Core Insight 4: Scattering Modeled as Evolution

“GLS theory proposes: The universe might not be ‘moving,’ but ‘scattering.’”

🎯 Core Idea

In traditional physics, we describe system evolution with differential equations:

$$\frac{d\psi }{dt}=\text{(some operator)}\cdot \psi$$

But GLS theory proposes a deeper perspective:

GLS theory proposes: System evolution might be essentially a scattering process, and the scattering matrix $S(\omega )$ encodes all dynamical information!

In other words:

In GLS framework: Evolution ⟺ Scattering, Dynamics ⟺ S-Matrix

🏔️ Starting from Valley Echoes: Scattering Reveals Structure

The Valley Echo Analogy

Imagine you shout in a valley:

You emit sound wave → Sound wave hits mountain wall → Reflects back (echo)

By analyzing the echo, you can infer:

• Valley shape (geometry)
• Wall material (absorption/reflection coefficient)
• Valley size (delay time)

graph LR
    I["Incident Wave<br/>|in⟩"] --> S["Scattering Region<br/>(Valley, Particle, Black Hole)"]
    S --> O["Outgoing Wave<br/>|out⟩"]
    S --> R["Reflected Wave<br/>|ref⟩"]

    O -.-> |"Analyze"| G["Infer Geometric Structure"]
    R -.-> |"Analyze"| G

    style I fill:#e1f5ff
    style S fill:#fff4e1
    style O fill:#ffe1e1
    style R fill:#e1ffe1

Scattering in physics is like this:

• Emit particles/waves (input)
• Interact with target (scattering)
• Detect outgoing states (output)
• Infer internal structure from input-output relations!

Why is Scattering So Fundamental?

Because in quantum theory, we can never directly see the “interior”, only:

1. Prepare initial state $|\text{in}⟩$ (at past infinity)
2. Measure final state $|\text{out}⟩$ (at future infinity)
3. The relation between them is the S-matrix!

$$|\text{out}⟩=S|\text{in}⟩$$

Key insight:

S-matrix is considered the only observable thing! The “real description” of intermediate processes might be redundant, even misleading!

🌀 S-Matrix: Essence of Evolution

What is S-Matrix?

S-matrix (Scattering matrix) is a unitary operator:

$$S:{\mathcal{H}}_{\text{in}}\to {\mathcal{H}}_{\text{out}}$$

Satisfying:

1. Unitarity: $S^{†}S=S{S}^{†}=\mathbb{I}$ (probability conservation)
2. Causality: Only connects past asymptotic states with future asymptotic states
3. Lorentz covariance: Covariant under relativistic framework

Physical meaning:

S-matrix element $S_{fi}=⟨f|S|i⟩$ is the transition amplitude from initial state $|i⟩$ to final state $|f⟩$.

Scattering cross-section (observable):

$$\sigma \propto |S_{fi}{|}^2$$

S-Matrix Contains All Information

Heisenberg’s S-matrix program (1943) proposed:

“The task of physics is not to describe ‘processes,’ but to calculate S-matrix elements.”

Why?

• Initial and final states are observable (prepared and measured in laboratory)
• Intermediate processes are unobservable (Heisenberg uncertainty principle)
• What is theoretically observable is the S-matrix!

graph TB
    subgraph "Observable"
        I["Initial State |in⟩<br/>t → -∞"]
        O["Final State |out⟩<br/>t → +∞"]
    end

    subgraph "Unobservable (Black Box)"
        P["'Real Process'?<br/>Path Integral?<br/>Field Equation?"]
    end

    I --> |"S-Matrix"| O
    I -.-> |"Interpretation, Not Ontology"| P
    P -.-> O

    style I fill:#e1f5ff
    style O fill:#ffe1e1
    style P fill:#f0f0f0,stroke-dasharray: 5 5

⏱️ Wigner-Smith Time Delay Matrix

Group Delay: Time Wave Packet Stays in Scattering Region

Consider a wave packet incident on a scattering region:

Question: How long does it “stay” in the scattering region?

Answer: Given by Wigner-Smith time delay matrix $Q(\omega )$!

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

Physical meaning:

• Eigenvalues ${\tau }_n(\omega )$ of $Q(\omega )$ are time delays of each channel
• $\text{tr}Q(\omega )={\sum }_n{\tau }_n(\omega )$ is total delay

Key formula (Eisenbud-Wigner formula):

$${\tau }_W(\omega )=\frac{\partial \Phi (\omega )}{\partial \omega }$$

where $\Phi (\omega )=-i\ln \det S(\omega )$ is the total scattering phase.

Time Delay = Density of States

More remarkably, through Birman-Kreĭn formula:

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

where ${\rho }_{\text{rel}}(\omega )$ is the relative density of states (quantum state density that scattering system has more than free system).

This means:

Time delay is mathematically equivalent to density of states! The system’s “complexity” (how many states) determines the wave packet’s “stay time”!

graph LR
    S["Scattering Matrix<br/>S(ω)"] --> |"Frequency Derivative"| Q["Time Delay Matrix<br/>Q(ω) = -iS†∂_ωS"]
    Q --> |"Take Trace"| T["Total Delay<br/>τ_W = tr Q"]
    T --> |"Birman-Kreĭn"| R["Density of States<br/>ρ_rel = τ_W/2π"]

    style S fill:#e1f5ff
    style Q fill:#fff4e1
    style T fill:#ffe1e1
    style R fill:#e1ffe1

🔄 Evolution = Scattering: Unified Perspective

Why Say Evolution is Scattering?

In GLS theory, any physical process might be viewed as scattering:

Core principle:

As long as there is “input” and “output,” whatever happens in between can theoretically be described by S-matrix!

Deep Meaning of Unitary Evolution

Quantum mechanical evolution operator:

$$U(t_2,t_1)=e^{-iH(t_2-t_1)/\hslash }$$

In the limit $t_1\to -\infty$, $t_2\to +\infty$, it becomes the S-matrix:

$$S=\underset{t_1\to -\infty ,t_2\to +\infty }{\lim }U(t_2,t_1)$$

But GLS theory reverses this logic:

GLS theory argues: Maybe not “first have evolution U, then define S,” but “first have S, evolution U is a projection of S at finite time”!

graph TB
    S["Fundamental: S-Matrix<br/>(Asymptotic)"] --> |"Finite Time Projection"| U["Evolution Operator<br/>U(t₂,t₁)"]
    U --> |"Infinitesimal"| H["Hamiltonian<br/>H = iℏ∂_t U"]

    H -.-> |"Traditional Path"| U
    U -.-> |"Traditional Path"| S

    style S fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style U fill:#e1f5ff
    style H fill:#e1ffe1

🧮 Birman-Kreĭn Formula: Unification of Spectrum, Phase, Delay

This is one of the core mathematical tools of GLS theory.

Birman-Kreĭn formula connects:

1. Spectral shift function $\xi (\omega )$: Extra spectral weight that scattering system has compared to free system
2. Scattering determinant: $\det S(\omega )=e^{-2\pi i\xi (\omega )}$
3. Relative density of states: ${\rho }_{\text{rel}}(\omega )=-{\xi }^{\prime }(\omega )$

Complete formula chain:

$$\boxed{\det S(\omega )=e^{-2\pi i\xi (\omega )}\Rightarrow \frac{{\Phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )}$$

This is the mathematical source of the Unified Time Scale Identity!

graph TB
    D["Scattering Determinant<br/>det S(ω)"] --> |"Take Logarithm"| P["Total Phase<br/>Φ(ω) = -i ln det S"]
    P --> |"Derivative w.r.t. Frequency"| Pp["Phase Derivative<br/>Φ'(ω)"]

    D --> |"BK Formula"| X["Spectral Shift Function<br/>ξ(ω)"]
    X --> |"Derivative"| Xp["Relative Density of States<br/>ρ_rel = -ξ'"]

    Pp --> |"÷ π"| U["Unified Scale<br/>Φ'/π"]
    Xp --> U

    Q["Time Delay<br/>tr Q(ω)/2π"] --> U

    style D fill:#e1f5ff
    style U fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

🌌 Universe as Scattering Matrix

The Entire Universe is an S-Matrix!

In GLS’s matrix universe framework (which we will detail later), there is an amazing proposition:

GLS theory hypothesizes: The ontology of the universe might be a huge family of scattering matrices $\mathbb{S}(\omega )$!

• Each frequency $\omega$ corresponds to a unitary matrix $S(\omega )$
• All spacetime, gravity, particles are “emergences” of this matrix
• Evolution is the “flow” of the matrix

THE-MATRIX might not be science fiction, but a mathematical reality!

Causal Network = Scattering Network

In the language of causal structure:

• Each small causal diamond $D_{p,r}$ has an associated scattering matrix $S_{p,r}(\omega )$
• Diamond evolution is guaranteed by unitarity of $S$
• Markov property of diamond chains is guaranteed by locality of scattering

Causal propagation ⟺ Scattering propagation!

graph LR
    D1["D₁<br/>S₁(ω)"] --> |"Scattering Evolution"| D2["D₂<br/>S₂(ω)"]
    D2 --> D3["D₃<br/>S₃(ω)"]
    D3 --> D4["D₄<br/>S₄(ω)"]

    style D1 fill:#e1f5ff
    style D2 fill:#fff4e1
    style D3 fill:#ffe1e1
    style D4 fill:#e1ffe1

🔬 Experimental Verifiability

The beauty of scattering theory is: It is theoretically directly observable!

Experiment 1: Wigner-Smith Delay in Mesoscopic Conductors

In quantum dots, mesoscopic rings, etc., we can directly measure:

• Multi-port scattering matrix $S_{ij}(\omega )$ (using vector network analyzer)
• Time delay matrix $Q(\omega )$ (from frequency derivative of $S$)
• Verification: $\text{tr}Q/2\pi ={\rho }_{\text{rel}}$ (through density of states measurement)

Experiment 2: Shapiro Gravitational Time Delay

Delay of radar signal passing near the Sun:

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

This is the group delay of gravitational scattering!

Can be reinterpreted using $Q(\omega )$ framework: Gravitational field is a scattering potential, Shapiro delay is $\text{tr}Q$!

Experiment 3: Cosmological Redshift as Phase Rhythm

Redshift in FRW universe:

$$1+z=\frac{a(t_0)}{a(t_e)}$$

Can be rewritten as phase rhythm ratio:

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

Redshift can be interpreted as the phase evolution of cosmic scattering matrix!

🔗 Connections to Other Core Ideas

• Time is Geometry: Time scale $\tau$ is derived from $Q(\omega )$
• Causality is Partial Order: Scattering preserves causal order (S only connects past with future)
• Boundary is Reality: S-matrix is defined on boundary asymptotic states
• Entropy is Arrow: Scattering process entropy monotonically increases (unitarity + coarse-graining)

🎓 Further Reading

To understand more technical details, you can read:

• Theory document: unified-time-scale-geometry.md
• Boundary framework: boundary-time-geometry-unified-framework.md
• Previous: 03-boundary-is-reality_en.md - Boundary is Reality
• Next: 05-entropy-is-arrow_en.md - Entropy is Arrow

🤔 Questions for Reflection

1. Why do we say “intermediate processes” are unobservable? What properties of quantum mechanics lead to this?
2. Why is Wigner-Smith matrix defined as $Q=-i{S}^{†}{\partial }_{\omega }S$ rather than other forms?
3. If the universe is an S-matrix, what does “time evolution” mean?
4. What is the relationship between unitarity of scattering and probability conservation?
5. Under what conditions does Birman-Kreĭn formula hold? What is the physical meaning of spectral shift function?

📝 Key Formulas Review

$$\boxed{|\text{out}⟩=S|\text{in}⟩}\text{(S-Matrix Definition)}$$

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

$$\boxed{\det S(\omega )=e^{-2\pi i\xi (\omega )}}\text{(Birman-Kreı˘n Formula)}$$

$$\boxed{\frac{1}{2\pi }\text{tr}Q(\omega )={\rho }_{\text{rel}}(\omega )=\frac{{\Phi }^{\prime }(\omega )}{\pi }}\text{(Time Scale Identity)}$$

Next Step: After understanding “Scattering is Evolution,” we will see “Entropy is Arrow”—time’s directionality comes from entropy increase, which is closely connected with causality, scattering, and boundary!