0$$

where the effective potential is:

$$V_{\text{eff}} = \frac{\lambda}{4}(A^2 - \eta^2)^2 + g(x)A^2 + \alpha I_{\text{info}}$$

Phase Equation (Continuity-like)

$$\partial_t[A^2(\partial_t \phi - eA_0)] - c_1 \nabla \cdot [A^2(\nabla \phi - e\mathbf{A})] + J_{\text{topo}} = 0$$

• $J_{\text{topo}}$: topological source term (vortex creation/annihilation)

This equation exhibits the structure of a conservation law for the phase current $j^\mu = A^2 D^\mu \phi$.

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IV. Cross-Domain Applications and Testable Predictions

Quantum Systems

• Interpretation: $A = \sqrt{\rho}$