Why Boundary? Paradigm Shift from Bulk to Boundary

“Truly computable physical objects are often concentrated on the boundary, while the bulk is more like reconstruction or evolution from boundary data.”

🎯 Core Question

Question: Why must physics be defined on the boundary?

Answer Preview: Because all measurable physical quantities are theoretically realized through the boundary!

💡 Intuitive Image: Room Analogy

Imagine a scenario:

Traditional Physics View (Bulk-Centered):

• Physics happens inside the room
• Boundary (walls) is just a constraint
• To understand the room, must know what happens at every point inside

GLS Boundary View (Boundary-Centered):

• Physics essence is considered to be on the walls!
• Room interior is just “projection” of wall information
• To understand the room, only need to know data on walls

graph LR
    subgraph Traditional View
    BULK1["Bulk<br/>Main Character"] --> BOUND1["Boundary<br/>Supporting Role"]
    end

    subgraph GLS View
    BOUND2["Boundary<br/>Main Character"] --> BULK2["Bulk<br/>Reconstruction"]
    end

    style BULK1 fill:#e1f5ff
    style BOUND2 fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Key Insight:

• You can only measure the room through walls!
• Sound, light, temperature on walls theoretically completely determine room interior
• Interior is viewed as a necessary consequence of wall data

📜 Historical Evidence: Three Major Paradigm Shifts

1. Scattering Theory: $S$-Matrix at Asymptotic Boundary

History: 1940s-60s, Heisenberg, Wheeler proposed $S$-matrix theory

Core Concept:

• Experiments can only measure asymptotic particles ($t\to \pm \infty$)
• Scattering matrix $S$ defined at spacetime asymptotic boundary
• Bulk interaction details cannot be directly observed

Mathematical Expression:

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

where ${\mathcal{H}}_{in/out}$ are Hilbert spaces of asymptotic free states, defined on spacetime boundaries ${\mathcal{I}}^{\pm }$.

graph LR
    IN["Incoming Particles<br/>I⁻ (Past Boundary)"] --> INT["?<br/>Interaction Region"]
    INT --> OUT["Outgoing Particles<br/>I⁺ (Future Boundary)"]

    S["S-Matrix"] -.->|"Direct Connection"| IN
    S -.->|"Direct Connection"| OUT

    INT -.->|"Cannot Directly Measure"| S

    style IN fill:#e1f5ff
    style OUT fill:#e1f5ff
    style INT fill:#f0f0f0,stroke-dasharray: 5 5
    style S fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px

Boundary Nature of Birman-Kreĭn Formula:

In Unified Time chapter, we learned:

$$\det S(\omega )=\exp (-2\pi i\xi (\omega ))$$

New Understanding Now:

• $\xi (\omega )$: spectral shift function, bulk spectral change
• $\det S(\omega )$: scattering determinant, boundary data
• Birman-Kreĭn identity indicates: bulk spectral change can be read from boundary scattering data.

Physical Meaning: $$\boxed{\text{Bulk Information (Spectrum)}=\text{Function of Boundary Data (}S\text{-Matrix)}}$$

2. Quantum Field Theory: Modular Flow Localized on Regional Boundary

History: 1970s, Tomita-Takesaki modular theory; 2010s, boundary modular Hamiltonian

Core Concept:

• Given region $O$ and state $\omega$, there exists canonical one-parameter automorphism group (modular flow)
• Bisognano-Wichmann theorem: modular flow of vacuum state is Lorentz transformation on that region’s boundary
• Modular Hamiltonian can be written as local integral of boundary stress tensor

Bisognano-Wichmann Theorem (Wedge Region):

For Rindler wedge $W=\{(t,x,y,z):|t|<x\}$, modular flow of Minkowski vacuum $|0⟩$ restricted to $\mathcal{A}(W)$ is:

$${\sigma }_s^{{\omega }_0}(A)=U_{\mathrm{boost}}(s)A{U}_{\mathrm{boost}}^{-1}(s)$$

That is, hyperbolic rotation (Lorentz boost) along wedge boundary!

graph TB
    REGION["Causal Region O"] --> BOUNDARY["Boundary ∂O"]
    BOUNDARY --> FLOW["Modular Flow σₜʷ"]

    FLOW --> HAM["Modular Hamiltonian K_O"]
    HAM --> INTEGRAL["Boundary Integral"]

    INTEGRAL --> STRESS["Stress Tensor<br/>T_μν"]

    FORMULA["K_O = 2π ∫∂O ξᵘ T_μν n^ν dΣ"]

    style BOUNDARY fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style INTEGRAL fill:#e1f5ff

Mathematical Expression (Spherical Region):

For spherical causal diamond $D$, modular Hamiltonian is:

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

where ${\xi }^{\mu }$ is conformal Killing vector on boundary, $n^{\nu }$ is normal.

Physical Meaning: $$\boxed{\text{Algebraic Intrinsic Time (Modular Flow)}\cong \text{Boundary Geometric Generator}}$$

Null-Modular Double Cover (Deeper Boundary Structure):

For causal diamond $D=J^{+}(p)\cap J^{-}(q)$, boundary consists of two null hypersurfaces:

$$\partial D={\mathcal{N}}^{+}\cup {\mathcal{N}}^{-}$$

Modular Hamiltonian can be completely localized on these two null boundaries:

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

This is pure boundary expression, no bulk integral needed!

3. General Relativity: Necessity of GHY Boundary Term

History: 1977, Gibbons-Hawking-York discovered Einstein-Hilbert action is ill-defined

Problem Discovery:

Einstein-Hilbert action:

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

Computing variation:

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

Problem: Boundary term contains $n^{\rho }{\nabla }_{\rho }\delta g_{\alpha \beta }$ (normal derivative of metric)!

Consequences:

• Fixing boundary induced metric $h_{ab}$ is insufficient for well-defined variation
• Also need to fix $n^{\rho }{\nabla }_{\rho }{g}_{\alpha \beta }$ (unnatural boundary condition)
• Hamiltonian functional is not differentiable

graph TB
    SEH["S_EH = ∫√(-g) R"] --> VAR["Compute Variation δS_EH"]
    VAR --> BULK["✓ Bulk Term<br/>G_μν δg^μν"]
    VAR --> BOUND["✗ Boundary Term<br/>Contains n·∇δg"]

    BOUND --> BAD1["Need to Fix n·∇g"]
    BOUND --> BAD2["Hamiltonian Not Differentiable"]
    BOUND --> BAD3["Variational Principle Ill-Defined"]

    style BOUND fill:#ffe1e1,stroke:#cc0000,stroke-width:2px

GHY Solution:

Add boundary term:

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

where:

• $h_{ab}$: induced metric
• $K=h^{ab}{K}_{ab}$: trace of extrinsic curvature
• $\epsilon =n^{\mu }{n}_{\mu }\in \{\pm 1\}$: orientation factor

Magical Effect:

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

Boundary terms completely cancel!

graph LR
    SEH["S_EH"] --> NOTOK["✗ Ill-Defined"]
    SGHY["+ S_GHY"] --> OK["✓ Well-Defined"]

    OK --> EIN["Einstein Equations"]

    style NOTOK fill:#ffe1e1
    style OK fill:#e1ffe1

Physical Meaning: $$\boxed{\text{Differentiability of Gravitational Action}\leftarrow \text{Determined by Boundary Term}}$$

Deeper Understanding:

Why is boundary term needed? Because Einstein equations are second-order partial differential equations, integration by parts produces boundary terms. This is not a technical detail, but geometric necessity:

Gauss-Codazzi Equation:

$$R=\hat{R}+K^{ab}{K}_{ab}-K^2+2{\nabla }_{\mu }(n^{\nu }{\nabla }_{\nu }{n}^{\mu }-n^{\mu }{\nabla }_{\nu }{n}^{\nu })$$

The last term is a total divergence, integrated produces boundary term, exactly the source of GHY term!

🔗 Unification of Three Evidences

Now we see an astonishing unification:

Common Theme: $$\boxed{\text{Bulk}\approx \text{Function (or Reconstruction) of Boundary}}$$

graph TB
    SCATTER["Scattering Theory"] --> UNITY["Boundary Completeness"]
    QFT["Quantum Field Theory"] --> UNITY
    GR["General Relativity"] --> UNITY

    UNITY --> PRINCIPLE["Unified Principle"]

    PRINCIPLE --> P1["Bulk Objects<br/>Determined by Boundary Data"]
    PRINCIPLE --> P2["Computable Quantities<br/>Concentrated on Boundary"]
    PRINCIPLE --> P3["Time, Algebra, Geometry<br/>Unified on Boundary"]

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

🌟 Boundary Completeness Principle

Based on the three major evidences above, we propose:

Theoretical Postulate (Boundary Completeness):

Physical content of bulk region $\mathcal{M}$ can theoretically be completely reconstructed from some boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$ (within the applicable range of the given theory), i.e., time evolution and response operators are all determined by boundary one-parameter automorphism groups and state evolution.

Three Realizations:

1. Scattering Theory: Wave operators and $S$-matrix reconstruct Hamiltonian
2. AdS/CFT: Boundary CFT completely determines bulk AdS geometry
3. Hamilton-Jacobi: Boundary data reconstruct bulk Einstein equation solutions

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

    RECON --> BULK1["Bulk Geometry<br/>g_μν"]
    RECON --> BULK2["Bulk Fields<br/>φ"]
    RECON --> BULK3["Bulk Evolution<br/>Time Flow"]

    BULK1 -.->|"Uniquely Determined"| BOUNDARY
    BULK2 -.->|"Uniquely Determined"| BOUNDARY
    BULK3 -.->|"Uniquely Determined"| BOUNDARY

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

🔍 Why Is Traditional Physics “Bulk-Centered”?

Historical Reasons:

1. Newtonian Mechanics: Point particles move in space, space is the stage
2. Early Field Theory: Fields defined at each spacetime point
3. Mathematical Habit: Partial differential equations solved in regions

Measurement Reality:

• You can never measure “deep in bulk”
• All detectors are on some “boundary”
• Signals must propagate to observer (boundary)

Paradigm Lock:

• Textbooks continue “field at spacetime point” language
• But quantum field theory already shifted to operator algebras (boundary view)
• General relativity must add GHY term (boundary correction)

💎 New Role of Bulk: Phantom of Boundary Data

Traditional View: Bulk is real, boundary is additional

GLS View: Boundary is viewed as real, bulk is viewed as reconstructed

Analogy: Hologram

• 3D image you see (bulk)
• Actually stored on 2D film (boundary)
• Destroy a small piece of film, entire 3D image blurs but doesn’t disappear
• Information on boundary, displayed in bulk

graph LR
    HOLO["Holographic Film<br/>2D Boundary"] --> IMAGE["3D Image<br/>Bulk"]

    INFO["Information"] --> HOLO
    INFO -.->|"Not Directly In"| IMAGE

    style HOLO fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style IMAGE fill:#f0f0f0,stroke-dasharray: 5 5

Mathematical Analogy:

• Boundary Data: Cauchy data (initial values)
• Bulk Solution: Evolved fields
• Uniqueness Theorem: Appropriate boundary data uniquely determines bulk solution

🎯 Three Levels of Boundaries

According to physical content, boundaries have three levels:

Level 1: Geometric Boundary

Definition: Topological boundary $\partial \mathcal{M}$ of manifold $\mathcal{M}$

Examples:

• Boundary of finite spacetime region
• Black hole horizon
• Cosmological horizon
• Conformal boundary of AdS spacetime

Level 2: Causal Boundary

Definition: Asymptotic structure ${\mathcal{I}}^{\pm }$ (past/future null infinity)

Physical Meaning:

• Natural boundary of scattering theory
• Where light signals ultimately arrive
• Endpoints of null geodesics

Level 3: Observer Horizon

Definition: Boundary of observer’s accessible domain

Examples:

• Rindler horizon (accelerating observer)
• de Sitter horizon (cosmology)
• Causal diamond boundary (local observer)

graph TB
    BOUND["Boundary Concept"] --> GEO["Geometric Boundary<br/>∂M"]
    BOUND --> CAUSAL["Causal Boundary<br/>I±"]
    BOUND --> OBS["Observer Horizon"]

    GEO --> EX1["Finite Region Boundary"]
    CAUSAL --> EX2["Light Infinity"]
    OBS --> EX3["Accelerating Observer Horizon"]

    style BOUND fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Unified Understanding:

All these boundaries can be described by boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$:

• $\partial \mathcal{M}$: chosen geometric boundary
• ${\mathcal{A}}_{\partial }$: algebra of observables on that boundary
• ${\omega }_{\partial }$: state on boundary (defines “vacuum” or thermal state)

🔬 Experimental Evidence

Boundary view is not just theoretical elegance, but has experimental support:

1. Scattering Experiments

All high-energy physics experiments are boundary measurements:

• Particle accelerators: incoming particle preparation, outgoing particle detection
• Detectors: “spherical boundary” surrounding collision point
• Data: $S$-matrix elements, i.e., boundary-boundary amplitudes

2. Black Hole Thermodynamics

Hawking Radiation:

• Horizon as boundary
• Thermal radiation emitted from boundary
• Entropy $S=\mathcal{A}/(4G)$ proportional to boundary area (not volume!)

3. Cosmological Observations

CMB (Cosmic Microwave Background):

• What we see is “last scattering surface” (past light cone boundary)
• Cosmological parameters extracted from this 2D boundary
• Future observations limited by de Sitter horizon

4. AdS/CFT Correspondence

Theoretical Prediction, Numerical Verification:

• Strongly coupled plasma properties (RHIC experiments)
• Calculated using boundary CFT, matches experiments
• Holographic duals of condensed matter systems

🤔 Philosophical Reflection

Question: Why Does Nature Choose Boundary?

Possible Answers:

1. Causality: Information propagation takes time, ultimately reaches boundary
2. Measurement Theory: Measurement devices must be in finite region (some boundary)
3. Quantum Entanglement: Entanglement entropy on boundary determines bulk properties
4. Holographic Principle: Gravitational theories naturally have one less dimension (boundary one dimension lower)

Question: Does Bulk Still Have Meaning?

Answer: Yes, but role changes

• Traditional Role: Stage where physics happens
• New Role: Convenient representation of boundary data
• Analogy: Map (boundary) vs. territory (bulk), but map already contains all information!

Question: Does This Mean “Space Doesn’t Exist”?

Answer: No, rather “space is emergent”

• Fundamental Level: Boundary data (information theory)
• Emergent Level: Bulk geometry (classical description)
• Relation: Geometry emerges from entanglement structure

graph TB
    FUND["Fundamental Level: Boundary Data"] --> EMERGE["Emergent Level: Bulk Geometry"]

    FUND --> INFO["Information<br/>Entanglement"]
    EMERGE --> GEO["Metric<br/>Curvature"]

    INFO -.->|"Reconstruction"| GEO

    style FUND fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style EMERGE fill:#e1f5ff

📝 Chapter Summary

We answered the core question: Why must physics be defined on boundary?

Three Major Evidences

1. Scattering Theory: $S$-matrix at asymptotic boundary, Birman-Kreĭn formula connects bulk spectrum with boundary data
2. Quantum Field Theory: Modular flow localized on regional boundary, Bisognano-Wichmann theorem
3. General Relativity: GHY boundary term makes variation well-defined, boundary determines action differentiability

Core Insight

$$\boxed{\text{Computable Physics}\approx \text{Function of Boundary Data}}$$

Paradigm Shift

Boundary Completeness Principle

Bulk physical content can theoretically be completely reconstructed from boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$.

Next Step: Since boundary is so fundamental, next article we will define boundary data triple in detail.

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