07 Boundary as Stage: Where Does Physics Happen?

Core Idea

After completing unified time theory (Chapter 5), we now ask a more fundamental question:

Where does physics happen?

Traditional view holds: Physics happens in spatial interior. Particles move in space, fields evolve in space, forces act in space.

But GLS theory gives a subversive answer:

The place where physics truly happens is considered to be the boundary. The bulk interior is viewed as a “projection” or “holographic image” of boundary data.

This is the core idea of boundary as unified stage.

Everyday Analogy: Theater Stage

Imagine you’re watching a play:

graph TB
    subgraph "Traditional View: Stage is Interior"
        Stage1["🎭 Traditional Stage<br/>(Interior Space)"]
        Actors1["Actors Perform on Stage"]
        Audience1["👥 Audience<br/>(Viewing from Outside)"]

        Stage1 --> Actors1
        Actors1 -.->|"Watch"| Audience1
    end

    subgraph "GLS View: Stage is Boundary"
        Boundary["🎭 Boundary Stage<br/>(Boundary = Stage Itself)"]

        Boundary -->|"Actor 1"| Actor1["Scattering Process<br/>(Time Translation)"]
        Boundary -->|"Actor 2"| Actor2["Modular Flow Evolution<br/>(Algebra Automorphism)"]
        Boundary -->|"Actor 3"| Actor3["Geometric Evolution<br/>(Brown-York Energy)"]

        Actor1 -.->|"Same Essence"| Unity["☯️ Trinity<br/>Same Boundary Generator"]
        Actor2 -.-> Unity
        Actor3 -.-> Unity
    end

    style Stage1 fill:#ffe66d,stroke:#f59f00
    style Actors1 fill:#4ecdc4,stroke:#0b7285
    style Audience1 fill:#e9ecef,stroke:#495057
    style Boundary fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Actor1 fill:#4ecdc4,stroke:#0b7285
    style Actor2 fill:#4ecdc4,stroke:#0b7285
    style Actor3 fill:#4ecdc4,stroke:#0b7285
    style Unity fill:#ffe66d,stroke:#f59f00,stroke-width:4px

Key Insight:

• Traditional View: Stage is three-dimensional interior space, actors perform within it
• GLS View: Stage is viewed as the boundary, all “actors” (physical processes) perform on the boundary
• Three seemingly different “actors” (scattering, modular flow, geometry) are actually three guises of the same boundary generator

Boundary Triple: Unified Stage Setting

To define “boundary stage,” we need three basic elements:

graph TB
    Triple["Boundary Triple<br/>(∂ℳ, 𝒜_∂, ω_∂)"]

    Triple --> Element1["∂ℳ<br/>Geometric Boundary<br/>(Physical Space of Stage)"]
    Triple --> Element2["𝒜_∂<br/>Boundary Algebra<br/>(Set of Observables)"]
    Triple --> Element3["ω_∂<br/>Boundary State<br/>(Rule for Expectation Values)"]

    Element1 -.->|"Where"| Where["📍 Performance Venue"]
    Element2 -.->|"What"| What["🎬 Script"]
    Element3 -.->|"How"| How["🎭 Director's Instructions"]

    Where --> Stage["🎭 Complete Stage"]
    What --> Stage
    How --> Stage

    style Triple fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Element1 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Element2 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Element3 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Where fill:#ffe66d,stroke:#f59f00
    style What fill:#ffe66d,stroke:#f59f00
    style How fill:#ffe66d,stroke:#f59f00
    style Stage fill:#a9e34b,stroke:#5c940d,stroke-width:4px

Element 1: Geometric Boundary ∂ℳ

Physical Meaning: “Floor” of physical stage

Concrete Examples:

• Scattering Theory: Spacetime infinity boundary (incoming/outgoing particles come/go from here)
• Black Hole: Event horizon (final boundary of information)
• AdS Spacetime: Conformal boundary (where holographic CFT lives)
• Cosmology: Cosmological horizon (limit of what we can observe)

Element 2: Boundary Algebra 𝒜_∂

Physical Meaning: “Script”—what can be observed

Composition: $${\mathcal{A}}_{\partial }=\text{All Observables on Boundary}$$

Including:

• Creation/annihilation operators of scattering channels
• Boundary field operators
• Quasi-local energy operators
• Boundary stress tensor

Mathematical Structure: von Neumann algebra (operator algebra with $\ast$ operation)

Element 3: Boundary State ω_∂

Physical Meaning: “Director’s Instructions”—how to compute expectation values

Definition: $${\omega }_{\partial }:{\mathcal{A}}_{\partial }\to \mathbb{C}$$

Satisfying:

• Positivity: ${\omega }_{\partial }(A^{\ast }A)\ge 0$
• Normalization: ${\omega }_{\partial }(1)=1$
• Linearity: ${\omega }_{\partial }(aA+bB)=a{\omega }_{\partial }(A)+b{\omega }_{\partial }(B)$

Physical Examples:

• Vacuum state $|0⟩$
• KMS thermal equilibrium state (temperature $\beta$)
• Coherent states, squeezed states, etc.

Three Actors, One Stage

On the boundary stage, there are three “actors,” seemingly different, but essentially the same:

graph TB
    Stage["🎭 Boundary Stage<br/>(∂ℳ, 𝒜_∂, ω_∂)"]

    Stage -->|"Guise 1"| Actor1["Scattering Actor<br/>Time Delay Operator"]
    Stage -->|"Guise 2"| Actor2["Modular Flow Actor<br/>Algebra Automorphism"]
    Stage -->|"Guise 3"| Actor3["Geometric Actor<br/>Brown-York Hamiltonian"]

    Actor1 -->|"Time Measure"| Time1["dμ^scatt = (tr Q/2π)dω"]
    Actor2 -->|"Modular Time"| Time2["σ_t^ω(A) = Δ^it A Δ^-it"]
    Actor3 -->|"Boundary Time"| Time3["H_∂^grav = ∫√σ T_BY^ab ξ_b"]

    Time1 -.->|"Same Essence"| Unity["☯️ Unified Boundary Generator H_∂"]
    Time2 -.-> Unity
    Time3 -.-> Unity

    Unity -->|"Generates"| Evolution["Time Evolution on Boundary<br/>e^-itH_∂"]

    style Stage fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Actor1 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Actor2 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Actor3 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Time1 fill:#ffe66d,stroke:#f59f00
    style Time2 fill:#ffe66d,stroke:#f59f00
    style Time3 fill:#ffe66d,stroke:#f59f00
    style Unity fill:#a9e34b,stroke:#5c940d,stroke-width:4px
    style Evolution fill:#e9ecef,stroke:#495057

Actor 1: Scattering Time Delay (Microscopic Quantum)

Character Setting:

Measure incoming/outgoing particles at boundary (infinity)

Key Props:

• Scattering matrix $S(\omega )$
• Wigner-Smith time delay operator $Q(\omega )=-iS(\omega {)}^{†}{\partial }_{\omega }S(\omega )$
• Birman-Kreĭn spectral shift function $\xi (\omega )$

Lines (scale identity):

$$\frac{{\phi }^{\prime }(\omega )}{\pi }={\xi }^{\prime }(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

Physical Meaning:

Time particles “dwell” in scattering region = phase derivative = spectral density change

Actor 2: Modular Flow Automorphism (Algebraic Structure)

Character Setting:

Natural evolution of boundary algebra

Key Props:

• Tomita operator $S$
• Modular operator $\Delta$
• Modular flow ${\sigma }_t^{\omega }(A)={\Delta }^{it}A{\Delta }^{-it}$

Lines (modular flow formula):

$$K_{\partial }=-\log \Delta ,{\sigma }_t^{\omega }={\Delta }^{it}(\cdot ){\Delta }^{-it}$$

Physical Meaning:

Modular flow parameter = intrinsic time (naturally induced by algebra-state pair)

Actor 3: Brown-York Boundary Energy (Macroscopic Gravity)

Character Setting:

Quasi-local energy on boundary

Key Props:

• GHY boundary term $S_{\mathrm{GHY}}=\frac{1}{8\pi G}{\int }_{\partial M}\sqrt{|h|}K$
• Brown-York stress tensor $T_{\mathrm{BY}}^{ab}=\frac{2}{\sqrt{-h}}\frac{\delta S}{\delta h_{ab}}$
• Boundary Hamiltonian $H_{\partial }^{\mathrm{grav}}[\xi ]=\int \sqrt{\sigma }{u}_a{T}_{\mathrm{BY}}^{ab}{\xi }_b$

Lines (quasi-local energy):

$$H_{\partial }^{\mathrm{grav}}[\xi ]={\int }_B\sqrt{\sigma }{u}_a{T}_{\mathrm{BY}}^{ab}{\xi }_b{\mathrm{d}}^{2}x$$

Physical Meaning:

Generator of boundary time translation = quasi-local energy

Boundary Trinity Theorem

Now we reveal the secret of these three “actors”:

graph TB
    Question["🤔 Are Three Actors the Same Person?"]

    Question --> Theorem["Boundary Trinity Proposition"]

    Theorem --> Condition["Satisfy Matching Conditions:<br/>· Scattering→QFT Embedding<br/>· QFT→Gravity Holographic Correspondence<br/>· Thermal Time Normalization"]

    Condition --> Result["Unique Unified Boundary Generator H_∂ Exists"]

    Result --> R1["Scattering Time Delay ⟷ H_∂"]
    Result --> R2["Modular Flow Parameter ⟷ c₁ H_∂"]
    Result --> R3["Brown-York Time ⟷ c₂ H_∂"]

    R1 -.->|"Affine Equivalent"| Unity["☯️ Trinity"]
    R2 -.-> Unity
    R3 -.-> Unity

    Unity -->|"Mathematical Expression"| Formula["H_∂ = ∫ω dμ^scatt(ω)<br/>= c₁ K_D<br/>= c₂⁻¹ H_∂^grav"]

    style Question fill:#e9ecef,stroke:#495057
    style Theorem fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Condition fill:#4ecdc4,stroke:#0b7285
    style Result fill:#ffe66d,stroke:#f59f00,stroke-width:3px
    style R1 fill:#a9e34b,stroke:#5c940d
    style R2 fill:#a9e34b,stroke:#5c940d
    style R3 fill:#a9e34b,stroke:#5c940d
    style Unity fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Formula fill:#e9ecef,stroke:#495057

Proposition Content:

Under the premise of satisfying matching conditions, there theoretically exists a unique unified boundary time generator $H_{\partial }$ (up to affine transformation) such that:

$$\text{Scattering Time}⟺\text{Modular Flow Time}⟺\text{Brown-York Time}$$

Are equivalent in common domain.

Everyday Analogy:

• Three actors are three guises of the same person
• Change different costumes (scattering, modular flow, geometry)
• But essence is the same boundary generator
• Like Clark Kent = Superman = Kal-El

Null-Modular Double Cover: Special Stage of Null Boundary

For null boundaries (light cones), there is a particularly elegant structure:

graph TB
    Diamond["💎 Causal Diamond D<br/>(Future Vertex q, Past Vertex p)"]

    Diamond -->|"Boundary"| Null1["Null Hypersurface 𝒩⁺<br/>(Future Light Cone)"]
    Diamond -->|"Boundary"| Null2["Null Hypersurface 𝒩⁻<br/>(Past Light Cone)"]

    Null1 -->|"Remove Joint"| E1["Smooth Leaf E⁺"]
    Null2 -->|"Remove Joint"| E2["Smooth Leaf E⁻"]

    Cover["Null-Modular Double Cover<br/>Ẽ_D = E⁺ ⊔ E⁻"]

    E1 --> Cover
    E2 --> Cover

    Cover -->|"Modular Hamiltonian"| ModH["K_D = 2π Σ_σ ∫_{E^σ} g_σ T_σσ dλ d^(d-2)x"]

    ModH -.->|"Completely Defined on Boundary"| Boundary["✓ Modular Flow Localized on Null Boundary"]

    style Diamond fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Null1 fill:#4ecdc4,stroke:#0b7285
    style Null2 fill:#4ecdc4,stroke:#0b7285
    style E1 fill:#ffe66d,stroke:#f59f00
    style E2 fill:#ffe66d,stroke:#f59f00
    style Cover fill:#a9e34b,stroke:#5c940d,stroke-width:4px
    style ModH fill:#e9ecef,stroke:#495057
    style Boundary fill:#fff,stroke:#868e96,stroke-width:3px

Physical Image:

Imagine a diamond:

• Top vertex = some future time
• Bottom vertex = some past time
• Diamond surface = light cone (null hypersurface)

Null-Modular Double Cover:

Divide diamond surface into two “leaves”:

• $E^{+}$ = Future light cone (remove tip)
• $E^{-}$ = Past light cone (remove tip)

Modular Hamiltonian:

$$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }(\lambda ,x_{\perp })T_{\sigma \sigma }(\lambda ,x_{\perp })\mathrm{d}\lambda {\mathrm{d}}^{d-2}x$$

Where:

• $T_{++},T_{--}$ = Stress tensor components along two sets of null directions
• $g_{\sigma }$ = Geometric weight function (determined only by diamond shape)

Key: Modular Hamiltonian is completely defined on null boundary!

Concrete Example: Black Hole Horizon

Traditional View: Horizon is Mysterious

graph TB
    Far["🌍 Distant Observer"]
    Horizon["⚫ Event Horizon<br/>(Black Hole Boundary)"]
    Inside["❓ Interior<br/>(Unknown)"]

    Far -.->|"Cannot See"| Horizon
    Horizon -->|"Separates"| Inside

    Hawking["Hawking Radiation?<br/>Information Loss?"]

    Horizon -.-> Hawking

    style Far fill:#4ecdc4,stroke:#0b7285
    style Horizon fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Inside fill:#e9ecef,stroke:#495057,stroke-dasharray: 5 5
    style Hawking fill:#ffe66d,stroke:#f59f00,stroke-dasharray: 5 5

Boundary Stage View: Horizon is Complete Stage

graph TB
    Horizon2["⚫ Horizon = Boundary Stage<br/>(∂ℳ, 𝒜_∂, ω_∂)"]

    Horizon2 -->|"Actor 1"| Scatt["Scattering:<br/>Hawking Particles<br/>As Scattering Process"]

    Horizon2 -->|"Actor 2"| Mod["Modular Flow:<br/>Unruh Temperature<br/>T_U = κ/2π"]

    Horizon2 -->|"Actor 3"| Geom["Geometry:<br/>Quasi-Local Energy<br/>= Black Hole Mass M"]

    Trinity["Trinity"]

    Scatt --> Trinity
    Mod --> Trinity
    Geom --> Trinity

    Trinity -.->|"Unified Explanation"| Result["Bekenstein-Hawking Entropy<br/>S_BH = A/4G<br/>= Boundary Algebra Entropy"]

    style Horizon2 fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Scatt fill:#4ecdc4,stroke:#0b7285
    style Mod fill:#4ecdc4,stroke:#0b7285
    style Geom fill:#4ecdc4,stroke:#0b7285
    style Trinity fill:#ffe66d,stroke:#f59f00,stroke-width:4px
    style Result fill:#a9e34b,stroke:#5c940d,stroke-width:3px

Boundary Stage Interpretation:

1. Scattering Perspective:

   • Hawking radiation = Scattering process on horizon
   • Particle creation = Off-diagonal elements of $S$-matrix

2. Modular Flow Perspective:

   • Unruh temperature $T_U=\kappa /2\pi$ = Period of modular flow
   • KMS condition → Thermal equilibrium

3. Geometric Perspective:

   • Quasi-local energy = Black hole mass $M$
   • Brown-York tensor → Horizon stress

Unified Result:

Bekenstein-Hawking entropy: $$S_{\mathrm{BH}}=\frac{A}{4G}=\text{von Neumann Entropy of Boundary Algebra}$$

No need to know black hole interior! All information is on horizon (boundary).

GHY Boundary Term: Why Gravity Needs Boundary

Problem: Einstein-Hilbert Action is Incomplete

Consider Einstein-Hilbert action:

$$S_{\mathrm{EH}}=\frac{1}{16\pi G}{\int }_M\sqrt{-g}R{\mathrm{d}}^{4}x$$

Variation:

$$\delta S_{\mathrm{EH}}=\text{(Volume Integral)}+\text{(Boundary Term)}$$

Boundary term contains ${\partial }_n\delta g$ (normal derivative of metric variation)!

graph LR
    Problem["Problem:<br/>Boundary Term Contains ∂_n δg"]

    Problem -->|"Cannot Use"| Boundary["Boundary Induced Metric h_ab<br/>(Dirichlet Boundary Conditions)"]

    Problem -->|"Causes"| Issue["Variation Ill-Defined<br/>Hamiltonian Form Doesn't Exist"]

    Solution["Solution: Add GHY Boundary Term"]

    Issue --> Solution

    Solution --> GHY["S_GHY = (1/8πG)∫_∂M √|h| K"]

    GHY -.->|"Cancels ∂_n δg"| Fixed["✓ Total Variation Well-Defined<br/>δS_tot = δS_EH + δS_GHY"]

    style Problem fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Boundary fill:#ffe66d,stroke:#f59f00
    style Issue fill:#e9ecef,stroke:#495057
    style Solution fill:#4ecdc4,stroke:#0b7285,stroke-width:4px
    style GHY fill:#a9e34b,stroke:#5c940d,stroke-width:3px
    style Fixed fill:#fff,stroke:#868e96,stroke-width:3px

GHY Boundary Term:

$$S_{\mathrm{GHY}}=\frac{1}{8\pi G}{\int }_{\partial M}\sqrt{|h|}K{\mathrm{d}}^{3}x$$

Where:

• $h_{ab}$ = Boundary induced metric
• $K=K_{ab}{h}^{ab}$ = Trace of extrinsic curvature

After Adding GHY Term:

$$\delta (S_{\mathrm{EH}}+S_{\mathrm{GHY}})=\frac{1}{16\pi G}{\int }_M\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }+\frac{1}{16\pi G}{\int }_{\partial M}\sqrt{|h|}(K_{ab}-K{h}_{ab})\delta h^{ab}$$

• Volume term → Einstein equations
• Boundary term → Brown-York stress tensor

Deep Meaning:

Gravity is considered to be fundamentally a boundary theory! Without boundary term, even variation cannot be defined.

Philosophical Meaning: Stage Is Everything

graph TB
    Question["🤔 Where Does Physics Happen?"]

    Question -->|"Traditional View"| Bulk["Bulk Interior<br/>· Particles in Space<br/>· Fields in Space<br/>· Interactions in Space"]

    Question -->|"GLS View"| Boundary["Boundary Stage<br/>· Scattering on Boundary<br/>· Modular Flow on Boundary<br/>· Geometry on Boundary"]

    Boundary --> Insight["💡 Deep Revelations"]

    Insight --> I1["Bulk is 'Holographic<br/>Projection' of Boundary Data"]
    Insight --> I2["Three Physics<br/>(Quantum/Algebra/Gravity)<br/>Unified on Boundary"]
    Insight --> I3["Boundary is Stage<br/>Stage is Everything"]

    style Question fill:#e9ecef,stroke:#495057
    style Bulk fill:#ffe66d,stroke:#f59f00
    style Boundary fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Insight fill:#4ecdc4,stroke:#0b7285,stroke-width:4px
    style I1 fill:#a9e34b,stroke:#5c940d
    style I2 fill:#a9e34b,stroke:#5c940d
    style I3 fill:#a9e34b,stroke:#5c940d

Core Insights:

1. Mathematical Realization of Holographic Principle:

   • ’t Hooft/Susskind: Three-dimensional information can be encoded on two-dimensional surface
   • GLS: Boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$ completely determines bulk

2. Time-Algebra-Geometry Unification:

   • Not three independent theories
   • But three representations of the same boundary generator
   • $H_{\partial }=\int \omega \mathrm{d}{\mu }^{\mathrm{scatt}}=c_1{K}_D=c_2^{-1}{H}_{\partial }^{\mathrm{grav}}$

3. Boundary Priority Principle:

   • Define boundary first
   • Bulk is “extension” or “reconstruction” of boundary
   • All observables are on boundary

Chapter Summary

Core Insight:

The true stage of physics is considered to be the boundary, not the bulk. Scattering time delay, modular flow evolution, and Brown-York boundary energy are viewed as three guises of the same boundary generator, unified on boundary triple (∂ℳ, 𝒜_∂, ω_∂).

Key Formulas:

Boundary triple: $$(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$$

Boundary trinity: $$\frac{{\phi }^{\prime }(\omega )}{\pi }={\xi }^{\prime }(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )⟺K_D⟺H_{\partial }^{\mathrm{grav}}$$

Null-Modular double cover: $$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }{T}_{\sigma \sigma }\mathrm{d}\lambda {\mathrm{d}}^{d-2}x$$

GHY boundary term: $$S_{\mathrm{GHY}}=\frac{1}{8\pi G}{\int }_{\partial M}\sqrt{|h|}K{\mathrm{d}}^{3}x$$

Everyday Analogies:

• Theater stage: Actors perform on stage (boundary)
• Three guises: Same actor (boundary generator) in different costumes
• Diamond double face: Null-Modular double cover = two smooth leaves of diamond

Trinity:

Philosophical Revelation:

The universe is not a “box” (bulk), but a “stage” (boundary). The three-dimensional space we see is just a holographic projection of boundary data.

Connections to Other Chapters

graph TB
    Current["📍 This Chapter:<br/>Boundary as Stage"]

    Prev1["← 05 Unified Time<br/>Time Scale Identity"]
    Prev2["← 11 Boundary Language<br/>Three Boundary Axioms"]

    Next1["→ 08 Boundary Observer-Time<br/>Observer Defined on Boundary"]
    Next2["→ 09 Boundary as Clock<br/>Boundary Translation Operator"]

    Prev1 -->|"Time Scale<br/>Now on Boundary Stage"| Current
    Prev2 -->|"Boundary Language<br/>Now Has Concrete Stage"| Current

    Current -->|"Boundary Stage<br/>How Observer Defined"| Next1
    Current -->|"Boundary Generator<br/>How Becomes Clock"| Next2

    style Current fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Prev1 fill:#4ecdc4,stroke:#0b7285
    style Prev2 fill:#4ecdc4,stroke:#0b7285
    style Next1 fill:#ffe66d,stroke:#f59f00
    style Next2 fill:#ffe66d,stroke:#f59f00

Extended Reading

Source Theoretical Literature:

• docs/euler-gls-paper-bondary/boundary-as-unified-stage.md - Complete mathematical framework of boundary as unified stage

Related Chapters:

• 05 Unified Time - Theoretical foundation of time scale
• 11 Boundary Language - Three axioms of boundary language
• 01 Why Boundary - Motivation for boundary priority
• 04 Brown-York Energy - Detailed explanation of quasi-local energy

In the next chapter, we will explore boundary observer and time, seeing how observers are defined on the boundary stage.