Boundary Data Triple: Complete Encoding of Physics

“Three data, encoding everything.”

🎯 Core Idea

In the previous article, we learned: Physics is viewed as being on the boundary.

Now the question: What data is needed on the boundary?

Answer: Theoretically exactly three objects:

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

graph TB
    TRIPLE["Boundary Triple"] --> GEO["∂M<br/>Geometric Boundary"]
    TRIPLE --> ALG["A_∂<br/>Observable Algebra"]
    TRIPLE --> STATE["ω_∂<br/>State"]

    GEO --> EX1["Manifold Structure"]
    ALG --> EX2["Operator Algebra"]
    STATE --> EX3["Expectation Values"]

    GEO -.->|"Where"| PHYS["Physics"]
    ALG -.->|"What to Measure"| PHYS
    STATE -.->|"What Result"| PHYS

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

💡 Intuitive Image: Three Elements of Measurement

Imagine you’re doing an experiment:

First Element: Where to Measure?

Geometric Boundary $\partial \mathcal{M}$

Like the walls of a laboratory:

• Has position (space)
• Has shape (geometry)
• May have multiple pieces (piecewise boundary)

Examples:

• Particle detector position: sphere $r=R$
• Black hole horizon: null surface $r=2M$
• Cosmic horizon: past light cone

Second Element: What to Measure?

Observable Algebra ${\mathcal{A}}_{\partial }$

Like experimental instruments:

• Can measure energy $\hat{E}$
• Can measure momentum $\hat{p}$
• Can measure various fields $\hat{\varphi }(x)$

Key: Not arbitrary functions, but operators!

Third Element: What is the Measurement Result?

State ${\omega }_{\partial }$

Like experimental setup and results:

• Initial state preparation (vacuum? thermal state?)
• Measurement expectation value $⟨A⟩=\omega (A)$
• Statistical fluctuations $⟨A^2⟩-⟨A{⟩}^2$

graph LR
    WHERE["Where?<br/>∂M"] --> MEASURE["Measurement"]
    WHAT["What?<br/>A_∂"] --> MEASURE
    RESULT["Result?<br/>ω_∂"] --> MEASURE

    MEASURE --> DATA["Physical Data"]

    style MEASURE fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px

All three are indispensable!

• Without $\partial \mathcal{M}$: don’t know where
• Without ${\mathcal{A}}_{\partial }$: don’t know what to measure
• Without ${\omega }_{\partial }$: don’t know result

📐 First Component: Geometric Boundary $\partial \mathcal{M}$

Definition

$\partial \mathcal{M}$ is a three-dimensional manifold (when $\mathcal{M}$ is four-dimensional spacetime), decomposable as:

$$\partial \mathcal{M}=\bigcup _i{\mathcal{B}}_i$$

where each ${\mathcal{B}}_i$ can be:

• Timelike piece (timelike): $n^2=+1$
• Spacelike piece (spacelike): $n^2=-1$
• Null piece (null): ${\ell }^2=0$

graph TB
    BOUND["Boundary ∂M"] --> TIME["Timelike Piece<br/>Side"]
    BOUND --> SPACE["Spacelike Piece<br/>Initial/Final"]
    BOUND --> NULL["Null Piece<br/>Horizon"]

    JOINT["Corner C_ij = B_i ∩ B_j"]

    TIME --> JOINT
    SPACE --> JOINT
    NULL --> JOINT

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

Geometric Data

Each piece carries:

Non-Null Pieces $(n^2=\pm 1)$

• Induced Metric: $$h_{ab}=g_{ab}-\epsilon n_a{n}_b,\epsilon =n^2$$

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

• Intrinsic Curvature: $${\hat{R}}_{abcd}[\gamma ]$$

  (Riemann tensor computed with $h_{ab}$)

Null Pieces $({\ell }^2=0)$

• Transverse Metric: $${\gamma }_{AB}(A,B=2,3)$$

• Expansion: $$\theta ={\gamma }^{AB}{W}_{AB},W_{AB}={\gamma }_A{}^{\mu }{\gamma }_B{}^{\nu }{\nabla }_{\mu }{\ell }_{\nu }$$

• Surface Gravity: $$\kappa =-k_{\mu }{\ell }^{\nu }{\nabla }_{\nu }{\ell }^{\mu }(\ell \cdot k=-1)$$

Corners

Intersection ${\mathcal{C}}_{ij}={\mathcal{B}}_i\cap {\mathcal{B}}_j$ is a two-dimensional surface, carrying:

• Induced Metric: ${\sigma }_{ab}$ (inherited from $h_{ab}$)
• Angle: $\eta$ or logarithmic angle $a$

Example:

graph TB
    subgraph Piecewise Boundary Example
    INIT["Initial Spacelike Surface<br/>t=0"] --> CORNER1["Corner"]
    SIDE["Timelike Side<br/>r=R"] --> CORNER1
    SIDE --> CORNER2["Corner"]
    FINAL["Final Spacelike Surface<br/>t=T"] --> CORNER2
    end

    style CORNER1 fill:#ffe1e1
    style CORNER2 fill:#ffe1e1

🔬 Second Component: Observable Algebra ${\mathcal{A}}_{\partial }$

Definition

${\mathcal{A}}_{\partial }$ is a von Neumann algebra acting on Hilbert space $\mathcal{H}$, containing:

• Boundary field operators: $\hat{\varphi }(x),x\in \partial \mathcal{M}$
• Scattering operators: incoming/outgoing channels
• Quasi-local energy operators

Algebraic Structure

Closure: $$A,B\in {\mathcal{A}}_{\partial }\Rightarrow AB,A+B,\alpha A\in {\mathcal{A}}_{\partial }$$

Self-Adjointness: $$A\in {\mathcal{A}}_{\partial }\Rightarrow A^{†}\in {\mathcal{A}}_{\partial }$$

Weak Closure: $$A_n\to A\text{ weakly}\Rightarrow A\in {\mathcal{A}}_{\partial }$$

graph TB
    ALG["Algebra A_∂"] --> FIELD["Field Operators<br/>φ(x)"]
    ALG --> SCATTER["Scattering Operators<br/>S-matrix"]
    ALG --> ENERGY["Energy Operators<br/>H_∂"]

    PROP["Algebraic Properties"] --> CLOSE["Closure"]
    PROP --> SELF["Self-Adjoint"]
    PROP --> WEAK["Weak Closure"]

    style ALG fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px

Three Realizations

Depending on physical context, ${\mathcal{A}}_{\partial }$ has different realizations:

1. Scattering End

$${\mathcal{A}}_{\partial }^{\mathrm{scatt}}=\text{span}\{a_{\mathrm{in}}^{†}(k),a_{\mathrm{out}}^{†}(k^{\prime }),\dots \}$$

• Incoming/outgoing creation-annihilation operators
• $S$-matrix: $S:{\mathcal{H}}_{\mathrm{in}}\to {\mathcal{H}}_{\mathrm{out}}$
• Boundary time measure: $\mathrm{d}{\mu }_{\partial }^{\mathrm{scatt}}=\frac{1}{2\pi }\mathrm{tr}Q(\omega )\mathrm{d}\omega$

2. Regional Algebra End

$${\mathcal{A}}_{\partial }=\mathcal{A}(\partial O)$$

• Local algebra on boundary $\partial O$ of causal region $O$
• Satisfies causality: $[\mathcal{A}(O_1),\mathcal{A}(O_2)]=0$ when $O_1,O_2$ spacelike separated
• Modular flow automorphism group: ${\sigma }_t^{\omega }:{\mathcal{A}}_{\partial }\to {\mathcal{A}}_{\partial }$

3. Gravitational End

$${\mathcal{A}}_{\partial }^{\mathrm{grav}}=\{h_{ab},{\pi }^{ab},\dots \}$$

• Boundary metric $h_{ab}$ and its conjugate momentum ${\pi }^{ab}$
• Brown-York quasi-local energy operators
• Algebra generated by boundary Killing vectors

🌟 Third Component: State ${\omega }_{\partial }$

Definition

${\omega }_{\partial }$ is a state on ${\mathcal{A}}_{\partial }$, i.e., satisfying:

1. Positivity: $\omega (A^{†}A)\ge 0$ for all $A\in {\mathcal{A}}_{\partial }$
2. Normalization: $\omega (\mathbb{I})=1$
3. Linearity: $\omega (\alpha A+\beta B)=\alpha \omega (A)+\beta \omega (B)$

Physical Meaning: $\omega (A)$ gives the expectation value of operator $A$.

GNS Construction

Given $({\mathcal{A}}_{\partial },{\omega }_{\partial })$, there exists unique triple:

$$({\pi }_{\omega },{\mathcal{H}}_{\omega },{\Omega }_{\omega })$$

where:

• ${\mathcal{H}}_{\omega }$: GNS Hilbert space
• ${\pi }_{\omega }:{\mathcal{A}}_{\partial }\to \mathcal{B}({\mathcal{H}}_{\omega })$: representation
• ${\Omega }_{\omega }\in {\mathcal{H}}_{\omega }$: cyclic vector

such that: $$\omega (A)=⟨{\Omega }_{\omega },{\pi }_{\omega }(A){\Omega }_{\omega }⟩$$

graph TB
    STATE["State ω_∂"] --> GNS["GNS Construction"]

    GNS --> HIL["Hilbert Space<br/>H_ω"]
    GNS --> REP["Representation<br/>π_ω"]
    GNS --> VEC["Cyclic Vector<br/>Ω_ω"]

    EXPECT["Expectation Value<br/>⟨A⟩ = ω(A)"] --> GNS

    style STATE fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px

Three Important States

1. Vacuum State ${\omega }_0$

Definition: Poincaré-invariant state (Minkowski spacetime)

Properties:

• Lowest energy: ${\omega }_0(H)=0$
• Reeh-Schlieder property: ${\Omega }_0$ cyclic for regional algebras
• Modular flow = geometric symmetry: Bisognano-Wichmann theorem

2. KMS State ${\omega }_{\beta }$

Definition: Thermal equilibrium state at temperature $T=1/\beta$

KMS Condition: $${\omega }_{\beta }(AB)={\omega }_{\beta }(B{\sigma }_{i\beta }^{\omega }(A))$$

Examples:

• Hawking radiation: ${\beta }_H=8\pi M$ (Schwarzschild black hole)
• Unruh effect: $\beta =2\pi /a$ (acceleration $a$)

3. Relative State ${\omega }_{\rho }$

Definition: State given density matrix $\rho$

Relative Entropy: $$S({\omega }_{\rho }\Vert \omega )=\mathrm{Tr}(\rho \ln \rho )-\mathrm{Tr}(\rho \ln \sigma )$$

where $\sigma$ is the density matrix corresponding to $\omega$.

🔗 Intrinsic Connections of the Triple

The three components are not independent, but closely related:

1. Geometry Determines Algebra

Causality: Causal structure of $\partial \mathcal{M}$ is considered to determine locality of algebra

$$O_1\perp O_2(\text{spacelike separated})\Rightarrow [\mathcal{A}(O_1),\mathcal{A}(O_2)]=0$$

2. State Induces Modular Flow

Tomita-Takesaki Theory: $({\mathcal{A}}_{\partial },{\omega }_{\partial })$ uniquely determines modular flow

$${\sigma }_t^{\omega }(A)={\Delta }_{\omega }^{it}A{\Delta }_{\omega }^{-it}$$

where ${\Delta }_{\omega }$ is the modular operator.

3. Duality Between Geometry and State

Bisognano-Wichmann Theorem: Modular flow of vacuum state ${\omega }_0$ restricted to wedge $W$ is Lorentz boost along wedge boundary

$${\sigma }_s^{{\omega }_0}{|}_{\mathcal{A}(W)}=\mathrm{Ad}(U_{\mathrm{boost}}(2\pi s))$$

graph TB
    GEO["Geometry ∂M"] -.->|"Causal Structure"| ALG["Algebra A_∂"]
    ALG -.->|"State"| STATE["ω_∂"]
    STATE -.->|"Modular Flow"| MOD["σₜʷ"]
    MOD -.->|"BW Theorem"| GEO

    style GEO fill:#e1f5ff
    style ALG fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style STATE fill:#e1ffe1
    style MOD fill:#ffe1f5

🌟 Mathematical Formulation of Boundary Completeness

Now we can precisely formulate boundary completeness:

Proposition (Boundary Reconstruction):

Given boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$ with appropriate regularity conditions, there theoretically exists unique (up to natural equivalence) bulk theory $(\mathcal{M},\mathcal{A},\omega )$ such that:

1. ${\mathcal{A}}_{\partial }\subset \mathcal{A}$ is boundary subalgebra
2. $\omega {|}_{{\mathcal{A}}_{\partial }}={\omega }_{\partial }$
3. $\partial \mathcal{M}$ is causal boundary of $\mathcal{M}$

Proof Outline (different contexts):

Scattering Theory

• Given: $S(\omega )$ matrix (boundary data)
• Reconstruct: Hamiltonian $H$ and potential $V$ (bulk)
• Uniqueness: Inverse scattering theory

AdS/CFT

• Given: Boundary CFT data $(T^{ab},J^{\mu },\dots )$
• Reconstruct: Bulk AdS metric $g_{\mu \nu }$
• Uniqueness: Holographic renormalization group

Hamilton-Jacobi

• Given: Boundary $(h_{ab},{\pi }^{ab})$ (Cauchy data)
• Reconstruct: Bulk $g_{\mu \nu }$ satisfying Einstein equations
• Uniqueness: Uniqueness theorem for hyperbolic PDEs

📊 Triple in Three Contexts

Unified Scale:

In all three contexts, time scale density $\kappa (\omega )$ can be extracted from boundary data:

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

(See Unified Time chapter for details)

🎯 Minimality of Boundary Data

Question: Is the triple minimal? Can we use less data?

Answer: Theoretically no. Each of the three components is considered irreplaceable.

Why Do We Need $\partial \mathcal{M}$?

Counterexample: Without geometry, algebra cannot define locality

• Locality structure of algebra encoded in causal structure of $\partial \mathcal{M}$
• Without geometry, cannot distinguish “spacelike separated” from “timelike related”

Why Do We Need ${\mathcal{A}}_{\partial }$?

Counterexample: With only geometry, don’t know what to measure

• Geometry only gives “stage”
• Algebra gives “actors” (operators)
• Without operators, state cannot be defined

Why Do We Need ${\omega }_{\partial }$?

Counterexample: With only algebra, don’t know what state we’re in

• Algebra gives all possible operators
• State chooses a specific “realization”
• Without state, expectation values undefined

graph TB
    MIN["Minimality"] --> Q1["Can Remove ∂M?"]
    MIN --> Q2["Can Remove A_∂?"]
    MIN --> Q3["Can Remove ω_∂?"]

    Q1 --> NO1["✗ Lose Geometry"]
    Q2 --> NO2["✗ Lose Observables"]
    Q3 --> NO3["✗ Lose State"]

    style NO1 fill:#ffe1e1
    style NO2 fill:#ffe1e1
    style NO3 fill:#ffe1e1

🔍 Example: Boundary Triple of Schwarzschild Black Hole

Geometric Boundary

In region $r_{+}<r<\infty$ (exterior):

$$\partial \mathcal{M}={\mathcal{H}}^{+}\cup {\mathcal{I}}^{+}$$

where:

• ${\mathcal{H}}^{+}$: future horizon (null)
• ${\mathcal{I}}^{+}$: future null infinity (null)

graph TB
    EXTERIOR["Exterior Region"] --> HORIZON["Horizon H⁺<br/>r = 2M"]
    EXTERIOR --> INFINITY["Infinity I⁺<br/>r → ∞"]

    HORIZON --> NULL1["Null Surface"]
    INFINITY --> NULL2["Null Surface"]

    style HORIZON fill:#ffe1e1
    style INFINITY fill:#e1f5ff

Observable Algebra

$${\mathcal{A}}_{\partial }=\mathcal{A}({\mathcal{H}}^{+})\otimes \mathcal{A}({\mathcal{I}}^{+})$$

• Outgoing modes on horizon
• Scattering states at infinity

State

Hartle-Hawking State ${\omega }_{\mathrm{HH}}$:

• On past horizon: Minkowski vacuum
• On future horizon: thermal state at temperature $T_H=1/(8\pi M)$
• KMS condition holds

Hawking Radiation:

• Horizon emits thermal radiation
• Observer at infinity measures blackbody spectrum
• Entropy $S={\mathcal{A}}_{{\mathcal{H}}^{+}}/(4G)=4\pi M^2/G$

🤔 Exercises

1. Basic Understanding

Question: Why does boundary triple need exactly three components?

Hint: Try removing each one separately, see what physical information is lost.

2. Calculation Exercise

Question: For one-dimensional scattering problem, write boundary triple $(\partial \mathcal{M},{\mathcal{A}}_{\partial },{\omega }_{\partial })$.

Hint:

• $\partial \mathcal{M}=\{x=-\infty \}\cup \{x=+\infty \}$
• What operators does ${\mathcal{A}}_{\partial }$ contain?
• How to define expectation values for ${\omega }_{\partial }$?

3. Conceptual Deepening

Question: How is boundary triple embodied in AdS/CFT correspondence?

Hint:

• $\partial \mathcal{M}$: AdS boundary (conformal boundary)
• ${\mathcal{A}}_{\partial }$: Operator algebra of boundary CFT
• ${\omega }_{\partial }$: CFT vacuum or finite temperature state

4. Philosophical Reflection

Question: Does boundary triple imply “observer-dependent” physics?

Hint: Different boundaries correspond to different observers (e.g., Rindler vs. Minkowski), but physical laws unchanged.

📝 Chapter Summary

Core Definition

Boundary Triple:

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

• $\partial \mathcal{M}$: geometric boundary (where)
• ${\mathcal{A}}_{\partial }$: observable algebra (what to measure)
• ${\omega }_{\partial }$: state (what result)

Completeness Theorem

Boundary triple is considered to completely encode physical content:

• Bulk can theoretically be reconstructed from boundary
• Time evolution determined by boundary automorphism groups
• Expectation values of observables given by boundary state

Three Realizations

Minimality

All three components are indispensable:

• Remove geometry: lose causal structure
• Remove algebra: lose observables
• Remove state: lose expectation values

Next Step: With boundary triple, we need to make action well-defined on boundary. This is the role of GHY boundary term!

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