Scattering Phase and Group Delay: Operational Definition of Time

“Group delay can be interpreted as the reading of phase clock by scattering process.”

🎯 Core Proposition

Definition (Wigner-Smith Group Delay Operator):

For frequency-dependent unitary scattering matrix $S(\omega )$, define group delay operator:

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

Physical meaning:

• $Q(\omega )$ is self-adjoint matrix (Hermitian)
• Eigenvalues ${\tau }_j(\omega )$ correspond to time delays of each scattering channel
• Trace $\text{tr}Q(\omega )$ corresponds to total group delay
• Key relationship:

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

Where $\Phi (\omega )=\arg \det S(\omega )$ is total scattering phase.

💡 Intuitive Image: Echo Delay

Analogy: Valley Echo

Imagine you shout in a valley, sound propagation process:

You --Sound Wave--> Cliff --Reflection--> You
   t_out         Scattering      t_in

Time delay: How much slower is echo than direct propagation?

$$\Delta t=(t_{\text{in}}-t_{\text{out}})-t_{\text{free propagation}}$$

Scattering analogy:

• Free propagation → free particle ($H_0$)
• Cliff reflection → scattering potential ($H=H_0+V$)
• Time delay → group delay $Q(\omega )$
• Echo pitch change → phase shift $\Phi (\omega )$

Physical meaning: Group delay measures “how much slower interaction makes wave packet”.

Wave Packet Story

Consider a narrow wave packet incident on scattering center:

$${\psi }_{\text{in}}(x,t)=\int A(\omega )e^{i(kx-\omega t)}d\omega$$

Wave packet center position:

$$x_{\text{center}}(t)=\frac{\partial \Phi (\omega )}{\partial k}{|}_{{\omega }_0}$$

Wave packet center arrival time:

$$t_{\text{arrival}}=\frac{\partial \Phi (\omega )}{\partial \omega }{|}_{{\omega }_0}=\text{tr}Q({\omega }_0)$$

graph LR
    W["Wave Packet<br/>Incident"] --> S["Scattering Center<br/>V(r)"]
    S --> D["Delay<br/>Δt = tr Q"]
    D --> O["Outgoing Wave Packet<br/>Phase Shift Φ"]

    style W fill:#e1f5ff
    style S fill:#ffe1e1
    style D fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style O fill:#e1ffe1

Key: In the wave packet approximation, group delay corresponds to the actual time delay of wave packet center.

📐 Mathematical Derivation

Scattering Operator and S Matrix

In scattering theory, from initial state $|{\psi }_{\text{in}}⟩$ to final state $|{\psi }_{\text{out}}⟩$:

Scattering operator:

$$S=({\Omega }^{+}{)}^{†}{\Omega }^{-}$$

Where ${\Omega }^{\pm }$ are Møller wave operators:

$${\Omega }^{\pm }=\text{s-}\underset{t\to \pm \infty }{\lim }{e}^{iHt}{e}^{-i{H}_{0}t}$$

In energy representation:

For each frequency $\omega$, there is channel space ${\mathcal{H}}_{\omega }\simeq {\mathbb{C}}^{N(\omega )}$, on which unitary matrix $S(\omega )$:

$$S(\omega ):{\mathcal{H}}_{\omega }\to {\mathcal{H}}_{\omega },S(\omega {)}^{†}S(\omega )=I$$

Why unitary? Energy conservation, total probability unchanged before and after scattering.

Total Scattering Phase

Since $S(\omega )$ is unitary, can be written as:

$$S(\omega )=e^{iK(\omega )}$$

Where $K(\omega )$ is self-adjoint matrix.

Determinant:

$$\det S(\omega )=e^{i\text{tr}K(\omega )}=e^{i\Phi (\omega )}$$

Total phase:

$$\Phi (\omega )=\arg \det S(\omega )=\text{tr}K(\omega )$$

Physical meaning: Sum of phase shifts of all channels.

Wigner-Smith Operator Derivation

Question: What is derivative of phase with respect to frequency?

From $\det S(\omega )=e^{i\Phi (\omega )}$, differentiate both sides:

$$\frac{d}{d\omega }\det S=i{e}^{i\Phi }\frac{d\Phi }{d\omega }$$

Left side: Using matrix determinant derivative formula:

$$\frac{d}{d\omega }\det S=\det S\cdot \text{tr}\left(S^{-1}\frac{dS}{d\omega }\right)$$

Since $S$ is unitary, $S^{-1}=S^{†}$:

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

Combining:

$$\det S\cdot \text{tr}\left(S^{†}\frac{\partial S}{\partial \omega }\right)=i{e}^{i\Phi }\frac{d\Phi }{d\omega }$$

Canceling $\det S=e^{i\Phi }$:

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

Define group delay operator:

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

Obtain:

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

graph TB
    S["S Matrix<br/>Unitary S†S = I"] --> D["Determinant<br/>det S = e^(iΦ)"]
    D --> P["Total Phase<br/>Φ(ω) = arg det S"]
    P --> DER["Differentiate<br/>d/dω"]
    DER --> Q["Group Delay<br/>Q = -iS†∂_ωS"]
    Q --> T["Trace<br/>tr Q = ∂_ωΦ"]

    style S fill:#e1f5ff
    style Q fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style T fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

Q is Self-Adjoint

Prove $Q(\omega )$ is Hermitian matrix:

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

Using $S^{†}S=I$ and differentiating:

$$\frac{\partial S^{†}}{\partial \omega }S+S^{†}\frac{\partial S}{\partial \omega }=0$$

Therefore:

$$\frac{\partial S^{†}}{\partial \omega }S=-S^{†}\frac{\partial S}{\partial \omega }$$

Substituting:

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

Conclusion: $Q$ is self-adjoint. So eigenvalues are all real, can be interpreted as real time delays.

🧮 Single-Channel Scattering

One-Dimensional Potential Barrier

Simplest example: particle scattered by one-dimensional potential $V(x)$.

Single channel: $S(\omega )$ is $1\times 1$ matrix (complex number):

$$S(\omega )=e^{2i\delta (\omega )}$$

Where $\delta (\omega )$ is scattering phase shift.

Total phase:

$$\Phi (\omega )=\arg S=2\delta (\omega )$$

Group delay:

$$Q(\omega )=-i{S}^{†}\frac{\partial S}{\partial \omega }=-i{e}^{-2i\delta }\frac{\partial }{\partial \omega }{e}^{2i\delta }$$

$$=-i{e}^{-2i\delta }\cdot 2i\frac{d\delta }{d\omega }{e}^{2i\delta }=2\frac{d\delta }{d\omega }$$

Trace (single channel, trace is itself):

$$\text{tr}Q=2\frac{d\delta }{d\omega }$$

Verification:

$$\frac{\partial \Phi }{\partial \omega }=\frac{\partial (2\delta )}{\partial \omega }=2\frac{d\delta }{d\omega }=\text{tr}Q✓$$

Physical Interpretation

Wigner time delay theorem (1955):

For wave packet of width $\Delta \omega$, time delay after scattering:

$$\Delta t=\frac{d\delta }{d\omega }{|}_{{\omega }_0}$$

Physical image:

Near potential barrier, particle "stays" longer
→ More phase accumulation
→ Derivative of phase with respect to energy = delay time

Example: Resonance scattering

Near resonance energy $E_r$:

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

$$\frac{d\delta }{dE}\approx \frac{\Gamma /2}{(E-E_r{)}^2+(\Gamma /2{)}^2}$$

At resonance $E=E_r$:

$$\Delta t=\frac{d\delta }{dE}{|}_{E_r}=\frac{2}{\Gamma }={\tau }_{\text{lifetime}}$$

Results match: Group delay equals resonance lifetime.

🌀 Multi-Channel Scattering

Two-Channel Example

Consider two scattering channels (e.g., spin up/down):

$$S(\omega )=\left(\begin{array}{cc}{S}_{11}(\omega )& S_{12}(\omega )\\ S_{21}(\omega )& S_{22}(\omega )\end{array}\right)$$

Group delay operator:

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

Is $2\times 2$ self-adjoint matrix with two real eigenvalues ${\tau }_1(\omega ),{\tau }_2(\omega )$.

Trace:

$$\text{tr}Q={\tau }_1+{\tau }_2$$

Physical meaning:

• ${\tau }_1$: delay time of channel 1
• ${\tau }_2$: delay time of channel 2
• Total delay: sum of both

Channel Coupling

Diagonal case (no coupling):

$$S=\left(\begin{array}{cc}{e}^{2i{\delta }_1}& 0\\ 0& e^{2i{\delta }_2}\end{array}\right)$$

$$Q=\left(\begin{array}{cc}2d{\delta }_1/d\omega & 0\\ 0& 2d{\delta }_2/d\omega \end{array}\right)$$

$$\text{tr}Q=2\frac{d{\delta }_1}{d\omega }+2\frac{d{\delta }_2}{d\omega }$$

Off-diagonal case (with coupling):

Channel interference! Off-diagonal elements of $Q$ nonzero, physically corresponding to coherent delay between channels.

graph TB
    subgraph "Single Channel"
        S1["S = e^(2iδ)"] --> Q1["Q = 2dδ/dω"]
    end

    subgraph "Multi-Channel"
        S2["S Matrix<br/>N×N"] --> Q2["Q = -iS†∂_ωS"]
        Q2 --> E["Eigenvalues<br/>τ₁, τ₂, ..."]
        E --> TR["Trace<br/>∑τⱼ = ∂_ωΦ"]
    end

    style Q1 fill:#e1f5ff
    style Q2 fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style TR fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🔬 Experimental Verification

1. Microwave Cavity Experiment

Setup:

• Microwave cavity (resonant cavity)
• Vector network analyzer measures $S(\omega )$
• Multi-port configuration

Method:

1. Sweep frequency to measure $S_{ij}(\omega )$
2. Numerically differentiate $\partial S/\partial \omega$
3. Calculate $Q(\omega )=-i{S}^{†}{\partial }_{\omega }S$
4. Extract $\text{tr}Q(\omega )$

Results:

• At cavity resonance frequency, $\text{tr}Q$ shows peak
• Peak $\approx Q_{\text{factor}}/{\omega }_0=1/\Gamma$
• Highly consistent with theoretical prediction.

Reference: Fyodorov & Sommers, J. Math. Phys. 38, 1918 (1997)

2. Optical Delay Measurement

Setup: Light pulse through medium (e.g., optical fiber, atomic gas)

Measurement:

• Input pulse: $E_{\text{in}}(t)=E_0{e}^{-t^2/(2{\sigma }^2)}{e}^{-i{\omega }_{0}t}$
• Output pulse: $E_{\text{out}}(t)$

Group delay:

$${\tau }_g=-\frac{d\varphi }{d\omega }{|}_{{\omega }_0}$$

Where $\varphi (\omega )$ is transmission phase.

Experiments:

• Slow light (EIT): ${\tau }_g\sim 1\text{ ms}$ (atomic medium)
• Fast light (anomalous dispersion): ${\tau }_g<0$ (negative delay!)

Relationship with $Q$:

• Transmission $T(\omega )=|t(\omega ){|}^2$
• Transmission amplitude $t(\omega )=e^{i\varphi (\omega )}$
• $Q_{\text{trans}}=\partial \varphi /\partial \omega ={\tau }_g$

3. Shapiro Delay (Gravitational)

In weak gravitational field, time delay of photon propagation:

Outside Schwarzschild metric:

$$\Delta t\approx \frac{4GM}{c^3}\ln \frac{4r_E{r}_R}{b^2}$$

Where:

• $M$: central mass (Sun)
• $r_E,r_R$: Earth, radar target distances
• $b$: minimum distance

Frequency dependence: In plasma, gravity + dispersion:

$$\Phi (\omega )={\Phi }_{\text{geo}}(\omega )+{\Phi }_{\text{plasma}}(\omega )$$

$$\text{tr}Q(\omega )=\frac{\partial {\Phi }_{\text{geo}}}{\partial \omega }+\frac{\partial {\Phi }_{\text{plasma}}}{\partial \omega }$$

Observation: Cassini spacecraft radar experiment, precision $10^{-5}$.

Physical meaning: Gravitational time delay = derivative of gravitational “scattering” phase.

graph LR
    MW["Microwave Cavity<br/>Q_factor"] --> V["Verification<br/>tr Q = ∂_ωΦ"]
    OPT["Optics<br/>Slow/Fast Light"] --> V
    GR["Shapiro Delay<br/>Gravitational Field"] --> V
    V --> U["Unified Understanding<br/>Delay = Phase Derivative"]

    style MW fill:#e1f5ff
    style OPT fill:#ffe1e1
    style GR fill:#fff4e1
    style U fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

📊 Properties of Group Delay

Property 1: Hermitianity

$$Q^{†}=Q$$

Meaning: Eigenvalues real, correspond to real time delays.

Property 2: Trace Formula

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

Meaning: Total delay equals derivative of total phase.

Property 3: Positivity (Generally Not True)

Note: $Q$ is not necessarily positive definite!

Possible: ${\tau }_j(\omega )<0$ (negative delay)

Physical interpretation:

• Anomalous dispersion region
• Fast light effect
• Tunneling time (controversial)

Causality: Although negative delay exists, signal front still satisfies causality (Sommerfeld-Brillouin theorem).

Property 4: High-Frequency Asymptotic

Theorem (Levinson):

$$\underset{\omega \to \infty }{\lim }\Phi (\omega )=0$$

(Under appropriate normalization)

Corollary:

$${\int }_0^{\infty }\text{tr}Q(\omega )d\omega =\Phi (\infty )-\Phi (0)=\text{finite}$$

Physical meaning: Total time delay integral converges.

💡 Profound Meaning

Operational Definition of Time

Traditional view: Time is external parameter $t$.

Scattering view: Time is measurable delay.

Operational definition:

1. Prepare narrow wave packet ($\Delta \omega$ small)
2. Measure phase $\Phi (\omega )$ before and after scattering
3. Calculate derivative $\partial \Phi /\partial \omega$
4. Obtain time delay $\Delta t=\text{tr}Q$

Philosophical meaning:

• Time is not a priori existence
• Time can be viewed as record of scattering process
• Time is closely related to rate of phase change

Connecting Quantum and Classical

Quantum side:

• Phase $\Phi (\omega )$
• Scattering matrix $S(\omega )$
• Unitary evolution $U(t)=e^{-iHt/\hslash }$

Classical side:

• Delay time $\Delta t$
• Wave packet trajectory $x(t)$
• Proper time $\tau$

Bridge:

$$\boxed{\Delta t=\text{tr}Q(\omega )=\frac{\partial \Phi }{\partial \omega }=\frac{\hslash \partial \Phi }{\partial E}}$$

Semiclassical limit: $\hslash \to 0$, phase $\Phi /\hslash \to S/\hslash$ (action), stationary phase method gives classical orbit.

Connection with Time Scale Identity

Recall unified time scale formula:

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

Scattering phase part:

$$\frac{1}{2\pi }\text{tr}Q(\omega )=\frac{1}{2\pi }\frac{\partial \Phi }{\partial \omega }$$

Next article will prove:

$$\Phi (\omega )=-2\pi \xi (\omega )$$

Where $\xi (\omega )$ is Birman-Kreĭn spectral shift function!

Therefore:

$$\text{tr}Q=2\pi \frac{\partial \xi }{\partial \omega }=-2\pi {\rho }_{\text{rel}}$$

Theoretical framework is self-consistent.

graph TB
    S["Scattering Matrix<br/>S(ω)"] --> PH["Total Phase<br/>Φ = arg det S"]
    PH --> Q["Group Delay<br/>Q = -iS†∂_ωS"]
    Q --> TR["Trace<br/>tr Q = ∂_ωΦ"]

    XI["Spectral Shift Function<br/>ξ(ω)"] --> BK["Birman-Kreĭn<br/>Φ = -2πξ"]
    BK --> PH

    XI --> RHO["Relative Density of States<br/>ρ_rel = -∂_ωξ"]
    RHO --> ID["Time Scale Identity<br/>tr Q/2π = ρ_rel"]
    TR --> ID

    style S fill:#e1f5ff
    style Q fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style ID fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

📝 Key Formulas Summary

🎓 Further Reading

• Original paper: E.P. Wigner, Phys. Rev. 98, 145 (1955)
• Group delay: F.T. Smith, Phys. Rev. 118, 349 (1960)
• Microwave experiment: Fyodorov & Sommers, J. Math. Phys. 38, 1918 (1997)
• Gravitational delay: I.I. Shapiro, Phys. Rev. Lett. 13, 789 (1964)
• GLS theory: unified-time-scale-geometry.md
• Previous: 01-phase-and-proper-time_en.md - Phase and Proper Time
• Next: 03-spectral-shift_en.md - Spectral Shift Function and Birman-Kreĭn Formula

🤔 Exercises

1. Conceptual understanding:

   • Why must $Q$ be Hermitian?
   • Does negative group delay violate causality?
   • What is the difference between group delay and phase delay?

2. Calculation exercises:

   • For $S(\omega )=e^{2i\delta }$, prove $Q=2d\delta /d\omega$
   • Calculate group delay for $2\times 2$ diagonal S matrix
   • Resonance scattering: $\delta =\arctan [\Gamma /(2(E-E_r))]$, find $Q(E_r)$

3. Physical applications:

   • What is the relationship between quality factor $Q_{\text{factor}}$ of microwave cavity and group delay?
   • How does Shapiro delay experiment verify general relativity?
   • In slow light experiments, does information propagate faster than light?

4. Advanced thinking:

   • Tunneling time problem: How long does quantum tunneling take?
   • In multi-channel scattering, can we have ${\tau }_i<0,{\tau }_j>0$?
   • How to reconstruct $S(\omega )$ from $Q(\omega )$?

Next step: We have understood phase-time equivalence (Article 1) and scattering delay (Article 2). Next article will reveal spectral shift function $\xi (\omega )$ and prove Birman-Kreĭn formula connecting scattering and spectrum!