Boundary Theory Summary: Boundary Nature of Physics

“From the GLS perspective, the bulk is viewed as a projection of boundary data; physical objects are primarily defined on the boundary.”

🎯 Core Review

In this chapter, we explored a paradigm shift from bulk to boundary. Let’s review this theoretical journey.

Core Insight

graph TB
    BULK["Bulk Physics<br/>Seems Complete"] -->|"Careful Check"| ISSUE["Discover Problems"]

    ISSUE --> S1["Scattering Theory<br/>S-Matrix Only Defined on Boundary"]
    ISSUE --> S2["Quantum Field Theory<br/>Modular Flow Localized on Boundary"]
    ISSUE --> S3["General Relativity<br/>EH Action Ill-Defined"]

    S1 --> BOUND["Boundary Physics<br/>True Foundation"]
    S2 --> BOUND
    S3 --> BOUND

    BOUND --> TRIPLE["Boundary Triple<br/>(∂M, A_∂, ω_∂)"]

    TRIPLE --> TIME["Unified Time Scale<br/>κ(ω)"]
    TRIPLE --> GEO["GHY Boundary Term<br/>K_{ab}"]
    TRIPLE --> ALG["Modular Flow Generator<br/>K_ω"]

    TIME -.->|"Same Object"| GEO
    GEO -.->|"Same Object"| ALG
    ALG -.->|"Same Object"| TIME

    style BULK fill:#ffe1e1,stroke:#cc0000
    style BOUND fill:#e1f5ff,stroke:#0066cc,stroke-width:4px
    style TRIPLE fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Step 1: Three Major Historical Evidences

In 01-Why Boundary, we saw three independent historical evidences, all pointing to the same conclusion:

Evidence 1: Scattering Theory (Quantum Mechanics)

Analogy: Imagine you’re throwing balls in a completely dark room, you can only measure balls entering/exiting at the door. What happens inside the room? You’ll never see it!

Mathematics:

$$\boxed{S\text{-Matrix Defined at Asymptotic Boundary}{\mathcal{I}}^{\pm }}$$

Key Formula:

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

where ${\Omega }_{\pm }$ are wave operators, their definition requires asymptotic boundary conditions.

Insight: Bulk spectral information completely determined by boundary scattering data!

Evidence 2: Quantum Field Theory (Modular Flow Theory)

Analogy: Room’s thermometer is not in the room, but nailed on the wall. Wall is where measurement happens!

Mathematics:

$$\boxed{\text{Modular Hamiltonian}{K}_{\omega }\text{Localized on Regional Boundary}\partial O}$$

Key Formula (Bisognano-Wichmann Theorem):

$${\sigma }_t^{\omega }(A)=e^{2\pi i{K}_{\mathrm{boost}}t}A{e}^{-2\pi i{K}_{\mathrm{boost}}t}$$

For wedge $W$, modular flow equals Lorentz boost, its generator localized on wedge boundary (Rindler horizon).

Insight: Relative entropy, entanglement entropy and other information-theoretic quantities all defined on boundary!

Evidence 3: General Relativity (Variational Principle)

Analogy: Imagine defining energy for a box with a lid. If you only look inside the box, energy definition is incomplete—you must consider the lid (boundary)!

Mathematics:

$$\boxed{\delta S_{\mathrm{EH}}=\text{Volume Term}+\text{Boundary Term (Contains Normal Derivatives)}}$$

Einstein-Hilbert action alone cannot give well-defined variational principle!

GHY Rescue:

$$S=S_{\mathrm{EH}}+S_{\mathrm{GHY}}$$

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

Now boundary terms completely cancel!

Insight: Gravitational action principle is essentially a boundary variational principle!

Step 2: Boundary Data Triple

In 02-Boundary Data Triple, we constructed unified boundary description:

Definition (Boundary Triple)

$$\boxed{(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })}$$

Three Components:

graph LR
    TRIPLE["Boundary Triple"] --> GEO["∂M<br/>Geometric Boundary<br/>(Where)"]
    TRIPLE --> ALG["A_∂<br/>Observable Algebra<br/>(What to Measure)"]
    TRIPLE --> STATE["ω_∂<br/>Boundary State<br/>(What Result)"]

    GEO --> METRIC["Induced Metric h_{ab}"]
    GEO --> NORMAL["Normal Vector n^μ"]
    GEO --> EXTRINSIC["Extrinsic Curvature K_{ab}"]

    ALG --> FIELD["Boundary Field Operators"]
    ALG --> SCATTER["Scattering Channels"]
    ALG --> MODULAR["Modular Algebra"]

    STATE --> VACUUM["Vacuum State"]
    STATE --> THERMAL["Thermal/KMS State"]
    STATE --> ENTANGLE["Entanglement Structure"]

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

Boundary Completeness Principle

Proposition (Boundary Completeness Hypothesis): Given boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$, bulk physical content can theoretically be completely reconstructed.

Three Levels of Reconstruction:

Boundaries Can Be Piecewise

Importance: Real boundaries are often not smooth!

graph TB
    BOUNDARY["Piecewise Boundary ∂M"] --> TL["Timelike Piece<br/>Timelike"]
    BOUNDARY --> SL["Spacelike Piece<br/>Spacelike"]
    BOUNDARY --> NL["Null Piece<br/>Null"]

    TL --> C1["Corner C<br/>Corner"]
    SL --> C1
    NL --> C1

    C1 --> CORNER["Corner Term<br/>S_corner"]

    style BOUNDARY fill:#e1f5ff,stroke:#0066cc,stroke-width:3px
    style CORNER fill:#ffe1e1

Necessity of Corner Terms: At intersections (corners/joints) of different boundary pieces, additional corner terms are needed to ensure action differentiability.

Step 3: Deep Meaning of GHY Boundary Term

In 03-GHY Boundary Term, we revealed necessity of GHY boundary term:

Core Theorem (GHY Cancellation Mechanism)

Proposition (GHY Cancellation Mechanism): For Einstein-Hilbert action plus GHY boundary term:

$$S_{\mathrm{total}}=S_{\mathrm{EH}}+S_{\mathrm{GHY}}$$

Under variation fixing boundary induced metric $h_{ab}$, boundary terms completely cancel:

$$\boxed{\delta (S_{\mathrm{EH}}+S_{\mathrm{GHY}})=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }}$$

Physical Meaning:

• Left side: only bulk Einstein tensor
• Right side: boundary term is zero
• Conclusion: variational principle well-defined!

Geometric Meaning of Extrinsic Curvature

Intuitive Understanding: Extrinsic curvature $K_{ab}$ measures how boundary “curves outward”.

graph LR
    SURFACE["Boundary Surface"] --> INTRINSIC["Intrinsic Curvature<br/>Curvature of Surface Itself"]
    SURFACE --> EXTRINSIC["Extrinsic Curvature<br/>Curvature in Embedding Space"]

    INTRINSIC --> R["Ricci Scalar R(h)"]
    EXTRINSIC --> K["Extrinsic Curvature K_{ab}"]

    K --> DEF["Definition: K_{ab} = h_a^μ h_b^ν ∇_μ n_ν"]

    style EXTRINSIC fill:#e1f5ff,stroke:#0066cc

Formula:

$$K_{ab}=h_a{}^{\mu }{h}_b{}^{\nu }{\nabla }_{\mu }{n}_{\nu }$$

where:

• $h_a{}^{\mu }$: embedding map from boundary to bulk
• $n^{\nu }$: unit normal vector
• ${\nabla }_{\mu }$: bulk covariant derivative

Trace:

$$K=h^{ab}{K}_{ab}$$

Special Nature of Null Boundaries

For null boundaries (e.g., black hole horizon), GHY term has different form:

$$S_{\mathcal{N}}=\frac{1}{8\pi G}{\int }_{\mathcal{N}}\sqrt{\gamma }(\theta +\kappa )\mathrm{d}\lambda {\mathrm{d}}^{2}x$$

where:

• ${\gamma }_{AB}$: transverse 2D metric
• $\theta$: expansion
• $\kappa$: surface gravity
• $\lambda$: affine parameter

Physical Intuition: Null geodesics cannot be described by extrinsic curvature (because normal vector is null vector!), must use expansion and surface gravity.

Step 4: Brown-York Quasi-Local Energy

In 04-Brown-York Energy, we defined generator of boundary time:

Definition (Brown-York Stress Tensor)

$$\boxed{T_{\mathrm{BY}}^{ab}=\frac{1}{8\pi G}(K^{ab}-K{h}^{ab})}$$

This is boundary stress-energy tensor, completely determined by boundary geometric data!

Quasi-Local Energy

Given time-like unit vector $u^a$ on boundary and 2D cross-section $\mathcal{S}$:

$$\boxed{E_{\mathrm{BY}}={\int }_{\mathcal{S}}\sqrt{\sigma }{u}_a{u}_b{T}_{\mathrm{BY}}^{ab}{\mathrm{d}}^{2}x}$$

Meaning of “Quasi-Local”:

• Not truly local (depends on boundary choice)
• But more local than ADM mass (doesn’t need asymptotically flat)
• Is “best possible” local energy definition

Asymptotic Limit

Property: In asymptotically flat spacetime, Brown-York energy converges to ADM mass:

$$\underset{r\to \infty }{\lim }{E}_{\mathrm{BY}}({\mathcal{S}}_r)=M_{\mathrm{ADM}}$$

Example: Schwarzschild Black Hole

For Schwarzschild black hole, on 2D sphere at radius $r$:

$$E_{\mathrm{BY}}(r)=\frac{r}{2G}\left(1-\sqrt{1-\frac{2GM}{r}}\right)$$

Asymptotic Behavior:

• $r\to \infty$: $E_{\mathrm{BY}}\to M$ (ADM mass)
• $r\to 2GM$: $E_{\mathrm{BY}}\to M$ (horizon energy)

Key Insight: $$\boxed{\text{Brown-York Energy}=\text{Generator of Boundary Time Translation}}$$

Step 5: Unification of Boundary Observers

In 05-Boundary Observers, we achieved ultimate unification:

Core Theorem (Boundary Observer Unification)

Proposition: The following three “boundary observers” are essentially equivalent within the theoretical framework:

$$\boxed{{\kappa }_{\mathrm{scatt}}\sim {\kappa }_{\mathrm{mod}}\sim {\kappa }_{\mathrm{geom}}}$$

graph TB
    OBSERVER["Boundary Observer<br/>(Abstract Concept)"] --> THREE["Three Realizations"]

    THREE --> SCATTER["Scattering Observer<br/>At Asymptotic Boundary I±"]
    THREE --> MODULAR["Modular Flow Observer<br/>At Regional Boundary ∂O"]
    THREE --> GEOMETRIC["Geometric Observer<br/>At Timelike Boundary B"]

    SCATTER --> S1["Measurement: S-Matrix"]
    SCATTER --> S2["Time Scale: tr Q/(2π)"]

    MODULAR --> M1["Measurement: Modular Hamiltonian K_ω"]
    MODULAR --> M2["Time Scale: Modular Flow Parameter"]

    GEOMETRIC --> G1["Measurement: Brown-York Energy"]
    GEOMETRIC --> G2["Time Scale: Boundary Time Generator"]

    S2 -.->|"Equivalent"| M2
    M2 -.->|"Equivalent"| G2
    G2 -.->|"Equivalent"| S2

    style OBSERVER fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style THREE fill:#e1f5ff,stroke:#0066cc,stroke-width:3px

Details of Three Observers

Unified Scale Equivalence Class

Definition: Time scale equivalence class $[\kappa ]$ is equivalence class of all scale densities differing by constant factors and allowed rescalings.

Proposition: In boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$, there exists unique scale equivalence class $[\kappa ]$ such that:

$$\boxed{{\kappa }_{\mathrm{scatt}}={\kappa }_{\mathrm{mod}}={\kappa }_{\mathrm{geom}}=\kappa (\omega )}$$

(within same energy window)

Perfect Example: Rindler Observer

Scenario: Uniformly accelerating observer (acceleration $a$)

Trinity:

1. Scattering End: Scattering phase shift of Rindler modes
2. Modular Flow End: Modular Hamiltonian of Rindler wedge
3. Geometric End: Brown-York energy of Rindler horizon

Common Temperature:

$$\boxed{T=\frac{a}{2\pi }}$$

This is Unruh temperature, which is simultaneously:

• Scattering phase derivative
• Modular flow inverse temperature
• Boundary time scale

🌟 Boundary Trinity: Ultimate Unification

Now we can show complete unified picture:

graph TB
    TRINITY["Boundary Trinity"] --> SCALE["Unified Time Scale<br/>κ(ω)"]
    TRINITY --> TRIPLE["Boundary Triple<br/>(∂M, A_∂, ω_∂)"]
    TRINITY --> GENERATOR["Boundary Time Generator<br/>H_∂"]

    SCALE --> PHASE["Scattering Phase Derivative<br/>φ'(ω)/π"]
    SCALE --> SPECTRAL["Spectral Shift Density<br/>ρ_rel(ω)"]
    SCALE --> DELAY["Group Delay<br/>tr Q/(2π)"]

    TRIPLE --> GEOMETRY["Geometric Boundary<br/>∂M"]
    TRIPLE --> ALGEBRA["Algebraic Boundary<br/>A_∂"]
    TRIPLE --> STATE["Boundary State<br/>ω_∂"]

    GENERATOR --> SCATTER["Scattering Generator<br/>∫ ω dμ_scatt"]
    GENERATOR --> MODULAR["Modular Hamiltonian<br/>K_ω"]
    GENERATOR --> GHY["Brown-York Hamiltonian<br/>H_BY"]

    PHASE -.->|"Birman-Kreĭn"| SPECTRAL
    SPECTRAL -.->|"Trace Formula"| DELAY
    DELAY -.->|"Cycle"| PHASE

    SCATTER -.->|"Matching"| MODULAR
    MODULAR -.->|"Holographic"| GHY
    GHY -.->|"Asymptotic"| SCATTER

    style TRINITY fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style SCALE fill:#e1ffe1,stroke:#00cc00,stroke-width:3px
    style TRIPLE fill:#e1f5ff,stroke:#0066cc,stroke-width:3px
    style GENERATOR fill:#ffe1f5,stroke:#cc00cc,stroke-width:3px

Main Results Summary

Proposition A (Scattering End Scale Identity):

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

Proposition B (Modular Flow End Localization): Modular Hamiltonian can be completely localized on boundary:

$$K_{\partial }=2\pi {\int }_{\partial O}{\xi }^{\mu }{T}_{\mu \nu }{n}^{\nu }\mathrm{d}\Sigma$$

Proposition C (GHY Well-Definedness):

$$\delta (S_{\mathrm{EH}}+S_{\mathrm{GHY}})=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }$$

Boundary terms completely cancel!

Proposition D (Boundary Trinity): There exists unified boundary time generator $H_{\partial }$ such that:

$$H_{\partial }=\int \omega \mathrm{d}{\mu }_{\partial }^{\mathrm{scatt}}(\omega )=c_1{K}_D=c_2{H}_{\partial }^{\mathrm{grav}}$$

📊 Core Formulas Summary

Boundary Data Triple

$$\boxed{(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })}$$

GHY Boundary Term

Timelike/Spacelike Boundaries:

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

where $\epsilon =n^{\mu }{n}_{\mu }\in \{\pm 1\}$, $K=h^{ab}{K}_{ab}$

Null Boundaries:

$$\boxed{S_{\mathcal{N}}=\frac{1}{8\pi G}{\int }_{\mathcal{N}}\sqrt{\gamma }(\theta +\kappa )\mathrm{d}\lambda {\mathrm{d}}^{2}x}$$

Brown-York Stress Tensor and Energy

$$\boxed{T_{\mathrm{BY}}^{ab}=\frac{1}{8\pi G}(K^{ab}-K{h}^{ab})}$$

$$\boxed{E_{\mathrm{BY}}={\int }_{\mathcal{S}}\sqrt{\sigma }{u}_a{u}_b{T}_{\mathrm{BY}}^{ab}{\mathrm{d}}^{2}x}$$

Unified Time Scale

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

Extrinsic Curvature

$$\boxed{K_{ab}=h_a{}^{\mu }{h}_b{}^{\nu }{\nabla }_{\mu }{n}_{\nu }}$$

$$\boxed{K=h^{ab}{K}_{ab}}$$

🔗 Connections to Other Chapters

Looking Back: Unified Time Chapter (Chapter 5)

In Unified Time chapter, we proved Time Scale Identity:

$$\kappa (\omega )={\phi }^{\prime }(\omega )/\pi ={\rho }_{\mathrm{rel}}(\omega )=\mathrm{tr}Q(\omega )/(2\pi )$$

Now we see: This unified scale is entirely a boundary phenomenon!

• Scattering phase ${\phi }^{\prime }(\omega )$: defined at asymptotic boundary
• Spectral shift function ${\rho }_{\mathrm{rel}}(\omega )$: determined by boundary scattering data
• Group delay $\mathrm{tr}Q(\omega )$: sum of boundary channel delays

Looking Back: IGVP Framework Chapter (Chapter 4)

IGVP variational principle needs boundary theory to be complete:

graph LR
    IGVP["IGVP Variational Principle"] --> BOUNDARY["Boundary Well-Definedness"]
    BOUNDARY --> GHY["GHY Boundary Term"]
    GHY --> ENTROPY["Generalized Entropy"]
    ENTROPY --> EINSTEIN["Einstein Equations"]

    EINSTEIN --> QNEC["QNEC/QFC"]
    QNEC --> BY["Brown-York Energy"]
    BY --> BOUNDARY

    style IGVP fill:#fff4e1,stroke:#ff6b6b
    style BOUNDARY fill:#e1f5ff,stroke:#0066cc,stroke-width:3px

Key Insight:

• Generalized entropy defined on boundary of small causal diamond
• First-order extremum → Einstein equations
• Second-order variation → QNEC/QFC
• Boundary term → Brown-York energy

Looking Forward: Causal Structure Chapter (Chapter 7)

Boundary theory provides foundation for causal structure:

Causal Diamond:

$$D(p,q)=J^{+}(p)\cap J^{-}(q)$$

Its boundary consists of future null hypersurface and past null hypersurface!

Null-Modular Double Cover:

$${\tilde{E}}_D=E^{+}\bigsqcup E^{-}$$

where $E^{\pm }$ are two null boundary pieces of causal diamond.

Modular Hamiltonian:

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

Completely defined on null boundaries!

Looking Forward: Matrix Universe Chapter (Chapter 10)

Boundary theory reveals essence of holographic principle:

AdS/CFT Correspondence:

$$\boxed{\text{Boundary CFT}⟺\text{Bulk AdS Geometry}}$$

This is ultimate manifestation of boundary completeness principle!

Heart-Universe Equivalence: Observer’s “inner experience” (heart) equivalent to “external universe” (universe), because both are different projections of boundary data.

💡 Deep Philosophical Implications

1. Boundary Nature of Physics

Traditional View:

• Bulk is fundamental
• Boundary is technical supplement
• Observation happens in bulk

Boundary Revolution:

• Boundary is fundamental
• Bulk is reconstruction from boundary data
• All observation essentially happens on boundary

2. Boundary Interpretation of Measurement

Quantum Measurement: Measurement is not “observing bulk”, but projecting on boundary!

graph LR
    SYSTEM["Quantum System<br/>(Bulk)"] -->|"Evolution"| STATE["Quantum State"]
    STATE -->|"Measurement"| BOUNDARY["Boundary Projection"]
    BOUNDARY --> OUTCOME["Measurement Result<br/>(Boundary Data)"]

    OUTCOME -.->|"Reconstruction"| INFER["Infer Bulk"]

    style BOUNDARY fill:#e1f5ff,stroke:#0066cc,stroke-width:4px

Insight:

• We never see “true bulk”
• We only see “boundary projection”
• Bulk reconstruction is indirect

3. Essence of Observers

Viewpoint: Any observer can theoretically be modeled as a boundary observer.

Three Levels:

1. Scattering Observer: Measures incoming/outgoing particles at asymptotic boundary
2. Modular Flow Observer: Defines relative entropy on regional boundary
3. Geometric Observer: Measures quasi-local energy on timelike boundary

Unification: These three observers are essentially equivalent, just different projections of same boundary structure!

🎓 Thought Questions

Question 1: Why Can’t We Define “Local Energy” in Minkowski Space?

Hint: Energy conservation requires time translation symmetry. Minkowski space has global time translation, but energy density $T_{00}$ is not a tensor…

Answer: In curved spacetime, there’s no global time translation symmetry, hence no global energy. Brown-York quasi-local energy is best alternative, defined on boundary, doesn’t depend on global symmetry.

Question 2: What’s the Difference Between Extrinsic Curvature $K_{ab}$ and Ricci Curvature $R_{ab}$?

Hint: One is “intrinsic”, one is “extrinsic”.

Answer:

• Ricci Curvature $R_{ab}$: Intrinsic curvature, only depends on metric of surface itself
• Extrinsic Curvature $K_{ab}$: Extrinsic curvature, depends on how surface embeds in surrounding space

Analogy: A piece of paper can be rolled into cylinder (extrinsic curvature ≠ 0), but paper itself is flat (intrinsic curvature = 0).

Question 3: Why Do Null Boundaries Need Special Treatment?

Hint: What condition does normal vector $n^{\mu }$ satisfy?

Answer: For null boundaries, normal vector is null vector: $n^{\mu }{n}_{\mu }=0$. Therefore:

• Cannot normalize $n^{\mu }$
• Definition of extrinsic curvature needs modification
• Must use expansion $\theta$ and surface gravity $\kappa$ instead of $K$

Question 4: What Are the Limits of Boundary Completeness Principle?

Hint: Under what circumstances is boundary data insufficient to reconstruct bulk?

Open Questions:

• Topological changes (e.g., baby universe formation)
• Quantum gravity effects
• Near singularities
• Beyond cosmological horizon

These are current research frontiers!

🌟 Unique Contributions of This Chapter

Compared to traditional general relativity and quantum field theory textbooks, unique aspects of this chapter:

1. Unified Perspective

Traditional:

• GHY boundary term (GR)
• Modular flow (QFT)
• Scattering theory (QM)

Taught separately in different courses, unrelated.

This Chapter: Unifies three as boundary trinity, reveals they are different projections of same object.

2. Boundary Completeness

Traditional: Boundary is technical boundary condition.

This Chapter: Boundary is physical essence, bulk is reconstruction from boundary data.

3. Accessible Explanations

Traditional: Pure technical derivation, hard to understand physical intuition.

This Chapter: Multi-level explanations

• Analogy Level: Room, wall, measurement
• Concept Level: Boundary, observer, time scale
• Mathematical Level: Formulas, theorems, proofs
• Source Theory Level: Links to original literature

4. Coherence

Traditional: Topics taught in isolation.

This Chapter:

• Follows Unified Time Chapter (time scale)
• Serves IGVP Framework (variational principle)
• Leads to Causal Structure Chapter (causal diamond)
• Supports Matrix Universe Chapter (holographic principle)

🔮 Future Prospects

Boundary theory provides foundation for many frontier problems:

1. Black Hole Information Paradox

Problem: When black hole evaporates, where does information go?

Boundary Perspective:

• Information never in black hole “interior”
• Information always encoded on horizon (boundary)
• Black hole evaporation is boundary evolution

2. Holographic Principle

AdS/CFT:

$$\text{(d+1)-Dimensional Gravity}⟺\text{d-Dimensional CFT}$$

This is ultimate version of boundary completeness: all bulk degrees of freedom on boundary!

3. Quantum Gravity

Path Integral:

$$Z=\int \mathcal{D}g{e}^{iS[g]}$$

Under boundary theory framework, should be rewritten as:

$$Z=\int \mathcal{D}(\text{Boundary Data})e^{i{S}_{\mathrm{boundary}}}$$

Bulk metric $g$ is just function of boundary data!

4. Cosmological Horizon

de Sitter Universe: Has cosmological horizon.

Question: What happens beyond horizon?

Boundary Answer: Horizon is true boundary! There’s no “beyond”. All observable physics defined on horizon (boundary).

📚 Further Learning

Core Literature

1. GHY Boundary Term:

   • Gibbons & Hawking (1977): Action integrals and partition functions
   • York (1972): Role of conformal three-geometry

2. Brown-York Energy:

   • Brown & York (1993): Quasilocal energy and conserved charges

3. Modular Flow Theory:

   • Bisognano & Wichmann (1975): On the duality condition
   • Casini et al. (2011): Towards a derivation of holographic entanglement entropy

4. Boundary CFT:

   • Maldacena (1998): The large N limit of superconformal field theories
   • JLMS (Jensen et al.): Entropy in AdS/CFT

Source Theory Documents

This tutorial is based on following source theories:

• docs/euler-gls-paper-bondary/boundary-as-unified-stage.md
• docs/euler-gls-paper-bondary/trinity-master-scale-boundary-time-geometry-null-modular-unification.md
• docs/euler-gls-extend/ghy-boundary-terms-variational-completeness.md

Recommend deep reading for more rigorous mathematical derivations.

✨ Summary: Revolution from Bulk to Boundary

Let’s review core insights of this revolution:

graph TB
    START["Physics in Bulk?"] -->|"Scattering Theory"| SCATTER["Can Only Measure Boundary"]
    START -->|"Quantum Field Theory"| QFT["Modular Flow on Boundary"]
    START -->|"General Relativity"| GR["Action Ill-Defined"]

    SCATTER --> INSIGHT["Physics on Boundary!"]
    QFT --> INSIGHT
    GR --> INSIGHT

    INSIGHT --> TRIPLE["Boundary Triple<br/>(∂M, A_∂, ω_∂)"]

    TRIPLE --> TIME["Unified Time Scale κ"]
    TRIPLE --> GEO["GHY Boundary Term K"]
    TRIPLE --> ALG["Modular Flow Generator K_ω"]

    TIME -.->|"Same Object"| GEO
    GEO -.->|"Same Object"| ALG
    ALG -.->|"Same Object"| TIME

    TIME --> COMPLETE["Boundary Completeness"]
    GEO --> COMPLETE
    ALG --> COMPLETE

    COMPLETE --> HOLO["Holographic Principle"]
    COMPLETE --> QUANTUM["Quantum Gravity"]
    COMPLETE --> COSMO["Cosmology"]

    style START fill:#ffe1e1,stroke:#cc0000
    style INSIGHT fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style TRIPLE fill:#e1f5ff,stroke:#0066cc,stroke-width:4px
    style COMPLETE fill:#e1ffe1,stroke:#00cc00,stroke-width:4px

Core Propositions

Boundary Trinity Proposition:

$$\boxed{\begin{array}{rl}& \text{Scattering Time Scale}={\phi }^{\prime }(\omega )/\pi \\ & \text{Modular Flow Time Scale}={\rho }_{\mathrm{rel}}(\omega )\\ & \text{Geometric Time Scale}=\mathrm{tr}Q(\omega )/(2\pi )\\ & \Downarrow \\ & \text{Unified Boundary Time Scale }\kappa (\omega )\end{array}}$$

GHY Well-Definedness Proposition:

$$\boxed{\delta (S_{\mathrm{EH}}+S_{\mathrm{GHY}})=\frac{1}{16\pi G}{\int }_{\mathcal{M}}\sqrt{-g}{G}_{\mu \nu }\delta g^{\mu \nu }}$$

Boundary Completeness Proposition:

$$\boxed{\text{Bulk Physics}=F[\text{Boundary Triple }(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })]}$$

where $F$ is some reconstruction functional.

Physics is not in bulk, but on boundary.

This is not a metaphor, but a profound physical insight. All computable physical objects—scattering phases, entanglement entropy, energy-momentum—are concentrated on boundary. Bulk is viewed as a “reconstruction” projected from boundary data.

Boundary theory reveals true nature of physics:

• Time is not background, but boundary scale
• Space is not container, but boundary projection
• Observer is not in bulk, but on boundary

Next chapter, we’ll see how boundaries organize through causal structure, forming exquisite structure of causal diamonds and Null-Modular double cover.

Completion: This chapter (06-Boundary Theory) completed ✅

Next Chapter: 07-Causal Structure Chapter - Causal Organization of Boundaries

Back: Boundary Theory Chapter Overview | Complete GLS Theory Tutorial