Time Scale Identity: Proof of Four-in-One

“Multiple concepts of time might be just different manifestations of the same physical entity.”

🎯 Core Theorem

Theorem (Time Scale Identity):

Under appropriate scattering-spectral-geometric conditions, the following four quantities are mathematically equivalent:

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

Where:

• $\phi (\omega )$: normalized scattering phase ($\phi =\Phi /2$)
• ${\rho }_{\text{rel}}(\omega )$: relative density of states
• $Q(\omega )$: Wigner-Smith group delay operator
• $\kappa (\omega )$: unified time scale density

Physical meaning:

• Phase derivative ${\phi }^{\prime }/\pi$: rate of change of quantum phase
• Relative density of states ${\rho }_{\text{rel}}$: density of energy level shifts
• Group delay trace $\text{tr}Q/(2\pi )$: density of wave packet delay
• Conclusion: They can be viewed as three projections of the same time scale.

graph TB
    K["Unified Time Scale<br/>κ(ω)"] --> P["Phase Projection<br/>φ'/π"]
    K --> R["Spectral Projection<br/>ρ_rel"]
    K --> Q["Scattering Projection<br/>tr Q/2π"]

    P --> V["Observable<br/>Interference Phase"]
    R --> V
    Q --> V

    style K fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style V fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

💡 Intuitive Image: Three Mirrors

Analogy: Three Views of the Same Mountain

Imagine a mountain:

        *
       /|\
      / | \
     /  |  \
    /   |   \
   /____|____\

Viewed from three directions:

• Phase angle: mountain’s outline (${\phi }^{\prime }$)
• Spectral angle: mountain’s height distribution (${\rho }_{\text{rel}}$)
• Scattering angle: time needed to climb ($\text{tr}Q$)

Implication: They describe the same physical object from different angles.

Identity says: These three views give consistent information.

Music Analogy

Imagine a musical piece:

Three notation methods:

1. Phase spectrum (Fourier analysis): frequency components
2. Energy level spectrum (resonance peaks): main frequencies
3. Time delay (reverberation): sound duration

Identity says: These three analysis methods extract the same time structure.

📐 Complete Proof

Proof Structure

We will prove the identity in two steps:

Step 1: Prove ${\phi }^{\prime }/\pi ={\rho }_{\text{rel}}$ (phase-spectral equivalence)

Step 2: Prove ${\rho }_{\text{rel}}=\text{tr}Q/(2\pi )$ (spectral-scattering equivalence)

Conclusion: The three are equal.

graph LR
    PH["Phase Derivative<br/>φ'/π"] --> BK["Birman-Kreĭn<br/>Φ = -2πξ"]
    BK --> RHO["Relative Density of States<br/>ρ_rel = -∂_ωξ"]

    RHO --> LOG["Logarithmic Derivative<br/>∂_ω ln det S"]
    LOG --> Q["Group Delay<br/>tr Q"]

    PH --> ID["Identity<br/>φ'/π = ρ_rel = tr Q/2π"]
    RHO --> ID
    Q --> ID

    style BK fill:#e1f5ff
    style LOG fill:#ffe1e1
    style ID fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px

Step 1: Phase-Spectral Equivalence

Proposition 1: ${\phi }^{\prime }(\omega )/\pi ={\rho }_{\text{rel}}(\omega )$

Proof:

From Birman-Kreĭn formula (Article 3):

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

Taking logarithm (choosing continuous branch):

$$\ln \det S(\omega )=-2\pi i\xi (\omega )$$

Taking imaginary part, define total phase:

$$\Phi (\omega )=\arg \det S(\omega )=\text{Im}[\ln \det S(\omega )]$$

From Birman-Kreĭn:

$$\Phi (\omega )=\text{Im}[-2\pi i\xi (\omega )]=-2\pi \xi (\omega )$$

(Because $\xi$ is real function)

Define half-phase:

$$\phi (\omega ):=\frac{\Phi (\omega )}{2}=-\pi \xi (\omega )$$

Differentiating with respect to $\omega$:

$$\frac{d\phi }{d\omega }=-\pi \frac{d\xi }{d\omega }$$

From relative density of states definition (Article 3):

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

Substituting:

$$\frac{d\phi }{d\omega }=\pi {\rho }_{\text{rel}}(\omega )$$

Dividing by $\pi$:

$$\boxed{\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )}$$

QED: First equality holds.

Step 2: Spectral-Scattering Equivalence

Proposition 2: ${\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$

Proof:

Path 1: Starting from scattering matrix

Differentiate logarithm of scattering matrix. Using matrix identity:

$$\frac{d}{d\omega }\ln \det S(\omega )=\text{tr}\left[\frac{d\ln S(\omega )}{d\omega }\right]$$

(This is because $\ln \det A=\text{tr}\ln A$)

Further:

$$\frac{d\ln S}{d\omega }=S^{-1}\frac{dS}{d\omega }$$

From unitarity of $S$, $S^{-1}=S^{†}$:

$$\frac{d\ln S}{d\omega }=S^{†}\frac{\partial S}{\partial \omega }$$

Taking trace:

$$\frac{d}{d\omega }\ln \det S=\text{tr}\left(S^{†}\frac{\partial S}{\partial \omega }\right)$$

Introducing Wigner-Smith operator (Article 2):

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

Therefore:

$$S^{†}\frac{\partial S}{\partial \omega }=iQ(\omega )$$

Substituting:

$$\frac{d}{d\omega }\ln \det S=i\text{tr}Q(\omega )$$

Path 2: Starting from Birman-Kreĭn

From Birman-Kreĭn formula:

$$\ln \det S(\omega )=-2\pi i\xi (\omega )$$

Differentiating:

$$\frac{d}{d\omega }\ln \det S=-2\pi i\frac{d\xi }{d\omega }$$

From ${\rho }_{\text{rel}}=-d\xi /d\omega$:

$$\frac{d}{d\omega }\ln \det S=2\pi i{\rho }_{\text{rel}}(\omega )$$

Combining the two paths:

$$i\text{tr}Q(\omega )=2\pi i{\rho }_{\text{rel}}(\omega )$$

Canceling $i$:

$$\text{tr}Q(\omega )=2\pi {\rho }_{\text{rel}}(\omega )$$

Dividing by $2\pi$:

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

QED: Second equality holds.

Complete Identity

Combining Proposition 1 and Proposition 2:

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

Define unified time scale density:

$$\kappa (\omega ):=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Physical meaning: $\kappa (\omega )d\omega$ can be interpreted as the “time interval” corresponding to frequency range $[\omega ,\omega +d\omega ]$.

graph TB
    S["Scattering Matrix<br/>S(ω)"] --> DET["Determinant<br/>det S"]
    DET --> BK["Birman-Kreĭn<br/>det S = e^(-2πiξ)"]
    BK --> XI["Spectral Shift<br/>ξ(ω)"]

    DET --> LOG["Logarithm<br/>ln det S"]
    LOG --> DIFF1["Derivative (Path 1)<br/>= tr(S†∂_ωS) = i tr Q"]
    LOG --> DIFF2["Derivative (Path 2)<br/>= -2πi∂_ωξ = 2πiρ_rel"]

    DIFF1 --> EQ["Equate<br/>i tr Q = 2πiρ_rel"]
    DIFF2 --> EQ

    XI --> RHO["Density of States<br/>ρ_rel = -∂_ωξ"]
    XI --> PH["Half-Phase<br/>φ = -πξ"]
    PH --> PHD["Derivative<br/>φ' = πρ_rel"]

    EQ --> ID["Time Scale Identity<br/>φ'/π = ρ_rel = tr Q/2π"]
    PHD --> ID
    RHO --> ID

    style S fill:#e1f5ff
    style BK fill:#ffe1e1
    style EQ fill:#fff4e1
    style ID fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

🧮 Explicit Example: Single-Channel Scattering

One-Dimensional Potential Barrier

Setup: $S(k)=e^{2i\delta (k)}$, $k=\sqrt{2mE}/\hslash$

Verify identity:

1. Phase derivative:

$$\phi =\frac{\Phi }{2}=\delta (k)$$

$$\frac{{\phi }^{\prime }(E)}{\pi }=\frac{1}{\pi }\frac{d\delta }{dE}$$

2. Spectral shift:

From Birman-Kreĭn: $\xi (E)=-\delta (k)/\pi$

$${\rho }_{\text{rel}}(E)=-\frac{d\xi }{dE}=\frac{1}{\pi }\frac{d\delta }{dE}$$

3. Group delay:

$$Q(E)=2\frac{d\delta }{dE}$$

$$\frac{\text{tr}Q}{2\pi }=\frac{1}{\pi }\frac{d\delta }{dE}$$

Verification:

$$\frac{{\phi }^{\prime }}{\pi }=\frac{1}{\pi }\frac{d\delta }{dE}={\rho }_{\text{rel}}=\frac{\text{tr}Q}{2\pi }$$

Conclusion: Results verify the validity of the identity.

Resonance Scattering

Near resonance $E_r$:

$$\delta (E)={\delta }_{\text{bg}}+\arctan \frac{\Gamma /2}{E-E_r}$$

Calculation:

$$\frac{d\delta }{dE}=\frac{1}{1+{\left(\frac{\Gamma /2}{E-E_r}\right)}^2}\cdot \frac{-\Gamma /2}{(E-E_r{)}^2}$$

$$=\frac{\Gamma /2}{(E-E_r{)}^2+(\Gamma /2{)}^2}$$

Three expressions:

$$\frac{{\phi }^{\prime }}{\pi }={\rho }_{\text{rel}}=\frac{\text{tr}Q}{2\pi }=\frac{1}{\pi }\cdot \frac{\Gamma /2}{(E-E_r{)}^2+(\Gamma /2{)}^2}$$

Lorentzian line shape.

Integral:

$${\int }_{-\infty }^{\infty }\kappa (E)dE={\int }_{-\infty }^{\infty }{\rho }_{\text{rel}}(E)dE=1$$

Meaning: One resonance contributes unit “time”.

graph LR
    DELTA["Phase Shift<br/>δ(E)"] --> D1["Derivative<br/>dδ/dE"]
    D1 --> L["Lorentz Peak<br/>~ Γ/[(E-E_r)² + (Γ/2)²]"]
    L --> K["Time Scale<br/>κ(E)"]
    K --> INT["Integral<br/>∫κ dE = 1"]

    style DELTA fill:#e1f5ff
    style L fill:#ffe1e1
    style K fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style INT fill:#e1ffe1

🌀 Profound Meaning

1. Three Faces of Time

Quantum face (phase):

• Phase $\phi$ changes with energy
• ${\phi }^{\prime }$ measures “sensitivity of phase to energy”
• Measurable via quantum interference

Spectral face (energy levels):

• Density of states ${\rho }_{\text{rel}}$ describes energy level distribution
• Interaction “shifts” energy levels
• Measurable via spectroscopy experiments

Scattering face (time delay):

• Group delay $Q$ describes wave packet delay
• Time is directly measurable delay
• Measurable via scattering experiments

Identity says: These three are highly unified in mathematical structure.

2. Unified Time Scale

Define time integral:

$$T(\omega )={\int }_{{\omega }_0}^{\omega }\kappa ({\omega }^{\prime })d{\omega }^{\prime }$$

$$={\int }_{{\omega }_0}^{\omega }\frac{{\phi }^{\prime }({\omega }^{\prime })}{\pi }d{\omega }^{\prime }=\frac{\phi (\omega )-\phi ({\omega }_0)}{\pi }$$

$$={\int }_{{\omega }_0}^{\omega }{\rho }_{\text{rel}}({\omega }^{\prime })d{\omega }^{\prime }=-[\xi (\omega )-\xi ({\omega }_0)]$$

$$={\int }_{{\omega }_0}^{\omega }\frac{\text{tr}Q({\omega }^{\prime })}{2\pi }d{\omega }^{\prime }$$

Physical meaning:

• $T(\omega )$ is “accumulated time” from ${\omega }_0$ to $\omega$
• Can be calculated from phase, spectral shift, or group delay
• They give consistent answers.

3. Operational Definition of Time

Traditional view: Time is a priori parameter $t$

GLS view: Time can be extracted from scattering data.

Operational steps:

1. Measure scattering matrix $S(\omega )$
2. Calculate $Q(\omega )=-i{S}^{†}{\partial }_{\omega }S$
3. Extract time scale $\kappa =\text{tr}Q/(2\pi )$
4. Integrate to get time $T=\int \kappa d\omega$

Or:

1. Measure phase $\phi (\omega )$
2. Differentiate ${\phi }^{\prime }$
3. Normalize $\kappa ={\phi }^{\prime }/\pi$

Same result.

graph TB
    OBS["Observation<br/>Scattering Experiment"] --> S["S Matrix<br/>S(ω)"]
    OBS --> PHI["Phase<br/>φ(ω)"]

    S --> Q["Group Delay<br/>Q = -iS†∂_ωS"]
    Q --> K1["Time Scale<br/>κ = tr Q / 2π"]

    PHI --> D["Derivative<br/>φ'"]
    D --> K2["Time Scale<br/>κ = φ' / π"]

    K1 --> T["Time<br/>T = ∫κ dω"]
    K2 --> T

    K1 -.->|"Identity"| K2

    style OBS fill:#e1f5ff
    style K1 fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style K2 fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style T fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

🔑 Uniqueness and Equivalence Class

Theorem (Local Uniqueness of Time Scale)

Statement: Given scattering data $(S(\omega ))$ satisfying time scale identity, there exists unique (locally) time parameter $\tau (\omega )$ such that:

$$\frac{d\tau }{d\omega }=\kappa (\omega )=\frac{\text{tr}Q(\omega )}{2\pi }$$

Any other time parameter $t(\omega )$ satisfying same physical requirements must have:

$$t=\alpha \tau +\beta$$

Where $\alpha >0,\beta \in \mathbb{R}$ are constants.

Proof idea:

Assume two times $\tau ,t$ both satisfy:

$$\frac{d\tau }{d\omega }=\frac{dt}{d\omega }=\kappa (\omega )$$

Then:

$$\frac{d(t-\tau )}{d\omega }=0$$

Integrating:

$$t-\tau =\text{constant}=\beta$$

So $t=\tau +\beta$.

If rescaling allowed: $dt=\alpha d\tau$, then $t=\alpha \tau +\beta$.

Physical meaning: Time scale is unique up to affine transformation.

Time Scale Equivalence Class

Definition:

$$[\tau ]:=\{t\mid t=\alpha \tau +\beta ,\alpha >0\}$$

Members include:

• Proper time $\tau$
• Coordinate time $t$
• Killing time $t_K$
• ADM lapse $N$
• Null affine parameter $\lambda$
• Conformal time $\eta$
• Frequency inverse ${\omega }^{-1}$
• Redshift parameter $z$
• Modular time $s_{\text{mod}}$

They are converted to each other through monotonic rescaling.

graph TB
    TAU["Unified Time Scale<br/>[τ]"] --> T1["Proper Time<br/>τ"]
    TAU --> T2["Killing Time<br/>t_K"]
    TAU --> T3["ADM Time<br/>N·t"]
    TAU --> T4["Null Parameter<br/>λ"]
    TAU --> T5["Conformal Time<br/>η"]
    TAU --> T6["Frequency Scale<br/>ω^(-1)"]
    TAU --> T7["Redshift Scale<br/>1+z"]
    TAU --> T8["Modular Time<br/>s_mod"]

    T1 -.->|"Affine Transformation"| T2
    T2 -.->|"Affine Transformation"| T3
    T3 -.->|"Affine Transformation"| T4

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

📊 Derivation Chain Summary

🌟 Connection with Previous Articles

Article 1: Phase and Proper Time

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

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

Connection: Phase grows linearly with proper time, frequency ${\omega }_C$ is the “time scale”.

Article 2: Scattering Phase and Group Delay

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

Connection: Group delay is derivative of phase with respect to frequency, direct measurement of “time scale”.

Article 3: Spectral Shift Function

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

$$\Phi =-2\pi \xi$$

Connection: Relative density of states describes energy level shifts, also spectral manifestation of “time scale”.

Article 4 (This Article): Four-in-One Unification

$$\boxed{\text{Phase}\equiv \text{Spectrum}\equiv \text{Scattering}\equiv \text{Time}}$$

Theoretical logic closure.

graph TB
    P1["Article 1<br/>φ = (mc²/ℏ)∫dτ"] --> ID["Article 4<br/>Time Scale Identity"]
    P2["Article 2<br/>tr Q = ∂_ωΦ"] --> ID
    P3["Article 3<br/>ρ_rel = -∂_ωξ"] --> ID

    ID --> U["Unified Time Scale<br/>κ(ω)"]
    U --> APP["Applications<br/>Gravity/Cosmology/Quantum"]

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

🤔 Exercises

1. Conceptual understanding:

   • Why is time scale identity important?
   • What are the physical meanings of phase, spectral, and scattering perspectives?
   • How do members of time scale equivalence class convert to each other?

2. Calculation exercises:

   • For $S=e^{2i\delta }$, verify identity
   • For resonance $\delta =\arctan [\Gamma /(2(E-E_r))]$, calculate $\kappa (E)$
   • Verify ${\int }_{-\infty }^{\infty }\kappa dE=1$

3. Derivation exercises:

   • From $\det S=e^{-2\pi i\xi }$ derive ${\phi }^{\prime }/\pi ={\rho }_{\text{rel}}$
   • From $\ln \det S=\text{tr}\ln S$ derive $\text{tr}Q=2\pi {\rho }_{\text{rel}}$
   • Prove local uniqueness of time scale

4. Advanced thinking:

   • How to generalize identity in multi-channel scattering?
   • In non-Hermitian systems, is $\kappa$ still real?
   • What is topological meaning of time scale identity?

Next step: We have completed complete derivation of time scale identity! Next article will explore geometric times (Killing, ADM, null, conformal) and show how they fit into unified scale.

Navigation:

• Previous: 03-spectral-shift_en.md - Spectral Shift Function
• Next: 05-geometric-times_en.md - Geometric Times
• Overview: 00-time-overview_en.md - Unified Time Chapter Overview
• GLS theory: unified-time-scale-geometry.md