Causal Structure Emerges from QCA: Birth of Partial Order and Light Cone

In previous section, we established five-tuple axioms of QCA. Now we arrive at one of most stunning conclusions:

Relativistic causal structure is not pre-assumed “background spacetime”, but mathematical necessity naturally emerging from finite propagation property of QCA!

This section will strictly prove: How discrete QCA derives causal light cone of continuous relativity.

Problem Statement: Where Does Causality Come From?

Causal Structure of Traditional Relativity

In standard Minkowski spacetime $({\mathbb{R}}^4,\eta )$, causal structure is defined by light cone:

Event $p$ can affect event $q$ if and only if $q$ is in future light cone of $p$: $$q\in I^{+}(p)⟺\eta (q-p,q-p)<0\text{ and }{t}_q>t_p$$

Light Cone Equation: $$-c^2(t_q-t_p{)}^2+|{\mathbf{x}}_q-{\mathbf{x}}_p{|}^2=0$$

graph TD
    A["Minkowski Spacetime"] --> B["Metric η"]
    B --> C["Light Cone Structure<br/>I⁺(p), I⁻(p)"]
    C --> D["Causal Partial Order<br/>p≺q"]

    E["Pre-Given"] --> A
    E --> B

    style A fill:#d4a5a5
    style E fill:#ff6b6b
    style D fill:#ffd93d

Problems:

• Light cone is pre-given (metric $\eta$ is background)
• Causality depends on existence of continuous spacetime
• In quantum gravity spacetime itself fluctuates, how to define causality?

QCA Challenge: How Causality in Discrete?

In QCA:

• No pre-existing spacetime: Only discrete lattice points $\Lambda$
• No pre-existing metric: Only graph distance $\text{dist}(x,y)$
• No pre-existing light cone: Only finite propagation radius $R$ of evolution $\alpha$

Core Question:

Can we derive relativistic causal partial order from discrete structure of QCA?

Answer is yes! And derived causal structure completely agrees with relativity in continuous limit.

Event Set and Discrete Light Cone

Definition of Events

Definition 2.1 (Event Set): Event set of QCA universe is defined as: $$E:=\Lambda \times \mathbb{Z}$$ where:

• $\Lambda$: Spatial lattice points
• $\mathbb{Z}$: Discrete time steps

Element $(x,n)\in E$ means “event occurring at lattice point $x$ at time step $n$”.

Projections: $$\text{sp}:E\to \Lambda ,\text{sp}(x,n)=x(\text{spatial coordinate})$$ $$\text{tm}:E\to \mathbb{Z},\text{tm}(x,n)=n(\text{time coordinate})$$

graph LR
    A["Event (x,n)"] --> B["Spatial Coordinate<br/>sp(x,n)=x∈Λ"]
    A --> C["Time Coordinate<br/>tm(x,n)=n∈ℤ"]

    B --> D["Lattice Position"]
    C --> E["Discrete Moment"]

    D --> F["Event Set E=Λ×ℤ"]
    E --> F

    style A fill:#ffd93d
    style F fill:#6bcf7f

Geometric Light Cone: Derived from Finite Propagation

Recall QCA Axiom QCA-2: There exists finite propagation radius $R$ such that operator $A$ supported on $F$, after evolution supported within $B_R(F)$.

Corollary: Operator $A_y$ supported on single point $\{y\}$, after $n$ steps of evolution: $$\text{supp}({\alpha }^n(A_y))\subset B_{nR}(y)$$

Intuitive Explanation: “Signal” at point $y$ at time step $0$ can at most propagate to points within distance $nR$ at time step $n$.

Definition 2.2 (Geometric Reachability Relation): Define binary relation ${\le }_{\text{geo}}$ on $E$: $$(x,n){\le }_{\text{geo}}(y,m)⟺m\ge n\text{ and }\text{dist}(x,y)\le R(m-n)$$

Physical Meaning: Event at $(x,n)$ can causally affect event at $(y,m)$ if and only if time is long enough ($m\ge n$) and spatial distance is within “light speed × time” range.

graph TD
    A["Event (x,n)"] --> B["Reachable Event (y,m)"]

    B --> C["Time Condition<br/>m≥n"]
    B --> D["Spatial Condition<br/>dist(x,y)≤R(m-n)"]

    C --> E["Causal Arrow<br/>Future→"]
    D --> F["Finite Speed<br/>v_max=R"]

    E --> G["Geometric Reachability<br/>(x,n)≤_geo(y,m)"]
    F --> G

    style A fill:#ffd93d
    style G fill:#6bcf7f

Shape of Discrete Light Cone

Fix event $(x_0,n_0)$, its future geometric light cone is defined as: $$I_{\text{geo}}^{+}(x_0,n_0):=\{(y,m)\in E:(x_0,n_0){\le }_{\text{geo}}(y,m)\}$$

Explicit Characterization: $$I_{\text{geo}}^{+}(x_0,n_0)=\{(y,m):m>n_0,\text{dist}(y,x_0)\le R(m-n_0)\}$$

Similarly define past geometric light cone: $$I_{\text{geo}}^{-}(x_0,n_0)=\{(y,m):m<n_0,\text{dist}(y,x_0)\le R(n_0-m)\}$$

Example (one-dimensional $\Lambda =\mathbb{Z}$, $R=1$):

Future light cone of event $(0,0)$: $$I_{\text{geo}}^{+}(0,0)=\{(x,n):n>0,|x|\le n\}$$

This is exactly a discrete cone!

graph TD
    A["n=3<br/>x∈{-3,-2,-1,0,1,2,3}"] --> B["n=2<br/>x∈{-2,-1,0,1,2}"]
    B --> C["n=1<br/>x∈{-1,0,1}"]
    C --> D["n=0<br/>x=0"]

    E["Future Light Cone<br/>|x|≤n"] --> A

    style D fill:#6bcf7f
    style E fill:#ffd93d

From Geometry to Statistics: Causality of Correlation Functions

Statistical Causality

Pure geometric relation ${\le }_{\text{geo}}$ is derived from kinematics of QCA evolution (finite propagation). But physically, causality should manifest in statistical correlations of observables.

Definition 2.3 (Statistical Causality): For events $(x,n)$ and $(y,m)$, define statistical causality ${⪯}_{\text{stat}}$: $$(x,n){⪯}_{\text{stat}}(y,m)⟺\text{exists local operators }{A}_x,B_y\text{ such that correlation function non-zero}$$

More precisely, under state ${\omega }_0$: $$C_{AB}(n,m):={\omega }_0({\alpha }^{-m}(B_y){\alpha }^{-n}(A_x))-{\omega }_0({\alpha }^{-m}(B_y)){\omega }_0({\alpha }^{-n}(A_x))\ne 0$$

Physical Meaning: $(x,n)$ can causally affect $(y,m)$ if and only if measurement at point $x$ at time $n$ can affect measurement result at point $y$ at time $m$.

Support Properties of Correlation Functions

Lemma 2.4 (Light Cone Constraint of Correlation Functions): If $A\in {\mathcal{A}}_{\{x\}}$ supported on $\{x\}$, $B\in {\mathcal{A}}_{\{y\}}$ supported on $\{y\}$, and $m>n$, then:

If $\text{dist}(x,y)>R(m-n)$, necessarily: $$C_{AB}(n,m)=0$$

Proof: By finite propagation property: $$\text{supp}({\alpha }^{-n}(A_x))\subset B_{nR}(x)$$ $$\text{supp}({\alpha }^{-m}(B_y))\subset B_{mR}(y)$$

If $\text{dist}(x,y)>R(m-n)$, then two supports disjoint: $$B_{nR}(x)\cap B_{mR}(y)=\varnothing$$

Therefore ${\alpha }^{-m}(B_y)$ and ${\alpha }^{-n}(A_x)$ commute: $$[{\alpha }^{-m}(B_y),{\alpha }^{-n}(A_x)]=0$$

Thus correlation function factorizes: $${\omega }_0({\alpha }^{-m}(B_y){\alpha }^{-n}(A_x))={\omega }_0({\alpha }^{-m}(B_y)){\omega }_0({\alpha }^{-n}(A_x))$$ i.e. $C_{AB}(n,m)=0$.

graph LR
    A["dist(x,y)>R(m-n)"] --> B["Supports Disjoint"]

    B --> C["Operators Commute<br/>[A,B]=0"]

    C --> D["Correlation Vanishes<br/>C_AB=0"]

    E["No Causal Influence"] --> D

    style A fill:#ff6b6b
    style D fill:#6bcf7f

Core Theorem: Equivalence of Geometric and Statistical Causality

Equivalence Theorem

Theorem 2.5 (Equivalence of Causal Structures): In QCA universe ${\mathfrak{U}}_{\text{QCA}}$, for any events $(x,n),(y,m)\in E$: $$(x,n){\le }_{\text{geo}}(y,m)⟺(x,n){⪯}_{\text{stat}}(y,m)$$

Proof:

($\Rightarrow$): Assume $(x,n){\le }_{\text{geo}}(y,m)$, i.e. $m\ge n$ and $\text{dist}(x,y)\le R(m-n)$.

Choose local operators $A_x\in {\mathcal{A}}_{\{x\}}$, $B_y\in {\mathcal{A}}_{\{y\}}$, e.g. Pauli operators ${\sigma }_x^z,{\sigma }_y^z$.

Since $\text{dist}(x,y)\le R(m-n)$, supports of evolved operators ${\alpha }^{-n}(A_x)$ and ${\alpha }^{-m}(B_y)$ may intersect, therefore generally: $$[{\alpha }^{-m}(B_y),{\alpha }^{-n}(A_x)]\ne 0$$

Thus correlation function $C_{AB}(n,m)\ne 0$ (at least for some choices of $A,B$). Therefore $(x,n){⪯}_{\text{stat}}(y,m)$.

($\Leftarrow$): Assume $(x,n){⪯}_{\text{stat}}(y,m)$, i.e. exists $A_x,B_y$ such that $C_{AB}(n,m)\ne 0$.

By contrapositive of Lemma 2.4, if $\text{dist}(x,y)>R(m-n)$ or $m<n$, then necessarily $C_{AB}(n,m)=0$, contradiction!

Therefore must have $m\ge n$ and $\text{dist}(x,y)\le R(m-n)$, i.e. $(x,n){\le }_{\text{geo}}(y,m)$.

QED.

graph TD
    A["Geometric Reachability<br/>(x,n)≤_geo(y,m)"] <==> B["Statistical Causality<br/>(x,n)⪯_stat(y,m)"]

    A --> C["m≥n and<br/>dist≤R(m-n)"]
    B --> D["Correlation Function<br/>C_AB≠0"]

    C --> E["Finite Propagation"]
    D --> E

    E --> F["QCA Axiom<br/>Consistency"]

    style A fill:#ffd93d
    style B fill:#ffd93d
    style F fill:#6bcf7f

Physical Interpretation

This theorem is very profound:

Geometric Causality = Statistical Causality

Means:

• Geometric light cone derived from QCA kinematics (finite propagation $R$)
• Statistical causality defined from quantum correlation measurements
• Completely consistent!

This is not coincidence, but intrinsic self-consistency of QCA axioms.

Verification of Partial Order Properties

Partial Order Axioms

Definition 2.6 (Partial Order): Binary relation $⪯$ is called partial order if satisfies:

1. Reflexivity: $e⪯e,\forall e\in E$
2. Transitivity: $e_1⪯e_2,e_2⪯e_3\Rightarrow e_1⪯e_3$
3. Antisymmetry: $e_1⪯e_2,e_2⪯e_1\Rightarrow e_1=e_2$

Proposition 2.7 (${\le }_{\text{geo}}$ is Partial Order): Geometric reachability relation ${\le }_{\text{geo}}$ is partial order on $E$.

Proof:

(1) Reflexivity: $(x,n){\le }_{\text{geo}}(x,n)$ because $n\ge n$ and $\text{dist}(x,x)=0\le R(n-n)=0$.

(2) Transitivity: Assume $(x,n){\le }_{\text{geo}}(y,m)$ and $(y,m){\le }_{\text{geo}}(z,k)$.

Then:

• $m\ge n$ and $\text{dist}(x,y)\le R(m-n)$
• $k\ge m$ and $\text{dist}(y,z)\le R(k-m)$

By triangle inequality: $$\text{dist}(x,z)\le \text{dist}(x,y)+\text{dist}(y,z)\le R(m-n)+R(k-m)=R(k-n)$$

And $k\ge m\ge n$, therefore $(x,n){\le }_{\text{geo}}(z,k)$.

(3) Antisymmetry: Assume $(x,n){\le }_{\text{geo}}(y,m)$ and $(y,m){\le }_{\text{geo}}(x,n)$.

Then:

• $m\ge n$ and $n\ge m$, therefore $m=n$
• $\text{dist}(x,y)\le R(m-n)=0$, therefore $x=y$

Hence $(x,n)=(y,m)$.

QED.

graph TD
    A["Partial Order ≤_geo"] --> B["Reflexivity<br/>(x,n)≤(x,n)"]
    A --> C["Transitivity<br/>≤→≤→≤"]
    A --> D["Antisymmetry<br/>≤+≤→="]

    B --> E["Each Event<br/>Reachable to Itself"]
    C --> F["Causal Chain Transitive"]
    D --> G["Causal Direction Unique"]

    E --> H["Causal Poset<br/>(E,⪯)"]
    F --> H
    G --> H

    style A fill:#ffd93d
    style H fill:#6bcf7f

Local Finiteness: Key Property of QCA

Definition

Definition 2.8 (Locally Finite Partial Order): Poset $(E,⪯)$ is called locally finite if for any $e\in E$: $$|I^{+}(e)|<\infty \text{ and }|I^{-}(e)|<\infty$$ where $I^{+}(e):=\{e^{\prime }:e⪯e^{\prime }\}$ is future of $e$, $I^{-}(e):=\{e^{\prime }:e^{\prime }⪯e\}$ is past of $e$.

Theorem 2.9 (Local Finiteness of QCA Causal Set): $(E,{\le }_{\text{geo}})$ is locally finite poset.

Proof: Fix $(x_0,n_0)\in E$.

Future light cone: $$I_{\text{geo}}^{+}(x_0,n_0)=\{(y,m):m>n_0,\text{dist}(y,x_0)\le R(m-n_0)\}$$

For fixed $m>n_0$, $y\in \Lambda$ satisfying $\text{dist}(y,x_0)\le R(m-n_0)$ are finite (because $|B_{R(m-n_0)}(x_0)|<\infty$, local finiteness assumption).

Therefore each time slice $\{m\}\times \Lambda$ has only finitely many events in future light cone.

But problem: $m$ can take infinitely many values!

Correction: Local finiteness should be understood as for any finite time interval $[n_0,n_0+T]$, intersection of future light cone with interval is finite: $$|I_{\text{geo}}^{+}(x_0,n_0)\cap (\Lambda \times [n_0,n_0+T])|<\infty$$

Similarly, past light cone is finite in finite time interval.

This is called local finiteness in temporal sense.

graph TD
    A["Event (x₀,n₀)"] --> B["Future Light Cone I⁺"]

    B --> C["Any Time Slice n>n₀"]
    C --> D["Finite Lattice Points<br/>|B_R(m-n₀)(x₀)|<∞"]

    D --> E["Local Finiteness"]

    F["QCA Assumption<br/>|B_R(x)|<∞"] --> D

    style A fill:#ffd93d
    style E fill:#6bcf7f

Connection with Causal Set Theory

In Sorkin’s causal set theory, spacetime ontology is defined as locally finite poset $(E,⪯)$.

Causal Set Axioms:

1. $(E,⪯)$ is poset
2. Local finiteness: $|I^{+}(e)\cap I^{-}(e^{\prime })|<\infty$

QCA naturally provides a realization of causal set!

Philosophical Meaning:

QCA universe automatically satisfies causal set axioms!

Discrete quantum evolution → causal set spacetime structure.

Alexandrov Topology: Reconstructing Topology from Partial Order

Double Cone Open Sets

Definition 2.10 (Causal Diamond): For $(x,n)\ll (y,m)$ ($\ll$ means strict partial order), define causal diamond: $$A((x,n),(y,m)):=I^{+}(x,n)\cap I^{-}(y,m)$$ Set of events simultaneously in future of $(x,n)$ and past of $(y,m)$.

Alexandrov Topology: Topology generated with all causal diamonds $\{A((x,n),(y,m))\}$ as topological basis is called Alexandrov topology ${\tau }_A$.

Theorem 2.11 (Existence of Alexandrov Topology): $(E,{\tau }_A)$ is topological space.

graph TD
    A["Event Pair<br/>(x,n)≪(y,m)"] --> B["Causal Diamond<br/>A((x,n),(y,m))"]

    B --> C["Topological Basis<br/>{A(·,·)}"]

    C --> D["Alexandrov Topology<br/>τ_A"]

    D --> E["Topological Space<br/>(E,τ_A)"]

    style A fill:#ffd93d
    style E fill:#6bcf7f

Continuous Limit and Manifold Reconstruction

Theorem 2.12 (Continuous Limit Gives Lorentzian Manifold) (informal statement): Under appropriate continuous limit $(a\to 0,\Delta t\to 0,R=1)$, Alexandrov topology of QCA causal set $(E,{⪯}_{\text{geo}})$ converges to standard topology of Minkowski spacetime.

Proof Idea:

1. In continuous limit, events $(x,n)\to (\mathbf{x},t)\in {\mathbb{R}}^{d+1}$
2. Geometric reachability ${\le }_{\text{geo}}$ becomes Minkowski causal partial order ${\le }_M$ in limit
3. Alexandrov basis $A(p,q)$ converges to standard double cone $I^{+}(p)\cap I^{-}(q)$
4. Topology convergence theorem (requires finer analysis)

Physical Meaning:

Starting from discrete QCA causal set, continuous limit automatically recovers topological structure of continuous manifold!

Spacetime topology emerges from causal partial order.

graph LR
    A["Discrete QCA<br/>(E,≤_geo)"] -->|Continuous Limit| B["Minkowski Spacetime<br/>(ℝ⁴,≤_M)"]

    A --> C["Alexandrov Topology<br/>τ_A"]
    B --> D["Standard Topology<br/>τ_std"]

    C -.->|Convergence| D

    E["Causality Emerges Spacetime"] --> B

    style A fill:#ffd93d
    style B fill:#6bcf7f
    style E fill:#6bcf7f

Lieb-Robinson Bound: Strict Bound on Effective Light Speed

Lieb-Robinson Theorem

Theorem 2.13 (Lieb-Robinson Bound, Simplified Version): Consider time evolution generated by QCA evolution $\alpha$ under local Hamiltonian $H$. For local operators $A,B$ supported on $X,Y⋐\Lambda$ with $\text{dist}(X,Y)=r$: $$\Vert [{\alpha }^n(A),B]\Vert \le C\Vert A\Vert \Vert B\Vert e^{-\mu (r-vn)}$$ where:

• $C$: Constant
• $v$: Effective light speed, $v=O(R)$
• $\mu >0$: Exponential decay rate

Physical Meaning: Even in discrete time, propagation speed of quantum correlations still has exponentially strict upper bound.

Connection with Relativity: In continuous limit, Lieb-Robinson bound gives: $$v_{\text{eff}}\to c$$ This is exactly speed of light!

graph TD
    A["QCA Evolution α"] --> B["Lieb-Robinson Bound"]

    B --> C["Correlation Propagation<br/>||[α^n(A),B]||"]

    C --> D["Exponential Decay<br/>e^{-μ(r-vn)}"]

    D --> E["Effective Light Speed v<br/>v=O(R)"]

    E --> F["Continuous Limit<br/>v→c"]

    F --> G["Relativistic Causality"]

    style A fill:#ffd93d
    style G fill:#6bcf7f

Conical Constraint of Information Propagation

Another corollary of Lieb-Robinson bound:

Corollary 2.14 (Information Cone): After time $n$, information originally localized at single point $\{x_0\}$ can only be localized in $B_{vn}(x_0)$ with probability $\ge 1-ϵ$, where $ϵ\sim e^{-\mu n}$ exponentially small.

Proof Idea: By Lieb-Robinson bound, ${\alpha }^n(A_{x_0})$ and $B_y$ ($\text{dist}(y,x_0)>vn$) almost commute, therefore measurement of $B_y$ almost unaffected by $A_{x_0}$.

This forms conical constraint of information propagation, similar to light cone of relativity!

Summary: Complete Picture of Causal Structure Emergence

Logical chain from QCA axioms to relativistic causal structure:

graph TD
    A["QCA Axioms<br/>Finite Propagation R"] --> B["Geometric Reachability<br/>(x,n)≤_geo(y,m)"]

    A --> C["Statistical Causality<br/>(x,n)⪯_stat(y,m)"]

    B <==> C

    C --> D["Causal Partial Order<br/>(E,⪯)"]

    D --> E["Partial Order Properties<br/>Reflexive Transitive Antisymmetric"]
    D --> F["Local Finiteness<br/>|I⁺∩Interval|<∞"]

    E --> G["Alexandrov Topology<br/>τ_A"]
    F --> G

    G --> H["Continuous Limit<br/>Lorentzian Manifold"]

    H --> I["Minkowski Spacetime<br/>Relativistic Causality"]

    style A fill:#ffd93d
    style I fill:#6bcf7f

Five Levels of Emergence:

1. QCA Finite Propagation → 2. Geometric Reachability Relation → 3. Statistical Causality → 4. Causal Poset → 5. Alexandrov Topology → 6. Lorentzian Manifold

Core Insight:

Causal light cone of relativity is not “background stage” pre-drawn by God,

but mathematical necessity emerging from finite speed of information propagation in discrete QCA!

Popular Analogy: Information Like Water Wave Diffusion

Water Wave Analogy

Imagine a pond, throw a stone creating ripples:

Classical Continuous Picture:

• Water surface is continuous ${\mathbb{R}}^2$
• Ripples propagate outward at speed $v$
• Causal cone: $r\le vt$

QCA Discrete Picture:

• Water surface is lattice $\Lambda ={\mathbb{Z}}^2$
• Each lattice point vibrates at discrete time steps $n\in \mathbb{Z}$
• Vibration propagates by local rules: at most $R$ lattice points per step

Emergence: When lattice spacing $a\to 0$, propagation of discrete vibrations looks like continuous ripples at macroscopic scale, speed $v=Ra/\Delta t$.

graph TD
    A["Discrete Water Surface<br/>Lattice Vibrations"] --> B["Local Propagation<br/>R Lattice Points Per Step"]

    B --> C["Macroscopic Scale<br/>Lattice Spacing a→0"]

    C --> D["Emergent Continuous Wave<br/>Speed v=Ra/Δt"]

    D --> E["Water Wave Causal Cone<br/>r≤vt"]

    style A fill:#ffd93d
    style E fill:#6bcf7f

Analogy to QCA:

Core Analogy:

Just as vibrations on discrete lattice points emerge continuous water waves at macroscopic scale,

discrete QCA emerges continuous spacetime and relativistic causality in long-wavelength limit!

Next Step: Terminal Object in 2-Category

Next section is core climax of this chapter: We will construct 2-category ${\mathbf{Univ}}_{\mathcal{U}}$, define terminal object ${\mathfrak{U}}_{\text{phys}}^{\ast }$, and prove terminal object uniquely exists under four axioms:

1. Unified Time Scale: $\kappa (\omega )={\phi }^{\prime }(\omega )/\pi ={\rho }_{\text{rel}}(\omega )=(2\pi {)}^{-1}\text{tr}Q(\omega )$
2. Generalized Entropy Monotonicity: ${\delta }^2{S}_{\text{gen}}\ge 0$
3. Topological Anomaly-Free: $[K]=0$
4. Causal Locally Finite: $(E,⪯)$ is locally finite poset

Section 2 (this section) has proven: QCA automatically satisfies Axiom 4!

Next section will prove: Objects satisfying Axioms 1-4 are unique terminal object in 2-category → Uniqueness of physical laws guaranteed by category theory!

This is categorical pinnacle of entire unified theory!