Chapter 21 Section 1: Geometric Foundations of Causal Diamonds

Introduction

In the overview of the previous section, we established the overall picture of causal diamond chain and null-modular double cover theory. Now, let’s delve into the first core concept: Geometric Foundations of Causal Diamonds.

Imagine standing at a crossroads, watching the traffic light change from red to green. In those few seconds, all possible events between the red light turning on and the green light turning on form a “causal diamond.” It contains:

• Pedestrians you might see
• Vehicles that might pass
• Light propagation paths
• Sound transmission trajectories

All these events are constrained by causal relations within a finite spacetime region—this is the causal diamond.

In this section, we will rigorously define the geometric structure of causal diamonds and reveal their deep connections with unified time scale, boundary time geometry, and quantum entanglement.

1. Causal Structure in Minkowski Spacetime

1.1 Future Light Cone and Past Light Cone

In Minkowski spacetime ${\mathbb{R}}^{1,d-1}$ of special relativity ($d\ge 2$, typically $d=4$), the metric is:

$$d{s}^2=-d{t}^2+d{x}_1^2+\cdots +d{x}_{d-1}^2$$

For any event $p\in {\mathbb{R}}^{1,d-1}$, define:

Future light cone:

$$J^{+}(p)=\{q:(q-p{)}^2\le 0,t_q\ge t_p\}$$

Past light cone:

$$J^{-}(p)=\{q:(q-p{)}^2\le 0,t_q\le t_p\}$$

where $(q-p{)}^2$ is the square of spacetime interval, with negative sign indicating timelike or null separation.

Everyday Analogy: Range of Light Propagation

• All locations that light emitted from point $p$ can reach at time $t$ form the future light cone
• All source positions that can reach point $p$ at time $t_p$ form the past light cone

graph TB
    subgraph "Spacetime Diagram"
        P["Event p<br/>(t_p, x_p)"]
        F1["Future Event<br/>t > t_p"]
        F2["Future Event<br/>t > t_p"]
        F3["Future Event<br/>t > t_p"]
        PAST1["Past Event<br/>t < t_p"]
        PAST2["Past Event<br/>t < t_p"]
        PAST3["Past Event<br/>t < t_p"]

        PAST1 -->|"Past Light Cone J⁻"| P
        PAST2 --> P
        PAST3 --> P
        P -->|"Future Light Cone J⁺"| F1
        P --> F2
        P --> F3
    end

    style P fill:#ffeb3b
    style F1 fill:#c8e6c9
    style F2 fill:#c8e6c9
    style F3 fill:#c8e6c9
    style PAST1 fill:#ffcdd2
    style PAST2 fill:#ffcdd2
    style PAST3 fill:#ffcdd2

1.2 Definition of Causal Diamond

Given two events $p_{\text{past}},p_{\text{future}}$ satisfying $p_{\text{past}}\in J^{-}(p_{\text{future}})$ (i.e., $p_{\text{past}}$ is in the past light cone of $p_{\text{future}}$), define:

$$D(p_{\text{past}},p_{\text{future}})=J^{+}(p_{\text{past}})\cap J^{-}(p_{\text{future}})$$

This is the causal diamond, also called causal diamond or Alexandrov set.

Geometric Intuition:

• $J^{+}(p_{\text{past}})$: All events reachable from $p_{\text{past}}$
• $J^{-}(p_{\text{future}})$: All events that can reach $p_{\text{future}}$
• Intersection $D$: All intermediate events that are both reachable from $p_{\text{past}}$ and can reach $p_{\text{future}}$

graph TB
    subgraph "Composition of Causal Diamond"
        P_past["Past Vertex<br/>p_past<br/>(0, 0)"]
        M1["Intermediate Event<br/>(t₁, x₁)"]
        M2["Intermediate Event<br/>(t₂, x₂)"]
        M3["Intermediate Event<br/>(t₃, x₃)"]
        P_future["Future Vertex<br/>p_future<br/>(T, 0)"]

        P_past -->|"Timelike Path"| M1
        P_past --> M2
        P_past --> M3
        M1 --> P_future
        M2 --> P_future
        M3 --> P_future

        M1 -.->|"Spacelike Separation"| M2
        M2 -.-> M3
    end

    subgraph "Boundary Structure"
        E_in["Incoming Boundary ∂⁻D<br/>Entering Diamond from Outside"]
        E_out["Outgoing Boundary ∂⁺D<br/>Exiting Diamond to Outside"]
    end

    style P_past fill:#ff6b6b
    style P_future fill:#4ecdc4
    style M1 fill:#ffe66d
    style M2 fill:#ffe66d
    style M3 fill:#ffe66d
    style E_in fill:#a8dadc
    style E_out fill:#f1faee

1.3 Special Cases: Spherical and Wedge Causal Diamonds

In $d$-dimensional Minkowski spacetime, the two simplest types of causal diamonds are:

(1) Spherical Causal Diamond

Take $p_{\text{past}}=(0,\vec{0})$, $p_{\text{future}}=(T,\vec{0})$, then:

$$D_{\text{sphere}}(T)=\{(t,\vec{x}):|\vec{x}|\le t,|\vec{x}|\le T-t,0\le t\le T\}$$

This is a double-cone structure centered at the origin with “height” $T$.

(2) Wedge Causal Diamond

Take the Rindler wedge in half-space $\{x_1\ge 0\}$:

$$W=\{(t,x_1,{\vec{x}}_{\perp }):|t|<x_1,x_1>0\}$$

The wedge is an “infinitely extended” causal diamond, closely related to the horizon of accelerated observers.

2. Double-Layer Decomposition of Null Boundaries

2.1 What Is a “Null Boundary”?

The boundary $\partial D$ of a causal diamond is very special—it consists of null hypersurfaces. The spacetime interval between any two points on a null hypersurface is zero, i.e.:

$$d{s}^2{|}_{\partial D}=0$$

This means observers on the boundary move at light speed, their “proper time” is zero. Therefore, the boundary is called a null boundary.

Everyday Analogy: Worldline of Light

Imagine a beam of light emitted from the past vertex, propagating along the boundary to the future vertex. For this beam of light, the “own time” of the entire journey is zero—this is the meaning of “null.”

2.2 Double-Layer Structure $\tilde{E}=E^{+}\bigsqcup E^{-}$

The null boundary actually has two layers:

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

where:

• $E^{+}$: Outgoing layer (outgoing null surface), null hyperplane along $u=\text{const}$
• $E^{-}$: Incoming layer (ingoing null surface), null hyperplane along $v=\text{const}$

Here $(u,v)$ are null coordinates:

$$u=t-r,v=t+r$$

where $r=|\vec{x}|$ is the spatial radial coordinate.

Physical Meaning of Double-Layer Decomposition:

graph TB
    subgraph "Double-Layer Boundary of Causal Diamond"
        Core["Diamond Interior<br/>D_interior"]

        Eplus1["E⁺ Layer<br/>u = 0"]
        Eplus2["E⁺ Layer<br/>u = T/2"]

        Eminus1["E⁻ Layer<br/>v = 0"]
        Eminus2["E⁻ Layer<br/>v = T/2"]

        Eminus1 -->|"Information Inflow"| Core
        Eminus2 --> Core
        Core -->|"Information Outflow"| Eplus1
        Core --> Eplus2
    end

    subgraph "Action of Modular Conjugation J"
        Jmap["Modular Conjugation J<br/>Exchanges Two Layers<br/>and Reverses Time Direction"]
        Eplus1 <-->|"J"| Eminus1
        Eplus2 <-->|"J"| Eminus2
    end

    style Core fill:#fffacd
    style Eplus1 fill:#add8e6
    style Eplus2 fill:#add8e6
    style Eminus1 fill:#ffb6c1
    style Eminus2 fill:#ffb6c1
    style Jmap fill:#d3d3d3

Key Insights:

1. $E^{+}$ layer is the interface where information leaves the diamond
2. $E^{-}$ layer is the interface where information enters the diamond
3. The two layers are related by modular conjugation $J$: $J(E^{+})=E^{-}$
4. Modular conjugation not only exchanges the two layers but also reverses time direction

2.3 Affine Parameters and Transverse Coordinates

On each boundary layer, we introduce coordinates:

$$(\lambda ,x_{\perp })\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{d-2}$$

where:

• $\lambda$: Affine parameter, like “distance along a light ray”
• $x_{\perp }=(x_2,\dots ,x_{d-1})$: Transverse coordinates

Example: Boundary Coordinates of Spherical Diamond

For spherical causal diamond $D_{\text{sphere}}(T)$, the outgoing layer $E^{+}$ can be parameterized as:

$$E^{+}:(t,r,\Omega )=(\lambda ,\lambda ,\Omega ),\lambda \in [0,T/2],\Omega \in S^{d-2}$$

where $\Omega$ are angular coordinates (transverse coordinates) on the $(d-2)$-dimensional sphere.

3. Modular Hamiltonian: Energy Conservation on Boundaries

3.1 Null Components of Energy-Momentum Tensor

In quantum field theory, the energy-momentum tensor $T_{ab}$ characterizes the distribution of energy, momentum, and stress. On null boundaries, the most important are the null-null components:

$$T_{++}=T_{vv},T_{--}=T_{uu}$$

They characterize energy flow density along null directions.

Physical Meaning:

• $T_{++}(\lambda ,x_{\perp })$: Energy flow along outgoing null direction ${\partial }_v$
• $T_{--}(\lambda ,x_{\perp })$: Energy flow along incoming null direction ${\partial }_u$

3.2 Geometric Decomposition of Modular Hamiltonian

Given causal diamond $D$, define the modular Hamiltonian:

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

where:

• $\sigma =+$ corresponds to outgoing layer $E^{+}$, $\sigma =-$ to incoming layer $E^{-}$
• $g_{\sigma }(\lambda ,x_{\perp })$ is the geometric weight function
• Integration measure $d\lambda d^{d-2}{x}_{\perp }$ is the standard measure on the boundary

Theorem (Double-Layer Geometric Decomposition, Exact Equality in CFT):

In conformal field theory (CFT), the geometric weight function for spherical causal diamond is:

$$g_{\sigma }(\lambda )=\lambda (1-\lambda /T)$$

satisfying boundary conditions: $g_{\sigma }(0)=g_{\sigma }(T)=0$.

Everyday Analogy: Weighted Integral of Boundary Energy

Imagine the boundary as a thin film, with energy flow density $T_{\sigma \sigma }$ at each point. The modular Hamiltonian is a weighted integral of this energy flow, where the weight $g_{\sigma }$ reflects geometric factors (such as “distance” from the vertex).

3.3 Connection Between Modular Hamiltonian and Unified Time Scale

Recall the unified time scale master formula:

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

The connection between modular Hamiltonian $K_D$ and unified time scale density $\kappa (\omega )$ is given by the Wigner-Smith group delay formula:

$$⟨K_D⟩\sim {\int }_{{\Omega }_D}\kappa (\omega )\rho (\omega )d\omega$$

where ${\Omega }_D$ is the frequency window corresponding to causal diamond $D$, and $\rho (\omega )$ is the density of states.

Physical Meaning:

• Modular Hamiltonian is the total time-energy “locked” on the boundary
• Unified time scale density is the time flow density in frequency domain
• The two are unified through scattering phase–group delay relation

4. Quadratic Form Framework and Natural Domain

4.1 What Is a “Quadratic Form”?

In quantum mechanics, observables correspond to Hermitian operators $A$, with expectation values:

$$⟨A{⟩}_{\psi }=⟨\psi |A|\psi ⟩$$

When $A$ is not a bounded operator, the above is not defined for all states $|\psi ⟩$. We need to introduce the framework of quadratic forms.

Definition: A quadratic form $\mathfrak{k}$ is a mapping:

$$\mathfrak{k}:\mathcal{D}(\mathfrak{k})\to \mathbb{R},\mathfrak{k}[\psi ]=⟨\psi |K|\psi ⟩$$

where $\mathcal{D}(\mathfrak{k})$ is the form domain, a dense subspace of the full Hilbert space $\mathcal{H}$.

4.2 Quadratic Form Definition of Modular Hamiltonian

For causal diamond $D$, the quadratic form of modular Hamiltonian is:

$${\mathfrak{k}}_D[\psi ]=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }(\lambda ,x_{\perp })⟨\psi |T_{\sigma \sigma }(\lambda ,x_{\perp })|\psi ⟩d\lambda d^{d-2}{x}_{\perp }$$

Assumption (Lower Bound of Quadratic Form):

There exists a real number $a_D\in \mathbb{R}$ such that for all $|\psi ⟩\in \mathcal{D}({\mathfrak{k}}_D)$:

$${\mathfrak{k}}_D[\psi ]\ge a_D\Vert \psi {\Vert }^2$$

This guarantees that modular Hamiltonian $K_D$ is a self-adjoint operator bounded below.

4.3 Shifted Graph Norm and Completeness

To discuss closure of quadratic forms, introduce the shifted graph norm:

For any $c_D>-a_D$, define:

$$\Vert \psi {\Vert }_{{\mathfrak{k}}_D,c_D}^2:=\Vert \psi {\Vert }^2+({\mathfrak{k}}_D[\psi ]+c_D\Vert \psi {\Vert }^2)$$

Theorem (Completeness of Form Domain):

The form domain $\mathcal{D}({\mathfrak{k}}_D)$ equipped with shifted graph norm $\Vert \cdot {\Vert }_{{\mathfrak{k}}_D,c_D}$ is a complete Hilbert space.

Physical Meaning:

• Shifted graph norm simultaneously controls the “size” $\Vert \psi {\Vert }^2$ and “modular energy” ${\mathfrak{k}}_D[\psi ]$ of states
• Completeness guarantees that limit states remain in the form domain, which is crucial for defining closure of inclusion-exclusion identity

5. Null Localization and QNEC Vacuum Saturation

5.1 QNEC (Quantum Null Energy Condition)

QNEC is an inequality about energy density, which holds for null hyperplanes in vacuum state.

QNEC Inequality:

For null half-space $R_V=\{u=0,v\ge V(x_{\perp })\}$ (where $V\in C^2$ is a smooth function), we have:

$$⟨T_{vv}(v,x_{\perp })⟩\ge \frac{1}{4\pi }\frac{{\partial }^2{S}_{\text{gen}}(R_V)}{\partial V^2}$$

where $S_{\text{gen}}$ is generalized entropy (including quantum entanglement entropy and classical area term).

Vacuum Saturation:

In vacuum state $|\Omega ⟩$, QNEC takes equality:

$$⟨\Omega |T_{vv}|\Omega ⟩=\frac{1}{4\pi }\frac{{\partial }^2{S}_{\text{vac}}}{\partial V^2}$$

This relation allows the modular Hamiltonian to be precisely expressed as a quadratic form integral on the boundary.

5.2 Modular Hamiltonian of Null Half-Space

For null half-space $R_V$, the explicit form of modular Hamiltonian is:

$$K_V=2\pi \int d^{d-2}{x}_{\perp }{\int }_{V(x_{\perp })}^{\infty }(v-V(x_{\perp }))T_{vv}(v,x_{\perp })dv$$

Geometric Interpretation:

• Weight $(v-V(x_{\perp }))$ represents the “depth” from boundary $v=V(x_{\perp })$
• The deeper into the interior, the larger the weight, the greater the contribution to modular Hamiltonian

graph LR
    subgraph "Null Half-Space R_V"
        Boundary["Boundary<br/>v = V(x_⊥)"]
        Interior1["Interior Point 1<br/>v₁ > V"]
        Interior2["Interior Point 2<br/>v₂ > v₁"]
        Interior3["Interior Point 3<br/>v₃ > v₂"]

        Boundary -->|"Weight v₁-V"| Interior1
        Interior1 -->|"Weight v₂-V"| Interior2
        Interior2 -->|"Weight v₃-V"| Interior3
    end

    subgraph "Modular Hamiltonian Contribution"
        Weight["g(v,x_⊥) = v - V(x_⊥)<br/>Linear Growth with Depth"]
        Energy["T_vv(v,x_⊥)<br/>Energy Flow Density"]
        Integral["K_V = 2π ∫ g·T_vv dv dx_⊥<br/>Weighted Integral"]

        Weight --> Integral
        Energy --> Integral
    end

    style Boundary fill:#ffd700
    style Interior1 fill:#98fb98
    style Interior2 fill:#7cfc00
    style Interior3 fill:#00ff00
    style Weight fill:#87ceeb
    style Energy fill:#ff6347
    style Integral fill:#dda0dd

5.3 Bisognano-Wichmann Property

In quantum field theory, the Bisognano-Wichmann property (BW property) is the cornerstone of modular theory for causal diamonds.

BW Property (Rindler Wedge):

For Rindler wedge $W=\{|t|<x_1,x_1>0\}$, its modular group ${\Delta }_W^{it}$ geometrizes as Lorentz boost:

$${\Delta }_W^{it}\cdot (t,x_1,{\vec{x}}_{\perp })=(e^{2\pi t}t,e^{2\pi t}{x}_1,{\vec{x}}_{\perp })$$

Modular conjugation $J_W$ corresponds to reflection $t\to -t$ plus CPT transformation.

Physical Meaning:

• Modular group ${\Delta }^{it}$ describes evolution of the system in “modular time” $t$
• BW property shows this evolution corresponds to acceleration (boost)
• This connects the deep unity between quantum entanglement and spacetime geometry

6. Exact Results in Conformal Field Theory

6.1 Conformal Transformations and Conformal Images

In conformal field theory (CFT), causal diamonds are mapped to simpler geometric shapes under conformal transformations.

Conformal Transformation:

In two-dimensional CFT, conformal transformations are given by analytic mappings $z\to w(z)$. In $d>2$ dimensions, conformal transformations preserve the form of the metric:

$$g_{ab}\to {\Omega }^2(x)g_{ab}$$

where $\Omega (x)$ is a positive local factor (called Weyl factor).

Example: Conformal Image of Spherical Diamond

Spherical causal diamond in ${\mathbb{R}}^{1,d-1}$ can be conformally mapped to a static patch on hyperbolic space ${\mathbb{H}}^{d-1}$. In this case, the metric becomes:

$$d{s}^2=-d{\tau }^2+{\sinh }^2\tau d{\Omega }_{d-2}^2$$

The modular Hamiltonian has a simple form in this conformal image.

6.2 Modular Hamiltonian in CFT

In two-dimensional CFT, the modular Hamiltonian of spherical diamond (interval) $I=[u_1,u_2]\times [v_1,v_2]$ is:

$$K_I={\int }_I\frac{(u-u_1)(u_2-u)}{u_2-u_1}{T}_{uu}+\frac{(v-v_1)(v_2-v)}{v_2-v_1}{T}_{vv}$$

The weight functions are exactly:

$$g_{+}(u)=\frac{(u-u_1)(u_2-u)}{u_2-u_1},g_{-}(v)=\frac{(v-v_1)(v_2-v)}{v_2-v_1}$$

Boundary Conditions:

• $g_{+}(u_1)=g_{+}(u_2)=0$ (weight zero at boundary endpoints)
• Weight reaches maximum at interval midpoint: $g_{+}\left(\frac{u_1+u_2}{2}\right)=\frac{u_2-u_1}{4}$

6.3 Entanglement Entropy Formula in CFT

In CFT, spherical diamonds have explicit entanglement entropy formulas. The result for two-dimensional CFT is:

$$S_I=\frac{c}{3}\log \frac{L}{ϵ}$$

where:

• $c$ is the central charge
• $L$ is the interval length
• $ϵ$ is the short-distance cutoff (regularization parameter)

Results for higher-dimensional CFT involve area law:

$$S_D=\frac{\text{Area}(\partial D)}{4G}+S_{\text{quantum}}$$

where the first term is classical Bekenstein-Hawking entropy, and the second term is quantum correction.

7. Totally-Ordered Approximation Bridge: From General Diamonds to Half-Spaces

7.1 Why Do We Need “Approximation”?

General causal diamonds $D$ may have complex boundary shapes. To rigorously define their modular Hamiltonian, we adopt the method of totally-ordered approximation bridge:

1. Approximate $D$ with a family of null half-spaces $\{R_{V_{\alpha }}^{\pm }{\}}_{\alpha }$
2. Modular Hamiltonian $K_{V_{\alpha }}$ of each half-space is given by QNEC
3. Recover $K_D$ in the limit $\alpha \to \infty$

7.2 Construction of Monotonic Approximation Family

Definition (Monotonic Approximation Family):

For each transverse coordinate $x_{\perp }$, construct monotonic function families:

$$V_{\alpha }^{+}(x_{\perp })\downarrow V^{+}(x_{\perp }),V_{\alpha }^{-}(x_{\perp })\uparrow V^{-}(x_{\perp })$$

such that:

• $R_{V_{\alpha }^{+}}\supset R_{V_{\alpha +1}^{+}}$ (monotonic contraction)
• $R_{V_{\alpha }^{-}}\subset R_{V_{\alpha +1}^{-}}$ (monotonic expansion)
• ${\lim }_{\alpha \to \infty }{R}_{V_{\alpha }^{\pm }}=E^{\pm }$ (limit equals diamond boundary)

Everyday Analogy: Approximating Curves with Steps

Imagine drawing a smooth curve, but only using horizontal line segments. You can approximate the curve with more and more, shorter and shorter segments. Totally-ordered approximation bridge is this idea applied to spacetime geometry.

7.3 Dominated Convergence and Path Independence

Assumption (Uniform Integrability of Null Energy Flow):

For any $|\psi ⟩\in {\mathcal{D}}_0$ and geometrically bounded monotonic approximation family $\{R_{V_{\alpha }}^{\pm }\}$, there exists $H_{\sigma }\in L_{\text{loc}}^1(E^{\sigma }\times {\mathbb{R}}^{d-2})$ such that:

$$|g_{\sigma }^{(\alpha )}(\lambda ,x_{\perp })⟨\psi |T_{\sigma \sigma }(\lambda ,x_{\perp })|\psi ⟩|\le H_{\sigma }(\lambda ,x_{\perp })$$

holds almost everywhere, and:

$$\underset{\alpha }{\sup }{\int }_{\mathcal{K}}{H}_{\sigma }<\infty$$

for any compact set $\mathcal{K}\subset E^{\sigma }\times {\mathbb{R}}^{d-2}$.

Theorem (Totally-Ordered Approximation Bridge Lemma):

Under the above assumption, there exists a monotonic half-space family $\{R_{V_{\alpha }}^{\pm }\}$ such that:

$$⟨\psi |K_D|\psi ⟩=\underset{\alpha \to \infty }{\lim }\sum _{\sigma =\pm }2\pi {\int }_{E^{\sigma }}{g}_{\sigma }^{(\alpha )}⟨\psi |T_{\sigma \sigma }|\psi ⟩$$

and the limit is independent of the chosen ordered approximation.

Proof Strategy:

1. Dominated convergence theorem guarantees pointwise limit exists
2. Quadratic form closure guarantees limit is continuous on form domain
3. Geometric monotonicity guarantees limit is path-independent

8. Interface with Unified Time Scale

8.1 Scattering Phase–Modular Hamiltonian Correspondence

In the scattering theory framework, modular Hamiltonian is related to phase increments of scattering matrix.

Birman-Krein Formula:

$$\int \frac{\partial }{\partial E}\arg \det S(E)h(E)dE=-2\pi \int {\xi }^{\prime }(E)h(E)dE$$

where $\xi (E)$ is the spectral shift function, related to density of states difference.

Combining with Wigner-Smith formula:

$$\text{tr}Q(E)=-i\text{tr}(S^{†}{\partial }_{E}S)=2\pi {\xi }^{\prime }(E)$$

we obtain unified time scale density:

$$\kappa (E)=\frac{1}{2\pi }\text{tr}Q(E)={\xi }^{\prime }(E)$$

8.2 Frequency Domain–Time Domain Correspondence

Through Fourier transform, unified time scale density $\kappa (\omega )$ in frequency domain corresponds to “local time flow” in time domain:

$$\tau (t)={\int }_{-\infty }^t\int \kappa (\omega )e^{-i\omega s}d\omega ds$$

Modular Hamiltonian $K_D$ can be viewed as total time-energy “accumulated” within causal diamond $D$:

$$⟨K_D⟩\sim {\int }_{{\Omega }_D}\kappa (\omega ){\rho }_D(\omega )d\omega$$

where ${\rho }_D(\omega )$ is the density of states within the diamond.

Summary

In this section, we established the rigorous geometric foundations of causal diamonds:

1. Causal Diamond Definition: $D(p_{\text{past}},p_{\text{future}})=J^{+}(p_{\text{past}})\cap J^{-}(p_{\text{future}})$
2. Double-Layer Decomposition of Null Boundaries: $\tilde{E}=E^{+}\bigsqcup E^{-}$, the two layers connected by modular conjugation $J$
3. Modular Hamiltonian: $K_D=2\pi {\sum }_{\sigma =\pm }\int g_{\sigma }{T}_{\sigma \sigma }d\lambda d{x}_{\perp }$
4. Quadratic Form Framework: Completeness of form domain guaranteed through shifted graph norm
5. QNEC Vacuum Saturation: Exact formula for modular Hamiltonian of null half-spaces
6. Totally-Ordered Approximation Bridge: Monotonic approximation from half-spaces to general diamonds
7. Unified Time Scale Interface: Correspondence between modular Hamiltonian and scattering phase–group delay

In the next section, we will construct the null-modular double cover, introduce ${\mathbb{Z}}_2$ parity invariants, and reveal their deep connections with self-referential scattering networks and fermion double-valuedness.

References

This section is mainly based on the following theoretical literature:

1. Null-Modular Double Cover Theory - euler-gls-extend/null-modular-double-cover-causal-diamond-chain.md, §2.1-2.3, §3.1-3.2
2. Bisognano-Wichmann Theorem - Classical quantum field theory literature
3. QNEC and Modular Theory - Faulkner et al. (2016), Casini-Teste-Torroba (2017)
4. Unified Time Scale - Chapter 5 unified-time/, especially scattering phase–group delay identity
5. Boundary Time Geometry - Chapter 6 boundary-theory/, null boundary structure
6. Conformal Field Theory - CFT classical textbooks, entanglement entropy and modular Hamiltonian

In the next section, we will delve into the construction of null-modular double cover and the topological structure of ${\mathbb{Z}}_2$ holonomy.