What is Boundary?

“We think reality exists in the ‘interior’ of space. But what if true reality actually exists on the ‘surface’?”

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Starting from a Balloon

Take a balloon, blow it up. The balloon expands.

Now ask: Is the balloon’s “volume” real?

🎈 Two Understandings of a Balloon

Understanding 1 (Common Sense): Volume is real

Balloon interior has air (matter)
Larger volume, more air
Volume is “truly existing” three-dimensional space

Understanding 2 (Holographic View): Surface is real

What you can directly observe is only the balloon’s surface
Surface tension and deformation determine internal pressure
You squeeze balloon, surface deforms, “interior” shape changes accordingly
Surface contains all information about interior

graph TB
    subgraph "Traditional View"
        V1["Volume<br/>(Real)"] --> S1["Surface<br/>(Boundary)"]
    end

    subgraph "Holographic View"
        S2["Surface<br/>(Real)"] --> V2["Volume<br/>(Reconstructed)"]
    end

    style V1 fill:#a8e6cf
    style S2 fill:#ff6b6b,color:#fff

💡 GLS Theory Proposes: The universe might be like this balloon. True “reality” might exist on the boundary (surface), and physics in “volume” is reconstructed from boundary data!

What is Boundary?

In mathematics and physics, the definition of boundary is clear:

Boundary = “Interface” of a region, separating inside and outside

📦 Example: Box

A cubic box:

Interior: Three-dimensional space, volume $V = L^{3}$
Boundary: Six faces (two-dimensional), area $A = 6 L^{2}$
Dimension Difference: Volume is 3D, boundary is 2D

graph TD
    Box["Cubic Box<br/>3D Interior"] --> Face1["Top Face<br/>2D"]
    Box --> Face2["Bottom Face<br/>2D"]
    Box --> Face3["Four Side Faces<br/>2D"]

    Face1 --> Total["Total Boundary<br/>6 Faces"]
    Face2 --> Total
    Face3 --> Total

    style Box fill:#a8e6cf
    style Total fill:#ff6b6b,color:#fff

Key Observation: Boundary dimension is always 1 less than interior!

Amazing Property of Boundary: Bekenstein-Hawking Entropy

🕳️ Mystery of Black Hole Entropy

In the 1970s, physicists discovered a shocking fact:

Black hole entropy is proportional to surface area, not volume!

$S_{BH} = \frac{A}{4 G ℏ}$

where:

$S_{BH}$ = Black hole entropy (information content)
$A$ = Black hole event horizon area
$G$ = Gravitational constant
$ℏ$ = Planck constant

🤯 Why Is This Shocking?

Usually, entropy should be proportional to volume:

Larger room, more ways to be disordered
Double volume → Entropy roughly doubles

But black holes violate this rule:

Black hole radius doubles → Area becomes 4 times → Entropy becomes 4 times
But volume becomes 8 times!

What Does This Mean?

Black hole “information” is all encoded on its surface (horizon), not interior!

graph TB
    subgraph "Ordinary Object"
        V["Volume<br/>V ∝ r³"] -->|determines| S1["Entropy<br/>S ∝ V ∝ r³"]
    end

    subgraph "Black Hole"
        A["Surface Area<br/>A ∝ r²"] -->|determines| S2["Entropy<br/>S ∝ A ∝ r²"]
    end

    style S2 fill:#ff6b6b,color:#fff

Like a hologram: A three-dimensional image encoded on a two-dimensional film.

Holographic Principle: Universe is a Hologram

🌌 ’t Hooft and Susskind’s Conjecture

Inspired by black hole entropy, physicists proposed the Holographic Principle:

Physics of any finite region can be completely encoded on its boundary.

Like a holographic card:

graph LR
    Card["Holographic Card<br/>(2D)"] -.contains.-> Image["3D Image<br/>(Looks 3D)"]

    style Card fill:#ff6b6b,color:#fff
    style Image fill:#a8e6cf

🧮 AdS/CFT Correspondence

The most famous holographic example is AdS/CFT correspondence (Maldacena, 1997):

AdS: Anti-de Sitter space (special type of gravitational spacetime)
CFT: Conformal Field Theory (a quantum field theory)

Core Idea:

A $(d + 1)$ -dimensional gravity theory = A $d$ -dimensional quantum field theory (on boundary)

graph TB
    subgraph "AdS Space (Gravity)"
        Bulk["(d+1)D<br/>Spacetime with Gravity"]
    end

    subgraph "Boundary (Quantum Field Theory)"
        Boundary["dD<br/>Quantum Field Theory Without Gravity"]
    end

    Bulk -.equivalent.-> Boundary

    style Bulk fill:#a8e6cf
    style Boundary fill:#ff6b6b,color:#fff

Example:

5D AdS space gravity theory $\Leftrightarrow$ 4D boundary quantum field theory
Black hole in gravity $\Leftrightarrow$ Thermal equilibrium state in quantum field theory

Why Important?

This suggests:

Gravity is not fundamental: Gravity may be an “emergent” phenomenon of boundary quantum theory
Dimensions can emerge: Extra spatial dimensions come from reorganization of boundary information
Holography is universal: Universe’s “interior” may be a holographic projection of “surface”

GLS Theory: Boundary Priority Principle

GLS unified theory elevates holographic principle to a fundamental axiom:

Axiom: Boundary Priority

Physical reality first manifests as boundary observable algebra and its spectral data; bulk dynamics is extension of boundary data.

📏 Boundary Spectral Triple

In GLS theory, the mathematical description of boundary is a Boundary Spectral Triple:

$(A_{\partial}, H_{\partial}, D_{\partial})$

Three Components:

$A_{\partial}$ = Observable algebra on boundary
$H_{\partial}$ = Hilbert space on boundary
$D_{\partial}$ = Dirac operator on boundary (determines geometry)

💡 Key Idea: Boundary geometry (metric) doesn’t need to be given a priori, but is defined by spectral structure of $D_{\partial}$ !

graph TB
    Spectral["Boundary Spectral Data<br/>Eigenvalues of D_∂"] --> Metric["Boundary Metric<br/>Distance, Curvature"]
    Metric --> Bulk["Bulk Geometry<br/>Reconstruct Interior Spacetime"]

    style Spectral fill:#ff6b6b,color:#fff
    style Bulk fill:#a8e6cf

🎯 Brown-York Stress Tensor

There’s a special physical quantity on boundary: Brown-York Stress Tensor $T_{BY}^{ab}$

Its definition:

$T_{BY}^{ab} = \frac{2}{- h} \frac{δ S _{grav}}{δ h _{ab}}$

Translation:

$h_{ab}$ = Metric on boundary (geometry)
$S_{grav}$ = Gravitational action
$T_{BY}^{ab}$ = Response to metric variation (stress-energy)

Physical Meaning:

Brown-York stress tensor tells you: Energy-momentum density on boundary.

It generates “time translation” on boundary, so boundary has its own time evolution!

graph LR
    BoundaryMetric["Boundary Metric<br/>h_ab"] -->|variation| BY["Brown-York<br/>Stress Tensor<br/>T^ab_BY"]
    BY -->|generates| TimeEvolution["Boundary Time Evolution<br/>∂/∂t"]

    style BY fill:#ff6b6b,color:#fff

Why Is Boundary the Starting Point of Reality?

🔍 Observers Are Always on Boundary

Think: How do you observe a physical system?

Example: Observing gas in a box

graph TB
    Observer["Observer<br/>(You)"] -.observe.-> Boundary["Box Surface<br/>Temperature, Pressure"]
    Boundary -.infer.-> Inside["Box Interior<br/>Molecular Motion"]

    style Observer fill:#ffe66d,stroke:#f59f00,stroke-width:2px
    style Boundary fill:#ff6b6b,color:#fff
    style Inside fill:#e0e0e0

What you can do:

✓ Measure temperature on box surface
✓ Measure pressure on box surface
✓ See inside through transparent wall (light propagates from boundary)

What you cannot do:

✗ Directly “see” position of a molecule inside (unless light carries information to boundary)

Conclusion: All observational data comes from boundary (or signals propagating from boundary)!

🌍 Where Is the Universe’s Boundary?

If the universe is infinitely large, does it have a boundary?

Answer: Yes! But not a spatial boundary, but a temporal boundary.

Cosmological Horizon: Due to cosmic expansion, some places’ light will never reach us
- Our “observable universe” has a boundary (horizon)
- Radius about 46 billion light-years
Causal Boundary: In GLS theory, boundary can be null boundary (light-speed boundary)
- Past null boundary: Big Bang (beginning of time)
- Future null boundary: Heat death (maximum entropy)

graph TD
    BigBang["Big Bang<br/>Past Boundary"] -->|time| Universe["Universe<br/>(Interior)"]
    Universe -->|time| HeatDeath["Heat Death<br/>Future Boundary"]

    Observer["Observer"] -.receive info.-> PastBoundary["Past Light Cone<br/>(Boundary)"]
    Observer -.send info.-> FutureBoundary["Future Light Cone<br/>(Boundary)"]

    style BigBang fill:#ffd3b6
    style HeatDeath fill:#ffaaa5
    style Observer fill:#ffe66d,stroke:#f59f00,stroke-width:2px

GLS Theory’s View:

The universe’s “interior” (3D space we live in) may be a holographic projection encoded on these temporal boundaries!

Boundary and Entropy: Area Law

📐 Area Law of Entanglement Entropy

In quantum field theory, divide space into two regions A and B:

+-------------------+
|| Region A |Region B|
||          |        |
||    \∂A/ |        |
+-------------------+
     Boundary

Entanglement Entropy: Degree of quantum entanglement between regions A and B

$S_{A} = - tr (ρ_{A} ln ρ_{A})$

Surprising Discovery:

$S_{A} \propto \frac{Area ( \partial A )}{ϵ ^{d - 2}}$

Entanglement entropy is proportional to boundary area, not volume!

$\partial A$ = Boundary of region A
$ϵ$ = UV cutoff (short-distance cutoff)
$d$ = Spatial dimension

💡 Another Holographic Signal: Quantum entanglement information is mainly encoded on boundary!

🔬 Ryu-Takayanagi Formula

In AdS/CFT correspondence, Ryu-Takayanagi formula gives precise holographic form:

$S_{A} = \frac{Area ( γ _{A} )}{4 G ℏ}$

where:

$S_{A}$ = Entanglement entropy of region A in boundary field theory
$γ_{A}$ = Minimal surface in bulk connecting $\partial A$ (extremal surface)
$G$ = Gravitational constant

graph TB
    subgraph "Boundary (CFT)"
        A["Region A"] --- Bdry["Boundary ∂A"]
    end

    subgraph "Bulk (AdS)"
        Surface["Minimal Surface γ_A<br/>Area = 4Gℏ S_A"]
    end

    Bdry -.corresponds to.-> Surface

    style Bdry fill:#ff6b6b,color:#fff
    style Surface fill:#a8e6cf

Meaning: Entanglement entropy on boundary corresponds to area of extremal surface in bulk!

Profound Insights of Boundary Theory

💎 No Fundamental Forces Theorem

GLS theory derives a core conclusion:

Theorem: No Fundamental Forces

All “forces” (gravity, electromagnetic, strong, weak) are not fundamental objects, but different projections of unified boundary connection curvature.

Imagine a unified boundary connection:

$Ω_{\partial} = ω_{Levi-Civita} \oplus A_{Yang-Mills} \oplus Γ_{resolution}$

Three Parts:

$ω$ = Levi-Civita connection (gravity)
$A$ = Yang-Mills connection (gauge fields: electromagnetic, strong, weak)
$Γ$ = Resolution connection (coarse-graining effects)

All “forces” come from curvature of this unified connection:

$R_{\partial} = R_{\partial} \oplus F_{\partial} \oplus R_{res}$

graph TD
    Unified["Unified Boundary Connection<br/>Ω_∂"] --> Curvature["Curvature<br/>R_∂"]

    Curvature --> Gravity["Gravity<br/>Projection of R_∂"]
    Curvature --> EM["Electromagnetic Force<br/>Projection of F_∂"]
    Curvature --> Strong["Strong Force<br/>Projection of F_∂"]
    Curvature --> Weak["Weak Force<br/>Projection of F_∂"]

    style Unified fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px,color:#fff

💡 Key Insight: GLS theory suggests that the 4 fundamental forces might be different aspects of the same boundary geometric structure!

Summary: Revolutionary Perspective of Boundary

Traditional View	GLS Boundary Theory
Volume is real, boundary is interface	Boundary is real, volume is reconstructed
Entropy proportional to volume	Entropy proportional to boundary area
Gravity is fundamental force	Gravity is emergence of boundary geometry
Observer in volume	Observer always on boundary
Dimensions are a priori	Dimensions emerge from boundary

🎯 Key Points

Holographic Principle: Physics in volume can be completely encoded on boundary
Black Hole Entropy: $S \propto A$ (area), not $V$ (volume)
AdS/CFT: $(d + 1)$ D gravity = $d$ D quantum field theory
Boundary Priority: Physical reality first manifests on boundary
No Fundamental Forces: All forces are projections of boundary connection curvature

💡 Most Profound Insight

The universe may not be a “solid” three-dimensional space, but a huge holographic projection—true information exists on the “surface” (boundary).

We feel we live in three-dimensional space, but this may be an illusion. True degrees of freedom may be on a lower-dimensional boundary.

What’s Next

We understand the importance of boundary. But there’s one more key concept: Scattering.

Why can particle collisions tell us about time?
What is the S-matrix?
What’s the relationship between scattering delay and time?

Answers to these questions are in the next article:

Next: What is Scattering? →

Remember: Boundary is not an irrelevant “shell,” but the source of reality. Understanding boundary, you understand the secret of the holographic universe.

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