Boundary Theory: Overview

“Physics is not in the bulk, but on the boundary.”

🎯 Core Ideas of This Chapter

In GLS theory, boundary is viewed as essence. This chapter will present a core perspective:

All computable physical objects are proposed to concentrate on the boundary, the bulk can be viewed as reconstruction of boundary data.

graph TB
    PHYS["Physical Reality"] --> BOUND["Boundary"]
    BOUND --> TIME["Time"]
    BOUND --> GEO["Geometry"]
    BOUND --> ALG["Algebra"]

    TIME --> KAPPA["Unified Time Scale κ"]
    GEO --> GHY["GHY Boundary Term"]
    ALG --> MOD["Modular Flow"]

    KAPPA -.->|"Same Object"| GHY
    GHY -.->|"Same Object"| MOD

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

📚 Chapter Content Map

This chapter consists of 11 articles, revealing the complete picture of boundary physics:

Article 1: Why Boundary?

Core Question: Why must physics be defined on the boundary?

Three Major Evidences:

1. Scattering Theory: $S$-matrix defined at asymptotic boundary
2. Quantum Field Theory: Modular flow localized on region boundary
3. General Relativity: Einstein-Hilbert action alone ill-defined, must add GHY boundary term

Key Insight: Bulk can be viewed as a “phantom” of boundary data!

Article 2: Boundary Data Triplet

Core Object:

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

where:

• $\partial \mathcal{M}$: Geometric boundary (can be piecewise, contains null pieces)
• ${\mathcal{A}}_{\partial }$: Boundary observable algebra
• ${\omega }_{\partial }$: Boundary state

Unified Framework: All boundary physics is theoretically encoded by this triplet!

Article 3: GHY Boundary Term

Core Formula:

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

where:

• $\epsilon =n^{\mu }{n}_{\mu }\in \{\pm 1\}$ (orientation factor)
• $K$: Extrinsic curvature
• $h_{ab}$: Induced metric

Physical Meaning:

• Variational Well-Definedness: Only with GHY term, Einstein-Hilbert action is well-defined for variations fixing boundary metric
• Corner Terms: Piecewise boundaries need additional corner terms
• Null Boundaries: Null geodesic boundaries need $(\theta +\kappa )$ structure

graph LR
    EH["Einstein-Hilbert<br/>Bulk Action"] --> VAR["Variation"]
    VAR --> BAD["✗ Ill-Defined<br/>Contains Normal Derivative Terms"]

    EH2["EH + GHY"] --> VAR2["Variation"]
    VAR2 --> GOOD["✓ Well-Defined<br/>Only Bulk Equations"]

    style BAD fill:#ffe1e1
    style GOOD fill:#e1ffe1

Article 4: Brown-York Quasi-Local Energy

Core Definition:

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

Physical Meaning:

• Quasi-Local Energy: $E_{\mathrm{BY}}=\int \sqrt{\sigma }{u}_a{u}_b{T}_{\mathrm{BY}}^{ab}{\mathrm{d}}^{2}x$
• Asymptotic Limit: $E_{\mathrm{BY}}\to M_{\mathrm{ADM}}$ (ADM mass)
• Differentiability: GHY boundary term makes Hamiltonian differentiable

Deep Connection: $$\text{Brown-York Energy}⟺\text{Boundary Time Generator}⟺\text{Modular Flow Parameter}$$

Article 5: Boundary Observer

Core Concept: Observer is theoretically essentially a boundary observer!

Three Realizations:

1. Scattering Observer: Measures scattering phase at asymptotic boundary
2. Modular Flow Observer: Defines modular Hamiltonian on region boundary
3. Geometric Observer: Measures Brown-York energy on timelike boundary

Unified Scale: All observers are considered to share the same time scale equivalence class $[\kappa ]$!

Article 6: Boundary Theory Mid-Summary

Summary of First 6 Articles: Basic framework of boundary data triplet

Article 7: Boundary as Stage

Core Idea: Physics is considered to truly happen on boundary, bulk is just “projection” or “holographic image” of boundary data

Boundary Triplet: $(\partial M,{\mathcal{A}}_{\partial },{\omega }_{\partial })$ unifies all boundary physics

• $\partial M$: Geometric boundary (physical space of stage)
• ${\mathcal{A}}_{\partial }$: Boundary algebra (set of observables/script)
• ${\omega }_{\partial }$: Boundary state (rules for expectation values/director’s instructions)

Three Actors, Same Stage:

1. Scattering Actor: Time translation $=\int \omega d{\mu }^{\text{scatt}}(\omega )$
2. Modular Flow Actor: Modular Hamiltonian $K_D=-\log \Delta$
3. Geometric Actor: Brown-York Hamiltonian $H_{\partial }^{\text{grav}}$

Boundary Trinity Proposition: $$H_{\partial }=\int \omega d{\mu }^{\text{scatt}}=c_1{K}_D=c_2^{-1}{H}_{\partial }^{\text{grav}}$$

Null-Modular Double Cover:

• Causal diamond boundary decomposes as $E^{+}\bigsqcup E^{-}$ (future/past null pieces)
• Modular Hamiltonian localized on null boundary
• ${\mathbb{Z}}_2$ holonomy characterizes topological structure

Daily Analogy: Theater stage (boundary) is where performance truly happens, three actors can be viewed as three guises of the same person

Article 8: Boundary, Observer, and Time

Core Idea: Time axis = geodesic chosen by observer attention on boundary section family

Three Profound Questions:

1. What is the boundary like without an observer?
2. What mathematical object is the world that the observer “sees”?
3. Is time a product of the observer’s “attention”?

Observer Triplet: $\mathcal{O}=(\gamma ,\Lambda ,{\mathcal{A}}_{\gamma ,\Lambda })$

• $\gamma$: Worldline (observer trajectory)
• $\Lambda$: Resolution (minimum scale)
• ${\mathcal{A}}_{\gamma ,\Lambda }$: Observable algebra (measurable physical quantities)

World Section: ${\Sigma }_{\tau }=(\gamma (\tau ),{\mathcal{A}}_{\gamma ,\Lambda }(\tau ),{\rho }_{\gamma ,\Lambda }(\tau ))$ The world the observer “sees” at time $\tau$

Core Propositions:

• No-Observer Time Inference: No observer → No time, only scale field $\kappa (\omega )$
• Attention Geodesic Hypothesis: Time axis $\tau$ must satisfy:
  1. Scale condition: $\frac{d\tau }{d\lambda }=\int \kappa (\omega )w_{\lambda }(\omega )d\omega$
  2. Generalized entropy geodesic: Section family $\{\sigma (\tau )\}$ makes $S_{\text{gen}}$ stationary
• Section Universe $\mathfrak{S}$: Space of all possible sections, observer experience = a path in it

New Interpretation of Double-Slit Interference: With/without detector = different paths in section universe, quantum superposition = path superposition!

Daily Analogy: Film reel (all frames exist simultaneously), projector (attention) selects frame sequence to produce time flow

Article 9: Boundary Clock

Core Idea: Boundary clock = directly measure scale master $\kappa (\omega )$ using windowed spectral readings

Physical Challenge: Ideal clock is impossible

• Requires infinite time $t\in (-\infty ,+\infty )$ to run
• Requires infinite bandwidth $\omega \in \mathbb{R}$
• Requires infinite energy

Ideal vs Windowed Reading:

• Ideal Reading: ${\mathcal{R}}_{\text{ideal}}={\int }_{-\infty }^{+\infty }\kappa (\omega )f(\omega )d\omega$ (impossible)
• Windowed Reading: ${\mathcal{R}}_{\text{window}}={\int }_{-W}^{+W}W(\omega )\kappa (\omega )f(\omega )d\omega$ (practically feasible)

PSWF/DPSS Optimal Window Functions (Slepian Theorem):

• Prolate Spheroidal Wave Functions are the optimal window function family
• Optimal energy concentration under constraints of time window $[-T,T]$ and frequency band $[-W,W]$
• Effective degrees of freedom: $N_{\text{eff}}\approx 2WT/\pi$
• Eigenvalues ${\lambda }_n$ drop sharply: ${\lambda }_0\approx 1$, ${\lambda }_{2WT/\pi }\approx 0$

Windowed Clock Formula: $${\Theta }_{\Delta }(\omega )=({\rho }_{\text{rel}}\ast P_{\Delta })(\omega )$$ Solves negative delay problem, ensures causality!

Experimental Applications:

• Atomic clock networks (GPS/optical clocks)
• Microwave cavity scattering experiments
• Fast Radio Burst (FRB) time delay
• δ-ring scattering standard source

Daily Analogy: Finite precision watch vs ideal infinite precision clock, minimize error using optimal window functions

Article 10: Trinity Master Scale

Core Idea: Unification of three time definitions might not be coincidence, but profound necessity of boundary geometry

Ultimate Question: Why do three completely different definitions give the same time scale? $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Scale Equivalence Class: $[\kappa ]$ - uniqueness under affine transformation

• Two scales equivalent: ${\tau }_2=a{\tau }_1+b$ (allowing rescaling and translation)
• All affinely related scales form equivalence class $[\kappa ]$
• Different units measure the same length! (e.g., meters, feet, light-seconds)

Three Definitions of Trinity Master Scale:

1. Scattering Phase Derivative (scattering theory): $${\kappa }_{\text{scatt}}(\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }$$ Physical picture: Wavefunction phase change when particles scatter

2. Modular Flow Time Parameter (operator algebra): $${\kappa }_{\text{mod}}(\omega )={\rho }_{\text{rel}}(\omega )$$ Physical picture: Evolution parameter induced by quantum state entanglement structure

3. Brown-York Boundary Energy (general relativity): $${\kappa }_{\text{grav}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$$ Physical picture: Time translation generated by boundary quasi-local energy

Core Propositions:

• Affine Uniqueness Proposition (Proposition 3.1): ${\kappa }_{\text{scatt}}\sim {\kappa }_{\text{mod}}\sim {\kappa }_{\text{grav}}$ Three scales belong to the same equivalence class!
• Topological Class Equivalence (Theorem 3.2): Null-Modular ${\mathbb{Z}}_2$ class $[K]$ equivalent to:
  • Half-phase transition $\Delta \phi =\pi \mathrm{mod}2\pi$
  • Fermion statistics sign $(-1{)}^F$
  • Time crystal period doubling
• Generalized Entropy Variation (Theorem 3.3): $${\delta }^2{S}_{\text{gen}}=\int \kappa (\omega )\Psi (\omega )d\omega +C{\delta }^2{\Lambda }_{\text{eff}}$$ Time scale is the weight of generalized entropy second-order variation!

Null-Modular ${\mathbb{Z}}_2$ Class: $[K]\in H^2(Y,\partial Y;{\mathbb{Z}}_2)$

• Topological DNA of time
• Encodes global topological information on boundary null surfaces
• Determines fundamental properties like fermion statistics, half-integer spin

Daily Analogy: Three blind men touching an elephant (deepened version)

• Blind man A touches trunk (scattering), B touches leg (modular flow), C touches tail (gravity)
• Reported “lengths” $L_1,L_2,L_3$ must be equal
• Reason: They might all be intrinsic scales of the elephant, determined by intrinsic geometry!

Article 11: Boundary Theory Summary

Complete Picture:

graph TB
    BOUNDARY["Boundary<br/>(∂M, A_∂, ω_∂)"] --> THREE["Trinity"]

    THREE --> TIME["Time Scale<br/>κ(ω)"]
    THREE --> GEO["Geometric Boundary<br/>GHY + BY"]
    THREE --> ALG["Algebraic Boundary<br/>Modular Flow"]

    TIME --> SCATTER["Scattering Phase<br/>φ'(ω)/π"]
    TIME --> WIGNER["Group Delay<br/>tr Q/(2π)"]
    TIME --> SPEC["Spectral Shift<br/>ρ_rel"]

    GEO --> K["Extrinsic Curvature<br/>K"]
    GEO --> ENERGY["Quasi-Local Energy<br/>E_BY"]

    ALG --> MOD["Modular Hamiltonian<br/>K_ω"]
    ALG --> ENT["Relative Entropy<br/>S(ρ‖ω)"]

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

🔗 Connections with Other Chapters

Following Unified Time Chapter (Chapter 5)

In Unified Time chapter, we proved:

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

Now we will see: This unified scale is theoretically determined by boundary data!

Leading to Causal Structure Chapter (Chapter 7)

Boundary theory provides foundation for causal structure:

• Causal Diamond: Defined by boundary null surfaces
• Null-Modular Double Cover: Natural structure of null boundaries
• Modular Hamiltonian: Localized on boundary null surfaces

Connecting IGVP Framework (Chapter 4)

Boundary theory completes IGVP variational principle:

• Generalized Entropy: Extremum on small causal diamond boundary
• Einstein Equation: First-order condition from boundary variation
• QNEC/QFC: Second-order conditions from boundary variation

💡 Learning Roadmap

graph TB
    START["Start Boundary Theory"] --> PART1["Part 1: Basic Framework<br/>(Articles 1-6)"]

    PART1 --> WHY["01-Why Boundary<br/>Three Historical Evidences"]
    WHY --> TRIPLE["02-Boundary Triplet<br/>(∂M, 𝒜_∂, ω_∂)"]
    TRIPLE --> GHY["03-GHY Boundary Term<br/>Variational Well-Definedness"]
    GHY --> BY["04-Brown-York Energy<br/>Quasi-Local Energy"]
    BY --> OBS["05-Boundary Observer<br/>Three Observer Unifications"]
    OBS --> SUM1["06-Mid-Summary"]

    SUM1 --> PART2["Part 2: Deepening Unification<br/>(Articles 7-10)"]

    PART2 --> STAGE["07-Boundary as Stage<br/>Boundary Trinity"]
    STAGE --> TIME["08-Boundary Observer and Time<br/>Attention Geodesic"]
    TIME --> CLOCK["09-Boundary Clock<br/>PSWF Optimal Window Functions"]
    CLOCK --> MASTER["10-Trinity Master Scale<br/>Affine Uniqueness"]

    MASTER --> FINAL["11-Boundary Theory Summary<br/>Complete Picture"]

    GHY -.->|"Mathematical Details"| TECH["Technical Appendix"]

    style START fill:#e1f5ff
    style PART1 fill:#fff4e1
    style PART2 fill:#ffe1e1
    style FINAL fill:#e1ffe1,stroke:#00cc00,stroke-width:3px

Recommended Reading Order

Quick Path (grasp core):

1. 01-Why Boundary (intuition)
2. 03-GHY Boundary Term (core formula)
3. 07-Boundary as Stage (unified perspective)
4. 10-Trinity Master Scale (ultimate unification)
5. 11-Summary (complete picture)

Complete Learning (deep understanding): Read 01-11 in order, divided into two stages:

• Stage 1: 01-06 (basic framework)
• Stage 2: 07-10 (deepening unification)

Technical Research (rigorous derivation): Focus on:

• Appendix of 03-GHY Boundary Term (variational calculation)
• 09-Boundary Clock (PSWF/DPSS mathematics)
• 10-Trinity Master Scale (topological ${\mathbb{Z}}_2$ class)

🎓 Core Conclusions Preview

After completing this chapter, you will understand:

1. Boundary Completeness Hypothesis

Proposition: Bulk physics content can theoretically be completely reconstructed from boundary triplet.

Evidence:

• Scattering theory: Wave operators and $S$-matrix
• AdS/CFT: Boundary CFT reconstructs bulk geometry
• Hamilton-Jacobi: Boundary data reconstructs bulk solution

2. Boundary Time Trinity Proposition

Proposition: The following three “boundary times” are proposed to be equivalent:

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

Unified Generator:

$$H_{\partial }=\int \omega \mathrm{d}{\mu }_{\partial }^{\mathrm{scatt}}(\omega )=c_1{K}_D+c_2^{-1}{H}_{\partial }^{\mathrm{grav}}$$

3. GHY Necessity Argument

Argument: On non-null boundaries, after adding

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

for variations fixing induced metric $h_{ab}$:

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

Boundary terms completely cancel!

4. Quasi-Local Energy Convergence Property

Property: Brown-York quasi-local energy converges to ADM mass in asymptotically flat limit:

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

And is conserved under spacetime evolution (under appropriate boundary conditions).

🤔 Thinking Questions (Chapter Preview)

Question 1: Why is Einstein-Hilbert Action Ill-Defined?

Hint: Calculate $\delta S_{\mathrm{EH}}$, see what uncontrollable derivatives appear in boundary terms.

Answer in: 01-Why Boundary, 03-GHY Boundary Term

Question 2: What is “Quasi-Local” Energy?

Hint: Why can’t we define “local” energy in curved spacetime? What is the best alternative?

Answer in: 04-Brown-York Energy

Question 3: How Do Boundary Observers Measure Time?

Hint: Recall time scale identity from Unified Time chapter, now all on boundary!

Answer in: 05-Boundary Observer

Question 4: How Does AdS/CFT Reflect Boundary Completeness?

Hint: Boundary CFT completely determines bulk AdS geometry.

Answer in: 06-Summary, and future advanced topics chapters

📖 Notation Conventions

This chapter uses the following core symbols:

Geometric Symbols

• $\mathcal{M}$: Spacetime manifold (4-dimensional)
• $\partial \mathcal{M}$: Boundary (3-dimensional, can be piecewise)
• $g_{\mu \nu }$: Bulk metric (signature $-+++$)
• $h_{ab}$: Induced metric
• $n^{\mu }$: Unit normal vector
• $\epsilon :=n^{\mu }{n}_{\mu }\in \{\pm 1\}$: Orientation factor

Curvature Symbols

• $R$: Ricci scalar
• $K_{ab}$: Extrinsic curvature
• $K:=h^{ab}{K}_{ab}$: Trace of extrinsic curvature

Boundary Objects

• $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$: Boundary triplet
• $T_{\mathrm{BY}}^{ab}$: Brown-York stress tensor
• $E_{\mathrm{BY}}$: Brown-York quasi-local energy
• $S_{\mathrm{GHY}}$: Gibbons-Hawking-York boundary term

Null Boundaries

• ${\ell }^{\mu }$: Null generator vector ($\ell \cdot \ell =0$)
• $\theta$: Expansion
• $\kappa$: Surface gravity
• ${\gamma }_{AB}$: Transverse two-dimensional metric

🔍 Unique Contributions of This Chapter

Compared to traditional general relativity textbooks, this chapter:

1. Unifies Three Perspectives

   • Traditional: Separately discuss GHY term, Brown-York energy, modular flow
   • This chapter: Unified as boundary trinity

2. Emphasizes Boundary Completeness

   • Traditional: Boundary is technical supplement
   • This chapter: Boundary is physical essence

3. Connects Time Scale

   • Traditional: Isolated discussion of various times
   • This chapter: All times unified by boundary scale

4. Intuitive Explanations

   • Traditional: Pure technical derivation
   • This chapter: Multi-level explanations (analogy → concept → formula → source)

🌟 Why Is This Chapter Important?

Boundary Theory is one of the pillars of GLS theory, because:

Theoretical Level

• Reveals boundary essence of physics
• Unifies three perspectives: time, geometry, algebra
• Provides foundation for causal structure and topological constraints

Application Level

• Black hole thermodynamics: Horizon is boundary
• AdS/CFT: Core of holographic principle
• Quantum gravity: Boundary degrees of freedom

Philosophical Level

• Paradigm shift from bulk to boundary
• Boundary essence of observer
• Measurement as boundary projection

Ready?

Let’s begin this paradigm revolution from bulk to boundary!

Next Article: 01-Why Boundary - Revealing why physics must be defined on boundary

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