What is Scattering?

“Echoes tell you the shape of a cave. Scattering tells you properties of particles, even tells you what time itself is.”

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Starting from Echoes

Stand in a valley and shout: “Hello—”

A few seconds later, you hear the echo: “Hello—”

🗻 What Does Echo Tell You?

graph LR
    You["You<br/>Emit Sound"] -->|sound wave propagation| Mountain["Mountain Wall<br/>Reflects"]
    Mountain -->|echo| You2["You<br/>Hear Echo"]

    style You fill:#ffe66d,stroke:#f59f00,stroke-width:2px
    style Mountain fill:#a8e6cf

Through echo, you can infer:

1. Distance: Delay time × Sound speed ÷ 2 = Distance to mountain wall
2. Shape: Direction of echo → Shape of mountain wall
3. Material: Timbre change of echo → Properties of rock

💡 Key Insight: You “emit” sound waves into valley, mountain wall “scatters” sound waves, you receive scattered sound waves, thus learning about the valley.

This is the basic idea of scattering!

What is Scattering?

In physics, scattering refers to:

Process where objects (particles, waves) interact with another object and change direction or properties.

📊 Three Elements of Scattering

graph LR
    Input["Incident<br/>Initial State |ψ_in⟩"] -->|interaction| Scattering["Scattering Region<br/>Potential V(r)"]
    Scattering -->|after scattering| Output["Outgoing<br/>Final State |ψ_out⟩"]

    style Scattering fill:#ff6b6b,color:#fff

Three Key Parts:

1. In-State $|{\psi }_{\text{in}}⟩$: State before scattering
2. Scattering Region: Region where interaction occurs
3. Out-State $|{\psi }_{\text{out}}⟩$: State after scattering

🎱 Classical Example: Billiard Ball Collision

On a billiard table, white ball hits red ball:

graph LR
    White1["White Ball<br/>Velocity v"] -->|collision| Collision["Collision Point"]
    Collision -->|scatter| White2["White Ball<br/>Velocity v'"]
    Collision -->|scatter| Red["Red Ball<br/>Velocity u"]

    style Collision fill:#ff6b6b,color:#fff

Scattering Results:

• White ball changes direction and speed
• Red ball gains momentum
• Total momentum conserved

By measuring scattered velocities and angles, can infer:

• Mass of balls
• Elastic coefficient of collision
• Interaction time

Quantum Scattering: S-Matrix

In quantum mechanics, scattering is described by the S-matrix (scattering matrix).

📐 Definition of S-Matrix

S-matrix connects in-state and out-state:

$$|{\psi }_{\text{out}}⟩=S|{\psi }_{\text{in}}⟩$$

Properties of S-Matrix:

1. Unitarity: $S^{†}S=I$ (probability conservation)
2. Energy Dependent: $S=S(\omega )$ ($\omega$ is energy)
3. Symmetry: Reflects system symmetries (time reversal, parity, etc.)

graph TB
    In["In-State<br/>|ψ_in⟩"] --> SMatrix["S-Matrix<br/>Unitary Evolution"]
    SMatrix --> Out["Out-State<br/>|ψ_out⟩ = S|ψ_in⟩"]

    SMatrix --> Info1["Amplitude<br/>Probability"]
    SMatrix --> Info2["Phase<br/>Time Delay"]
    SMatrix --> Info3["Channels<br/>Possible Out-States"]

    style SMatrix fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px,color:#fff

🔍 What Can We Learn from S-Matrix?

Example: Particle Physics Experiments

In large colliders (like LHC):

1. In-State: Two protons collide
2. Scattering: Protons collide at extremely high energy, produce various particles
3. Out-State: Particles observed by detectors (electrons, muons, photons, etc.)

By measuring S-matrix, physicists discovered:

• Quarks
• W/Z bosons
• Higgs boson

graph LR
    Proton1["Proton 1"] -->|collide| Collision["Collision<br/>Extremely High Energy"]
    Proton2["Proton 2"] -->|collide| Collision
    Collision -->|produce| Higgs["Higgs Boson<br/>(Predicted via S-Matrix)"]
    Higgs -->|decay| Photons["γ Rays<br/>(Detector Observation)"]

    style Collision fill:#ff6b6b,color:#fff
    style Higgs fill:#4ecdc4,color:#fff

💡 S-Matrix is the “Dictionary” of Particle Physics: It tells you “given in-state, possible out-states and their probabilities.”

Scattering Delay: Where Does Time Come From

⏱️ Wigner-Smith Time Delay

In scattering process, how long does a particle “stay” in the scattering region?

This is described by Wigner-Smith Time Delay Matrix:

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

Physical Meaning of Q-Matrix:

• Eigenvalues of $Q$ = Time delays of different scattering channels
• $\text{tr}Q$ = Total time delay (average over all channels)

graph LR
    Particle["Particle<br/>Incident"] -->|enter| Zone["Scattering Region<br/>Stay Time τ"]
    Zone -->|leave| Out["Particle<br/>Outgoing"]

    Zone -.measure.-> Delay["Time Delay<br/>τ = tr Q(ω)"]

    style Zone fill:#ff6b6b,color:#fff
    style Delay fill:#4ecdc4,color:#fff

Example: Tunneling Effect

Quantum tunneling: Particle passes through potential barrier, even if energy insufficient

     Energy
       ↑
       |    ╭──Potential Barrier──╮
       |    │                     │
  E ──|────┼───→  │   Particle Tunnels
       |    │                     │
       └────┴─────────────────────┴─→ Position

Time Delay Tells You: How long particle “stays” in barrier

• Thicker barrier → Longer delay
• Lower energy → Longer delay

🌊 Phase and Time

In scattering process, wave function acquires a phase:

$${\psi }_{\text{out}}=e^{i\phi (\omega )}{\psi }_{\text{in}}$$

Key Relationship:

$$\text{Time Delay}=\frac{\partial \phi }{\partial \omega }$$

That is: Derivative of phase with respect to energy is time delay!

💡 GLS Theory Proposes: This might be the quantum origin of time—time might not be an external parameter, but the derivative of scattering phase!

GLS Theory: Scattering is Evolution

One of GLS unified theory’s core insights:

GLS theory argues: Scattering is not just particle collisions, it might be the essence of time evolution.

🔄 Evolution = Scattering

Imagine universe as a huge scattering system:

graph LR
    State1["Universe at t=0<br/>Initial State"] -->|time evolution| State2["Universe at t=T<br/>Final State"]

    State1 -.equivalent to.-> Scatter1["Scattering In-State"]
    State2 -.equivalent to.-> Scatter2["Scattering Out-State"]
    Scatter1 -->|S-Matrix| Scatter2

    style State1 fill:#ffd3b6
    style State2 fill:#a8e6cf

Unitary Evolution Operator $U(t)=e^{-iHt/\hslash }$ can be seen as “scattering matrix” in energy eigenstate basis:

$$U(t)\leftrightarrow S(\omega )$$

Part of Unified Time Scale Identity:

$$\text{Scattering Delay}=\frac{1}{2\pi }\text{tr}Q(\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }$$

Translation:

• Scattering Delay ($\text{tr}Q$) = Time particle stays in scattering region
• Phase Derivative (${\phi }^{\prime }$) = Rate of change of wave function phase with respect to energy

In the GLS framework, they are mathematically equivalent!

Birman-Kreĭn Formula: Phase and Spectrum

📊 Spectral Shift Function

When you add a perturbation to system (like potential $V$), energy levels shift:

    No Perturbation         With Perturbation
    ------   →    ------  E₃' (shifted up)
    ------   →    ------  E₂' (almost unchanged)
    ------   →    ------  E₁' (shifted down)

Spectral Shift Function $\xi (\omega )$ tells you: Total shift of energy levels near $\omega$

Birman-Kreĭn Formula:

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

That is:

• Phase of S-matrix determinant = Spectral shift function
• Derivative of spectral shift function = Relative density of states

$${\rho }_{\text{rel}}(\omega )=-{\xi }^{\prime }(\omega )$$

🎯 Complete Form of Unified Time Scale

Now we can write the complete unified time scale identity:

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

Four Equivalent Quantities:

1. $\kappa (\omega )$ = Scattering time delay (how long particle stays)
2. ${\phi }^{\prime }(\omega )/\pi$ = Phase derivative (how fast wave function rotates)
3. ${\rho }_{\text{rel}}(\omega )$ = Relative density of states (how many energy levels)
4. $\text{tr}Q(\omega )/2\pi$ = Trace of Wigner-Smith delay

graph TD
    Center["Time<br/>(Unified Scale)"] --> Delay["Scattering Delay<br/>tr Q(ω)"]
    Center --> Phase["Phase Derivative<br/>φ'(ω)"]
    Center --> Density["Density of States<br/>ρ_rel(ω)"]
    Center --> Spectral["Spectral Shift<br/>ξ'(ω)"]

    Delay -.Wigner-Smith.-> Center
    Phase -.Schrödinger.-> Center
    Density -.Birman-Kreĭn.-> Center
    Spectral -.Birman-Kreĭn.-> Center

    style Center fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px,color:#fff

💡 Core Insight of GLS Theory: Time might not be an external clock, but an intrinsic emergent property of scattering-phase-spectrum structure.

Scattering Experiments: How to Measure S-Matrix

🔬 Classical Scattering Experiment

Rutherford Scattering (1909):

• α particles shot at gold foil
• Observe scattering angle distribution
• Discovered atomic nucleus

graph LR
    Alpha["α Particles<br/>(Helium Nuclei)"] -->|shot at| Gold["Gold Foil<br/>Atomic Nuclei"]
    Gold -->|large angle scattering| Detector["Detector<br/>Unexpected Large Angles!"]

    style Gold fill:#ff6b6b,color:#fff
    style Detector fill:#ffe66d,stroke:#f59f00,stroke-width:2px

Conclusion: Atom is not “plum pudding,” but has an extremely small, dense nucleus!

⚛️ Modern Scattering Experiments

Large Hadron Collider (LHC):

1. Accelerate protons to near light speed (99.9999991% of c)
2. Collide two proton beams
3. Observe scattering products (various particles)
4. Reconstruct S-matrix, search for new physics

graph TB
    LHC["LHC<br/>Collider"] --> Collision["Proton Collision<br/>Energy 13 TeV"]
    Collision --> Products["Products<br/>Quarks, Leptons, Bosons"]
    Products --> Detector["Detectors<br/>ATLAS, CMS"]
    Detector --> SMatrix["Reconstruct S-Matrix<br/>Discover New Particles"]

    style LHC fill:#a8e6cf
    style Collision fill:#ff6b6b,color:#fff
    style SMatrix fill:#4ecdc4,color:#fff

Major Discoveries:

• 2012: Higgs boson ($H$)
• Verified standard model predictions
• Nobel Prize (2013)

Scattering and Boundary

What’s the relationship between scattering and boundary?

In GLS theory, scattering occurs on boundary!

🎭 Boundary Scattering Picture

Imagine a causal diamond:

      Future Vertex
       /|\
      / | \
     /  |  \  Scattering Region
    /   |   \  (Boundary)
   /____|____\
   Boundary  |  Boundary
        |
      Past Vertex

Two Understandings of Scattering:

1. Bulk View: Particles move in interior, affected by potential
2. Boundary View: Particles scatter on boundary, interior is “empty”

GLS Theory Advocates: Boundary view might be more fundamental!

• Scattering data (S-matrix) defined on boundary
• “Evolution” in bulk is reconstructed from boundary data

graph TB
    Boundary1["Boundary: In-State<br/>Initial Data"] -->|Bulk Evolution<br/>(Reconstruction)| Boundary2["Boundary: Out-State<br/>Scattering Data"]

    Boundary1 -.S-Matrix.-> Boundary2

    style Boundary1 fill:#ffd3b6
    style Boundary2 fill:#a8e6cf

This is another manifestation of holographic principle!

Summary: Multiple Faces of Scattering

🎯 Key Points

1. S-Matrix: Unitary operator connecting in-state and out-state
2. Wigner-Smith Delay: $Q(\omega )=-i{S}^{†}{\partial }_{\omega }S$
3. Time Delay: $\text{tr}Q$ = Time particle stays in scattering region
4. Birman-Kreĭn Formula: $\det S(\omega )=e^{-2\pi i\xi (\omega )}$
5. Unified Scale: Scattering delay = Phase derivative = Density of states

💡 Most Profound Insight

GLS theory proposes: Time might not be an external clock, but intrinsic delay of scattering processes. Evolution of universe is essentially a huge scattering process.

Just as echo tells you the shape of a cave, scattering tells you the structure of universe—even what time itself is.

What’s Next

We understand scattering. The last fundamental concept is: Entropy.

• Why does time have direction?
• Why does entropy always increase?
• What’s the relationship between entropy and causality, scattering?

Answers to these questions are in the next article:

Next: What is Entropy? →

Remember: Scattering is not simple “collision,” but the fundamental process by which physical world acquires information, evolves, and even defines time. Understanding scattering, you understand how the universe “runs.”

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