Unified Time: The Essence of Time

“Time may not be a single entity, nor unrelated fragments—GLS theory proposes that various time concepts might be different projections of the same mathematical object.”

🎯 Chapter Goals

In physics, “time” appears in many forms:

• Relativity: Proper time $\tau$, Killing time, conformal time
• Quantum Mechanics: Phase $\varphi =-Et/\hslash$, evolution operator $U(t)$
• Scattering Theory: Wigner-Smith group delay $Q(\omega )$
• Cosmology: Cosmic time $t$, redshift $z$
• Information Theory: Modular time (Tomita-Takesaki)

Are these “times” different?

GLS Theory Perspective: They can be viewed as projections of the same “unified time scale” in different physical contexts.

graph TB
    UT["Unified Time Scale<br/>[τ]"] --> P["Phase Projection<br/>φ'(ω)/π"]
    UT --> R["Spectral Projection<br/>ρ_rel(ω)"]
    UT --> Q["Scattering Projection<br/>tr Q(ω)/2π"]
    UT --> G["Geometric Projection<br/>d Proper Time"]
    UT --> M["Modular Projection<br/>s_mod"]
    UT --> C["Cosmic Projection<br/>1+z"]

    style UT fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style P fill:#e1f5ff
    style R fill:#e1ffe1
    style Q fill:#ffe1e1

🌟 Core Formula: Time Scale Identity

$$\boxed{\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )}$$

This equation indicates:

1. Phase derivative ${\phi }^{\prime }(\omega )/\pi$ (quantum)
2. Relative density of states ${\rho }_{\text{rel}}(\omega )$ (spectral theory)
3. Group delay trace $\text{tr}Q(\omega )/(2\pi )$ (scattering)
4. Normalized scale $\kappa (\omega )$ (geometry)

Exhibit profound unity in mathematical structure.

🗺️ Chapter Roadmap

Station 1: Phase and Proper Time

Proposition: For a particle of mass $m$, phase along worldline can be expressed as:

$$\varphi =\frac{m{c}^2}{\hslash }\int d\tau$$

Physical Meaning: Phase can be viewed as the “quantum counter” of proper time.

Station 2: Scattering Phase and Group Delay

Wigner-Smith Matrix:

$$Q(\omega )=-iS(\omega {)}^{†}\frac{\partial S(\omega )}{\partial \omega }$$

Trace:

$$\text{tr}Q(\omega )=\frac{\partial \Phi (\omega )}{\partial \omega }$$

where $\Phi =\arg \det S$ is the total scattering phase.

Physical Meaning: Group delay measures time delay of wave packet in scattering process.

Station 3: Spectral Shift Function

Birman-Kreĭn Formula:

$$\det S(\omega )=e^{-2\pi i\xi (\omega )}$$

Relative Density of States:

$${\rho }_{\text{rel}}(\omega )=-\frac{d\xi }{d\omega }$$

Physical Meaning: Spectral shift function reflects energy level shifts caused by interactions.

Station 4: Time Scale Identity (⭐ Core)

Proof Outline:

From Birman-Kreĭn formula:

$$\Phi (\omega )=\arg \det S=-2\pi \xi (\omega )$$

Taking derivative:

$$\frac{d\Phi }{d\omega }=-2\pi \frac{d\xi }{d\omega }=2\pi {\rho }_{\text{rel}}(\omega )$$

Also because:

$$\text{tr}Q=\frac{d\Phi }{d\omega }$$

Therefore:

$$\frac{1}{2\pi }\text{tr}Q={\rho }_{\text{rel}}$$

Establishes an identity relation.

Station 5: Geometric Time

In curved spacetime, there exist various “time” definitions:

Key: They can all be classified into the equivalence class of unified scale.

Station 6: Modular Time

Tomita-Takesaki Theory:

For state $\omega$ and algebra $\mathcal{A}$, define modular flow:

$${\sigma }_t(A)={\Delta }^{it}A{\Delta }^{-it}$$

Connes-Rovelli Thermal Time Hypothesis: Modular flow parameter $t$ might correspond to physical time.

Relation to Geometric Time: Under appropriate conditions, modular time is proven to be affinely equivalent to Killing time.

Station 7: Cosmological Redshift

$$1+z=\frac{a(t_0)}{a(t_e)}=\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}$$

Physical Meaning: Redshift can be interpreted as the shear effect of cosmic scale factor evolution on “phase rhythm”.

Station 8: Time as Optimal Path of Generalized Entropy

Core Idea: Time evolution can be modeled as the optimal path chosen by system in generalized entropy landscape.

$$\delta S_{\text{gen}}=0$$

Physical Meaning: Time’s “choice” makes generalized entropy of causal history stationary.

Station 9: Unification of Time-Geometry-Interaction

Core Formula: Total connection attempts to unify all “forces”

$$\Omega ={\omega }_{\text{LC}}\oplus A_{\text{YM}}\oplus {\Gamma }_{\text{res}}$$

Physical Meaning: Gravity, gauge forces, resolution forces might all be projections of unified time geometry in different directions.

Station 10: Topological Invariants and Time

Topological “DNA” of Time:

1. Time scale master $\kappa (\omega )$
2. ${\mathbb{Z}}_2$ holonomy ${\nu }_{\sqrt{S}}(\gamma )$
3. Relative topological class $[K]\in H^2(Y,\partial Y;{\mathbb{Z}}_2)$

Physical Meaning: Fermion statistics might originate from topological invariants of time structure.

Station 11: Boundary Language Unified Framework

Boundary Three Axioms:

• A1: Conservation and Flux (bank statement)
• A2: Time Generation (revolving door)
• A3: Monotonicity and Consistency (second law of thermodynamics)

Physical Meaning: Time dynamics are manifested on the boundary.

Station 12: Time Domains and Solvable Models

Three Major Definition Domains of Scale Identity:

1. Elastic-Unitary Domain: Standard scattering theory
2. Non-Unitary-Absorption Domain: Open systems
3. Long-Range Potential Domain: Gravity and cosmology

Windowed Clock: ${\Theta }_{\Delta }(\omega )=({\rho }_{\text{rel}}\ast P_{\Delta })(\omega )$ is used to solve negative delay problem.

📊 Physical Picture of Unified Time Scale

graph TB
    subgraph "Quantum Layer"
        P["Phase φ = ∫ mc²dτ/ℏ"]
        Q["Group Delay Q(ω)"]
        S["Scattering Phase Φ(ω)"]
    end

    subgraph "Spectral Layer"
        X["Spectral Shift Function ξ(ω)"]
        R["Relative Density of States ρ_rel"]
    end

    subgraph "Geometric Layer"
        T["Proper Time τ"]
        K["Killing Time"]
        N["ADM Lapse"]
        L["Null Parameter λ"]
    end

    subgraph "Information Layer"
        M["Modular Flow σ_t"]
        E["Generalized Entropy S_gen"]
    end

    subgraph "Cosmic Layer"
        C["Conformal Time η"]
        Z["Redshift 1+z"]
    end

    P --> UT["Unified Time Scale<br/>κ(ω)"]
    Q --> UT
    S --> UT
    X --> UT
    R --> UT
    T --> UT
    K --> UT
    N --> UT
    L --> UT
    M --> UT
    E --> UT
    C --> UT
    Z --> UT

    style UT fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px

🎨 Analogy: Projection of a Cube

Imagine a four-dimensional cube (tesseract):

        * ---- *
       /|     /|
      * ---- * |
      | * ---|-*
      |/     |/
      * ---- *

Viewed from different angles, get different three-dimensional projections:

• From “phase direction” → Phase time $\varphi$
• From “spectral direction” → Density of states ${\rho }_{\text{rel}}$
• From “scattering direction” → Group delay $Q$
• From “geometric direction” → Proper time $\tau$

They can be understood as different projections of the same high-dimensional object (unified time scale).

🔑 Three Axioms

GLS theory is based on three axiomatic assumptions about time:

Axiom I: Causal Ordering

Assume there exists a strictly increasing time function that makes fundamental equations local (hyperbolic) form.

$$p\prec q⟺\tau (p)<\tau (q)$$

Axiom II: Unitary Evolution

Assume there exists a strongly continuous unitary group $U(t)=e^{-iHt}$, phase-time relation determined by stationary phase approximation.

$$\varphi =\int \frac{m{c}^2}{\hslash }d\tau$$

Axiom III: Entropy Monotonicity/Extremum

Generalized entropy $S_{\text{gen}}$ along time evolution satisfies monotonicity and extremum conditions.

$$\frac{d{S}_{\text{gen}}}{d\lambda }\ge 0,\delta S_{\text{gen}}=0$$

Theorem (mutually imply):

In semiclassical-holographic window:

$$\text{Axiom I + II}\iff \text{Time Scale Identity}\Rightarrow \text{Axiom III}\Rightarrow \text{Einstein Equation}$$

📐 Mathematical Structure

Equivalence Class Definition

Definition (Unified Time Scale Equivalence Class):

$$[\tau ]\sim \{\tau ,t,t_K,N,{\lambda }_{\text{null}},u,v,\eta ,{\omega }^{-1},z,s_{\text{mod}}\}$$

Members are converted to each other through monotonic rescaling, aiming to make dynamics local, causally ordered, and entropy structure simplest.

Affine Transformation

Within equivalence class, different times are related by affine transformation:

$$t_1=\alpha t_2+\beta$$

where $\alpha >0$ (preserving time arrow).

Examples:

• Killing time $\leftrightarrow$ Proper time: $\tau =\sqrt{-g_{tt}}\cdot t$
• ADM time $\leftrightarrow$ Proper time: $\tau =N\cdot t$
• Conformal time $\leftrightarrow$ Cosmic time: $\eta =\int dt/a(t)$

🌊 Core Insight: Unification of Time and Geometry

Traditional View:

• Time is external parameter
• Geometry is dynamics
• Two are independent

GLS View:

• Time is viewed as intrinsic structure of geometry
• $\varphi =(m{c}^2/\hslash )\int d\tau$
• Equivalence of Time, Phase, and Geometry

$$\boxed{\text{Time}\equiv \text{Phase}\equiv \text{Proper Time}\equiv \text{Unified Scale}}$$

🎓 Chapter Content Overview

Core Articles (Required Reading)

1. 01-phase-and-proper-time.md - Phase and Proper Time

   • Argument for $\varphi =(m{c}^2/\hslash )\int d\tau$
   • Worldline path integral

2. 04-time-scale-identity.md - Time Scale Identity (⭐⭐⭐)

   • Complete derivation of identity
   • Birman-Kreĭn formula
   • Proof of four-way unification

3. 08-time-summary.md - Unified Time Summary

   • Complete picture review

In-Depth Articles (Recommended)

4. 02-scattering-phase.md - Scattering Phase and Group Delay

   • Wigner-Smith matrix
   • Operational definition of time delay

5. 03-spectral-shift.md - Spectral Shift Function

   • Birman-Kreĭn theorem
   • Relative density of states

6. 05-geometric-times.md - Geometric Times

   • Killing, ADM, null, conformal
   • Unification of four times

7. 06-modular-time.md - Modular Time

   • Tomita-Takesaki theory
   • Thermal time hypothesis

8. 07-cosmological-redshift.md - Cosmological Redshift

   • Redshift as phase shear
   • Time structure of FRW universe

🚀 Learning Paths

Quick Path (Understand Core Ideas)

Read: 00-time-overview (this article) → 01-phase-and-proper-time → 04-time-scale-identity → 08-time-summary

Gain: Understand core idea of “time is geometry”.

Solid Path (Master Derivation)

Read all articles in order, complete exercises.

Gain: Able to derive time scale identity.

Research Path (Deep Technical)

Read this chapter + original paper: unified-time-scale-geometry.md

Gain: Research-level understanding, able to apply to new problems.

📝 Key Terms Chinese-English Glossary

🤔 Preview Questions

Before starting, think about these questions:

1. Conceptual Questions:

   • What is “time”? Is there absolute time?
   • Is “time” in quantum mechanics the same as in relativity?
   • What does “delay” mean in scattering theory?

2. Mathematical Questions:

   • What is phase? What is the relationship between phase and frequency?
   • What is spectrum? What does spectral shift mean?
   • What is unitary matrix? Why is $S(\omega )$ unitary?

3. Physical Questions:

   • Why does time slow down near black holes?
   • What is the physical cause of cosmological redshift?
   • Is there a relationship between quantum entanglement and time?

Answers to these questions are all in the unified time scale!

🌟 Why Is This Chapter Important?

Unified Time is the heart of GLS theory:

1. Connects Quantum and Geometry: $\varphi =(m{c}^2/\hslash )\int d\tau$
2. Unifies All Time Concepts: Time scale identity
3. Provides Foundation for IGVP: Time is parameter of entropy evolution
4. Reveals Essence of Time: Time is geometry, geometry is phase

Unified time scale is the cornerstone of GLS theory.

graph TB
    T["Unified Time Scale"] --> I["IGVP<br/>δS_gen = 0"]
    T --> B["Boundary Theory<br/>Spectral Triplet"]
    T --> C["Causal Structure<br/>Partial Order"]
    T --> Q["QCA Universe<br/>Category Theory"]
    T --> M["Matrix Universe<br/>Equivalence"]

    I --> U["GLS<br/>Unified Theory"]
    B --> U
    C --> U
    Q --> U
    M --> U

    style T fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style U fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

Ready? Let’s begin exploring the essence of time!

Next Article: 01-phase-and-proper-time.md - Equivalence of Phase and Proper Time